New solution generating transformations for the stationary axially-symmetric Einstein-Maxwell
equations
by Terry Lee Lemley
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by Terry Lee Lemley (1983)
Abstract:
Exact solution generating methods for the stationary axially-symmetric Einstein-Maxwell equations are
investigated in this thesis. In particular, the attempt has been made to extend Cosgrove's vacuum
symmetry groups, Q and Q~, to the electrovac case. The primary method used is that of invariant
groups, but due to the close relationship to Backlund transformations, a brief review of the latter is
given.
The symmetry groups Q and Q~ are found to exist for the electrovac case, but as of yet their
application has met with minimal success.
It is still felt by the author that more of the vacuum results pertaining to Q and Q~ can be extended to
the electrovac case. NEW SOLUTION GENERATING TRANSFORMATIONS
FOR THE STATIONARY AXIALLY-SYMMETRIC
EINSTEIN-MAXWELL EQUATIONS
by
T e r r y Lee LemIey
A t h e s i s s u b m itte d i n p a r t i a l f u l f i l l m e n t
o f t h e re q u ir e m e n t s f o r t h e degree
of
D o c to r o f P h ilo s o p h y
in
P h ysics
MONTANA STATE UNIVERSITY
Bozeman,Montana
May 1983
L 543
a
ii
APPROVAL
o f a t h e s i s s u b m itte d by
T e r r y Lee Lemley
T h is t h e s i s has been re a d by each member o f t h e t h e s i s com m ittee
and has been found t o be s a t i s f a c t o r y r e g a r d i n g c o n t e n t , E n g lis h usage,
f o r m a t , c i t a t i o n s , b i b l i o g r a p h i c s t y l e , and c o n s is t e n c y , and i s read y
f o r subm ission t o t h e C o lle g e o f G rad u ate S t u d i e s .
/ cJ . /
Date
/
/
/a
5 _____
C h a ir p e r s o n ,
G rad u ate Commit t e e
Approved f o r t h e M a jo r Department
Approved f o r t h e C o lle g e o f G rad u ate S tu d ie s
5- - Z - I
Date
G rad u ate Dean
iii
STATEMENT OF PERMISSION TO USE
In p r e s e n t i n g t h i s t h e s i s
f o r a do cto ral
in p a r t i a l
f u l f i l l m e n t o f t h e re q u ire m e n ts
deg ree a t Montana S t a t e U n i v e r s i t y ,
L i b r a r y s h a l l make i t
a v a i l a b l e t o b o rro w e rs under r u l e s o f t h e L i b r a r y .
I f u r t h e r ag re e t h a t co p yin g o f t h i s t h e s i s
s c h o l a r l y p u rp o ses,
U .S .
C o p y r ig h t Law.
I ag re e t h a t t h e
c o n s is te n t w ith
is
a llo w a b le o n ly f o r
" f a i r use" as p r e s c r i b e d i n t h e
Requests f o r e x t e n s i v e copying o r r e p r o d u c t io n o f
t h i s t h e s i s should be r e f e r r e d t o U n i v e r s i t y M i c r o f i l m s
In te rn a tio n a l,
300 N o rth Zeeb Road, Ann A r b o r , M ic h ig a n 4 8 1 0 6 , t o whom I have g ra n te d
" t h e e x c l u s i v e r i g h t t o re p ro d u c e and d i s t r i b u t e c o p ie s o f t h e d i s s e r ­
ta tio n
i n and from m i c r o f i l m and t h e r i g h t t o rep ro d u ce and d i s t r i b u t e
by a b s t r a c t i n any f o r m a t . "
S ig n a t u r e __
Date
M a y
oa
A u iA/
I
I
iv
TABLE OF CONTENTS
Page
ABSTRACT............................ ............. ....................... ....................... .................. ................
vi
1.
INTRODUCTION...................... ............. ...................................................... ..
I
2.
THE FIELD EQUATIONS................................................................................... ..
3
3.
4.
5.
6.
N o t a t i o n . ......................................................................................................................
The Maxwell E q u a t io n s ........................... ................................... ....................... ..
The E i n s t e i n E q u a t i o n s . . . . . ...................................... ..................................
6
9
10
TRANSFORMATION GROUPS............................................... ..................
14
P r o p e r t i e s o f T r a n s f o r m a t io n Groups...........................
Appl i c a t i o n s .
............................................................................... ..
A M e t h o d . . . . . . ............................. ................................................ ..
One P a ra m e te r S u b g r o u p s . . . . . ...........................................
The N eugebauer-K ram er Mapping.................................. .......................
14
18
20
24
26
BACKLUND TRANSFORMATIONS...................................... ..............................
28
P r o p e r t i e s o f t h e Backlund T r a n s f o r m a t i o n ................. ............................
N e u g e b a u e r's Backlund T r a n s f o r m a t i o n . . . ...................... ....................... ..
N e u g e b a u e r's Commutation T h e o r e m . . . ..............................
28
29
33
REVIEW OF THE VACUUM C A S E . . . . . . . .................................... ................ ..
35
F i e l d s , P o t e n t i a l s , and G e n e r a t in g F u n c t io n s ................. ..
The Symmetry G r o u p s . . . . . . . ...............................
The Groups G and H.................................. ..................... .......................................
The Group K . .......... ............................................... ......................... ..
The Groups Q and Q...................................................................... .........................
A p p l i c a t i o n s . ...............................
Backlund T r a n s f o r m a t i o n s ................................................................
Summary..................................
35
37
37
38
39
39
41
42
REVIEW OF THE ELECTROVAC CASE.................................. ....
43
F i e l d s , P o t e n t i a l s , and G e n e r a t in g F u n c t io n s ...............
The Symmetry G r o u p s . . . . . . . . . . . . . . ......................................
The Groups G' and H 1. . . , ................................ ......................... .......................
The Group K 1............................
43
46
46
47
V
TABLE OF CONTENTS— C o n tin u ed
7.
S0
THE EXTENDED Q AND Q GROUP...
o e e e e o e o o e o o o e o o p o o e o d e e e o
XtiG Extended Q Gnoup©# ©• ©• « • ©• • • ©«© ©©«© ©©©©©©©* ©©©©«© ©©©©©? ©©
The F i n i t e Q T n sn sf Pnrnst I on ©©©©oo©©©©©©©©©*©©©©©©©©©©©*©©**©©
Sesnch Ton Q©©©©©©©*©©©©©©©©©©©©©©©©^©©©©©©©©©©©©©©©©©©©©©©©©
51
53
54
APPLICATION OF THE Q GROUP©
60
A c t io n on t h e M e t r i c Components f ^ i and
A p p l i c a t i o n o f Q t o R e i ssnen-Nond$tnom©©©
9.
eo © o ©©
S
U
M
M
REFERENCES C l TED
A
R
Y
..
o p o o e o e o © © © ^ © © © © © ©
© © © • o o o o © © o o o © © o © p o ©
...................................
...................
64
66
70
APPENDICES.
A ppendix
Appendix
Appendix
Appendix
A ppendix
60
62
A
B
C
D
E
- C o n s is te n c y o f E q u a tio n s ( 7 . 1 - 6 ) . . . . . . . . ............... ..
- Example C a l c u l a t i o n f o r E q u a tio n ( 7 . 1 2 ) . , . ? , . . . .
- C o n s is te n c y o f E q u a tio n ( 7 . 1 6 ) ............... ..
- I n t e g r a t i o n o f E q u a tio n s ( 7 . 1 7 , 1 8 ) . . . . , 6 . . . . . . . .
- I n t e g r a t i o n o f E q u a tio n s ( 7 . 4 6 , 4 8 ) . . . . ....................
71
78
81
84
87
vi
ABSTRACT
E x a c t s o l u t i o n g e n e r a t i n g methods f o r t h e s t a t i o n a r y a x i a l l y sym m etric E i n s t e i n - M a x w e l l e q u a t io n s a re i n v e s t i g a t e d i n t h i s t h e s i s .
In p a r t i c u l a r , t h e a t t e m p t has been made t o e x te n d C o s g ro v e 's vacuum
symmetry g ro u p s , Q and Q, t o t h e e l e c t r o v a c c a s e .
The p r im a r y method
used i s t h a t o f i n v a r i a n t g ro u p s , b u t due t o t h e c lo s e r e l a t i o n s h i p
t o Backlund t r a n s f o r m a t i o n s , a b r i e f r e v i e w o f t h e l a t t e r i s g i v e n .
The symmetry groups Q and Q a r e fo u n d t o e x i s t f o r t h e e l e c t r o v a c
c a s e , b u t as o f y e t t h e i r a p p l i c a t i o n has met w it h m in im al success.
I t i s s t i l l f e l t by t h e a u t h o r t h a t more o f t h e vacuum r e s u l t s
p e r t a i n i n g t o Q and Q can be e x te n d e d t o t h e e l e c t r o v a c c a s e .
I
CHAPTER I
INTRODUCTION
T h is t h e s i s
in v e s tig a te s
a method o f f i n d i n g e x a c t s o l u t i o n s
f o r th e s t a t io n a r y a x ia lly -s y m m e tr ic E in s te in -M a x w e lT e q u a tio n s .
a d je c tiv e s
p r e c e d in g t h e words E in s t e i n - M a x w e l l
a x ia l-s y m m e try ,
in v e s tig a te d .
u n i f o r m l y r o t a t i n g about i t s
The words E in s t e i n - M a x w e l l
th e ir
im p ly t h a t a system w it h
a x i s o f symmetry, i s b e in g
reveal
belongs t o t h e s u b j e c t a r e a s o f g r a v i t a t i o n
The
th a t th is
th e s is
and e le c t r o d y n a m ic s w it h
r e l a t e d e q u a t io n s .
B ein g n o n l i n e a r and c o u p le d , t h e E in s t e i n - M a x w e l l
d evelo p ed i n C h a p te r 2 a r e v e ry d i f f i c u l t t o s o l v e .
e q u a tio n s
B e f o r e s p e c ia l
methods were in t r o d u c e d f o r f i n d i n g new s o l u t i o n s , o n ly a h an d fu l o f
'
s o l u t i o n s were known.
Two methods which have been found y s e f u l in
f i n d i n g new s o l u t i o n s a r e t h e methods o f group t r a n s f o r m a t i o n s anc|
••
Backlund t r a n s f o r m a t i o n s .
These a r e re v ie w e d i n C h ap ters 3 and 4 ,
.
re s p e c tiv e ly .
The case where no e x t e r i o r e l e c t r o m a g n e t i c f i e l d s
has been i n v e s t i g a t e d by numerous w o rkers i n
recent y e a rs .
grow ing consensus among t h e s e w o rkers t h a t t h i s
s o lv e d .
A b rie f
re v ie w o f t h i s
vacuum case i s
A llo w a n c e f o r e l e c t r o m a g n e t i c f i e l d s
problem i s
found p r e v io u s t o t h i s
th e s is f o r t h is
T h ere i s a
n e a rly
g iv e n i n C h a p te r 5,
w it h t h e g iv e n symmetries
in t r o d u c e s a so u rce te rm i n t o t h e E i n s t e i n e q u a t i o n s .
re s u lts
a r e p re s e n t
case.
C h a p te r 6 re v ie w s
2
C h a p te r 7 ex te n d s t h e Q and Q t r a n s f o r m a t i o n s
(1979a)
f o r vacuum t o t h e case where s t a t i o n a r y a x i a l l y - s y m m e t r i c
e le c tro m a g n e tic f i e l d s
o rig in a l
found by Cosgrove
m a te ria l
is
are p re s e n t.
T h is
is th e f i r s t
p resen ted .
C h a p t e r 8 uses t h e new t r a n s f o r m a t i o n s
g e n e r a t e a new s o l u t i o n .
C h a p te r 9.
c h a p t e r in which
The r e s u l t s
of th is
in an a t t e m p t t o
t h e s i s a r e g iv e n in
3
CHAPTER 2
THE FIELD EQUATIONS
At t h e t u r n o f t h e c e n t u r y o n ly two f o r c e s
known, t h e g r a v i t a t i o n a l
b e a u tifu lly
e q u a tio n s .
f o r c e and t h e e l e c t r o m a g n e t i c f o r c e .
p o r t r a y e d by two p h y s ic a l
Law o f U n i v e r s a l
G ra v ita tio n
E in s te in
and t h e l a t t e r
by M a x w e ll's f i e l d
founded t h e t h e o r y o f s p e c i a l
In c lu d in g th e g r a v it a t io n a l
E i n s t e i n t o t h e t h e o r y o f g e n e ra l
fie ld s
p la y e d i m p o r t a n t r o l e s
G en eral
e q u a t io n s
re la tiv ity
e le c tro m a g n e tic f i e l d s
Maxwell
e v e n tu a lly
c o n s is ts p a r t i a l l y
fie ld ,
on what
le d
Thus, b oth c l a s s i c a l
in t h e d evelopm ent o f g e n e r a l
re la tiv ity .
in th e s o lv in g o f th e
t h e E i n s t e i n e q u a t io n s .
a r e p r e s e n t , th en g e n e ra l
e q u a t io n s t o i n c l u d e t h e e f f e c t s
e q u a t io n s a r e w r i t t e n
re la tiv ity
e q u a t io n s o v e r Newton's
fie ld
re la tiv ity .
g o v e rn in g t h e g r a v i t a t i o n a l
Both were
t h e o r i e s , t h e f o r m e r by Newton's
has amounted t o t h e cho o sin g o f M a x w e ll's f i e l d
laws o f m o tio n .
i n n a t u r e were
re la tiv ity
ip o d ifie s t h e
o f cu rved s p a c e t im e .
The f i e l d
i n t h e d e c e p t i v e l y s im p le t e n s o r fo rm :
Ri k - I Z Z g i k R = (-SirGZc1O T 1k
are th e E in s te in
Maxwell
(2 .1 )
e q u a t io n s and
F l k ;k = " 4irsl
a r e t h e Maxwell
If
;
e q u a tio n s .
Fi k , l +Fl i , k + F k l , i
(2 .2 ,3 )
T o g e th e r t h e y a re c a l l e d t h e E i n s t e i n -
e q u a tio n s .
'
The E i n s t e i n - M a x w e l l
f o l l o w i n g manner.
= 0
.
e q u a t io n s a r e u s u a l l y t h o u g h t o f in t h e
On t h e l e f t - h a n d
s id e o f E q u a tio n s
(2 .1 )
are th e
4
terms r e l a t e d t o t h e c u r v a t u r e o f s p a c e t im e .
. T h is c u r v a t u r e has as i t s
so u rce t h e energy-momentum t e n s o r T1-Jc which stands on t h e r ig h t - h a n d
s id e o f E q u a tio n s
(2 .1 ).
th e m atter f i e l d s ,
T h is so u rce te rm in c lu d e s t h e e f f e c t s
e x c lu d in g g r a v i t y ,
and i s
b a s i c a l l y t h e s t a te m e n t o f
m ass-energy e q u iv a le n c e in t h e g e n e r a t io n o f t h e g r a v i t a t i o n a l
In p a r t i c u l a r ,
it
w ill
e le c tro m a g n e tic f i e l d s
The Maxwell
usual
p re s e n t.
e q u a t io n s a r e m o d if ie d d i r e c t l y
in th e l i t e r a t u r e ,
and t h e s e m i- c o lo n
e ffe c ts
o f th e g r a v ita tio n a l
a comma r e p r e s e n t s p a r t i a l
fie ld
fie ld ,
(As
d iffe re n tia tio n ,
In d ire c t
a r e c o n t a in e d i n t h e c u r r e n t d e n s it y
r i g h t - h a n d s i d e o f E q u a tio n s
Taken t o g e t h e r ,
d iffic u lt
by t h e c o v a r i a n t
rep resen ts c o v a ria n t d i f f e r e n t i a t i o n . )
o f the g r a v it a t io n a l
te rm Si on t h e
fie ld .
c o n t a i n t h e en erg y and momentum o f t h e
d e r i v a t i v e which c o n t a in s t h e e f f e c t s
is
of a ll
(2 .2 ).
th e E in s te in -M a x w e ll
s e t o f e q u a t io n s t o s o l v e .
e q u a t io n s a r e 9 very
They a r e h i g h l y co u p le d and
n o n lin e a r.
In t h i s
F irs t,
th a t
th e s is ,
s im p lif y in g c o n d itio n s w i l l
t h e re g io n s o f space c o n s id e re d w i l l
is ,
th e c u rre n t d e n s ity
t e n s o r T 1Jc w i l l
w ill
s1 w i l l
be added.
be e x t e r i o r t o t h e s o u rc e s ;
be z e r o ,
and t h e energy-momentum
c o n t a i n term s which depend o n ly on t h e f r e e e l e c t r o ­
m ag n etic f i e l d s .
fie ld s
several
T h is s i t u a t i o n
is c a lle d e le c tro v a c .
be assumed s t a t i o n a r y .
a x ia lly -s y m m e tric .
F in a lly ,
the f i e l d s
t h e s im u lta n e o u s t r a n s f o r m a t i o n
The f i r s t
T h ir d , th e f i e l d s
w ill
be assumed
be assumed i n v a r i a n t under
of the c o o rd in a te s ,
c o n d itio n s im p lif ie s
R . k = (-S T rG Z c ^ T i k
w ill
Second, t h e
th e f i e l d
(t,*)+ (-t,^ & ).
e q u a t io n s t o
(2 .1 a )
5
and
( 2 .2 a ,3)
A x ia l- s y m m e t r y means t h e e x i s t e n c e o f an a x i s o f symmetry
around which any r o t a t i o n
le a v e s t h e f i e l d s
th is
c o o r d i n a t e * which measures t h e d eg ree o f
d e f i n e s an a z im u t h a l
ro ta tio n
s t a t e o f motion such t h a t a l l
M a th e m a tic a lly , t h is
o f t h e sources have reached a u n ifo rm
the f i e l d s
a r e in d e p e n d e n t o f t i m e .
d e f i n e s t h e t im e c o o r d i n a t e t
o f which a l l
the
a r e in d e p e n d e n t .
F in a lly ,
t h e symmetry, t r a n s f o r m a t i o n
m eaning.
r o t a t i n g ab o u t i t s
a x i s , t h e n t h e above t r a n s f o r m a t i o n
v e lo c ity d is trib u tio n
If
(t,*)+ (-t,-*)
f o l l o w i n g p h y s ic a l
back i n t o
has changed w i t h i n
has t h e
t h e o b j e c t p ro d u c in g t h e f i e l d s
its e lf.
is
b r in g s t h e
T h is means t h a t t h e e n e r g y -
momentum t e n s o r and c u r r e n t d e n s i t y go i n t o t h e m s e lv e s .
fie ld s
M a th e m a tic a lly ,
ab o u t t h e a x i s .
S t a t i o n a r y means t h a t a l l
fie ld s
in v a ria n t.
S in c e n o th in g
t h e so u rce under t h e t r a n s f o r m a t i o n , t h e e x t e r n a l
produced by t h e so u rce should be t h e same.
T h is
te n s o r g ^ .
la s t
c o n d itio n
g re a tly
s im p lifie s
t h e form o f t h e m e t r i c .
B e f o r e a d d in g t h e l a s t c o n d i t i o n , t h e l i n e
ele m e n t had t h e
form
= g 11d t d t + 2 g 12 dt<4 + 2 g 13 d t d x 3+ 2 g i 4 d t d x 4 +
g2 2 d<i>d<|>+2g23d<j>dx3 +2g24 d<i>dx4 +g 33d x3dx3+
2g 34d x 3dx4 +g4 4 dx4 dx4 .
(H e re t
= x 1 , <j> = x2 , and g ^
= g ^ ( x 3 ,x 4 ) . )
(2 .4 )
6
Im posing t h e l a s t c o n d i t i o n
24 components.
r i d s t h e . g i k o f t h e 1 3 , 1 4 , 2 3 , and
The l i n e e le m e n t can now be w r i t t e n
i n t h e box d ia g o n a l
form
ds2 = f ABdxAdxB” hMNdxMdxN;
A ,B = 1 ,2 ;
(2 .5 )
M9N = 3 , 4 .
.
,
N o ta tio n
A b rie f
th e f i e l d
re v ie w o f t h e n o t a t i o n t o be used i n t h e development o f
e q u a t io n s w i l l
now be g iv e n .
T h is n o t a t i o n f o l l o w s K in n e r s le y
(1 9 7 7 a ).
To r a i s e
e ith e r
in d ic e s
i n a tw o - d im e n s io n a l
(T a b ) ' 5 i n v e r s e m a t r i x o r e
- £ 2 i = I ; E11 = £22 = e22 = E u
-
AB
space w i t h m e t r i c f ^ g ,
may be used,
0 .)
( e 12 = e 12 = r e 21
T h is can be seen by t h e
f o l l o w i n g a rg u m e n t.
F o r a g e n e ra l
The in v e r s e i s e a s i l y
A '1 =
I
(a d -b c )
two by two m a t r i x .
shown t o be
/d
-b
\-c . a
(ad -b c)
e aV
( 2. 6)
.
w ith
E =
0
s
and t h e s u p e r s c r i p t t
where t h e m a t r i x
(f^ g )
( f AB^ 1 '
in d ic a t in g m a trix tr a n s p o s itio n .
In t h e case
i s s y m m e tric , t h e in v e r s e m a t r i x i s
d e t ( f AB) e ^f AB
g iv e n by
(2 .7 )
'
7
or in
in d e x n o t a t i o n
by
. p - 2cACeBDf r
= ( f A B )"1
-P
(2 .7 a )
~ det(f^g).
T h e r e f o r e , d e f i n i n g f ^ B by
.AB
it
AC BD_
E e f,
( 2, 8)
-P
i s seen t h a t
f ABf ,
S tric tly
fAB i s
■p W
2 ,A
“p 6
.
s p e a k in g , s in c e e
AC
is
(2 ,9 )
C°
a t e n s o r d e n s i t y o f w e ig h t minus one^
a t e n s o r d e n s i t y o f w e ig h t minus two ( W e in b e rg , 1 9 7 2 : 9 9 ) .
c o o r d i n a t e t r a n s f o r m a t i o n s which have t h e J a c o b ia n equal
are s t r i c t
te n s o rs .
T h is d e t a i l
w ill
be ig n o re d as i t
For
t o ope, th ey
does n o t a f f e c t
t h e f o l l o w i n g d e v e lo p m e n t.
G iven t h e v e c t o r V ^,
r a i s i n g t h e in d e x i s d e f i n e d by
V8 = eBAVA.
(2 .1 0 )
I
It
is
im p o r t a n t t o n o te t h a t t h e c o n t r a c t e d in d e x i s t h e second in d e x i n
t h e e BA.
W ith t h i s
E
it
AC
c o n v e n t io n and t h e i d e n t i t y
(2. 11)
eCB
i s seen t h a t t o lo w e r t h e in d e x on Vb r e q u i r e s m u l t i p l y i n g by e ^ q .
I
8
That i s ,
V0 -
S g /.
(2 .1 2 )
T h is i s c o n s i s t e n t w i t h t h e
o rig in a l
r a is in g o p e ra tio n
l e a d i n g back t o t h e
v e c to r.
R e tu rn t o t h e s t a t i o n a r y a x i a l l y - s y m m e t r i c m e t r i c o f E q u atio n
(2 .5 ).
The i n d i c e s w i l l
b lo c k and t h e i n v e r s e o f
in stead o f r a is in g
be r a i s e d u s in g
(h ^ ^ ) i n t h e
in d ic e s in t h i s
a r e u s e d , th e n t h e r a i s e d
tild e .
in th e
(t,<|>) c o o r d i n a t e
(x 3 ,x 4 ) c o o rd in a te b lo c k .
manner, t h e i n v e r s e o f
in d ic e s w i l l
( f f t s ) and
be marked w i t h a s u p e r s c r i p t
Fo r exa m p le , '
M
V
w ith
(2 .1 3 )
-G 43 = -Glt3 = I
e 34
The tw o - d im e n s io n a l
w ill
If,
be d e s ig n a t e d v .
and G33 = G(
c o v a r i a n t d e r i v a t i v e a s s o c i a t e d w i t h h^w
The d iv e r g e n c e o f a v e c t o r f i e l d
V = (V g .V *)
is
g iven by
VV
= h
' (H1^ K k n Vm) j n ,
h b ein g t h e d e t e r m in a n t o f h^^.
(2 .1 4 )
Xk hMN i n tw o The e x p r e s s io n h " h
dim ensions i s c o n f o r m a l l y i n v a r i a n t .
"MN
tran sfo rm ed to h ' " \
tra n s fo rm a tio n
w it h
h^
= e ^^h^,
if
the m e tric
th e n h ^ h ^
and V ^ + e ^ V ^ .
( h 1 / 2 hMNVM) jN s th e n v . V = h ' 1 / 2
,
.
T h e re fo re ,
(h 1 / 2 hM\
)
.
if
jN =
is
= h ^ h^
i s n o t a c o o r d i n a t e t r a n s f o r m a t i o n ; un d er t h i s
■
tra n s fo rm a tio n
That i s ,
V»V = 0 -
T h is
/
9
h
'
(h z h
c o n fo rm a lly
1 9 0 9 :9 2 -9 3 ),
V^) ^ = 0.
S in c e e v e r y tw o - d im e n s io n a l
m e t r i c space i s
r e l a t e d t o t h e E u c lid e a n tw o - d im e n s io n a l
it
is
p o s s i b l e t o p ic k h ^
= 6^
.
space,
In t h i s
case, the
c o v a r i a n t d e r i v a t i v e V reduces t o t h e o r d i n a r y g r a d i e n t i n
c o o rd in a te s .
The l i n e
ele m e n t i s
(E is e n h a rt,
re c ta n g u la r
now w r i t t e n
d s2 = f ABdxAdxB- e 2 r 6MNdxMdxN ;
A ,B -
1 ,2 ;
M,N = 3 , 4 .
(2 .5 a )
. A n o t h e r common form o f t h e l i n e e l e m e n t , Lewis
(1 9 3 2 ),
is
r e l a t e d t o t h e above by t h e r e l a t i o n s
f 11 = f , f
= " f w * ^22 = ^
™p*"f ^ ,
e 2 r = T 1 B2 y .
In t h i s
(2 .1 5 )
p a r a m e t e r i z a t i o n , t h e l i n e ele m e n t i s w r i t t e n
ds2 = f (dt-a)d<j> J2 - T 1 B^y ( ( d x 3 ) 2 + ( d x 4 ) 2 ) + r 1p2 d(|>2 e
(2 .1 6 )
A g a in , - p 2 = d e t ( f AB) and i s a f u n c t i o n o f x 3 and x4 .
The Maxwell
U sin g t h e l i n e
w ill
now be w r i t t e n
E q u a tio n s
e lem en t o f E q u a tio n
( 2 . 5 a ) , th e f i e l d
b u t , b e g in n in g w i t h . t h e Maxwell
In tro d u c in g th e e le c tro m a g n e tic p o t e n t ia l
F i k " Ak , i " Ai , k »
E q u a tio n s
a r e in d e p e n d e n t o f
(t,<f>).
(2 .3 )
(t,<j>), and t h e Ai w i l l
The Maxwell
e q u a t ip n s
( 2 .2 a ,3 ).
A1- by t h e e q u a tio n
a re a u to m a tic a ly s a t i s f i e d .
e q u a t io n s
The Fi k
a l s o be assumed in d ep en d en t o f
H ence, t h e o n ly s u r v i v i n g components l e f t
f MN = a N,M- a M,.N-
e q u a tio n s
(2 .2 a )
are
-
- A ^ ^ and
may be w r i t t e n ,
(W e in b e rg , 1 9 7 2 : 1 2 5 ) ,
(/^g F1k) >k = o.
(2 .2 b )
10
These f a l l
i n t o two s e ts o f e q u a t i o n s .
The f i r s t
0 = (^ g Fmn) iN == (pe2rFMN) jN
s et is
(2 .1 7 )
( pFV , N #
which has s o l u t i o n s
FMN = Cp- I eMNi
(2 .1 8 )
T h is co rresp o n d s t o a m ag n etic f i e l d
o f f as p
-I
, which would be produced by an i n f i n i t e
t h e symmetry a x i s .
S in c e t h i s
source t h e c o n s t a n t C w i l l
The f i n a l
s o lu tio n
is
c u r r e n t a lo n g
not p h y s ic a l
f o r a bounded
is
th e re e x is ts a s c a la r p o te n tia l
(2.19)
, th e n one has t h e f o l l o w i n g
s c a l a r f u n c t i o n s U, v * ( vU ) = 0.
(2 .1 9 )
fa llin g
lin e
= V .(p fABVAB) = Vo (P-1TabVAb).
D e f i n e v by V = ( d / d x 4 , - 3 / a x 3 )
E q u a tio n s
d ire c tio n
be s e t t o z e r o .
s e t o f e q u a t io n s
0 = (v^g FAM)
For a ll
i n t h e a z im u th a l
C o n v e r s e ly ,
U such t h a t V = VU.
if
id e n tity .
v»V = 0, th en
T h e re fo re ,
in d ic a t e th e e x is te n c e o f s c a la r f i e l d s
such t h a t
VBa = P - 1 Ta 6VAb .
(2 .2 0 )
U sin g t h e r e l a t i o n s V = - V
and f ABf BC = - P 2 S ^
, an i n v e r s e
re la tio n
can
be d e r i v e d y i e l d i n g
VAa = - P - 1 Ta 8VBb .
(2 .2 0 a )
Forming t h e complex c o m b in a tio n <&A = AA+ i B A , t h e $ A a r e seen t o s a t i s f y
V@A = - i p ~ 1 f ABv $ B.
(2 .2 1 )
The E i n s t e i n E q u a tio n s
The E i n s t e i n e q u a t io n s
(2 .1 a )
a r e n e x t c o n s id e r e d .
It
can be
shown t h a t
2RAc = P- 1 V - ( P - 1V mV f , , , . )
(2 .2 2 )
11
and
4 * T Ab = P- 1V- ( p " 1 ( f ACABVAc - 1/2 6ABf CDAc VAD ) )
Hence, t h e E i n s t e i n e q u a t io n s a r e ,
.
(2 ,2 3 )
0.
(2 .2 4 )
s e ttin g G = c = I ,
V - ( p - ^ f ^ V f ^ ^ f ^ A g V A ^ A g f 00 Ac VAd ) )
=
U sing t h e i d e n t i t y
vW
th is
= ^B^C'
(2 .2 5 )
can be w r i t t e n
v - (p- 1 ( f ACV fBC- 2 f ACABVAc - 2 f BCAAVAc ) ) = 0.
T h is ,
th a t
in t u r n ,
(2 .2 6 )
i m p l i e s t h e e x i s t e n c e o f t h e s c a l a r p o t e n t i a l s ^ c such
'
’ *AC ’
I f A6 v f B r 2 f A
V
V
2 f CV
AB> '
■
( 2 - 27)
One n otes t h a t
.
v /A =
= i^ p - V f X )
= -P-1 Vp2 = -2Vp
(2.28)
or
V *Aa =
-2V P .
(2 .2 8 a )
T h is i m p l i e s t h a t
V-Vp =
0.
(2 .2 9 )
As a consequence o f E q u a tio n
x 3 1 = p , x4 ' = z
, where z i s
(2 .2 9 ),
one can choose new c o o r d in a t e s
d e f in e d by
Vp = Vz ,
and th e n t h e l i n e
(2 .3 0 )
e lem en t
(2 .1 6 )
becomes
ds2 = f(dt-a)d<|))2 - f * e 2Y (dp2 +dz2 ) + f * p 2 d<t>2
(2 .3 1 )
12
w i t h e2y
= e 2Y/ ( p , 3 2+p ^ 2 ) .
T h is form o f t h e l i n e e le m e n t i s
t h e l i n e ele m e n t in Weyl c a n o n ic a l
An i n v e r s e
re la tio n
7 f Ac =
c o o rd in a te s .
can be found t o E q u a tio n
V
v
known a?
2V
(2 .2 7 )
and is
(2 .3 2 )
bC >
w i t h t h e symmetry c o n d i t i o n on t h e f^Q r e q u i r i n g
fAB(, 1'BA+2V
(2 .3 3 )
BB+2flB7BA) = °-
Let
O
= V
(2 .3 4 )
2a A
A8C,
and d e f i n e
HAC = f AC+1 flA C " W
eACK»
^2 e 3 5 ^
w ith
VK = $ X* V » X.
Then i t
(2 .3 6 )
has been shown, K i n n e r s l e y
7HAC -
(1 9 7 7 a )s th a t
- ^ ' l f AB7HBC >
which i s t h e same form as E q u a tio n s
D e fin in g G =
e q u a t io n s
. <2 - 3 7 >
(2 .2 1 ).
and @ = * i ,
a f t e r some a l g e b r a , t h e f o l l o w i n g
re s u lt.
f n V2$ = (VG+2@*V$)«V$,
(2 .3 8 )
f n V2G = (VG+2@ *V$)'VG.
(2 .3 9 )
These e q u a t io n s ar,e t h e E r n s t e q u a t i o n s , E r n s t
( 1 9 6 8 b ) , and c o n t a in a l l
t h e i n f o r m a t i o n n e c e s s a ry f o r t h e s o l u t i o n o f t h e E in s t e i n - M a x w e l l
e q u a tio n s .
(V 2 i s t h e L a p l a c i a n o p e r a t o r a s s o c ia t e d w i t h t h e t h r e e -
d im e n s io n a l
m e tric
E q u a tio n
(2 .2 9 )
n o n ca n o n ic a l
( d x 3 ) 2 + ( d x ‘t ) 2 +p2d<|>2 ; t h a t i s , 182 / d p 2 + 3 2 / 3 z 2 + l / p 3 / 3 p . )
must a ls o be added t o
c o o r d i n a t e s a r e chosen.
E q u a tio n s
(2 .3 8 ,3 9 )
if
13
N o th in g th u s f a r has been s a id about t h e
e q u a tio n s .
The f i r s t
s e t o f th e s e a r e i d e n t i c a l l y
e q u a t io n s a l l o w f o r t h e y i n
E q u a tio n
t h e f o l l o w i n g two e q u a t i o n s .
:
-I
P Y»3 =
and R ^
zero.
fie ld
The R ^
( 2 . 1 6 ) t o be found by i n t e g r a t i n g
f _2
%
[ G , 3 G * , lt+G,ltG * , 3 + 2 6 ( G , 3 » * , 4 +G, 4 4>*,3 ) +
2 $ * ( G * , 3 * , 4 + G * , 4 @ , 3 ) + 4 * * $ ( $ , 3 @ * , 4 +@,4 $ * , 3 ) ]
-
r \ $ , 3 $ * , 4+$*,3*,4),
(2.40)
and
-2
P ^Ysi+ = -
%
[ G ,4 G * , 4 -G,3G*,3+2@(G,4 $ * , 4 - G , 3 $ * , 3 ) +
2 $ * ( G * 94» , 4 - G * $3 » , 3 ) + 4 * * $ ( 6 , 4» * , 4 - $ , 3» * s 3 ) ]
+
(2 .4 1 )
If
s o lu tio n s
t o E q u a tio n s
(2 .2 1 ,3 7 )
r i g h t - h a n d s id e s o f E q u a tio n s
(2 .2 1 ,3 7 )
or
(2 .3 8 ,3 9 )
or (2 .3 8 ,3 9 )
(2 .4 0 ,4 1 )
a r e s o lv e d f i r s t ,
a re known, th e n t h e
a r e known.
H ence, E q u a tio n s
and th en E q u a tio n s
(2 .4 0 ,4 ).)
are in te g r a te d .
The f i e l d
e q u a t io n s have now been i n t r o d u c e d .
t h e e q u a t io n s e x i s t ,
necessary.
but t h e s e s h a l l
E q u a tio n s ( 2 . 2 1 , 3 7 )
suggest t h a t t h e g r a v i t a t i o n a l
be i n t r o d u c e d o n ly where
or (2 .3 8 ,3 9 )
fie ld
O t h e r forms o f
a r e v e ry e l e g a n t and
and e l e c t r o m a g n e t i c f i e l d
have much
i n common.
The n e x t two c h a p t e r s w i l l
usefu l
in f in d in g
d e v e lo p some m a th e m a tic a l machinery
new s o l u t i o n s t o t h e s e e q u a t i o n s .
14
CHAPTER 3
TRANSFORMATION GROUPS
In t h i s
c h a p te r, tra n s fo rm a tio n
groups w i l l
be re v ie w e d and a
method p re s e n te d by example t o f i n d them f o r a g iv e n s e t o f p a r t i a l
d iffe re n tia l
e q u a tio n s .
c lo s e ly th a t
g iven by Hamermesh (1 9 6 2 :
64)
and i s
s u ffic ie n t
The p r e s e n t a t i o n g iv e n i n t h i s
2 7 9 ^ 3 2 1 ) and Cosgrove
f o r t h e u n d e r s t a n d in g o f t h i s t h e s i s ,
d e t a i l e d e x p o s i t i o n c o n c e rn in g t r a n s f o r m a t i o n
E is e n h a rt
c h a p te r fo llo w s
groups i s
(1 9 7 9 a :
37-
A more
g iv e n i n
(1933),
P ro p e rtie s
o f T r a n s f o r m a t io n Groups
C o n s id e r t h e t r a n s f o r m a t i o n on t h e v a r i a b l e s
x \
i
= l,2 ,,,,,n ,
gi ven by
z
x1 +X1 = Ti ( x 1 ,X2 , . . . , x n ; a 1 , a 2' , . . . ^at )
(3 .1 )
o r s y m b o l i c a l l y by
Xi - X i
= Ti (X ja ).
The f u n c t i o n s f 1 a r e a n a l y t i c
th a t
is ,
in t h e p a ra m e te rs aa , w i t h a = l , 2 , . . . , t ;
t h e y a r e e x p a n d a b le in a c o n v e rg e n t T a y l o r s e r i e s .
f o r th e tra n s fo rm a tio n s
c e rta in
(3 .1 a )
(3 .1 )
In o r d e r
t o form a g ro u p , t h e f i must s a t i s f y
c o n d itio n s .
Group c l o s u r e r e q u i r e s t h a t
p a ra m e te rs a “ and ba a r e p erfo rm ed i n
p a ra m e te rs ca such t h a t
if
two t r a n s f o r m a t i o n s w it h
s u c c e s s io n , th e n t h e r e e x i s t s
15
f 1 (x ;c )
= f 1 (x ; b) -
fi(f(x ;a );b ),
(3 ,2 )
T h is means t h a t t h e c01 may be e x p ress ed i n terms o f t h e a ° and b” as
c" =
The »a w i l l
(3 .3 ),
(3 .3 )
be assumed a n a l y t i c
E q u a tio n s
(3 .2 )
f 1 (x ,» (a ;b ))
in
its
may be w r i t t e n
arg u m en ts.
U sin g E q u a tio n s
as
= f 1 (f(x ;a );b ).
(3 .4 )
E x is t e n c e o f t h e group i d e n t i t y
im p lie s th e e x is te n c e o f th e
u n iq u e p a r a m e t e r s e t e ° such t h a t
X1 = f 1 (x ;e )
fo r a ll
=
x1
(3 .5 )
x.
Each t r a n s f o r m a t i o n must have an i n v e r s e .
T h is im p lie s , t h a t
f o r each p a r a m e t e r s e t a " , t h e r e i s a p a r a m e t e r s e t aa such t h a t
,
T h is
X1 = f 1 (x ;a )
= f 1(f(x ;a );a )
= X i .
im p lie s t h a t th e tra n s fo rm a tio n s
(3 .1 )
(3 .6 )
a r e s o l v a b l e f o r t h e Xi
in
term s o f t h e x 1 ; t h e c o n d i t i o n on t h e f 1* b e in g t h a t t h e J a c o b ia n o f t h e
t r a n s f o r m a t i o n must be d i f f e r e n t
E q u a tio n s
(3 .6 )
from z e r o .
Taken w i t h E q u a tio n s
im p lie s t h a t
e« = $ « ( a ; a ) ,
which i n t u r n
(s i? )
i m p l i e s t h a t t h e a“ can be w r i t t e n
s in c e t h e e® a r e f i x e d ;
th a t
in term s o f t h e a01,
is ,
Sa = o " ( a ) .
The fia ( a ) w i l l
(3 .3 ),
(3 .8 )
be assumed a n a l y t i c .
The an aI y t i c i t y
re q u ire m e n ts a r e
used t o make t h e group a L i e group.
U sin g E q u a tio n s
(3 .3 ,8 ),
it
is
p o s s i b l e t o show t h a t t h e
may be e x p ress ed in term s o f t h e a a and c“ .
In o t h e r w o rd s , t h e
a®
O
16
J a c o b ia n s
|3 » a /8 a ^ |
and
|3 » a / 3 b ^ |
3»a / 9 a p and 3$a / 3 b e a r e i n v e r t i b l e
A tte n tio n w i l l
In fin ite s im a l
id e n tity ;
im p ly in g t h e m a t r ic e s
m a tric e s .
now be s h i f t e d
( 3 .1 ) to th e in f in it e s im a l
(3 .1 ).
a r e n o n z e ro ,
from t h e f i n i t e
tra n s fo rm a tio n s
tra n s fo rm a tio n s
a s s o c ia t e d w i t h
tra n s fo rm a tio n s are tra n s fo rm a tio n s
t h e i r p a ra m e te rs a r e c lo s e t o t h e i d e n t i t y
E q u atio n s
c lo s e t o t h e
p a ra m e te rs ea .
L e t aa be t h e p a ra m e te rs t h a t t a k e t h e p o i n t s x 1 i n t o x 1.
n e ig h b o r in g p a ra m e te rs a^+da™ w i l l
Xi -Klxi s s in c e t h e f 1 a r e a n a l y t i c
t a k e t h e p o in t s
i n t h e aa .
x1 i n t o t h e p o in t s
Param ete rs t h a t a re c lo s e
t o t h e i d e n t i t y , Bot-^Saots can be found which w i l l
t h e p o i n t s X1V d x * .
The
ta k e th e p o in ts x 1 in t o
Two d i f f e r e n t p ath s from Xi t o X1V d x 1 a r e th u s
p o s s i b l e and a r e g iv e n by
X1 +dx1 = f i ( x ; a + d a )
(3 .9 )
o r t h e p a th
X1 = f 1 ( x ; a ) , and x1 +dx1 = f 1 ( x , e + S a ) .
(3 .1 0 )
Expanding t h e l a s t e q u a t io n i n a T a y l o r s e r i e s y i e l d s
d x 1 = ( d f 1 ( x ; a ) / 3 a S ) a _ gSa^ = F1 g (X )S a 9 .
E q u a tio n
(3 .1 1 )
(3 .3 ) y ie ld s
aa +daa = $a ( a ; e + S a )
(3 .1 2 )
so t h a t by exp an d in g t h e l a s t e q u a t io n y i e l d s
d a" = ( 3 $ “ ( a ; b ) / 3 b &) b ' = e 6a0
At a a = e a s Ga
p
= 0 a g ( a ) 6ae .
(3 .1 3 )
= Sct .
p
S in c e Ga ( a ) i s
in v e rtib le .
E q u a tio n s
(3 .1 3 )
a re s o lv a b le f o r
P
t h e Sact i n term s o f t h e d aa .
Sact = IjJctg (a ) d a 3 .
(3 .1 4 )
17
w i t h t h e i|»a 0 ( a )
d e f i n e d by
P
and * % ( * )
S u b s t i t u t i n g E q u a tio n s
(3 .1 4 )
= *"n-
O '" )
i n t o E q u a tio n s
(3 .1 1 ) y ie ld s
d x 1* = F1 g ( x ) / a ( a ) d a a
or,
(3 .1 6 )
e q u iv a le n tly ,
3 x 1 / 3 a a = F1 g ( x ) / a ( a ) .
If
t h e F1 ^ ( x )
and i|^a ( a )
(3 .1 7 )
a r e known, th e n t h e t r a n s f o r m a t i o n s
( 3 . 1 ) may
be fo u n d .
The i n f i n i t e s i m a l
tra n s fo rm a tio n s
(3 .1 0 )
change
th e fu n c tio n
A ( x ) as f o l l o w s ,
dA = O A Z 9Xi Jdxi
= ( 9AZaxi ) F 1'( x ) 5 a 3
P
= Sa3 F1 3 (X )(B A Z d x i ) =
w it h t h e o p e r a t o r X
.
Sa3 XgA
(3 .1 8 )
d e f i n e d by
3
(3 .1 9 )
Xg = F1 3 ( X ) O Z a x 1 ) .
The Xg a r e c a l l e d
op erato r
in fin ite s im a l
( ! + X flSa3 ) i s
o f t h e group because t h e
c lo s e t o t h e i d e n t i t y .
P
y ie ld s
o p e ra to rs
Using (1+X Sa^) on t h e x1
"
E q u a tio n s
P
(3 .1 1 ).
D e f i n i n g t h e commutator o f t h e i n f i n i t e s i m a l
!*« ,
it
X6 I -
w it h t h e c K
X6 ] '
(3 .2 0 )
XoA 6
- x „ 8x ,o ’
i s shown i n Hamermesh (1 9 6 2 :
l Xo -
o p e r a t o r s by
2 9 9 - 3 0 1 ) t h a t t h e commutators s a t i s f y
(3 .2 1 )
= V t -
in d e p e n d e n t o f t h e aa ; and t h e y a re c a l l e d
the s tru c tu re
010
co n stan ts.
It
has been shown, see Cohn (1 9 5 7 :
9 4 -1 0 6 ), th a t
if
the
s t r u c t u r e c o n s t a n t s a r e known, th e n t h e group t r a n s f o r m a t i o n s can be
found v i a t h e e q u a t io n s
18
Fj J a F 1 / a xJ ) - F J ( a F1 / a x J ) = cK
o'
T
7
t '
a
TO
F1 ,
(3 .2 2 )
K
and
(av|;Kij/aax)-(dij,Kx/3ay ) = ^
(3 .2 3 )
N eed less t o s a y , t h e e q u a t io n s above a r e d i f f i c u l t
can be s o l v e d , th e n E q u a tio n s
im p o r t a n t p o i n t
is th a t
if
(3 .1 7 )
to s o lv e .
a r e th e n used t o f i n d
If
f 1.
th e y
The
t h e Ck tq a r e found and c o rre s p o n d t o th o s e o f
a known g ro u p , t h e t r a n s f o r m a t i o n s a r e d e te rm in e d n e a r t h e i d e n t i t y .
A p p lic a tio n s
T r a n s f o r m a t io n , groups a r e used t o g e n e r a t e new s o l u t i o n s
g iv e n s o l u t i o n
w ith th e f l a t
m e tric ;
th a t
o f th e E in s te in -M a x w e ll
space m e t r i c ,
is ,
Fo r e x a m p le , s t a r t i n g
i s p o s s i b l e t o g e n e r a t e t h e S c h w a rz s c h ild
a t r a n s f o r m a t i o n e x i s t s which has t h e p h y s ic a l
o f p ro d u c in g mass.
m a th e m a tic a l
it
e q u a tio n s .
A lth o u g h i n t h e c l a s s i c a l
c u rio s ity ,
from a
th e o ry t h i s
i n a quantum t h e o r y i t
p ro p erty
i s o n ly a
c o u ld have m a jo r
im p o r t a n c e .
In i t s
most g e n e ra l
s e n s e , t h e symmetry t r a n s f o r m a t i o n changes
both t h e dependent and t h e in d e p e n d e n t v a r i a b l e s w h i l e l e a v i n g t h e f i e l d
e q u a t io n s i n v a r i a n t .
i n v a r i a n c e and i t
C la rific a tio n
i s w o r t h w h ile t o
is
needed as t o t h e meaning o f
re v ie w t h i s
b e f o r e e x a m in in g methods
o f f i n d i n g symmetry t r a n s f o r m a t i o n s .
Take t h e L a p la c e e q u a t io n i n t h r e e dim ensions as t h e f i r s t
e xam p le.
It
is w r itte n
i n m a n i f e s t l y c o v a r i a n t form
V2$ = 0.
T h is e q u a t io n holds t r u e
(3 .2 4 )
in a l l
c o o rd in a te s ;
however, t h e p a r t i a l
19
d iffe re n tia l
e q u a t io n
c o o rd in a te s .
it
r e p r e s e n t s has d i f f e r e n t forms i n d i f f e r e n t
In C a r t e s i a n c o o r d i n a t e s , t h e p a r t i a l
d iffe re n tia l
e q u a t io n i s
( 3 2» / 9 x 2 ) + ( 9 2» / 3 y 2 ) + ( 9 2$ / 3 z 2 ) = O9
in s p h e ric a l
c o o rd in a te s .
_9 ' r 2 _3$
dr
I
r2
3p
3 s i ne _3$
1
r 2 s i ne
.
p3$
_3
+
3r ;
and i n c y l i n d r i c a l
It
(3 .2 5 )
c o o rd in a te s ,
W
I
3p
r
90
its
92*
, 92$
)2 3<j>2
9Z2
I
92$
0 ,
(3 .2 6 )
r 2 s i n 2 0 3<j>2
fo rm i s
(3 .2 7 )
i s c l e a r t h a t t h e form o f t h e p a r t i a l
d iffe re n tia l
on t h e c o o r d in a t e s ch o sen , even though E q u a tio n
e q u a t io n depends
(3 .2 4 )
is
c o v a ria n t.
I n v a r i a n c e o f form i s p r e c i s e l y what i s meant when t h e e q u a t io n s a re
s a id t o be i n v a r i a n t under a g iv e n symmetry t r a n s f o r m a t i o n .
The t r a n s f o r m a t i o n
(3 .2 5 )
o f c o o r d in a t e s which do le a v e E q u atio n s
in v a r ia n t are th e r o ta tio n s
th e ro ta tio n
o rth o g o n a l
o f c o o rd in a te s ,
and i n v e r s i o n s .
(x ,y ,z )
tran sfo rm s
t r a n s f o r m a t i o n , and E q u a tio n
(3 .2 5 )
F o r i n s t a n c e , under
in to
(x ,y ,z )
by an
becomes
( 32 » / 3x2) + ( 92 $ / 9y2) + ( 92 * / 9 z2 ) =; 0.
T h is means t h a t
$ (x ,y ,z )
is
if
$ (x ,y ,z )
i s a s o l u t i o n t o E q u atio n
a ls o a s o l u t i o n t o E q u a tio n
exp ress ed i n term s o f
( 3 , 28 )
(3 .2 5 )
( 3 . 2 5 ) , th en
when ( x , y » z )
are
(x ,y ,z ).
As an example o f a n o n c o o r d in a t e t r a n s f o r m a t i o n , t a k e t h e
e q u a t io n t o be t h e vacuum E r n s t e q u a t i o n ,
(G +G *)V 2G = 2vG«VG .
see E q u a tio n
'
(2 .3 9 ),
(3 .2 8 )
20
The t r a n s f o r m a t i o n , c a l l e d t h e E h l e r ' s
tra n s fo rm a tio n ,
G->6
Y is
a real
(3 .2 8 )
is
d u a lity
g iv e n by
= G /(l+ iy G );
p aram eter.
(3 .2 9 )
It
is
s im p le t o v e r i f y t h a t G s a t i s f i e s
E q u atio n
and i s a new s o l u t i o n .
How a re t r a n s f o r m a t i o n s
tra n s fo rm a tio n
e a rly
or g r a v ita tio n a l
(3 .2 9 )
found?
in a p h y s ic is t's
o th e r tra n s fo rm a tio n s
in
in d e p e n d e n t o f i n t u i t i o n
in fin ite s im a l
O b v io u s ly , t h e r o t a t i o n s a r e e n c o u n tered
fe e lin g .
at fir s t
en co u n ter.
whereby such t r a n s f o r m a t i o n s
and t h e n t h e i n t e g r a t i o n
may be p e rfo rm e d t o y i e l d
th e f i n i t e
Many
symmetry t r a n s f o r m a t i o n s
in
is
needed
fin d in g the
d iffe re n tia l
(3 .2 2 ,2 3 ,1 7 )
A lth o u g h t h i ?
i s co m p lex, i t
(In tu itio n
(3 .2 9 ),
can be fo und.
o f E q u a tio n s
tra n s fo rm a tio n .
ad v a n ta g e o f not r e l y i n g on i n t u i t i o n .
it
A method i s
t r a n s f o r m a t i o n which le a v e s t h e p a r t i a l
method o f f i n d i n g
meaning.
E h le r 's tra n s fo rm a tio n
such a method e x i s t s and c o n s i s t s f i r s t
e q u a t io n s i n v a r i a n t ,
seldom i s
and t h e E h l e r ' s
p h y s ic s a r e modeled a f t e r t h e r o t a t i o n s and
not i n t u i t i v e
F o rtu n a te ly ,
th e ro ta tio n s
c a r e e r and have a c l e a r p h y s ic a l
have a s i m i l a r i n t u i t i v e
h o w ever, i s
lik e
a g a in has t h e
good t o h a v e , but
proved w i t h o u t e f f o r t . )
A Method
The vacuum E r n s t e q u a t io n
th is
method.
In p a r t i c u l a r ,
t r a n s f o r m a t i o n p a ra m e te r s .
I
be used t o d em o n strate
t h e E h le r ^ s t r a n s f o r m a t i o n
one o f t h e symmetry t r a n s f o r m a t i o n s
C o n s id e r a f i n i t e
(3 .2 8 ) w ill
(3 .2 9 ) w ill
be
fo u n d ,
tra n s fo rm a tio n ta k in g G in t o G w ith the
R e s t r i c t G t o be an e x p l i c i t
fu n c tio n o f
21
o n ly G, G *, and s , w r i t t e n
o u t as
G+G = G ( G , G * ; s ) .
S im ila rly
f o r G *,
,
i
G **G * = G * ( G , G * ; s )
G is
(3 .3 0 )
.
(3 .3 1 )
n o t assumed t o be t h e complex c o n ju g a t e o f G*.
(3 .3 0 ,3 1 )
Expand E q u atio n s
i n a M c C l a u r i n ' s s e r i e s w i t h t h e boundary c o n d i t i o n s G(s = 0)
and G *(s = 0) = G *.
To f i r s t
G = G + s A (G ,G *)iS A
o rd er in
s,
dG
ds
(3 .3 2 )
and
(3 .3 3 )
G* = G * + s B ( G , G * j , B = -T-—
It
i s n e c e s s a ry f o r G t o s a t i s f y E q u a tio n
(3 .2 8 )
and f o r G* t o s a t i s f y
t h e complex c o n ju g a t e e q u a t io n
(G +G *)V 2 G* = 2 V G *.V G *.
U sin g t h e c h a in
(3 .3 4 )
ru le o f c a lc u lu s ,
VG = vG+s(AgVG+Ag*vG*)
(3 .3 5 )
and
V2G = V2 G+s(AgV2 G+AG* V 2 G*+AGGVG«VG+
2Agg *VG.VG*+A g* g * V G * . V G * ) ,
w i t h Ag = 3A /9G , Agg4, = 32 A / 3GdG *, e t c .
'
A ls o ,
(G+G*) = (G + G *)+ s (A + B ) .
(3 .3 7 )
S i m i l a r e x p re s s io n s f o r vG* and V2 G* can a l s o be fo u n d .
re s u lts
i n t o E q u a tio n s
(3 .2 8 ,3 4 )
(3 .3 6 )
P u t t i n g th e s e
and u s in g t h e f a c t t h a t G and G*
22
a lr e a d y s a t i s f y th ese e q u a tio n s , th e f i r s t
equal t o z e r o .
o r d e r term s i n s a r e l e f t
The r e s u l t i n g e q u a t io n s a r e
(2(A+B)/(G+G*) - 2Aq + (G+G*)AGG)VG.VG +
( 2 (G + G *)A GG* -
4AG*)VG«VG* +
((G+G*)AG* G* + 2AG<r)vG*.vG* = 0 ,
(3.38)
and
(2 (A + B )/(G + G *)
-
( 2 (G + G *)B gg* -
4BG)VG.VG* + (( G + G * ) B GG + 2Bg ) v G.VG = 0.
The c o e f f i c i e n t s
2Bg* + (G+G*)BG* G* ) V G * .V G * +
o f VG-vG, v G * - v G * , and vG®yG* a r e s e t equal t o z e r o .
(G and G* a r e assumed f u n c t i o n a l l y
a fu n c tio n
of
d iffe re n tia l
(3 .3 9 )
(G -G *).)
e q u a t io n s
in d e p e n d e n t; t h e r e f o r e ,
(G+G*) i $
not
The f o l l o w i n g c o u p le d s e t o f l i n e a r p a r t i a l
re s u lts :
(G +G *)2A GG- 2 ( G + G * ) A G+2A .= - 2 B ,
(G+G*)Agg*-2AG£ - 0,
(G + G *)A g *G*+ 2 A G* = 0 ,
(G+G*)2BG* G* - 2 ( G + G * ) B G*+2B = - 2 A ,
( g+ g * ) b g* g - 2 b g = ° '
(G+G*)B gg +2B g = 0 .
F irs t
(G + G *).
(3 .4 0 a -f)
s o lv e E q u a tio n
(3 .4 0 c )
T h is y i e l d s t h e s o l u t i o n
A=
From E q u a tio n
- f (G )/(G + G *)+ g (G ).
(3 .4 0 b )
^ r( I n f )
it
is
= 4 / (G+G*)
found t h a t
.
u s in g t h e i n t e g r a t i n g f a c t o r
23
S in c e f
is a fu n c tio n
c o n fu s e t h e f
o f G o n l y , t h e above i m p l i e s t h a t f = 0.
h e re w i t h t h e m e t r i c f u n c t i o n
The o n ly s o l u t i o n p o s s i b l e i s th e n A =; g ( G ) .
E q u a tio n s
(3 .1 7 f,e ),
d iffe re n tia l
it
is
(Do not
in t r o d u c e d i n C h a p te r 2 . )
In a s i m i l a r manner, u s in g
shown t h a t B = h ( G * ) .
e q u a t io n s now r e s u l t from E q u a tio n s
Two co u p le d o r d i n a r y
(3 ,4 0 a ,d ).
They a r e
( G+G*) 2 g^Q- 2 ( G+G*) gQ+2g = -2h
and
(G + G *)2 h G* G* - 2 ( G + G * ) h G*+2h = - 2 g .
To s o lv e t h e s e two e q u a t i o n s , use t h e power s e r i e s e x p a n s io n s
g = I
Oi G1
i= 0 1
w i t h g-j, h-j c o n s t a n t s .
A = go+OiG+gaG2 ,
(3 .4 1 a )
B = - g 0+ g 1G * - g 2 ( G * ) 2 ,
(3 .4 1 b )
w ith g0 , g l f
of
is
h = I
h (G *)1 ,
i= 0 1
The s o l u t i o n i s
and
g2 a r b i t r a r y
G* = ( G ) * i s
re a l.
I f t h e added re q u ire m e n t
g i v e n , th e n t h e g0 and gg a r e p u r e l y i m a g i n a r y , and g^
The i n f i n i t e s i m a l
t r a n s f o r m a t i o n s a r e now w r i t t e n
G+G = G +s(g0+ g 1G+g2 G2 ) ,
(3 .4 2 a )
G*+G* = G * + s ( - g 0+ g 1G *-g 2 ( G * ) 2 ) .
(3 .4 2 b )
To work o n ly w i t h
a^ = i s g 2 .
real
p a r a m e t e r s , d e f i n e a * = i s g q , a2 = sg ^ , and
Then t h e i n f i n i t e s i m a l
fo rm o f E q u a tio n
t r a n s f o r m a t i o n s may be w r i t t e n
in t h e
( 3 .1 1 ) w ith
G = G - i a 1+a2 G - i a 3 G2 ,
(3 .4 3 a )
G* = G * + i a 1+a2 G * + i a 3 ( G * ) 2 .
(3 .4 3 b )
The i n f i n i t e s i m a l
E q u a tio n s
complex c o n s t a n t s .
(3 .1 9 ),
o perators
are,
com paring E q u a tio n s
( 3 . 4 3 a , b ) w it h
24
X3 = i 9 / 3 G * - i 3 / 3 G
,
X2 = G 3/3G +G *3 /3G * ,
X3 = i ( G * ) 2 3 / d G * - i G 2 d/dG ,
(3 .4 4 )
w it h t h e commutators
[X i*
E q u a tio n s
X2 ] -
X1 ,
(3 .4 5 ) y ie ld
i n t e g r a t e t h e g ro u p .
[ X 1 , X3 ] = - 2 X 2 ,
[X 2 , X3 ] = X3 .
(3 .4 5 )
t h e s t r u c t u r e c o n s t a n t s , and th e s e may be used t o
I n s t e a d , d e f i n e t h e f o l l o w i n g o p e r a t o r s by
J l = ! / E ( X 1H-X3 ) ,
J 2 = X2 ,
J3 = !/E fX 1 -
X3 ) .
(3 .4 6 )
Then t h e new commutators a r e g iv e n by
[ J 11J 2 ] = J 3 ,
[ ^ 2 *^ 3 ] = - ^ 1 »
[*^3 » ^ i] = - ^ 2 *
(3 .4 7 )
These a r e t h e same commutation r e l a t i o n s as f o r t h e L o r e n t z
tra n s fo rm a tio n s
in t h r e e d im e n s io n s , Hamermesh ( 1 9 6 2 :
t h e group S 0 ( 2 , l ) .
3 0 7 ) , and d e f i n e
T h u s, t h e symmetry group found above i s
lo c a lly
is o m o rp h ic t o 5 0 ( 2 , 1 ) .
One P a ra m e te r Subgroups
One p a ra m e te r subgroups o f E q u a tio n s
exam ined.
(3 .4 3 a ,b ) w i l l
An im p o r t a n t p o i n t t o n o te on one p a ra m e te r groups i s t h a t
t h e y a re a l l
lo c a lly
is o m o rp h ic t o t h e t r a n s l a t i o n
d im e n s io n , Hamermesh ( 1 9 6 2 :
2 9 5 ).
g ro u p s , t h e 6a o f E q u a tio n s
(3 .3 )
group i n one
T h is means t h a t f o r one p a ra m e te r
s a tis fy
<&(a;b) = a + b .
(3 .4 8 )
T h is i m p l i e s t h a t t h e tjj(a) i n E q u a tio n s
E q u a tio n
now be
(3 .1 7 )
(3 .1 7 )
i s equal
t o on e, meaning
can be w r i t t e n
I r = F 1 ( X) .
and t o g et t h e t r a n s f o r m a t i o n
s o lv e E q u a tio n ( 3 . 4 9 ) .
.
fu n c tio n s
(3 .4 9 )
Ti , i t
i s o n ly n e c e s s a ry t o
25
The one p a ra m e te r subgroups o f E q u a tio n s
fo llo w in g d i f f e r e n t i a l
= i ,
= G *,
d a2
da2
da3
= -iG 2 , -i® * =
da3
'
a1 = I ,
a2 = a3 = 0;
'
da1
= G,
These e q u a t io n s a r e e a s i l y
a2 = I ,
i(G *)2 ,
(3 .5 0 a )
a 1 = a3 = 0;
a3 = I ,
(3 .5 0 b )
a 1 = a2 = 0.
,
(3 .5 1 a )
(3 .5 1 b )
e a 2 G,
G = G / ( l + i a 3 G ).
y (3 .5 1 c )
As p ro m is e d , t h e E h l e r ' s t r a n s f o r m a t i o n
(3 .5 1 c )
(3 .5 0 c )
s o lv e d and have s o l u t i o n ;
G = G- i a 1 ,
G=
s a t is f y the
e q u a t io n s :
= - if
da1
(3 .4 3 a ,b )
has been reproduced i n E q u atio n
as a one p a ra m e te r subgroup o f t h e symmetry group o f form g iven
by E q u a tio n s
(3 .3 0 ,3 1 ).
fo llo w s .
E q u a tio n
E q u a tio n s
(2 .2 7 )
The o t h e r two subgroups a r e e x p l a i n e d as
(3 .5 1 a )
re s u lts
from t h e d e f i n i t i o n
up t o an a d d i t i v e c o n s t a n t , and E q u a tio n
co rresp o n d s t o t h e change o f c o o r d in a t e s
by two m a t r i x w it h d e t e r m in a n t equal
d )’
(3 .5 1 b )
( t , < | > ) > ( e " ^ ^ ^ t , e ^ a <ti).
The whole group o f t r a n s f o r m a t i o n s
A = (c
o f i|iAg i n
is
r e p r e s e n t e d by a r e a l
two
t o one;
ad-l)C = I ,
(3 .5 2 )
such t h a t
G = (d G -ic )/(a + ib G ).
See Cosgrove
(1 9 7 9 b :
E q u a tio n 2 . 3 ) )
(3 .5 3 )
f o r exam ple.
a r e p e r f o r m e d . i n s u c c e s s io n , t h e f i n a l
If
two t r a n s f o r m a t i o n s
tran sfo rm ed G w i l l
have th q same
26
form g iv e n i n
E q u a tio n
(3 .5 3 ),
but t h e e le m e n ts w i l l
r e p r e s e n t a t i o n m a t r ic e s m u l t i p l i e d
K in n e rs le y
(1 9 7 7 a :1 5 3 1 )
and (P )
to g eth er.
by Cosgrove
T h is
be th o s e o f t h e two .
group i s
named H by
(1979a:5 6).
The N eugebauer-K ram er Mapping
Only c o n tin u o u s t r a n s f o r m a t i o n s
p o in t.
w ill
At t h i s
have been c o n s id e re d up t o t h i s
p o i n t , a v e ry im p o r t a n t d i s c r e t e symmetry t r a n s f o r m a t i o n
be r e v ie w e d .
D e f i ne
2 f = G+G* and 21i|> = G -G *.
(3 .5 4 )
U sing t h e vacuum E r n s t e q u a t io n s y i e l d s
V3 O ( f “ 1V f + f ’ 2^V^) = 0 ,
(3 .5 5 )
V3* ( f ~ 2V<|>) = 0 ,
.
(3 .5 6 )
and t h e s e a r e e q u i v a l e n t t o t h e E r n s t e q u a t i o n .
E q u a tio n
(3 .5 6 )
i m p l i e s t h e e x i s t e n c e o f a p o t e n t i a l w such
th a t
Vio = p f " 2Vij).
(3.57)
E l i m i n a t i n g tJj i n f a v o r o f <o y i e l d s
.
V3 0 ( f ^ V f + p - ^ i o V i o ) =O ,
(3 .5 8 )
V3O(p"2f 2Vio) = O . ,
(3.59)
(E q u a tio n s
(3 .5 8 ,5 9 )
are
re s t a t e m e n t s
t h e Lewis p a r a m e t e r i z a t i o n .
r e s t a t e m e n t s o f E q u a tio n s
It
(3 .5 5 ,5 6 )
t h e f o l l o w i n g two e q u a t i o n s ,
L ik e w is e ,
o f E q u a tio n s
E q u a tio n s
(2 .2 6 )
(3 .5 5 ,5 6 )
f o r vacuum in
are
(2 .2 7 ).)
was noted t h a t t h e r e i s a mapping between E q u a tio n s
and ( 3 . 5 8 , 5 9 ) ,
f+pf
-I
g iv e n by
, lo-Hij), i|)-*—iio,
(3 .6 0 )
27
which sends one s e t o f e q u a t io n s i n t o t h e o t h e r and v i c e v e r s a .
mapping i s t h e N eu g eb au er-K ram er mapping and w i l l
be used i n
T h is
la tte r
ch ap ters.
The co n cep t o f group t r a n s f o r m a t i o n s
has now been re v ie w e d .
A
method has been p re s e n te d by which symmetry groups may be found f o r
g iv e n p a r t i a l
d iffe re n tia l
e q u a t io n s .
U sin g t h i s
method f o r t h e vacuum
E r n s t e q u a t i o n , a known symmetry group has been re p ro d u c e d .
symmetry group w i l l
b rie fly
r e v ie w e d .
T h is
re a p p e a r i n C h a p te r 5 where t h e vacuum case w i l l
be
28
CHAPTER 4
BACKLUND TRANSFORMATIONS
A n o th e r t r a n s f o r m a t i o n which has proved u s e f u l
new s o l u t i o n s
E is e n h a rt
p a rtia l
from o ld s o l u t i o n s
(1 9 0 9 :2 8 4 ).
d e riv a tiv e s .
s o lu tio n s .
In t h i s
in
g e n e r a t in g
i s t h e Backlund t r a n s f o r m a t i o n ,
see
The v a r i a b l e s t o be t r a n s f o r m e d a r e now t h e f i r s t
These,
i n t u r n , may be i n t e g r a t e d t o y i e l d
c h a p te r,
new
Backlund t r a n s f o r m a t i o n s a r e in t r o d u c e d and
th e n d e m o n s tra te d by an ex a m p le .
The p r e s e n t a t i o n
f o l l o w s c l o s e l y t h a t g iv e n by Lamb ( 1 9 7 4 )
in t h i s
ch ap ter
and Neugebauer (1 ,9 7 9 ).
••
P r o p e r t i e s o f t h e Backlund T r a n s f o r m a t io n
The n o t a t i o n used i n t h i s
p = 3 z /3 x ,
p = 3 z /3 x ,
s = 32 z / 3 x 3 y ,
q = d z /d y , q = 3 z /3 y ,
s = 32 z / 3 x d y ,
in d ic a te the p a r t ia l
s e c tio n
t
= d2 z / 3 y 2 , I
d e r i v a t i v e w ith
is :
z = z (x ,y ),
r = 32 z / 3 x 2 ,
= 32 z / 3 y 2 .
z = z (x ,y ),
r = d2 z / d x ? ,
S u b s c r ip t s
resp ect t o th e s u b s c rip te d
v a ria b le .
The Backlund t r a n s f o r m a t i o n t o be c o n s id e re d i s w r i t t e n
P = f(x ,y ,z ,z ,p ,q ),
q = h (x ,y ,z ,z ,p ,q ),
(4 .1 )
and
x = x, y = y.
In o r d e r t h a t t h e i n t e g r a b i l i t y
s a tis fy
(4 .1 a )
c o n d i t i o n py = qx h o l d s ,
f
and h must .
29
W -
Py - S x
— f — . - h - + f —□ - h -p + f Q - h p +
y
x
z
zK
zh
z k
(f p -
h -)s + f - t
= h -r = 0.
T h is e q u a t io n may be s a t i s f i e d
f-
= h - = 0 and f -
-
h- + f-q
(4. 2)
id e n tic a lly
-
in which case f -
h -p K+ f^ q -
h^p -
a ls o be c o n s id e r e d as a s e c o n d - o r d e r p a r t i a l
z.
If
th is
e q u a t io n i s
0.
- h- =
E q u a tio n
d iffe re n tia l
(4 .2 )
may
e q u a t io n f o r
o f a form known as Monge-Ampere, F o r s y t h e
( 1 9 5 9 : 2 0 0 ) , then th e tra n s fo rm a tio n s
(4 .1 )
a r e known as Backlund
tra n s fo rm a tio n s .
In p r a c t i c e ,
it
is th e p a r t i a l
d iffe re n tia l
e q u a t io n s
which a r e s t a r t e d w i t h and t h e Backlund t r a n s f o r m a t i o n s
(4 .2 )
( 4 , 1 ) which 9 re
so u g h t.
N e u g e b a u e r's Backlund T r a n s f o r m a t io n
The example w i l l
t h e case o f n o n can o n ical
E q u a tio n
(2 .2 9 ).
a g a in be t h e vacuum E r n s t e q u a t io n
c o o rd in a te s , th e f i e l d
W ritte n o u t, the f i e l d
(3 .2 8 ).
e q u a t io n s a l s o in c lu d e
e q u a t io n s a r e
(GtG*)[Gj 3j 3+G>4j 4 + i (p>i3Gi3tPi4G>4)]
- 2[(G
(G+G*)[G* j j +G^m + ^
3
3
=
2
'
+ {G . ) 2 ] ,
9^
9u
3 6
^
3
+ ^
0
(4.3)
4
)]
[(G*j ) + ( G ^ ) ) ,
3
2
4
(4.4)
2
(4.5)
P,3,3+»,4,4 = 0'
It
i s c o n v e n ie n t t o change v a r i a b l e s .
In
F irs t,
a r e changed t o X1 = x 3+ i x 4 and x2 = x3 - i x 4 .
t h e in d e p e n d e n t v a r i a b l e s
E q u atio n s
(4 .3 -5 )
become
30
(G+G* ) [ G , i >2 + 2 ^ ( p , l G , 2 +P , 2 G, l ) ]
(4.3a)
" 2G, 1 G,2»
( G + G * ) [ G * 9 l> 2 + ^ ( p $i G * i 2 + p $2 G * i 1 ) ]
= 26 *^ 6 *^ ,
(4 .4 a )
and
P 1 9 = O.
At t h i s
p o in t,
-
Neugebauer
(1 9 7 9 )
(4 .5 a )
changes t h e o r d e r o f t h e e q u a t io n s by
i n t r o d u c i n g t h e f o l l o w i n g dependent v a r i a b l e s :
M1 = G i / ( G + G * )
, M2 = G * y ( G + G * )
, M3 = p ^ / p ,
(4 .6 a )
, N2 = G 2 /( G + G * )
, N3 = p , / p .
(4 .6 b )
N1 = G* 2 / ( G + G * )
The E q u a tio n s
( 4 . 3 a - 5 a ) th e n become e q u i v a l e n t t o t h e system o f f i r s t
o r d e r co u p led p a r t i a l
Mi , 2 = c i
d iffe re n tia l
Mk Nl
» Ni , l
e q u a t io n s
= ci
Nk Ml
8
w i t h C 111 = c 222 = C3 33 = - C 1 12 = - C 221 = - I ,
c2 23 = - 1 / 2 .
be s a t i s f i e d
E q u a tio n s
and C 132 = C 1 13 = C2 31 =
These e q u a t io n s a l r e a d y c o n t a i n t h e i n t e g r a b i l i t y
c o n d itio n s G 1 2 = G 2 1
w ill
^ . 7)
and G* 1 2 = G* 2 j .
a u to m a tic a lly
if
T h erefo re,
th e tran sfo rm ed M / s
E q u a tio n
and N / s
(4 .2 )
s a tis fy
(4 .7 ).
The Backlund t r a n s f o r m a t i o n
is
assumed t o have t h e f o l l o w i n g
fo rm .
(4 .8 )
M1 - Aj kMk , N1 = ( A - 1 I 1 kNk ,
w it h
X
The r e q u ir e m e n t t h a t
0
0
0
0'
0
Y
(4 .9 )
«0 = Y •
and N^ s a t i s f y
f o l l o w i n g e q u a t io n s f o r a and y .
E q u a tio n s
(4 .7 )
re s u lts
in t h e
31
(4 .1 0 )
dy = Y ( Y - I ) M g d X i t ( Y - I ) N g d X g ,
d ( a Y ” 1^2 ) = [ - Y 1^ 2M2 + ( a Y " 1^2 )(M 2 -M 1 )+ (a Y '‘ 1^2 ) 2 M1Y 1/ 2 ] d x 1 +
[ - Y " 1 / 2 N1+ ( a Y " 1 / 2 ) ( N 1 -N 2 ) + ( a Y " 1 / 2 ) 2 N2Y‘ 1 / 2 ] d x 2 .
The. s o l u t i o n
o f t h e s e e q u a t io n s in tro d u c e s , i n t e g r a t i o n
e x a m p le , t h e s o l u t i o n t o E q u a tio n
(4 ,1 0 )
(4 .1 1 )
co n stan ts.
For
is
(4 .1 0 a )
where b i s an i n t e g r a t i o n
co n stan t, p s a t is f ie s
i s p ' s harmonic c o n j u g a t e .
a d i f f e r e n t tra n s fo rm a tio n
re p re s e n ts a l l
re p re s e n tin g
E q u a tio n
( I i ) a w ill
(4 .9 )
E q u a tio n s
Each s e t o f i n t e g r a t i o n
and w i l l
c o n stan ts.
be w r i t t e n
It
w ill
t h a t t h e p ro d u c t
( I i ) a(Ii)b
(XbOta
0
The 1%
and t h e a
( I i ) a(Ii)b
Ii
fo rm a group.
and ( I i ) a , i t
must be shown
tra n s fo rm a tio n .
is
By
r e p r e s e n t e d by t h e m a t r i x
0
(4 .1 3 )
' 1V a 1'
can be shown t h a t a ba g and Y bY a s a t i s f y
remembering t h a t t h e
(4 .8 ,9 ),
rep resen ts
The a a and y a must s a t i s f y
(Ii)b
1V a lllW
Then i t
( I i ) a.
and z
i n t h e same form as t h e m a t r i x o f
i s a l s o an I i
th e tra n s fo rm a tio n
constan ts
now be shown t h a t t h e
G iven two t r a n s f o r m a t i o n s
(4 .5 a ),
F o r ex a m p le , t h e m a t r i x
w i t h e lem en ts a a , 3 a ,Y a ,
(4 .1 0 ,1 1 ).
d e fin itio n ,
be l a b e l l e d
t h e t r a n s f o r m a t i o n s o f E q u a tio n
rep re s e n ts th e in t e g r a t io n
E q u a tio n
E q u a tio n s
( I i ) a t r a n s f o r m a t i o n a c t s qn t h e
(4 .1 0 ,1 1 )?
(Ii)b M -j,
( I l ) b Ni e
32
Hence, t h e
I 1 t r a n s f o r m a t i o n s e x h i b i t t h e group c l o s u r e p r o p e r t y .
t h e m a t r ic e s
r e p r e s e n t i n g t h e group t r a n s f o r m a t i o n s a r e d i a g o n a l , th e
tra n s fo rm a tio n s
I 1 a r e a l s o c o m m u ta tiv e .
A s s o c ia tiv ity ,
e x is te n c e o f th e i d e n t i t y ,
t r a n s f o r m a t i o n a r e th e n shown t o f o l l o w .
above p r o o f s
s a tis fie s
S in c e
and in v e r s e s f o r each
The m a jo r s t e p i n each o f t h e
i s t h e s te p showing t h a t t h e r e s p e c t i v e m a t r i x elem ent
E q u a tio n s
(4 .1 0 ,1 1 ).
The N eugebauer-K ram er mapping in t r o d u c e d in C h a p te r 3 now p la y s
an i m p o r t a n t r o l e .
In t h e
(M1 ,N 1- )
re a liz a tio n ,
th is
mapping i s
r e p r e s e n t e d by t h e m a t r ic e s U1-^ and V1- ^ 9 where
0
(U jk)
W ith t h i s
-I
0
-I
0
0
1/2 1
1/2
1
-I
0
t/
, and ( V 1 ) 1
J
I 0
0
-I
0
1/2
1/2
I
(4 .1 4 )
d i s c r e t e g ro u p , t o be c a l l e d S , d e f i n e a new group o f
tra n s fo rm a tio n s
Ig by S I 1S.
Ig i s a l s o a group o f Backlund
tra n s fo rm a tio n s
g iven i n t h e
(M19N1 ) r e a l i z a t i o n
by t h e m a t r ic e s W1^ and
Z1 ^ where
( W .k ) =
1$
0
0
0
a
0
(y - B ) / 2
(y -a )/2
Y
-
and
-Ot
(Z ik ) =
From t h e s e ,
but now
(4 .1 5 )
>
fo r
-I
0
0
b" 1
0
0
(2 a y )
-I.
\
1(a -y )
(4 .1 6 )
(2g Y )~ \e-y)
r"1
Ig t r a n s f o r m a t i o n s , y
i
s till
s a tis fie s
E q u a tio n
(4 .1 0 ),
33
d ( a y " 1 / 2 ) = [ - Y 1 / 2 (M3 / 2 - M 1 ) + ( a Y ' 1 / 2 )(M 2 -M 1 )+
( a Y 1 / 2 ) 2 (M3 / 2 - M 2 ) Y 1 / 2 ] d x 1+
[ - Y - 1 7 2 CN3 ^ - N 1 M a y " 1 7 2 ) (N2 - N 1 )+
.
(a Y - 1 7 2 ) 2 (N3/ 2 - N 2 )Y 1 7 2 ] d x 2 .
It
would seeip t h a t a f t e r each t r a n s f o r m a t i o n ,
d iffe re n tia l
firs t
firs t
(4 ,1 7 )
order p a r t ia l
e q u a t io n s would have t o be s o lv e d s in c e t h e M1- , N1 a re
d e riv a tiv e s .
But Neugebauer has produced a v e ry im p o r t a n t
th e o re m .
N e u g e b a u e r's Commutation Theorem
Given an i n i t i a l
tra n s fo rm a tio n
tra n s fo rm a tio n
( I 2 )b ,
(Il)c »
and y ' s
1V lfa 1
T y J v -TTT
Y b = 1T
d
(4 .1 7 ).
A ll
p a rtia l
is th e id e n t it y
• “c - ^
* ad
- y C = Ya
> Td = TT •
E q u a tio n
d
( 4 . 1 1 ) and t h e
s a tis fy
E q u a tio n
i s a v e ry im p o r t a n t th e o re m .
d iffe re n tia l
tra n s fo rm a tio n .
The
('a -')
. Ta
th e y 's
g e n e r a t i n g new s o l u t i o n s
Backlund
are
Here t h e a g and a c s a t i s f y
T h is
Backlund
( I 2 )d» such 1^ a t t h q p r o d u c t
(l2 )d (I l ) c ( I 2 )b (I l)a
c o r r e s p o n d in g a ' s
E q u a tio n
, N^ and an a r b i t r a r y
( I i ) a . th e n t h e r e a r e alw ays t h r e e n o n t r i v i a l
tra n s fo rm a tio n s
“b -
s o lu tio n
It
( 4 - 18
and
s a tis fy
(4 .1 0 ).
c o n s id e ra b ly s im p lif ie s
from o ld s o l u t i o n s w i t h o u t h a v in g t o i n t e g r a t e
e q u a t io n s a t each s t e p .
34
Now t h a t two m a jo r methods o f s o l u t i o n
g e n e r a t i o n have been
r e v ie w e d , t h e i r use w i t h t h e vacuum e q u a t io n s w i l l
next c h a p te r.
be summarized i n t h e
35
CHAPTER 5
REVIEW OF THE VACUUM CASE
In t h e l a s t two c h a p t e r s , two methods were p r e s e n t e d f o r
g e n e r a t i n g new s o l u t i o n s o f t h e vacuum E r n s t e q u a t i o n s .
b oth t h e e l e c t r o m a g n e t i c and g r a v i t a t i o n a l
e q u a tio n s .
T h is s i m i l a r i t y
e le c tro v a c ; t h e r e fo r e ,
it
(1 9 7 7 a ,b;
D e fin in g
it
s im ila r
is n a tu ra l
to
r e v ie w t h e vacuum case f i r s t .
c h a p t e r f o l l o w s t h a t g iv e n by K in n e r s le y
1 9 7 8 a ,b ) .
F ie ld s ,
VNAB = V
s a tis fy
has been used t o ex te n d vacuum r e s u l t s t o
The n o t a t i o n t o be used i n t h i s
and C h i t r e
fie ld s
In C h a p te r 2 ,
P o t e n t i a l s , and G e n e r a t in g F u n c tio n s
by
vhV
(5 -1 )
i s p o s s i b l e t o show t h a t
HAB = 1 ^ A B + h AXh B)
s a tis fie s
th e f i e l d
mn
m
E q u a tio n s
n»
^5 , 2 )
(2 .3 7 ).
C o n t in u in g i n t h i s
? NAB = HX A *? H B :
th e n i t
way, d e f i n e
( 5 '3 )
i s p o s s i b l e t o show t h a t
n+1
In
nY
HAB '
s a tis fie s
K 1W
h AXh
E q u a tio n s
(2 .3 7 ).
n
i
VHAB = - 1P
B>
„
A ll
o f th e H
(5 -4 )
s a tis fy
v~n
V
v h XB-
(5 .5 )
36
D e fin in g
(5 .6 )
'AB'
AB
im p lie s t h a t
n
-iH
On
NAB
(5 .7 )
AB
and t h e f i e l d s HAg a r e i n c o r p o r a t e d i n t o t h e h i e r a r c h y o f p o t e n t i a l s
mn
N^g ., T h is exten d s t h e range o f m and n t o a l l i n t e g e r v a l u e s .
K i n n e r s l e y and C h i t r e
mn
t h e Na b .
These a r e
(1977b)
mn
nY
nm
m
have g iv e n c e r t a i n
re la tio n s
s a tis fie d
by
(5 .8 )
NAB-NBfl* * HXA*H B * eAB5 0« 0
and
m ,n + l m + l,n
ml nY
m
n
(5.9)
NAB " NAB = lN AXH B- 3eAB6 - I 6 Oe
It
has proven u s e f u l
p o te n tia ls .
to d e f in e g e n e ra tin g fu n c tio n s
The two g e n e r a t i n g f u n c t i o n s
i n term s o f t h e
used i n t h e vacuum case a r e ,
somewhat chan g in g t h e n o t a t i o n o f K i n n e f s l e y and C h i t r e
. N
(s .t) = I
AB
I
W
(1 9 7 8 b ),
(s.io )
n
m=0 n=0
and
0
F a b (t )
-
I.
n=0
T h ere a r e a l g e b r a i c
fu n c tio n s ,
fo u n d .
„n
t " H AB
1R -
(5 .H )
1NlAB
n (0.t ) .
re la tio n s h ip s
between t h e s e two g e n e r a t i n g
such t h a t when t h e F / ^ t )
a r e known, t h e N ^ g ( s , t )
can be
37
The Symmetry Groups
In C h a p te r 3 , t h e group H was found s t a r t i n g w i t h t h e vacuum
E rn st e q u a tio n .
T h ere a r e o t h e r groups b e s id e H and t h e s e a r e now
r e v ie w e d .
The Groups G and H
K in n e rs le y
tra n s fo rm a tio n
(1977a)
s t a r t s w ith th e
(t,<|>) c o o r d i n a t e
group G, iso m o rp h ic t o t h e group o f 2 by 2 r e a l
w i t h u n i t d e t e r m in a n t S L ( 2 , R ) .
m a t r ic e s
G's t h r e e g e n e r a t o r s a r e chosen as
t'
= t+a<|>, <|>' = > ,
(5 .1 2 )
t'
= t,
f ' = <{>+bt.
I
(5 .1 3 )
t'
= C t , <j>' = c" <j>.
named (G1 ) a ,
(G2)b»
(G3)c»
(5 .1 4 )
re s p e c tiv e ly .
t h r e e g e n e r a t o r s on t h e f ^ B i s
The f i n i t e
a c t i o n o f th e s e
g iven by
fIl'
= fIV
f 1 2 ' = f 12"a f l l *
V
= f 2 2 - 2 a f 12+ a 2 f U ;
(5 .1 5 )
f 22l = f 22, f 12' = f i 2 - b f 2 2 >
f Il'
‘ h r Z b fK +b\ f >
fIl'
■ c 2 fI V
(5 .1 6 )
f I a ' = f 1 2 ’ ^22
- c2f 22.
(5 .1 7 )
B ein g a group o f c o o r d i n a t e t r a n s f o r m a t i o n s , G does not
p h y s i c a l l y a l t e r t h e s p a c e t im e .
However, by d e f i n i n g t h e group H = SGS
w i t h S b e in g t h e N eugebauer-K ram er map, H i s
E q u a tio n s
(3 .5 5 ,5 6 ).
Its
th re e g e n erato rs,
found t o be t h e group f o r
(H1 ) a = S(Gl ) „ 1- a S,
38
( H 2 ) b = S ( G 2) i b S , ( H 3 ) c = S ( G ) _ i S , produce t h e f o l l o w i n g a c t i o n on f
and
re s p e c tiv e ly .
f'
f,
\l>' = ^ - a ;
' (5 .1 8 )
T|)-b ( f 2 + ^ 2 )
f'
(l-b f+ b 2 f2 )
f'
= c2 f ,
These t h r e e f i n i t e
(5 .1 9 )
(l-bi|»2 +b2 f 2 )
ij,' = C2Ij,.
(5 .2 0 )
t r a n s f o r m a t i o n s a r e r e c o g n iz e d as t h e one p a ra m e te r
subgroups found i n C h a p te r 3 .
The Group K
N e x t, th e i n f i n i t e s i m a l
a c t i o n o f G (H) i s
^AB
T h is a llo w s t h e i n f i n i t e s i m a l
mn
on t h e H^g, and th e n on t h e
.
found on t h e
a c t i o n o f G and H t o be found
Forming commutators w it h t h e d i f f e r e n t
in fin ite s im a l
op erato rs
o f G and H, an i n f i n i t e - d i m e n s i o n a l
o p e r a t o r s th u s d e f i n e d w i l l
L ie a lg e b ra is g e n e ra te d ,
th e
k
be named T yv and d e f i n e t h e group K.
T h e ir
a c t i o n on t h e N . g i s g iv e n by
k mn A
m+k,n
m,n+k
k
ms
k -s ,n
NY)B
1 XYnAB “ eA (X NY )B " NA (Xe Y)B+ S=1 NA
"A (X in
Y)B *
(5 .2 1 )
where t h e p a r e n th e s e s i n d i c a t e s y m m e t r iz a t io n w it h r e s p e c t t o t h e
0
e n c lo s e d i n d i c e s .
The o p e r a t o r s Tvv a r e t h e i n f i n i t e s i m a l o p e r a t o r s o f
- 1 0 1
XY
G, and t h e ( T ^ , T ^ , T gg) a r e t h e i n f i n i t e s i m a l o p e r a t o r s o f H.
The
commutators by which t h e Tyy were o r i g i n a l l y
k
I
k+1
d e f in e d a re
k+1
(5 .2 2 )
*-TXY* t AB-* = gB (X t Y)A+ g A (X t Y )B 1
0
F o r ex a m p le , t h e commutator o f T ^
1
and
I
d e fin e s T ^ .
39
The Groups Q and Q
A n o th e r g ro u p , Q, l y i n g o u t s i d e t h e K group was found by
Cosgrove
(1 9 7 9 a ).
t h e c o o r d in a t e s
(p ,z )
T h is group changes n o t o n ly t h e p o t e n t i a l s ,
(p ,z ).
The i n f i n i t e s i m a l
but a ls o
a c t i o n o f Q on t h e c o o r d in a t e s
is
Qp = p z , Qz = l / 2 ( z 2 - p2 ) ,
mn
an,d on t h e p o t e n t i a l s N
is
mn
A ' m + l,n
m ,n + l ml n
QNab = l / 4 [ ( m + A - l ) Nab+ (n + B - 1 ) N ^ + i N ^ H ^ ] .
(5 .2 3 )
(5 .2 4 )
Given t h e N eugebauer-K ram er map S, a c o n ju g a t e group Q =
Its
in fin ite s im a l
its
i n f i n i t e s i m a l a c t i o n on N^g i s
.mn
m + l,n m ,n + l
QNab = l / 4 [ ( m + l ) NAB+n Nab ] .
From E q u a tio n s
a c t i o n on ( p , z )
(5 .2 4 ,2 5 )
Q-Q = -
it
is
id e n tic a l
SQS i s
t o E q u a tio n
d e fin e d .
( 5 . 2 3 ) , and
(5 .2 5 )
i s seen t h a t
I
1 / 4 T 12
(5 .2 6 )
h o ld s f o r t h e s e i n f i n i t e s i m a l
o p e ra to rs .
A p p lic a tio n s
The problem o f g e n e r a t i n g new s o l u t i o n s
from a g iv e n s o l u t i o n
now s e p a r a t e s
in to th ree p a rts .
Given a s o l u t i o n ; f o r example f l a t
mn
s p a c e t im e , t h e p o t e n t i a l s Nab must be fo u n d .
Second, t h e i n t e g r a t i o n
th e in fin ite s im a l
o p e r a t o r s must be p e rfo rm e d .
t r a n s f o r m a t i o n s which produce p h y s ic a l
p h y s ic a l
s o lu tio n ,
it
approaches i n f i n i t y ,
is
T h ird ,
of
o n ly th o s e
s o l u t i o n s s h o u ld be used.
By
u s u a l l y meant t h a t as t h e q u a n t i t y p2 +z2
t h e m e t r i c approaches t h e f l a t
s p a c e tim e m e t r i c .
F in d i n g t h e g e n e r a t i n g f u n c t i o n s
N / \ g ( s , t ) and F ^ g ( t ) has proved
mn
t h e most e f f e c t i v e way i n f i n d i n g t h e p o t e n t i a l s NAg.
An example o f a
g e n e r a t i n g f u n c t i o n which has been found i s t h e f l a t
s p a c e tim e
40
g e n e ra tin g fu n c tio n
fA
F^8 ( t ) ,
K i n n e r s l e y and C h i t r e
(1 9 7 8 a ),
t
Tm "
\
i
S W
(5 .2 7 )
B^
■i ( l - 2 t z + S ( t ) )
S (t)
(l-2 tz -S (t))
2 t$ (t)
/
w ith
S (t)2
= (l-2 tz )2 + (2 tp )2.
(5 ,2 8 )
O th e r g e n e r a t i n g f u n c t i o n s which have been found f o r a g iv e n spacetim e
are l i s t e d
b elo w .
1.
Z ip o y - V o o r h e e s ; K i n n e r s l e y and C h i t r e
2.
W e y l; H o e n s e la e r s , K i n n e r s l e y , and X an thopoulos
3.
K e r r -N U T ; Hauser and E r n s t
Cosgrove
tra n s fo rm a tio n s to f i n i t e
in th e in t e g r a t io n
tra n s fo rm a tio n s .
and FAB( t ) .
( Q ) ^ s and (Q )
Some o f t h e s e f i n i t e
Q and Q
t r a n s f o r m a t i o n s u sin g
tra n s fo rm a tio n s are
s
NAR( s + t 1 , s + t 2 )+ -------s + t2
s + t2
t2
^ 4 s ^ A B ^ 19 ^2 )
of
For exam ple,
( 1 9 8 0 a ) has s u c c e s s f u l l y i n t e g r a t e d t h e i n f i n i t e s i m a l
tra n s fo rm a tio n s to th e f i n i t e
N ^ g (s ,t)
(1979)
(1979a)
G e n e r a t in g f u n c t i o n s have p roved u s e f u l
in fin ite s im a l
(1 9 7 8 b )
(5 .2 9 )
It F 11(S n )
(5 .3 0 a )
(Q )4 s F n (t)
( s + t ) F u (s )
i F 12 ( s + t )
(5 .3 0 b )
^ U sF 1 2 ^ ^
F 1 2 (S)
More r e l a t i o n s
th ese,
it
of th is
t y p e a r e found i n t h e l a s t
must be remembered t h a t
(p ,z )
re fe re n c e c it e d .
a l s o changes.
In
41
O th e r f i n i t e
t r a n s f o r m a t i o n s which have been found a r e l i s t e d
be! ow.
k
The n u l l p a ra m e te r c a s e ; t h a t i s when t h e p a ra m e te rs q
in
k k
k
k
.
x”
t h e o p e r a t o r q xyT
s a tis fie s q ^ q
= 0„
K i n n e r s l e y and C h i t r e (1 9 7 8 a )
xy
k xy
1.
2.
the s t a t i c
k
vk
For s t a t i c
s p a c e t im e s .
3.
k+2
m e tric s ,
K i n n e r s l e y and C h i t r e
o perators
.......... p r e s e r v e s a s y m p t o t ic f l a t n e s s .
subgroup has been used t o
g e n e r a t e new s o l u t i o n s .
Its
a p p lic a b ility
K i n n e r s l e y and C h i t r e
4.
T h is
Fo r e x a m p le , K e r r - N u t
g e n e ra te d fro m t h e S c h w a r z s c h ild p o t e n t i a l s
subgroup.
a ll
(1 9 7 8 a )
The B sub g ro u p , h av in g i n f i n i t e s i m a l
B = Tn +T2 2 $ k =
was f i r s t
has been i n t e g r a t e d t o y i e l d
u s in g t h i s
does depend on t h e i n i t i a l
s o lu tio n .
(1 9 7 8 b ).
The HKX t r a n s f o r m a t i o n s may be used t o g e n e r a t e
a s y m p to tic a lly f l a t
s p a c e tim e w i t h a r b i t r a r y
H o e n s e la e r s , K i n n e r s l e y , and X anthopoulos
m u l t i p o l e moments,
(1 9 7 9 ).
BackTund T r a n s f o r m a t io n s
In C h a p te r 4 , t h e Backlund t r a n s f o r m a t i o n
was d e s c r i b e d .
Cosgrove
(1 9 7 9 b )
t o t h e two groups HQ and GQ.
p ro d u c ts o f t h e ele m e n ts
commute, o r d e r i s
has shown t h a t
I 1 and Ig a r e e q u i v a l e n t
(HQ s im p ly means t h e group c o n s i s t i n g o f
in t h e groups H and Q.
no p r o b le m .)
found by Neugqbauer
Neugebauer
S in c e t h e s e two groups
( 1 9 8 0 a , b) has a p p l i e d
I 1 and
Ig and t h a . commutation th eo rem t o g e n e r a t e a r b i t r a r y m u l t i p o l e moments
f o r a s y m p to tic a lly f l a t
s p a c e t im e .
Cosgrove
( 1 9 8 0 a ) , u s in g both
H a r r i s o n ' s Backlund t r a n s f o r m a t i o n and N eu g eb au er's commutation th e o re m .
I
42
has a ls o been a b le t o g e n e r a t e a s y m p t o t i c a l l y f l a t
s p acetim es w i t h
a r b i t r a r y m u l t i p o le moments.
Summary
O th e r methods e x i s t b e s id e s t h e group t r a n s f o r m a t i o n and t h e
Backlund t r a n s f o r m a t i o n .
B e lin s k y and Sakharov ( 1 9 7 8 , 1 9 7 9 )
i n v e r s e - s c a t t e r in g te c h n iq u e .
an i n t e g r a l
e q u a t io n method.
a s y m p to tic a lly f l a t
Hauser and E r n s t
(1 9 7 9 a ,b;
it
e s s e n tia lly
is
sp a c e tim e s w i t h a r b i t r a r y m u l t i p o l e moments.
no wonder t h a t th e s e w o rk e rs f e e l
s o lv e d .
1 9 8 0 a , b) use
Both a r e c a p a b le o f g e n e r a t i n g
so many methods c a p a b le o f g e n e r a t i n g e q u i v a l e n t r e s u l t s ,
(1 9 8 0 a ),
use an
With
see Cosgrove
t h e vacuum case t o be
43
CHAPTER 6
REVIEW OF THE ELECTROVAC CASE
Many o f t h e r e s u l t s
th e e le c tro v a c case.
from t h e vacuum case have been extended t o
These e l e c t r o v a c
re s u lts w i l l
be re v ie w e d i n t h i s
c h a p te r.
The n o t a t i o n a g a in f o l l o w s t h a t g iv e n by K in n e r s l e y and C h i t r e
( 1 9 7 7 a , b;
1 9 7 8 a ,b ) .
F ie ld s ,
The f i e l d s
P o te n tia ls ,
and G e n e r a t in g F u n c tio n s
n
mn
Hab and t h e p o t e n t i a l s Nftg were in t r o d u c e d in
C h a p te r 5 .
In a l i k e m anner, t h e f i e l d s HAg and <|>A and t h e p o t e n t i a l s
mn
mn mn mn
NA g , Ma , Lg , K a r e in t r o d u c e d f o r e l e c t r o v a c .
They a r e d e f i n e d
re c u rs iv e ly
by t h e f o l l o w i n g
re la tio n s h ip s :
\
I
I
$A = V
hAB = hAB s
mn m
n»
VK = $ x * V $ \
mn
VMa -
n+1
•a n+1
/.
HAB =
(6 .1 )
mn
m
nY
VLg = » X*VH g ,
m
n v
mn
m
nY
Hx a *V»
, VNab = Hx a *VH b , '
In
1I V
In
In
2V
(6 .2 )
(6 .3 )
ny
+V
In
(6 .4 )
>•
nY
n AB+2^A l B+ h AXh
B^
(6 .5 )
44
A ll
n
n
<&A and Hftg s a t i s f y t h e same f i e l d
the f i e l d s
e q u a tio n s .
I
v~n
a
i
Ii
-C
(6 .6 )
f A 7 4 X-
I
x~n
(6 .7 )
vhAB = - i p " f A v h XB1
D e fin in g
0
10
K = -l/2 i.
(6 .8 )
HAB = l e AB1
im p lie s
On
On
n
MA = - U A-
fie ld s
The r e l a t i o n s
mn nm
K -K *
a r e c o n t a in e d i n t h e h i e r a r c h y o f
found by K in n e rs T e y and C h i t r e
m nY
= $%*$ ,
m
11
*
CO
Z
I
CO
-J
*X *H B1
n AB- n BA*
nY
m
nm
-
(1977b) are
( 6. 10)
mn nm
mn
(6 .9 )
NAB = " th AB1
and t h a t t h e o r i g i n a l
p o te n tia ls .
n
( 6. 11)
n
„n
hxa* h
( 6. 12)
and
ml n,
m ,n + l m + l,n
iml I n
-K
'
= 21K K + i L x $ A+ ( i 6 m05 n 0 ) / 2 ,
m ,n + l m + l,n
L
B
m ,n + l m + l,n
Ma
-Ma
msn + l m + l,n
nAB
- nAB
ml I n
2iK
t
ml nv
Lg+1L, H*g ,
ml I n
ml
(6.14)
nY
=2iMA K +iNA/ ,
ml I n
(6.13)
, (6.15)
ml n Y
= 21MA 1 Bh n AXh B - 3eAB6 - I 6 O'
(6.16)
45
G e n e r a t in g f u n c t i o n s a r e d e f i n e d by
K (s ,t)
=
CO
CO
I
I
m „mn
s mt nK 9 K ( t )
°°
m=0 n=0
LA ( s , t )
I
=
I
S1W
»
o»
= Ji
J
m= O n=0
mn
flj Lf l ( t )
_ „mn
s t M ., @ .(t)
M
A
m
A
1t
S2 ( t )
B^
(6. 18)
t"L
oo
= %
n=0
n
(6 .1 9 )
t
mn
oo
"S B ' f ABtt ) -
I a t HflB.
T a k in g t h e g r a d i e n t o f E q u a tio n s
vfA
I
=
n=0
=
J0 J0
NAB( S > t )
(6 .1 7 )
n=0
m=0 n=0
M n (s ,t)
A
In
t nK s
I
=
(6 .4 ,5 )
n
( 6 . 20 )
le a d s t o t h e e q u a t io n s
[ ( ( l - 2 t z ) v H AX- 2 tp V H AX) FXb ( t )+
( 6. 21)
2 ( ( l - 2 t z ) V » A- 2 t p V » A ) L B( t ) ] ,
V$A ( t )
=
- L L
[ ( ( l - 2 t z ) 7 H AX- 2 t p 7 H AX) $ X ( t ) +
(6 . 2 2 )
2 ( ( l - 2 t z ) V * A- 2 t p V * A) K ( t ) ] ,
S f(t)
The f i r s t
(6 .2 3 )
= ( l - 2 t z ) 2+ ( 2 t p ) 2 .
in te g ra l
o f t h e above e q u a t io n s
2 K ( t ) D e t ( F ( t ) ) - 2 $ X ( t ) L Y ( t ) F XY( t )
Im p o r t a n t a l g e b r a i c
1 6 ).
re la tio n s h ip s
is
i
(6 .2 4 )
S (t)
can be d e r i v e d from E q u a tio n s
(6 . 10-
Some o f t h e s e a r e :
N,
2 itS (s ),
= T = T =AB+ ^ t l : t t 4 [ K ( = ) F X A (= )F B ( t ) +
(6 .2 5 )
( L A ( s ) F XB( t ) - L B ( t ) F XA( s ) ) $ X( s ) ] ,
MA ( s , t )
= ^ g i 5 l [ K ( s ) $ x ( t ) F XA( s ) - K ( t ) » A( s ) F XA( s )
(6 .2 6 )
46
LB ( s , t )
= ^
l j
[ D e t ( F ( t ) ) L B ( t ) - L Y ( s ) F XY( s ) F XB ( t ) ] e
K (s,t) =
+
(6 .2 7 )
[K (t)D et(F (t)) L Y ( s ) F XY( s ) » X ( t ) ] s
(6 .2 8 )
K *(s )
= - * X$ x * ( s ) - i S ( s ) D e t ( F ( s ) ) / 2 ,
(6 .2 9 )
La (S)
= -$ XFxa* ( s )+S( s )$ x ( s )FXa ( s ) / s ,
(6.30)
F c a M s ) = - 2 i S " 1 ( s ) ( ( l - 2 s z ) 6 c B+ 2 i s f cB) x
( K ( s ) F BA( s ) - * B( s ) L A ( s ) + $ B* * x ( s ) F XA( s ) ) ,
»c * ( s )
(6 .3 1 )
= s S ' 1 ( s ) ( ( l - 2 s z ) 6 c B+ 2 i s f c B) x
( F BX( s ) L X ( s ) - » B* D e t ( F ( s ) ) .
In t h e a b o ve, D e t ( F ( s ) )
(6 .3 2 )
i s t h e d e t e r m in a n t o f FAB( s ) .
The Symmetry Groups
The vacuum groups can be e n la r g e d t o t h e e l e c t r o v a c c a s e .
groups G1 and H1 a r e d e f i n e d and used t o f i n d
The
an e n la r g e d symmetry group
K '.
The Groups G1 and H1
The group G1 i s .c o m p r is e d o f t h e c o o r d i n a t e t r a n s f o r m a t i o n s
g iv e n in E q u a tio n s
(5 .1 2 -1 4 )
and t h e e l e c t r o m a g n e t i c and g r a v i t a t i o n a l
gage t r a n s f o r m a t i o n s
4 A ^ A +aA* hAB^h AB"<$A *a B"4 Aa b * ' a A * a B+1otABe
(6 .3 3 )
47
These gage t r a n s f o r m a t i o n s have a B complex and a AB r e a l
t h e y do n o t change t h e m e t r i c
and s y m m etric ;
o r th e e le c tro m a g n e tic f i e l d
Fam.
T h is now d e f i n e s G1 which i s an e i g h t p a ra m e te r group.
The group H 1 i s a l s o an e i g h t p a r a m e t e r group l e a v i n g E q u atio n s
(2 .3 8 ,3 9 )
in v a ria n t.
The g e n e r a t o r s o f H1 a r e t h e g r a v i t a t i o n a l
apd
e l e c t r o m a g n e t i c gage t r a n s f o r m a t i o n s :
H ii^ H n + ia ,
(6.34)
$^-*$i+a, H ii»H ii-2a*@ i-a*a;
(6.35)
th e g e n e ra liz e d E h le r 's tr a n s fo rm a tio n
*1
*1 *
H11
(!+Y H 1 1 ) '
t h e H a r r is o n
" ll+
(Ifiy H 11) ;
(8 .3 6 )
( 1 9 6 8 ) m ix in g t r a n s f o r m a t i o n
$ i+ c H ii
*1*
( I - ^
1- C a CH1 1 )
»
(8°37)
Hn
" il*
( 1 - 2 C * $ 1-C *C H 1 1 )
;
(6 .3 8 )
and t h e s c a l i n g and e l e c t r o m a g n e t i c d u a l i t y t r a n s f o r m a t i o n
• !+ B e 1® * ! ,
H11-^B2 H1 1 .
(6 .3 9 )
In t h e ab o ve, a , 3 , y » e' a r e r e a l
co n stan ts.
c o n s t a n t s w h i l e a and c a r e complex
T h e re i s no s im p le way t o g e t H'
from G1 s in c e t h e
N eugebauer-K ram er m ap p in g 's a c t i o n on t h e m e t r i c f AB has n o t been fo u n d ;
h o w ever, t h e m ap p in g 's a c t i o n on t h e group g e n e r a t o r s has been found and
w ill
be d e s c r ib e d l a t e r i n t h i s
ch a p te r.
The Group K'
A g a in ,
by f o rm in g a l l
p o s s i b l e commutators o f t h e I n f i n i t e s i m a l
o p e r a t o r s o f G' and H ' , an i n f i n i t e - d i m e n s i o n a l
g en erated .
T h is group i s c a l l e d K'
p a ra m e te r group i s
and i s t h e e l e c t r o v a c g e n e r a l i z a t i o n
48
o f K.
is
The i n f i n i t e s i m a l
g iv e n i n
K i n n e r s l e y and C h i t r e
t h e above r e f e r e n c e ,
a lre a d y im p lie d .
k
as cA i s
th e a c tio n
k
k
k
, Cx , Cx * , Z , and t h e i r p o t io n
( 1 9 7 7 b : 1 5 4 0 ) and rep ro d u ced h e r e .
is
(In
g iv e n w i t h t h e group p a ram eters
F o r e x a m p le , what K i n n e r s l e y and C h i t r e r e f e r t o
k»k
ky k
o p e r a t o r c ACx +cA*C x * .
the i n f i n i t e s i m a l
k
k
k
k
c o n f u s i o n , y Xy , c x , c x * ,
w ill
k
o perators a re T
.
a w ill
To a v o id
k
k
k
k
be t h e p a ra m e te rs and T Xy , Cx , Cx * , z
be t h e o p e r a t o r s . )
k
mn
m+k,n
m,n+k
k
ms
k ^ s ,n
1 XYnAB = eA (Y NX)B ‘
NA (XEY)B + J 1 NA (X NY )B '
k
k
mn
m+k,n
7 XYmA = eA (Y MX)
k
mn
m,n+k
t XY1B
= e B (Y l X)
k
mn
k
(6 .4 1 )
nA (X m Y)
k
+
ms k - s , n
I
(X n Y)B !
S=
m+k,n
CXNAB = 2 lG XALB -
(6 .4 3 )
k
21 I
S=I
CXLB =
21
'I
s=l
NAXK
(6 .4 5 )
k
-21
ms k - s .
I L- LB
S=I
*
(6 .4 4 )
AX B
m,n+k
k mn
m+k,n
+ N.v
Cv M, = E i e u -K
AX
X A
XA
k mn
(6 .4 2 )
ms k - s ,n
7 XYk = J 1 L (XMV)
k mn
(6 .4 0 )
»
(6 .4 6 )
49
k mn m , n + k - l
Cy K =
Ly -
k
mn
k - 1 ms k - s , n
£
Lv K
s=l
X
2i
m,n+k
k
(6 .4 7 )
ms k - s , n
CX * NAB = r 2 l e XBwA +21 J 1 MA NXB
k
mn
k
c X1mA
=- 2 i V
k
ms k - s , n
mX
a
S=I
mn
(6 .4 9 )
’
m,n+k . m + k - l , n
CX * LB = - 2 i 6 XBk
k
mn
+
m + k -l,n
Cy*K =
My
X
+2i
x
k mn
NXN
k
ms k - s , n
+2i J 1 K
NXB
.
(6 .5 0 )
9
k ms k -s » n
% K My
,
s=l
X
(6 .5 1 )
k ms k -s + 1 ,n
e n AB = - 2 ^ A
k mn
lB
m,n+k
XMa = -I
k mn
ms k - s + 1 , n
- 2J i
m+k,n
k
mA k
K in n e rs le y
-m
-
(1980)
0
ml k - s + 1 , n
_
„ .
K K
- I ^ S mnSnnSk 1 ,
(6 .5 5 )
in t e g e r v a lu e s .
K1 is o m o rp h ic t o S U ( 2 , 1 ) .
( m ,n )
(6 .5 4 )
9
lb
k mn
m+k,n
m,n+k
Z K = i
K - i K.
- Z l
w i t h k ru n n in g o v e r a l l
(6 .5 3 )
•
ms k - s + 1 , n
” 2 s^ 1 k
lb
(6 .5 2 )
•
k
Ma
1 lb = 1
G
(6 .4 8 )
»
has found e i g h t - p a r a m e t e r subgroups
m,n
( G ) of
They a r e
m
n
( Y m Y i a »^22 »C1 »
(m+n)
^2 *
(1-m -n )
Cl *
(1 -n )
» c2 *
0
» o }.
'
(6 .5 6 )
A ls o found were i n v o l u t i v e automorphisms which map t h e subgroups i n t o
one a n o t h e r .
AL
These a r e
( r ss )
I
: K '+ K ',
(r,s )
I
:
(m ,n ) ( r - m , s - n )
G +
G
(6 .5 7 )
9
(r,s ) k
(r,s )
1
Yn
I
k -r
= 1722,
(6 .5 8 )
(r,s )
I
k
= -Y iz ,
(6 .5 9 ) .
k
(r,s )
Y 12
I
( r 9s ) k
( T 8S)
k -r
= -iY n »
(6 .6 0 )
( r 8s ) k
( r 9s )
k + l-s
I
C1
I
= c2 * 9
(6 .6 1 )
( r 9s ) k
( r 9s )
k + l-r-s
I
c2
I
= i
C1 *
,
(6 .5 2 )
( r 9s ) k
( r 9s )
k+ s+ r-1
I
C1*
I
= -i
c2
,
(6 .6 3 )
I
Y22
1
( r 9s ) k
( r 9s )
k + s -1
I
c2*
I
=
C1 ,
(r,s ) k (r,s )
I
b
I
k
-or . .
( O 8O)
( I 8O)
G
and
G
a r e G1 and H' r e s p e c t i v e l y ,
The
( I 9O)
automorphism
I
i s t h e N eugebauer-K ram er map.
In vacuum, i t s
(6 .6 4 )
The subgroups
on t h e p o t e n t i a l s
has been f o u n d , but t h i s
a c tio n
does not h o ld t r u e i n t h e
e le c tro v a c case.
T h is com p letes t h e r e v ie w o f e l e c t r o v a c , t h e n e x t c h a p t e r w i l l
e x te n d C o s g ro v e 's vacuum Q and Q t r a n s f o r m a t i o n s t o t h e e l e c t r o v a c c a s e .
51
CHAPTER 7
THE EXTENDED Q AND Q GROUPS
In t h i s
c h a p t e r , t h e vacuum t r a n s f o r m a t i o n groups Q and Q w i l l
be e x ten d ed t o i n c l u d e t h e e l e c t r o m a g n e t i c f i e l d
2.
F irs t Q w ill
th e i n i t i a l
w ill
be ex te n d e d and th e n E q u a tio n
a n satz o f th e i n f i n i t e s i m a l
be m o d if ie d a l l o w i n g t h e f i n i t e
d e s c r ib e d i p C h a p te r
(5 .2 6 ) w ill
Q tra n s fo rm a tio n .
be used f o r
T h is a n s a t z
t r a n s f o r m a t i o n t o be i n t e g r a t e d ,
b r i n g i n g agreem ent w i t h t h e vacuum r e l a t i o n
_
( I 9O)
(1 ,0 )
Q = I Q i ;
( I 9O)
b e in g t h e d i s c r e t e automorphism d e s c r ib e d i n C h a p te r 6 .
A s s o c ia t e d w i t h Q i s
its
complex c o n ju g a t e group Q* found i n t h i s
( I 9I )
I
c h a p t e r t o be e q u i v a l e n t t o
Q
( I 9I ) .
~
I
.
T h is Q * t r a n s f o r m a t i o n
merges w i t h Q i n t h e vacuum case and i s ;
th e re fo re ,
a new f e a t u r e
b e lo n g in g t o t h e e l e c t r o v a c c ase.
The Extended Q Group
The vacuum t r a n s f o r m a t i o n
group Q was f i r s t e x te n d e d t o t h e
e l e c t r o v a c case by Cosgrove (1 9 8 0 b ) d u r in g a bout w i t h t h e f l u .
g r o u p 's a c t i o n on t h e f i e l d s
and p o t e n t i a l s was f i r s t
T h is
found by m y s e lf i n
V
a roundabout manner.
Cosgrove
A s im p lifie d
v e rs io n
is
( 1 9 8 0 ) has found t h e f i n i t e
p re s e n te d i n t h i s t h e s i s .
a c t i o n o f t h e vacuum Q
t r a n s f o r m a t i o n on t h e g e n e r a t i n g f u n c t i o n s F ^g ( t )
r e s u lts , th e in f in it e s im a l
a c tio n
o f Q is
=? i N ^ g ( Q , t ) .
found t o be
From h is
52
n
QHn
n+1
n
= l / 4 [ n H1 ^ I H 1 2 H1 1 ] ,
n
n+1
QH21 = l / 4 [ n
In
(7 .1 )
n
H2 i +1N2 l +1H2 2 Hl l ] ’
(7 .2 )
n
n+1
n
QH12 = l / 4 [ ( n + l ) H12+ i H 12H1 2 ] ,
n
n+1
QH22 = l / 4 [ ( n + l )
These r e l a t i o n s
In
(7 .3 )
n
H2 2 +1N22 +1H2 2 Hl'2] e /
(7 .4 )
a r e assumed t o h o ld u n a l t e r e d .
The o n ly c h o ic e
n
r e m a in in g i s t h e i n f i n i t e s i m a l a c t i o n o f Q on
S in c e t h e f i e l d
n
n
e q u a t io n s f o r t h e
a r e s i m i l a r t o t h e f i e l d e q u a t io n s f o r Hfl1, t h e
n
! n
a n s a t z i s made which t r a n s f o r m s » fl l i k e Hfl1.
H ence,
*
n
n+1
n
’
Q^1 = l / 4 [ n ^ 1 + I H 1^ 1 L
(7 .5 )
n
n+1
In
n
Q *2 = l / . 4 [ n $ 2 + IM2 + I 1H22^ 1 ] .
The c o n s is t e n c y o f E q u a tio n s
in fin ite s im a l
is
shown i n Appendix A.
The
a c t i o n on t h e p and z c o o r d i n a t e s i s assumed t h e same as
i n t h e vacuum c a s e ; t h a t
(6 .2 ,3 ;
(7 .1 -6 )
(7 .6 )
7 .1 -6 ),
is ,
Qp = p z , Qz -
th e in f in it e s im a l
a c tio n
l / 2 ( z 2 -p 2 ) .
U sin g E q u atio n s
o f Q on t h e p o t e n t i a l s can be
d e t e r m in e d .
The a c t i o n on t h e p o t e n t i a l s i s g iven by
mn
m ,n + l
ml n
ml n
ml
QK = l / 4 [ ( m + n )
K -Im L 1$ 2+ i ( m + l ) L 2 $ 1 -2imK
In
K ],
(7 .7 )
mn
m ,n + l
ml n
ml n
ml I n
QM1 = l / 4 [ ( m + n )
M1 -im N n » 2+ i ( m + l) N 12» 1 -2imM1 K ] ,
(7 .8 )
mn
m ,n + l
ml I n
ml n
QM0 = l / 4 [ ( m + n + l )
M0 - 2 i ( m + l ) M 0 K + i( m + 2 ) N O1 ^
ml n
.
i(m + l)N
0 ],
(7 .9 )
53
mn
m ,n + l
ml I n
ml n
ml. n
QL1 = l / 4 [ ( m + n )
Lj - 2 1 mK L j - I m L 1 H2 1 + i ( m + l ) L 2 H j j ] ,
(7 .1 0 ).
mn
m ,n + l
ml I n
ml n
ml n
QL2 = l / 4 [ ( m + n + l )
L2 -2imK
L3 - I m L j H22+ i ( m + l ) L 2 H j 3 ] ,
(7 .1 1 )
mn
m ,n + l
ml I n
ml n
ml n
QNj1 = l / 4 [ ( m + n )
Njj-21'mMj L j - t m N j j H g j + i f m + l j N j g H j j ] ,
(7 .1 2 )
n
l/4 [(m + n + l)
m ,n + l
ml I n
ml n
ml
N1 2 -2 im M j L2+ i ( m + l j N j g H j g - i m N j j H ^ ] ,
m ,n + l
ml I n
ml n
l/4 [(m + n + l)
N ,.-2 i(m + l)M , L .+ i(m + 2 )N „ H
ml n
11
i ( m + l ) N 2 1 H2 1 ] ,
(7 .1 3 )
(7 .1 4 )
mn
m ,n + i
ml I n
ml n
QN22 = l / 4 [ ( m + n + 2 )
N2 2 - 2 i ( m + l ) M 2 L 2+i.(m +2)N 22H j 2 i ( m + l ) N 2 1 H22] .
(7 ,1 5 )
As an example o f how t h e above r e l a t i o n s were o b t a i n e d . E q u a tio n
is
(7 .1 2 )
d e r i v e d i n Appendix B.
F i n a l l y , th e in f in it e s im a l
Y (s e e E q u a tio n
(2 .1 5 ,1 6 ))
is
a c t i o n o f Q on t h e m e t r i c f u n c t i o n
shown i n Appendix C t o be
Qy = 0
(7 .1 6 )
The F i n i t e
The f i n i t e
fu n c tio n s F ^ ( t )
a c tio n
on @ j ( t )
Q T r a n s f o r m a t io n
tra n s fo rm a tio n s
and <&A ( t )
and H j 2 ( t )
a r e found by u sin g t h e g e n e r a t in g
= iMA ( 0 , t ) .
is
found fro m E q u a tio n s
. Q $ j ( t ) = l / 4 [ D 1$ 1 ( t ) - $ 1 ( t ) / t
QF1 2 ( t )
w ith Dj = 3 / a t .
F o r exa m p le , t h e i n f i n i t e s i m a l
+ i H 12$ j ( t ) ] ,
= l / 4 [ D 1F 1 2 ( t ) + i H 12F 1 2 ( t ) ] ,
(7 ,3 ,5 )
t o be
(7 .1 7 )
(7 ,1 8 )
These e q u a t io n s a r e i n t e g r a t e d i n Appendix D t P y i e l d
54
th e f i n i t e
tra n s fo rm a tio n s .
The o t h e r f i n i t e
i n a s i m i l a r manner; a l l o f t h e f i n i t e
I t F n (s + t)
( Q ) 4 sF n ( t ) -
(s + t)p
(s)
12 '
t r a n s f o r m a t i o n s a r e found
tra n s fo rm a tio n s
are lis t e d
b elo w .
»
(7 .1 9 )
1F1 2 ( s + t )
(Q )4 s F l2 (t)
(7 .2 0 )
F io (S ) >
N2 2 ( s , s ) F 1 1 ( s + t )
( Q ) 4 s F2 IC t ) -
Noi ( s » s + t )
(s + t)
■.
F19(S)
is
(s + t)
(7 .2 1 )
N
(Q )4sF 2 2 ( t )
= 1
N22( S 1S H ) -
(s ,s )F ._ (s + t)
^
P
(s)
(7 .2 2 )
it*i(s + t)(7 .2 3 )
(Q )4s * i ( t ) = ( s + t)C 1 2 (s )
(7 .2 4 )
( Q ) 4 s$ 2 ( t ^ " ( s + t )
S in c e t h e a c t i o n o f t h e i n f i n i t e s i m a l
d ire c tly
from t h e vacuum c a s e ,
it
Q t r a n s f o r m a t i o n was o b t a in e d
sh o u ld be no s u r p r i s e t h a t t h e r e s u l t s
h e re a r e s i m i l a r .
The vacuum r e s u l t y ' ( p ' , z ' ) = y ( p , z )
im m e d ia t e ly from E q u a tio n
fu n c tio n y ' ( p , z )
(7 .1 6 ).
is
seen t o fo llo vy
T h is means t h a t t h e new m e t r i c
e q u a ls y ( p , z ) , where p = ( Q ) ^ 4sP and z = ( Q ) _ 4sz .
Search f o r Q
In E q u a tio n
in fin ite s im a l
(5 .2 6 ),
t h e r e l a t i o n s h i p between t h e vacuum
t r a n s f o r m a t i o n s Q and Q i s
ta k e n as t h e s t a r t i n g a n s a t z t p f i n d
f u r t h e r re q u ir e m e n t w i l l
g iv e n .
T h is e q u a t io n w i l l
Q f o r th e e le c tro v a c case.
be
A
be t h a t t h e Q group should be a c o n ju g a t e group
55
SQS where S i s one o f t h e i n v o l u t i v e automorphisms
w ith ,
( r»?)
I
.
To begin
d e f i n e an o p e r a t o r Qn by
I
Q-Q0 = - W
u
12.
(7 .2 5 )
Q0 has t h e f o l l o w i n g a c t i o n on t h e p o t e n t i a l s .
~ mn
m+1,n
m ,n + l
ml I n
Q0 Nab = l / 4 [ ( m + l )
NAB+n NAB+iMA L , ] ,
E q u a tio n
(7 .2 6 )
^ mn
m + l,n
m ,n + l
ml n .
Q0K = l / 4 [ ( m
K +n K - 1 / 2 1 Lx » X] ,
(7 .2 7 )
~ mn
m + l,n
m ,n + l
mn nY
Q0Ma = l / 4 [ ( m + l / 2 )
Ma +n
- l / 2 i N AX» X] ,
(7 .2 8 )
~ mn
m + l,n
m ,n + l
ml
Q0 La = l / 4 [ ( m + l / 2 )
La +n L_A + iK
(7 ,2 9 )
(7 .2 6 )
is
In
Lft] .
v e ry s i m i l a r t o t h e e x p r e s s io n found by Cosgrove
(1 9 8 0 : E q u a tio n ( 2 . 1 4 ) ) f o r vacuum.
The d i f f e r e n c e l i e s i n t h e te rm
ml In
iMA Lg.
I t i s t h i s te rm and t h e s i m i l a r term s i n E q u a tio n s ( 7 . 2 6 - 2 9 )
w hich p r e v e n t a s im p le i n t e g r a t i o n t o t h e f i n i t e
a lte rn a tiv e
f o r Q i s ; t h e r e f o r e , s o u g h t.
I
s t a g e , t h e Z t r a n s f o r m a t i o n o f E q u a tio n s
An
d e fin itio n
At t h i s
n o tic e d .
tra n s fo rm a tio n .
( 6 . 5 2 - 5 5 ) was
In p a r t i c u l a r .
mn
ml n„
K = Lx * x .
mn
m + l,n
LB = 1
mn
M b
mn
(7 .3 0 )
ml I n
LB ~2K
(7 ,3 1 )
LB *
i
i
m + l,n ml n Y
=
mB
+ V
ml I n
(7 .3 2 )
>
'
I
(7 .3 3 )
NAB = - 2MA LBe
I
56
D e f i n i n g Q as
- ' I
Q = Q0+ i Z / 8 ,
(7 .3 4 )
t h e a c t i o n on t h e p o t e n t i a l s i s much s i m p l e r .
,mn
m + l,n m ,n + l
QK = l / 4 [ m
K +n
K ],
~mn
QMa
=l / 4 [ ( m + l )
»mn
QLa
m + l,n
m ,n + l
= l/4 [m
La +n
Lft ] ,
(7 .3 5 )
m + l,n
m»n+l
Ma +n Ma
],
~mn
m + l,n
QNab = l / 4 [ ( m + l )
(7 .3 6 )
(7 .3 7 )
m ,n + l
Nab +n
Na b ] .
(7 .3 8 )
U n f o r t u n a t e l y , due t o t h e i in E q u a tio n ( 7 . 3 4 ) , t h i s Q i s n o t H q r m i t i a n .
.
I
Q and Q* = QQ- i z / 8 must be re g a rd e d as two d i f f e r e n t i n f i n i t e s i m a l
o p e ra to rs .
For t h i s
i s an e s s e n t i a l
a r e needed.
E q u a tio n s
re a s o n ,
c o m p lic a tio n ,
(Q A )* *
QA** b u t r a t h e r
(Q A )* = Q *A *.
s in c e t h e t r a n s f o r m s o f K * , LA* ,
U sing t h e complex c o n ju g a t e s o f E q u a tio n s
T h is
Ma * ,
Nab *
( 7 . 3 0 - 3 ? ) and
(6 .1 3 -1 6 ) y ie ld
-mn
m + l,n
m ,n + l
ml nv
QK *
=l / 4 [ m K *+n
K * + iL xV * ] ,
(7 = 3 9 )
-mn
QMa *
m + l,n
m ,n + l
ml
nx
=l / 4 [ m Ma *+n Ma * + i N AX* » A* ] ,
(7 .4 0 )
-mn
QLa *
m + l,n
m ,n + l
ml n Y
=l / 4 [ m L a * + ( n + l )
LA* + i L x* H \ * ] ,
(7 .4 1 )
-mn
QNab*
E q u a tio n s
m + l,n
=l/4 [m Nab* * ( „ +1)
(7 .3 5 -4 2 )
m ,n + l
E q u a tio n s
nY
Nab * +1'NAX*HB*t3EAB« - I s
and t h e i r complex c o n ju g a t e s
be s o lv e d u s in g g e n e r a t i n g f u n c t i o n s .
fu n c tio n s .
ml
( 7 . 3 5 - 3 8 ) ,are
( 7' 42>
(Q r e p la c e d by Q *) may
In term s o f t h e g e n e r a t i n g
.57
Q K tt1 st 2 ) = l / 4 C D 1+D2 - ( t 1 + t 2 ) / ( t 1 t 2 ) ] K ( t 1t 2 ) +
M t 2A 1
- t 1/ t 2) / 8 ,
(7 .4 .3 )
QMa (t ^ t 2 ) = l / 4 [ D 1+D2 - l / t 2 ]MA ( t 1 , t 2 ) ,
(7 .4 4 )
= l / 4 [ D 1+D2 - ( t 1 + t 2 ) / ( t 1t 2 ) ] L A ( t 1 , t 2 ) ,
(7 .4 5 )
(7.46)
^NAB( t l » t 2 ) = 1/ 4 [ D 1+D2 - l / t 2 ]NAB( t v t 2 ) + eAB/ t 2$
where D1 = 3 / 3 t 1$ D2 = 3 / 3 t 2 .
g e n e r a t i n g f u n c t i o n s FAB* ( t )
Q$A* ( t )
QFAB* ( t )
where D = 3 / 3 t .
e xam p les .
=
E q u a tio n s
(7 .4 0 ,4 2 ),
= - i N AB* ( 0 , t )
in term s o f th e
and » A* ( t )
= - i M A* ( 0 , t )
l/4[D$A* ( t ) - $ A* ( t ) / t + i H Ax * $ * ( t ) * ] ,
(7 .4 7 )
(7.48)
= l / 4 [ D F AB* ( t ) + i H AX* F xB ( t ) * ] ,
E q u a tio n s
The r e s u l t s
(Q)4sK(ti,t2)
(7 .4 6 ,4 8 )
are th e f i n i t e
are
are in te g ra te d
i n A ppendix E as
tra n s fo rm a tio n s :
= (Sft1)(Sft2) K(Stt1^tt2) +
■
is(t2-t1)(s+t1+t2)
— 2 ( s n 1 ) ( s + t 2T
“
•
(7"49)
( Q ) 4 s l A ^ i ' t Z^
= ( S H 1) ( S H 2) ^ A ^ M l'S + tg )
^ H s mA ^ I $t 2^
= ( s + t 2 ) MA ( s + t 1 , s + t 2 ) ,
•w
to
s
(7 .6 1 )
Sg ad
NAB^s+tl’S+t2^ + (SH2) ’
= 1(Ft"l(s))AXf*B‘(s+t)’
(Q)4sV t t)
(7 .5 0 )
=TriIr (F*"1(S,)AX*X*(S+t)-
t7"52)
(7.53)
'7.54)
58
The r i g h t - h a n d s id e s o f t h e s e e q u a t io n s c o n t a i n t h e o r i g i n a l
fu n c tio n s .
The t r a n s f o r m a t i o n s o f ( Q * ) 4s may be found by t a k i n g t h e
complex c o n ju g a t e o f E q u a tio n
It
(7 .4 9 -5 4 ).
remains t o be shown t h a t
( Q ) 4s by one o f the, d i s c r e t e
groups
t h e commutators o f t h e i n f i n i t e s i m a l
29)
and E q u a tio n s
_ k
CQqst Xy]
~
E q u a tio n
k
( Q ) 4g and ( Q * ) 4s a r e
( r ss)
I
.
re la te d to
T h is i s done by l o o k in g a t
p p e ra to rs .
U sing E q u a tio n s
= (2 k -l)
k + l,
Cx / 8
k
k+l
CQqsCx* ] = ( 2 k - l ) C x * / 8
,
(7 .5 5 )
(7 .2 5 ),
=
i t i s a l s o found t h a t
k+l
k
k+l
( k - l ) T n /4
s CQsT 2 2 ] = ( k + l ) T 2 2 / 4 ,
k
k+l
=
kT
12/ 4
W -T 12]
H
I—I
OJ
O
Cr
L-J
k
k
k
k+l
CQsC1] = ( k - 1 ) C1M ,
,
k+l
k C2 / 4
k
CQsC1 * ]
,
k+l
C Q .C ,* ] =
k
C * /4
,
k
CQsE ]
k+l
? ( k - l ) C 1* / 4 ,
k+l
= k E /4 .
an o p e r a t o r J i s t o be a c o n ju g a t e o p e r a t o r SQS, J ' s
be c o n ju g a t e t o Q 's com m u tato rs.
(7 .5 6 )
commutators must
That i s ,
CJ»A] = S [ Q ,A ] S ,
(7 .5 7 )
where S i s t o be one o f t h e
o f in te g e r
(7 .2 6 -
( 6 . 4 0 - 5 5 ) , t h e f o l l o w i n g commutators a r e o b t a in e d .
k+l
„ k
k+l
- kTXy / 4
,
CQ0E] ? k E / 4 ,
k
CQqsCx]
If
g e n e r a t in g
(r,s )
is t h is
D e fin in g th e o p erato rs
I
Qn = Q0+ i n z / 8 ,
(r,s )
I
.
It
i s found t h a t f o r no c o m b in a tio n
t r u e c o n c e rn in g t h e o p e r a t o r s Qq and Q0
(7 .5 8 )
59
w i t h n r a n g in g o v e r a l l i n t e g e r v a l u e s , u s in g t h e n o n v a n is h in g
I
commutators o f £ i n E q u a tio n s ( 7 . 5 5 , 5 6 ) and in
I k
k+1
[ £ ,C X]
then Qn =
i
Cv ,
:x
I k
[Z ,C * ]
(7 .5 9 )
- 1 CX *
(r,s)
(r,s)
I Q
I for the following integer values of (n,r,s) =
( 2 m + l , l , m + l ) ; m=0 , 1 , 2 , 3 , . . .
It
i s seen t h a t t h e o p e r a t o r s Q* and Q
( 1 , 0)
( 1 , 0)
defined previously in the chapter are
I Q I
re s p e c tiv e ly .
it
(6 .5 7 -6 0 )
(In
fin d in g th is
re s u lt,
( 1 , 1)
and
( 1 , 1)
I Q I . ,
must be n o ted t h a t E q u atio n s
h o ld f o r t h e p a ra m e te rs o f t h e t r a n s f o r m a t i o n .
It
i s a ls o
p o s s i b l e t o w r i t e t h i s i n term s o f t h e o p e r a t o r s ; f o r e x a m p le , i n
k
k+r
E q u a tio n ( 6 . 6 0 ) , T g E ^ l l • )
Now t h a t Q and Q* have been f o u p d , t h e n ext
s t e p i s t o a p p ly them.
60
CHAPTER 8
APPLICATION OF THE Q GROUP
In t h i s
c h a p te r, th e f i n i t e
tra n s fo rm a tio n
be found on t h e m e t r i c components f ^
and f ^ -
(Q )4 s l S a c tio n w i l l
I t w ill
t o t h e R e is s n e r - N o r d s t r o m m e t r i c and p o t e n t i a l s
th e n be a p p l i e d
found by T . C , Jones
(1 980).
A c t io n on t h e M e t r i c Components
E q u a tio n s
(6 .2 4 ,
2 9 -32)
and f^g
im p ly t h e f o l l o w i n g u s e f u l
re la tio n s .
( ! " 2 t z I=AB+ 2 i t f AB =
Sz ( t ) [ F * x * ( t ) F g * ( t )
F o r t h e m e t r i c components f ^
Z itfii
and f ^ *
+ 2I
th is
(8 .1 )
im p lie s
= S 2 ( t ) ( F 11* ( t ) F 1 2 ( t ) - F 12* ( t ) F 1 1 ( t )
|1
$ i*(t)
*i(t)),
(8 .2 )
Z i t f 12 = S 2 ( t ) ( F n * F 2 2 ( t ) - F 12* ( t ) F 2 1 ( t )
The f i n i t e
a c tio n
|1
$ i*(t)$ 2 (t))
-
of
( Q ) 4s on f ^
w ill
len g th of the c a l c u l a t i o n ,
+
+
(l-2 tz )
the f i n i t e
(8 .3 )
be d e r iv e d h e r e .
a c tio n
Due t o th e
o f ( Q ) 4s on f ^
w ill
o n ly
be s t a t e d .
The f o l l o w i n g u s e f u l
re la tio n s w ill
be used.
( Q ) 4 s P = p S“ 2 ( s ) ,
(8 .4 )
(Q )4s z = " k
(8 .5 )
"
(z - & ) S " 2 ( s ) ,
(Q )4sS f t ) = S ( s + t ) / S ( s ) .
(8 .6 )
61
Here and e ls e w h e re in t h i s
S 2 (t)
U sing E q u a tio n
= (l-2 tz )2
(8 .2 )
ch ap ter,
+ (2tp )2„
(8 .7 )
i n t h e form
2 i t f 11 = S 2 (t)(F 11* ( t ) F 12( t ) . F 12* ( t ) F 11(t ) + § l $ i * ( t ) * i ( t ) )
where the overhead bars indicate transformed quantities, and using
Equations ( 7 . 1 9 , 2 0 , 2 2 ) , i t is possible to show that
tS 2 (s + t)
2 itf i
( s + t ) S 2 ( s ) F 12* ( s ) F 1 2 ( s )
[ F i i * ( s + t ) F i 2 ( s + t ) - F 12* ( s t t ) F u ( s + t )
+
(8. 8 )
*i*(s + t)$ i(s + t)].
Comparing E q u a tio n
(8 .7 )
w i t h E q u a tio n
(8 .2 );
it
fo llo w s th a t
(8 .9 )
(Q )4 s fl1
T h is
is
E q u a tio n
S 2 ( s ) F 12* ( s ) F 1 2 ( s )
"
e x a c t l y t h e form found f o r t h e vacuum case by Cosgrove
(3 .1 c )),
It
(1 980:
can a ls o be shown t h a t , t h e f o l l o w i n g e q u a t io n i s
tru e .
( Q ) 4 Sf i2
i (z - l/2 s )
N2 2 ( s » s ) f j i
S2(S)
S2 ( s ) F 12* ( s ) F 1 ? ( s )
[ ( f 12+ i z ) [ F 1 2 ( s ) d e t ( F * ( s ) )
|1
( $ i * ( s ) F 22* ( s )
+
- $ 2 * ( s ) F 12* ( s ) ) $ 1 ( s ) ]
f i i [ F 22 ( s ) d et ( F * ( s )
+
*2 *(s)Fi2 *(s))*2 (s))]].
( # i* ( s ) F 22* ( s )
-
-
(8.10)
62
A p p lic a tio n
T .C .
re s u lts
Jones
(1 9 8 0 :
o f Q. t o R e is s n e r - N o r d s t r o m
E q u a tio n s
(6 .4 4 -4 7 )
found t h e f o l l o w i n g
f o r R e is s n e r - N o r d s t r o m .
*i(t)
o - 4 t 2 r 1 /2 .
F ll(t)
fll
(8 .1 1 )
'T x W
e ] [ i ” 4 t 2 ] " 1/ 2 .
(8 .1 2 )
3 ] [ l- 4 t 2r 1 /2 ,
(8 .1 3 )
= (X 2 ~ l ) / ( x + B ) 2 .
In th e s e e q u a t i o n s , B i s
a c o n s t a n t a s s o c ia t e d w it h t h e c h a rg e -to -m a s s
r a t i o , _x and y a re p r o l a t e
X = ^
y =
-
[ /(z+ k )2
2Y
[
(8 .1 4 )
s p h e r o id a l
c o o r d in a t e s d e f i n e d by
+ p2 + / ( z - k ) ?
v' ( z + k ) 2 +
p 2
-
+ p2],
/ (z-k)2 +
(8 .1 5 )
(8 .1 6 )
p 2 ] ,
which may be w r i t t e n
x = l/2 [S (-l/2 k )+ S (l/2 k )],
(8 .1 7 )
y = l/2 [S (-l/2 k )-S (l/2 k )].
(8 .1 8 )
and
From E q u a tio n
(8 .8 ),
f in ( x , y )
fii(x ,y )
(8 4 9 )
S2(x,y;s)F12 *(x,y;s)F12(x,y;s)
T h is i s e q u i v a l e n t t o
fii(x ',y ')
(8 . 20 )
f n (x ,y )
S2(s,,y';s)F12 *(x',y';s)F12 (x',y';s)
w it h
(x ',y ')
= ( ( Q ) _ 4 sx , ( Q ) _ 4 sy ) .
S (x ',y ';s )
and u sin g E q u a tio n s
;
( X i y ,
11
'
-
U sin g
= ( Q ) „ 4 sS ( x , y ; s )
(8. 21)
=. l / $ ( - s ) ,
( 8 . 1 0 - 1 2 , 1 7 , 1 8 , and 2 0 ) ,
it
fo llo w s t h a t
( l- 4 s ) 2[(S (-s -l/2 k )+ S (-s + l/2 k ))2 _ 4 S 2 (.s )]
[S (-s -l/2 k )(l-2 s )+ S (-s + l/2 k )(l+ 2 s )+ 2 $ ]2
(8. 22)
63
Use o f E q u a tio n
(8 .1 0 )
i s n o t needed in c a l c u l a t i n g
f l 2 » s in c e f o r R e is s n e r - N o r d s t r q m f ^
th is
is to
begin w it h E q u a tio n
(7 .9 )
is
zero.
( Q ) ^ 1S a c t i o n on
A n o th er way o f s e e in g
in K i n n e r s l e y
h e re as
(1977a)
reproduced
1
Vw = p f 2 (VS11+i((J.1*V(j.1- <(,1V<|)1* ) ) .
The f a c t t h a t F 1 1 ( I )
R e is s n e r - N o r d s t r o m ,
and S1 ( t )
and F 1 2 ( t )
combined w i t h E q u a tio n s
t h e t r a n s f o r m e d F 1 1 ( t ) , S1 ( t )
rig h t-h a n d
are real
s id e o f E q u a tio n
are
re a l.
(8 .2 3 )
is
(8 .2 3 )
pure im a g in a ry f o r
(7 .1 9 ,2 0 ,2 3 ),
T h is
zero .
is
in t u r n
im p l i e s t h a t
im p lie s t h a t the
T h is means t h a t t h e
t r a n s f o r m e d w i s a c o n s t a n t , which we a r e f r e e t o choose as z e r o ,
it
is
seen t h a t
( Q )4 S le a v e s t h e R e is s n e r - N o r d s t r o m s p a c e tim e s t a t i c .
In an argument s i m i l a r t o t h a t f o r f i n d i n g
w i t h E q u a tio n
(7 .2 3 ),
I 1 ( X 1V )
Thus,
it
is
( Q ) 4 Sf n »
b e g in n in g
p o s s i b l e t o show t h a t
= ------------- :-------------------- ^
------------------------ --------- ---
(8.24)
[S (-s -l/2 k )(l-2 s )+ S (-s + l/2 k )(l+ 2 s )+ 2 fj]
S in c e t h i s
E q u a tio n s
is
re a l, i t
(6 .6 )
im p ly ,
is
equal
t o A^, and
i s equal
to
zero.
Then
s in c e w = 0,
**
I
V t2 = - I p f
V *j,
which may be i n t e g r a t e d t o y i e l d
A
(8 .2 5 )
64
CHAPTER 9
SUMMARY
The main r e s u l t s
8.
In C h a p te r 7 ,
m e t r i c was fo u n d .
th is
p o in t,
t h e s i s were found in C h a p te rs 7 and
t h e vacuum groups found by
t h e e l e c t r o v a c c ase.
N ordstrom m e t r i c ;
of th is
In C h a p te r I ,
the Q tra n s fo rm a tio n
As, an ex a m p le , t h i s
however,
Cosgrove were extended t o
on an a r b i t r a r y
was a p p l i e d t o t h e R e i s s n e r -
no new s o l u t i o n was found in t h i s way.
Up t o
t h e e x t e n s io n o f t h e vacuum case t o e l e c t r o v a c has been
s u c c e s s fu l.
A problem t h a t has a r i s e n
Q and Q* found i n C h a p te r 7 .
tra n s fo rm a tio n s
The c a r e f u l
(7 .4 9 -5 4 ).
sen ten ce,
c o n s id e r t h e f o l l o w i n g .
2 fIl
They have been
a c tio n
o f Q (Q *) on t h e m e t r i c f u n c t i o n
i s t h e meaning o f
As an example t o what i s meant by t h e
not equal
to
E q u a tio n s
((Q j^ H ^ )*.
(7 .5 2 ,5 3 )
What th e n i s
( ( Q ) ^ H 1 1 )*?
la s t
<
im p ly t h a t
Now by d e f i n i t i o n ,
= Hi i +Hn * +2$ i * t i i ‘
To c a l c u l a t e t h e t r a n s f o r m e d f ^ ,
used.
a v o id a n c e o f t h e s e
Second, and most p r o b l e m a t i c ,
E q u a tio n s
is
q f the
d is c o v e re d .
the in f in it e s im a l
has not been fo u n d .
(Q )4 s I H u * )
in t h e a p p l i c a t i o n
in C h a p te r 8 was no a c c i d e n t .
tro u b le s o m e s in c e f i r s t
F irs t,
r e s id e s
(9 .1 )
(Q)^js (H 11* )
and ( Q ) ^ f a 1 * )
must be
T h is problem was a l r e a d y mentioned in
I
I
C h a p te r 7 and was f o r c e d upon t h e problem when t h e i z / 8 t e r m was added
65
to the tr a n s fo r m a tio n .
complex m e t r ic s
It
may be b e t t e r t o
le a v e out t h i s
te rm i f
a r e t o be a v o id e d .
The f o l l o w i n g
argument su ggests t h a t t h e vacuum case may be
f u r t h e r exten d ed t o e l e c t r o v a c .
T h ere a r e a group o f theorems known as
B aker-C am p b elI - H a u s d o r f th eo rem s.
e xp (sQ )exp (-sQ )
The one o f use is
= exp(sQ-sQ+|-2 [ Q sQ]+-|^ [ Q sCQ9Q ] ] +
Y2"
(9 .2 )
[[Q » Q ]sQ ]+ °• °
From C h a p te r 7 ,
CQ9Q ]
I
1
2
2
= l / 2 [ Q , T 12] + i [ Q , % ] / 8 = T 1 2 / 1 6 + i Z / 3 2 ,
(9 .3 )
S im ila rly ,
CQ9CQ9Q ] ]
CCQ9Q ] Q ]
What t h i s
seems t o
3
3
= T1 2 / 3 2 + i z / 6 4 ,
3
3
= T1 2 / 3 2 + i Z / 6 4 .
im p ly i s
ex p (s Q )e x p (-s Q )
Because a l l
(9 .6 )
(9 .4 )
(9 .5 )
th a t
«»
n
n
= exp ( I (a^ T + b ^ ) ) .
n=l
(9 .6 )
k
k
t h e T 12 and z commute, and as long as t h e t r a n s f o r m a t i o n
e x is ts ,
the th e tra n s fo rm a tio n s
commute w it h each o t h e r .
Backlund t r a n s f o r m a t i o n ;
o f t h e form e * p ( s Q ) e x p ( ^ s Q ) w i l l
T h is may be t h e e x t e n s io n o f H a r r i s o n ' s
see Cosgrove
(1 981).
The im p o rta n c e in s t u d y in g t h e Q (Q *) t r a n s f o r m a t i o n
e q u i v a l e n t modes o f e x p r e s s io n in o t h e r methods w i l l
e l e c t r o v a c case t o t h e s t a t e
s in g le
or t h e i r
p ro b a b ly b r in g t h e
o f success o f t h e vacuum c a s e .
66
REFERENCES CITED
67
REFERENCES CITED
VeAo Bel i n s k i i and V .E . S akh aro v. 1978« I n t e g r a t i o n o f t h e E i n s t e i n
E q u a tio n s by Means o f t h e In v e r s e S c a t t e r i n g Problem T ech n iq u e and
C o n s t r u c t io n o f Exact. S o l u t i o n s . Sov. Phys. JETPe 4 8 : 9 8 5 - 9 9 4 ,
V .A ,
B e l i n s k i i and VeEe S a kh aro v.
S o l i t o n s w it h A x ia l Symmetry.
1 979. S t a t i o n a r y G r a v i t a t i o n a l
Sov. Phys. JETP. 5 0 : 1 - 9 .
P0Me Cohn. 1957. L i e Groups, Cambridge T r a c t s
# 4 6 , Cambridge: Cambridge U n iv . P re s s .
C.M. Cosgrove.
1 979a .
PhD T h e s is .
in M a th e m a tic a l
Physics
U n i v e r s i t y o f Sydney.
C.M .
Cosgrove. 1979b. C on tin u o u s Groups and Backlund T r a n s f o r m a t io n s
G e n e r a t in g A s y m p t o t i c a l l y F l a t S o l u t i o n s . L e c t u r e p re s e n te d a t The
Second M arcel Grossmann M e e tin g on t h e Recent Development o f General
R e l a t i v i t y , T r i e s t e , I t a l y . 5-1 1 J u l y , 1 979.
C.M.
Cosgrove. 1980a . R e l a t i o n s h i p s Between t h e G r o u p - T h e o r e t ic and
S o l i t o n - T h e o r e t i c Tech n iq u es f o r G e n e r a t in g S t a t i o n a r y A x is y m m e tric
G r a v i t a t i o n a l S o l u t i o n s . JMP. 2 1 : 2 4 1 7 - 2 4 4 7 .
C.M.
Cosgrove.
1980b.
P r i v a t e c o m m u n icatio n .
C.M .
C osgrove. 1 981. Backlund T r a n s f o r m a t io n s in t h e Hauser-rErnst
Form alism f o r S t a t i o n a r y A x is y m m e tric S p a c e tim e s . JMP. 2 2 : 2 6 2 4 - 2 6 3 9 .
L .P .
E i s e n h a r t . 1 909. A T r e a t i s e on t h e D i f f e r e n t i a l
and S u r f a c e s . Boston: Ginn.
L.
Geometry o f Curves
P. E i s e n h a r t . 1 9 3 3 . Continuous Groups o f T r a n s f o r m a t io n s .
1961. New Y o rk: Dover P u b l i c a t i o n s , I n c .
R e p r in t e d
F .J .
E r n s t . 1 968a . New F o r m u la t io n o f t h e A x i a l l y Symmetric
G r a v i t a t i o n a l F i e l d Problem . Phys. R eview . 1 6 7 : 1 1 7 5 - 1 1 7 8 .
F .J .
E r n s t . 1968b. New F o r m u la t io n o f t h e A x i a l l y Symmetric
G r a v i t a t i o n a l F i e l d Problem . I I . Phys. Review. 1 6 8 : 1 4 1 6 - 1 4 1 6 .
A .R .
F o r s y t h e . 1 9 5 9 . Theory o f D i f f e r e n t i a l
New Y o rk: Dover P u b l i c a t i o n s , I n c .
E q u a tio n s ,
v.
6.
pg. 200.
M. Hamermesh. 1 962. Group Theory and I t s A p p l i c a t i o n t o P h y s ic a l
Pro b lem s. Second P r i n t i n g 1964. C a l i f o r n i a : A d d is o n -W e s le y
P u b lic a t io n s , In c .
68
I.
Hauser and F . J . E r n s t . 1979a . I n t e g r a l E q u a tio n Method f o r E f f e c t i n g
K i n n e r s l e y - C h i t r e T r a n s f o r m a t i o n s . Rhys. Rev, D. 2 0 : 3 6 2 * 3 6 9 .
I.
Hauser and F . J . E r n s t . 1979b. I n t e g r a l E q u a tio n Method f o r E f f e c t i n g
K i n n e r s l e y - C h i t r e T r a n s f o r m a t i o n s , I I . Rhys. Rev. D. 2 0 : 1 7 8 3 - 1 7 9 0 .
I.
Hauser and F . J . E r n s t . 1980a. A Homogeneous H i l b e r t Problem f o r t h e
K i n n e r s l e y - C h i t r e T r a n s f o r m a t i o n s . JMP. 2 1 : 1 1 2 6 - 1 1 4 0 .
I.
Hauser and F . J . E r n s t . 1980b. A Homogeneous H i l b e r t Problem f o r t h e
K i n n e r s l e y - C h i t r e T r a n s f o r m a t io n s o f E l e c t r o v a c S p a c e tim e s . JMP.
21:1418 -1422.
C. H o e n s e la e r s j B. X a n th o p o u lo s , W. K i n n e r s l e y . 1 979. Symmetries o f t h e
S t a t i o n a r y E in s t e i n - M a x w e l l E q u a t io n s . V I . T r a n s f o r m a t io n s which
G e n e ra te A s y m p t o t i c a l l y F l a t Spacetim es w it h A r b i t r a r y M u l t i p o l e
Moments. JMP. 2 0 : 2 5 3 0 - 2 5 3 6 .
T .C .
Jo n es. 1 9 8 0 . G r a v i t a t i o n a l
S t a t i o n a r y E in s t e i n - M a x w e l l
and E le c t r o m a g n e t ic P o t e n t i a l s o f t h e
F i e l d E q u a t io n s . JMP. 2 1 :17<)0 - 1 7 9 7 .
W. K in n e r s l e y .
E q u a t io n s .
1977a . Symmetries o f t h e S a t i o n a r y E i n s t e i n - M a x w e l l
I . JMP. 1 8 : 1 5 2 9 - 1 5 3 7 .
W. K i n n e r s l e y .
E q u a t io n s .
1977b. Symmetries o f t h e S t a t i o n a r y E i n s t e i n - M a x w e l l
I I . JMP. 1 8 : 1 5 3 8 - 1 5 4 2 .
W. K i n n e r s l e y . 1 9 8 0 . The Geroch Group and S o T ito n S o l u t i o n s o f th e
S t a t i o n a r y A x i a l l y Symmetric E i n s t e i n ' s E q u a t io n s .
W. K i n n e r s l e y and D.M. Chi t r e . 1 978a . Symmetries o f t h e S t a t i o n a r y
E i n s t e i n - M a x w e l l E q u a t io n s . I I I . JMP. 1 9 : 2 0 3 7 - 2 0 4 2 .
W. K i n n e r s l e y and D.M. C h i t r e . 1978b. Symmetries o f t h e S t a t i o n a r y
E in s t e i n - M a x w e l l E q u a t io n s . I V . T r a n s f o r m a t io n s which P re s e rv e
A s y m p to tic F l a t n e s s . JMP. 19: 2 0 3 7 - 2 0 4 2 .
t
••
G. Lamb, J r . 1 974. Backlund T r a n s f o r m a t io n s f o r C e r t a i n
E v o l u t i o n E q u a t io n s . JMP. 1 5 : 2 1 5 7 - 2 1 6 5 ,
T.
N o n lin e a r
L ew is. 1932. Some S p e c ia l S o lu t io n s o f t h e E q u atio n s o f A x i a l l y
Symmetric G r a v i t a t i o n a l F i e l d s . Pr o c . Royal S o c i e t y London A.
1 3 6:176 -192.
G. Neugebauer. 1 979. Backlund T r a n s f o r m a t io n s o f A x i a l l y Symmetric
S t a t i o n a r y G r a v i t a t i o n a l F i e l d s . J . Phys. A: Math. Gen, ( l e t t e r s )
12:L 67-L 70.
G.
Neugebauer. 1 980a . A General I n t e g r a l o f t h e A x i a l l y Symmetric
S t a t i o n a r y G r a v i t a t i o n a l F i e l d s . J . Phys. A: M ath. Gen. ( l e t t e r s )
13:L 19-L 22.
$9
G.
Neugebauer. 1980b. R e c u r s iv e C a l c u l a t i o n
S t a t i o n a r y E i n s t e i n E q u a t io n s . J . Phys.
S.
W ein b erg . 1972.
Sons, I n c .
G ra v ita tio n
o f A x i a l l y Symmetric
A: Math. Gen. 1 3 : 1 7 3 7 - 1 7 4 0 .
and Cosmology.
New Y o rk: John W ile y &
70
APPENDICES
71
APPENDIX A
CONSISTENCY OF EQUATIONS ( 7 . 1 - 6 )
I
The p r o o f f o r E q u a tio n s
e q u a t io n s a r e
n
= -ip
(7 .1 -6 )
is
g iven h e r e .
The f i e l d
-I,
X_n
f^
( 6. 6)
and
n
-I
VHAB = ™1p
y—H
(6 .7 )
f A VHXB*
In o r d e r f o r Q t o le a v e t h e f i e l d
e q u a t io n s
in v a ria n t,
q u a n titie s
(6 .6 ,7 ).
(It
must s a t i s f y
E q u a tio n s
t h a t V and v a r e a ls o t r a n s f o r m e d . )
o p erato r, th is
re q u ire s
t h e t ra n s fo rm e d
must not be f o r g o t t e n
In term s o f t h e i n f i n i t e s i m a l
Q
th a t
n y
v (Q*a )
= - i p " 1f AXv(Q»x ) + ip ‘’ 1z f AXv$x- i p " :l(QfAX) v * x
(A .l)
and
- I jr x;,™"
-i_ « x:,1
, - i,„ , x ,-n
V(QHa b ) = - i p - V / , ( Q H x B) + 1 p - ' z f A' , H x B- , p - i ( Q f A* ) , H x B.
It
is
seen t h a t t h e q u a n t i t i e s
fn
it
= 1 / 2 ( H i i +H h * )
QfAg a r e needed.
(A .2 )
U sing
+
(A .3 )
f 12 = l / 4 ( H i 2 + H i 2 * ^ H 2 i + H 2 i * + 2 $ i * $ 2 + 2 $ 2 * $ i ) »
(A .4 )
f 2 2 = 1 /2 (H 2 2 + H 2 2 * )
(A .5 )
+ *2**2»
can be shown t h a t
Qf I l
- f 11^12
= ------ 9------ + f I l Z ,
"
(A.6)
f 11^22
Qf 12 = ----- 9------- + f I l Z ,
(A .7)
and
f 22^12
Qf 22 =
2------ + f 2 2 z " f 12^22'
(A .8 )
W r i t i n g E q u a tio n s ( A . l , 2 ) o u t in f u l l y i e l d s t h e f o l l o w i n g e q u a t io n s ,
n
I
_ n
+ ip™ f i 2v ( Q * i )
_i
n
~n
»n
~n
„n
- ip
Cf l l V ( Q * 2 ) + Zf l 2 ^ * l - Z f i i V $ 2 - (Qf l 2 ) V * i + ( Q f i i ) V $ 2 ] ,
(A .9)
v(Q *i)
73
-9
M
V(Qs2 ) I
= -ip "
»
M
i p " f I 2 V(Qs2 )
Il
Mil
Mil
MM
MM
[ f 2 2 V ( Q S i ) - z f 2 2 v s 1+ z f 2 1 v s 2+ ( Q f 2 2 Jvs1- ( Q f 2 1 )VS2 ] ,
(AolO)
V(QH1 1 )+!'? ^ f 1Zv (QH1 1 )
-
ip
C f 11V(QH2 1 ) + z f ^ V H 11- Z f 11VH2 1 - ( Q f 1 2 )VH11+ ( Q f 1 1 )vH2 1 ] ,
(A .ll)
V(QH21 ) - i p
= -ip
1 ^ 2 V(QH2 1 )
1 C f 22v(.QH1 1 ) - z f 22VH11+ z f 21VH2 1 + ( Q f 2 2 )vH 1 1 - ( Q f 21 ) v H2 1 ] ,
( A . 1 2)
V(QH1 2 ) + i p
1^f12V(QH1 2 )
• - I r jC
ip
C f 11V(QH2 2 ) + z f 12VH12- z f ^ v H 2 2 - ( Q f 1 2 JvH12+ ( Q f 1 1 JvH2 2 ] ,
( A . 1 3)
V(QH2 2 ) - I p
f 12V(QH2 2 )
_1
n
_n
_n
_n
„n
. = -ip
C f22V(QH1 2 ) - z f 22VH12+ z f 21VH2 2 + ( Q f2 2 )v H 12- ( Q f 2 1 )VH2 2 ] .
(A .14)
A ll
o f t h e above e q u a t io n s f a l l
in to
p a i r s o f e q u a t io n s s i m i l a r t o
v ( Q A ) + i p " 1f 12v(QA)
i
_ n
_n
_n
„n
„n
= ip
C f 11V ( Q B ) + z f 12V A - z f 11V B - ( Q f 1 2 )v A + ( Q f 1 1 ) V B ] ,
and
V ( Q B ) - i p " 1f 12v(QB)
-
-ip
F u rth erm o re,
Cf 22V ( Q A ) - z f 22V A + zf21V B +(Q f2 2 J v A - ( Q f 2 1 ) V B ] .
n
n
a l l t h e A and B s a t i s f y t h e f i e l d e q u a t io n s
VA = - i p
( f 12V A - f 11VB)
( A . 1 6)
(A .1 7 )
and
.n
_i
~n
~n
VB = - i p
( f 22V A - f 2 1 V B ).
n
n
E q u a tio n s ( 7 . 1 - 6 ) im p ly t h a t t h e A and B s a t i s f y
n
n+1
n
QA = l / 4 [ n A + i H 12A ] ,
( A . 1 8)
( A . 1 9)
74
n
n+1 I n
n
n+1 I n
n
QB = l / 4 [ n B + i M 2 -HH2 2 A] o r l / 4 [ n B H N21H H2 2 A]
(A . 2 0)
n
n
n
n
n
n
f o r t h e A c o n s i s t i n g o f G1 o r H1 1 , and t h e B c o n s i s t i n g o f G2 o r H2 1 ,
and
QA =
n
QB =
n
when t h e A a r e
n+.l
n
l/4 [(n + l)A
-H H 12A] ,
n+1
In
n
l/4 [(n + l)B
+ i N 2 2 + iH 2 2 A]
n
n
n
H12 and t h e B a r e H2 2 .
(A .2 1 )
(A .2 2 )
The p r o o f beg in s by e x a m in in g t h e l e f t - h a n d
s id e o f E q u a tio n
(A .1 5 ),
n
,
» n
n+1
n
n
V ( Q A ) + i p ' 1f 12v(Q A ) = l / 4 [ ( n + l ) v A + i H 12VA + IA vH 12
I
_n+l
_h
n_
+ i p " f i z C f n t l ) ? A + i H 12vA +iA vH 1 2 ] ] ,
w ith
(n + 1 ) e q u a lin g n i f
(A .2 1 )
i s used.
E q u a tio n
U sin g E q u a tio n
( A . 19)
(A .1 7 )
i s used o r (n+%)
and E q u a tio n
(6 ,7 ),
if
E q u a tio n
the le ft-h a n d
s id e i s now
i
l/4 [(n + l)(-ip "
+ iH i2 (-ip
~n+l
_n+l
f 12V A + i p _ i f U V B )
i
~n
_I
~n
f 12VA+ip
f^ V B )
'
+ i A( - i p f 12VH12+ ip
fn y H 22)
i
_n+l
^n ! n_
+ i p ~ ^ f i 2 [ ( n + l ) V A + iH 12v A + iAvH1 2 ] ]
ip "
=
Now t h e
I
'n
~n+l
„n
n„
------ [ ( n + l ) V B + i H 12VB+iAvH2 2 ] .
r i g h t - h a n d s id e i s exam ined.
i
.
n
_n
_n
„n
_n
ip
11V ( Q B ) + z f 12V A - z f 11V B - ( Q f 1 2 ) v A + ( Q f 1 1 )v B ]
i
= ip
„
n+1
In In
In
n
[ f i i V l / 4 [ ( n + l ) B + i (M2 ,N2 1 ,N2 2 ) + i H2 2 A]
~n
„n
f 11^22
o
f 11^22
h
+ z f 12V A - z f n VB+( ■■ g— ■ , f 12z ) v A + ( g
+ ^ i 1 z JvB]
;
-l
75
Ip- Tn
~n+l
„ ln „ ln
^ ln
„n
n_
------ 2— - C ( n + l ) v B + i (VM2 ,VN2 I , V N 2 2 ) + i H22VA tiA vH 22
n
n
+Zfi2 2 VA -Zn12VB]
w
_n+l
„n
„n
_n
„n
^------ C ( n + l ) v B + 1 ( H 12*V » 2 -H 2 2 * V $ i ,H 12*VH2 1 -H 2 2 *v H n »
** n
n
#wn
n n*
Hi2*?H22-H22*VHi2)+(iH22+Zn22)vA-Zni2VB+iAvH22] .
S in c e t h e q u a n t i t i e s
i n p a re n th e s e s a l l
~n
_n
r e p r e s e n t H12 aVB-H2 2 * v A ,
ip -1 f n
_n+l
„n
„n
n
= ------ ^------ [ ( n + l ) v B + i ( H 12*v B -H 2 2 * v A ) + ( i H 2 2 +ZQ2 2 )vA
n n~
- Z n 12VB+!* AvH2 2 ]
Ip - ^ fi i
_n+l
n .
n
------ Jf------ [ ( n + l ) V B + ( i H 12* - Z n 2 2 ) v B + ( - i H 2 2 * + i H 2 2 +Zn2 2 )vA
n_
+ i AVH2 2 ]
ip -1 f i i
_n+l
■ n n_
------ ^------ [ ( n + l ) V B + i H 12vB+iAVH2 2 ] ,
s in c e i H 12* - Z n 12= i H 12 and O=- i H2 2 * + i H2 2 +Zn2 2 » ,
The l e f t - h a n d
fo r a ll
(7 .1 -6 ).
s im ila r,
s id e .
n.
T h is
N ext,
s id e and r i g h t - h a n d s id e o f E q u a tio n
im p lie s
E q u a tio n
E q u a tio n s
(A .9 ,1 1 ,1 3 )
a r e t r u e f o r E q u atio n s
( A . 1 6) must be checked.
no e x p l a n a t i o n w i l l
be g iv e n .
( A .1 5 ) match
S in c e t h e pro ced u re i s
F i r s t examine t h e l e f t - h a n d
76
n
i
~
n
n+1
V ( Q B ) - I p ^ f 12V(QB) = l / 4 { v [ ( n + l )
In
In
In
n
B +1 (M2 ,N21 ,N2 2 ) + i H2 2 A]
1
n+1
In In In
n
-ip f i 2V[(n+l) B + i (M2 ,N21,N22JtiH22A ]>
= l/4 [(n + l)v
n+1
n
n
n n
B t i ( H 12*VB -H 2 2 * v A ) + i H22VAtiAvH22
\
1
-ip "
'
-n +1
-n
-n
„n
n_
f l 2 [ ( n + l ) V - B + i (H12*VB-H22* v A )+ iH22VAtiAvH2 2 ] ]
*i
= l/4 {(n + l)(-ip "
_n+l
.
n+1
f 22V A + I p - ^ f 21V B )
+ i H i 2* ( - i p
f 22VA+ip ^ f 21VB)
- i H 22* ( - i p - 1f 12V A + if11p-^vB)
_i
+iH 2 2 ( - i p
,
-n
_n
f 12VA+ip
_n+l
n
n
f 1 j^vBj+i H22VAtiAvH22
„n
„n
„n
n„
-ip™ f i 2 [ ( n + l ) v B + i (H12*vB-H22*vA)+iH22vA+iAvH22]}
i
~n+l
-i
-n
I
«. n
= l / 4 { - i p " i f 2 2 ( n + l ) V A - i p " Af 2 2 ( i H 12* v A ) - i p ” i f 12( -2 iH 22*vA)
-ip
. -I
-ip
,• - I
-
(iH 22*VB-ip z f 12 (2iH 22VA)+(ip ^ f 1 1 (IH 22VB)
>
f 2 2 (iAvH12)}
-n +1
-n
{ f 2 2 ( n + l ) v A + ( i f 2 2 H12* - 2 i f 12H2 2 * + 2 i f 12H22 ) v A
\
77
~n
nr
+ ( I f l l H 2 2 * “ 1 ^ 1 1 ^ 2 2 ) V B + if2 2 A V H i2 )
N ext, th e
rig h t-h a n d s id e .
_i
-ip
.
n
„n
_n
f 22 ^ i 2
[ ] f 22V (QA)- z f 22^ A + Z f2 i VB+( ——
, f I l n ZZ
+(
= -ip
g—
~n
+ f 2 2 2 ~ t j2 ^ 2 2 )^A
f l 2 z)7B 3
1 f 22 .
n+1
n
{—
V [ ( n + 1 ) A + i H 12A3
^22^12
^11^22
+ (----- 2-------- f 12022 ) 9 A + --------- 2— VB}
-I
——
.
_n+l
-p
n_
{ >( n+l ) f 2 2 V A + i f 22 H i 2 V A + i f 2 2AvHi2
-P
-v
~n
+ ( 2 f 2 2 0 i 2 ” A f ^2^ 2 2 )V A + 4fjlO g ^ V B }
• -I
—
,
-n + 1
-n
{ ( n + l ) f 2 2 ^ A + ( i f 2 2 ^ 1 2 + 2 f 2 2 £21 2 ~ ^ f 12^22 ) 7 A
~n
n„
+ Z f 1 $222'^®^’^ ^22AVH12}
One s e e s , a f t e r a l i t t l e
th e le ft-h a n d
( A .1 0 ,1 2 ,1 4 )
6)
are tr u e .
a lg e b ra , th a t the
s id e o f E q u a tio n
(A .1 6 ).
a r e t r u e f o r E q u atio n s
r i g h t - h a n d s id e ag a in eq u a ls
T h is
(7 .1 -6 ).
im p lie s t h a t
T h e re fo re ,
Eq u atio n s
E q u atio n s
(7 .1 -
78
APPENDIX B
EXAMPLE CALCULATION FOR EQUATION ( 7 . 1 2 )
79
The o b j e c t o f t h i s
a p p en d ix i s t o show how E q u a tio n
( 7 . 1 2 ) was
o b ta in e d .
B egin w i t h t h e d e f i n i n g
mn m
n
m
n
vNI I =Hi i * V H 2 i - H 2 I * v H i i .
From t h i s
mn
it
mn
fo r N n »
re la tio n
(B .l)
can be shown t h a t t h e i n f i n i t e s i m a l
o p e r a t o r Q a c t i n g on
mn
N n , QNu , must satisfy
mn
v (QNi i
m
n
m
n
m
n
m
n
)=(QHi i * ) v H2 i -(QH2 i * ) v Hi i +Hi i * v (QH2 i )-H2i * v (QHh ).
m
m+1
m
The Q H n * = l / 4 [ m H i i * - i H i 2 * H i i * ] ,
ml
(B.2)
can be exp ress ed as
ml m
- i/4[mNii+2m$i*M1-Hn*(mH2i+2m»i*$2+(m -l)H i2* )
+ H2i* m ( H ii+ 2 $ i* $ i+ H ii* ) ],
a f t e r u s in g E q u a tio n s
(B,3)
(6 .4 ,5 ,1 0 ^ 1 1 ).
A s i m i l a r e x p re s s io n may be
m
o b t a in e d f o r QH2 i * ,
-
and i s
ml
ml m
i / 4 [ ( m + l ) N i 2+2m62 * M i - H u * ( ( m + l ) H 2 2 +2m$2 * » 2 + ( m - l ) H 2 2 * )
+ H2 i * ( ( m + l ) H i 2+2m$2 *$i+m H 2 i * ) ] .
U sin g t h e s e ,
it
mn
V ( Q N n )
can be shown, a f t e r some a l g e b r a , t h a t
m
m
n
=
i / 4 [ 2 ( m + n ) ( H n * ( f i 2 - i z ) - H 2 i * f n ) v H2 i +
.
,
m
m
m
2 ( m + n ) ( - H i i * f 22+H2 i * ( f i 2+ i z ) ) v H n +
ml
n
m l' n
ml I n
In In
( - m N n ) v H2 i + ( m+1) Ni 2 VHi i-2m M i VLi + 2 n l i VMi
n
ml
n
(B .4 )
+
ml
nH2i V N i i - ( n - l ) H i i Ni2 j .
(B.5)
80
U sing
m$n + l
VN11
it
is
m
n+1 m
n+1
H ii*V H 2 i-H 2 i* V H ii
m
m
n
i C 2 ( H 11* ( f 12- i z ) - H 2 1 * f 1 1 )vH21
m
m
ri
+ 2 ( - H 11* f 2 2 + H 2 1 * ( f 12 + i z ) ) v H 11
In ml n
ml
ml
+ ZL1V M ^ H 21VN1 1 - H 11VN1 2 ] ,
seen t h a t
mn
m ,n + l
ml In
ml n
V(QN1 1 ) = l / 4 [ ( m + n ) v
N1 1 - Z im v f M 1 L 1 J - Im v ( N 11H2 1 )
ml n
+ I(H W -I)V (N 12H1 1 ) ]
which proves E q u a tio n
(7 .1 2 ).
(B .6 )
•t
81
APPENDIX C
CONSISTENCY OF EQUATION ( 7 . 1 6 )
/
82
E q u a tio n
E q u a tio n s
( 7 . 1 6 ) 1i s d e r iv e d
(2 .4 0 ,4 1 )
X 1 = p + iz ,
E q u a tio n s
(2 .4 0 ,4 1 )
w ith
i
= 1 ,2 .
E q u a tio n s
and w it h
a p p e n d ix .
Begin w it h
th e v a r i a b l e change
x2 = p - iz .
(C .l)
th en become
^ f- l l I V
i 1V
(C - 2 )
Under t h e Q t r a n s f o r m a t i o n , t h e l e f t - h a n d
(0 .2 )
s id e o f
becomes
Y1 . - = Y
$J
,+ ((Q y )
@J
For th e tra n s fo r m a tio n
J
if
o f the
re la tio n s w ill
(0 .3 )
, I l x 1Y .■)
9J
where t h e p lu s s ig n a p p l i e s
fo llo w in g
in t h i s
9J
j= l
and t h e minus sig n a p p l i e s
r i g h t - h a n d s id e o f E q u a tio n s
if
j=i2.
( 0 . 2 ) , the
be u s e f u l .
Qp = p z , Qz = l / 2 ( z 2 - p 2 ) ,
9 / 3 x 1 l = ( I f i e x 1 )S Z ax1 ,
Qx 1 = - i x 12 / 2 ,
Qx2 = i x 2 2 / 2 ,
a Zax2 ' = ( l - i e x 2 )aZ 3x2 ,
(0 .4 )
(0 .5 )
2
QG = l Z 4 ( H u + i H I 2 G) = i ( N n + 2 $ 1 L1+GH2 1 ) Z 4 ,
(0 .6 )
2
QG* = l Z 4 ( H n * - i H 12* G * )
(0 .7 )
= - i (N n * + 2 » 1* L 1 *+G*H2 1 * ) Z 4 ,
2
Q»^ = l Z 4 ( $ ^ + i H ^ 2$ ^ )
= i(M^+2@^K+G@2 ) Z 4 ,
(0 .8 )
2
Q^1*
Then i t
is
= l Z 4 ( $ 1* - i H 1 2 * $ 1* )
= - i (M1* + 2 » 1*K *+ G *» 2 * ) Z 4 .
(0 .9 )
seen
4I
» P p U
(B12- Z )
(C .1 0 )
83
and
- E p f 1 + - E p r 1 - E p f - 1^ 1 2 .
(C .ll)
The b r a c k e t e d te rm on t h e r i g h t - h a n d
s id e o f E q u a tio n
( C .2 )
goes i n t o
cS
f
I +2V
jI V
1I t 2 V
13^
1M
+4V
4I V
j I 4I , ! ]
+
C -B jj+ U K -K 'J itZ ix ^ lp )] +
1^ f 6 j I 4 I e Ji - 1V
fG ‘ Ji 4 l . i + 2 ( K- K* ) f V
where t h e ± means t h e same as i n E q u a tio n
Ji l l , i ] l
(C .3 ).
■
<c - 1 2 >
F in a lly ,
4I t j I 4U i + l l t Ji 4l Ji +e{l/24l ‘ Ji 4l Ji C- a2 l+2l' ( K- K* ) l ( 41xi - 21p^
1 ( 4 2 GJi 4 l t Ji - V G* . i 4 l Ji ) /4>
(C 'l))
Combining term s on t h e r i g h t - h a n d s id e o f E q u a tio n s , ( C . 2 ) th e n
y ie ld s
± ix ..{th e
rig h t-h a n d
s id e o f E q u a tio n s
(C .2 )}
= I i x 1Y j .
From E q u a tio n s
(Qy)
(C .3 )
.= 0
9
J
(C .1 4 )
and ( C . 1 4 ) ,
it
can be concluded t h a t
Qy = a c o n s t a n t .
s in c e Y+0 as p2+ z 2 approaches i n f i n i t y ,
Qy = 0.
(C .1 5 )
t h e c o n c lu s io n
is
(C .1 6 )
84.
APPENDIX D
INTEGRATION OF EQUATIONS ( 7 . 1 7 , 1 8 )
85
E q u a tio n s
(7 .1 7 ,1 8 )
th e n become t h e p a r t i a l
a r e i n t e g r a t e d u s in g E q u a tio n
d iffe re n tia l
(3 .4 9 ).
They
e q u a t io n s
= [D1^ 1 ( t , s ) - ■ £ * i ( t , s ) + 1 H i 2 ( s ) * i ( t , s ) L
D2 S1 ( ^ s )
(D .l)
D2 F 1 2 ( t , s ) = [ D 1F 1 2 ( t , s ) + i ( D 1F 1 2 ( 0 , s ) ) F 1 2 ( t , s ) ] ,
w i t h D1 = 9 / a t ,
D2 = 9 / 9 s ,
Q4 g F1 2 ( t )
= F1 2 ( t , s ) ,
(D .2 )
and
(Q)4sM tO = » i ( t , s ) .
E q u a tio n
(D .2 )
is
s o lv e d by a s e p a r a t i o n
F i 2 (t» s )
= g (s )h (s + t).
S u b s titu tin g th is
i n t o E q u a tio n
of v a ria b le s
(D .3 )
(D .2 ) y ie ld s
■i
whose s o l u t i o n
g(s)
(D.4)
is t r i v i a l l y
c -
ih (s )
(D .5 )
>
T h erefo re,
F iz ft'S )
The i n i t i a l
a rb itra ry ,
h (s + t)
- c _ ih (s )
c o n d itio n
h (0)
(D .G )
*
F ig ft.O )
= F^gft)
re q u ire s
c = 0.
S in c e h ( s )
is
may be s e t equal t o one i m p ly in g
i F 12(s + t)
F i 2 C t',s)
=
(D .7 )
F i 2T s )
which f u r t h e r i m p l i e s
i
H i 2 (S)
E q u a tio n
.
(D .l)
F1 2 (S)
becomes
D2w ( t , s )
where w ( t , s )
S u b s titu tin g
|a
- D 1W f t , s)
= F12(S )^ 1( t , s ) .
w (t,s )
(D.8)
os
= —
w(t,s),
T h is
i s s o lv e d by s e p a r a t i n g v a r i a b l e s t o
= g (t)h (s + t).
i n t o E q u a tio n
= i t ♦ g = ct.
(D .9 )
(D .1 0 )
(C .9 ) y ie ld s
(D.ll)
86
T h erefo re,
c th (s + t)
F iz (S )
The i n i t i a l
c o n d itio n
h ( t , .
( D . 12)
‘
(t,0 )
= $ i(t)
,
re q u ire s
( D .1 3 )
or
it$ i(s + t)
= ( S H ) F 12 (S) *
(D .1 4 )
APPENDIX E
INTEGRATION OF EQUATIONS ( 7 . 4 6 , 4 8 )
88
E q u a tio n
(7 .4 6 )
rep resen ts th e p a r t i a l
/3 _
9
9s " W T " k " )
'I
^ (ti'tz .s )
ou2
=
d iffe re n tia l
(Na b ^ 1 $t 2 , s )
e q u a tio n
- eAB) ,
(E .l)
s b e in g t h e group p a r a m e t e r
T h is e q u a t io n
is
s o lv e d by t h e method o f c h a r a c t e r i s t i c s ,
the
e q u a t io n s b e in g e q u i v a l e n t t o
- t a d Nab
ds = - d t i = - d t 2 - w — ^
AB AB
The s o l u t i o n s a r e
(E ,2 )
s + t% = C l ,
(E .3 )
s + t 2 = c2 .
(E .4 )
AB
where gAg i s
(E .5 )
9ABt 2 + eAB9
d e te rm in e d by t h e i n i t i a l
NA g ( t 1 , t 2 , 0) = NAB( t 1 , t 2 ) .
NAg ( c i ,C a)
S e t s = 0 , then
AB
(E .6 )
r2
9A B =
T h erefo re,
- E
c o n d itio n
f o r Sj i 6 O,
NAg ( t i , t a , s )
=
(NAB( s + t l a s + t 2 )
'
eAB^ + e AB
tz
(E .7 )
= s + t 2 NAB^s + t l 9 S + t 2 ^ + s + ^ pAB*
E q u a tio n
(7 .4 8 )
re p re s e n ts th e p a r t i a l
d iffe re n ta l
e q u a t io n
(E .8 )
FAB*^t , S ^ = ^ A X * ( s ) F * g * ( t , s ) ,
&
- &
FV
(t,s )
= 1
l t FV
( t * s)
t = 0 FV
(t,s)i
(E .9 )
o r in m a t r i x n o t a t i o n
&
The a n s a t z i s
- it
F* ( t , s )
= i
ft
F* ( t , s )
t= 0 F * ( t , s ) .
(E .1 0 )
made t h a t
F *(t,s )
= g (s )h (s + t)
(E .U )
89
where g and h a r e 2 by 2 m a t r i c e s .
and b e in g c a r e f u l
•yg- = /ig
S -1 S
N ext,
P u ttin g th is
in to
E q u a tio n
(E .1 0 ),
t o m a in t a in t h e m a t r i x o r d e r , t h e e q u a t io n becomes
dh
S -1
g,
(E .1 2 )
i dh
1 3s
(.E.13)
use t h e m a t r i x i d e n t i t y
-I
dA
3s
=
(E ,1 4 )
to fin d th a t
If1 -
■i •
(E .1 5 )
3s *
T h erefo re,
g_1(s)
ih(s).
(E .1 6 )
where C i s a c o n s t a n t 2 b y . 2 m a t r i x .
F *(t,s )
S e ttin g t
C=0.
= (C -
T h is y i e l d s
i h ( s ) ) “ 1h ( s + t ) .
= 0 , and u s in g t h e i n i t i a l
(E .1 7 )
c o n d itio n F * ( 0 , s )
= i
r e q u ir e s
T h erefo re,
F *(t,s )
S et s = 0 ,
use t h e i n i t i a l
F *(t)
S in c e h ( t )
= i h '1(s )h (s + t).
(E .1 8 )
c o n d itio n
F *(t,0 )
= F *(t);
= ih ’ 1( 0 ) h ( t ) .
(E .1 9 )
i s th u s f a r a r b i t r a r y ,
it
is p o s s ib le to set h " 1 (0 )
= I so
th a t
h (t)
= -iF *(t).
(E .2 0 )
T h erefo re,
F *(t,s )
,-I
= i(-iF*(s))“
-iF *(s + t)
= iF * " 1(s )F *(s + t)
FAR* ( t , s ) . =
i ( F * - 1 ) AY ( s ) F XB* ( s +t ) .
(E .2 1 )
(E .2 2 ).
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