Uniformly accelerated solutions of Einsteins Equations
by Timothy James Henline
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Physics
Montana State University
© Copyright by Timothy James Henline (1984)
Abstract:
By examining the effect of Cosgrove's transformation, Q4s' when acting on some simple space-times,
the argument is put forward that the new solutions which are generated represent the space—times due
to linearly accelerated axially symmetric sources. The procedure used is to look at the effect of the
transformation on some simple space—times and compare the results with space-times which are
known to represent accelerated sources. The effect of the transformation on static solutions is carefully
examined. The author feels the transformation generates the metric of linearly accelerated sources. UNIFORMLY ACCELERATED SOLUTIONS
OF EINSTEIN'S EQUATIONS
by
Timothy James Henline
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Physics
MONTANA STATE UNIVERSITY
Bozeman, Montana
September 1983
APPROVAL
of a thesis
submitted by
Timothy James Henline
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Approved
for the Major Department
Approved for the College
Date
of Graduate Studies
Gradua te Dean
iii
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iv
TABLE OF CONTENTS
Page
LIST OF F I G U R E S . .. ..... ...... ............. ..............
ABSTRA CT. ............... ..................................
vi
vi i
INTRODUCTION....... ............ ............. . ........
I
Vacuum Field Equ ati ons .. ....... .....................
SL(2,R) Tensor N o t a t i o n .............................
Purpos.................................................
6
8
10
2.
FLAT SPACE-TIME FOR ACCELERATED OBSER VE RS ..........
12
3.
NEW SOLUTIONS VIA ACCELERATION F A C T O R S ..............
19
SIMPLE EXAMPLES ..... .................. ..............
29
5.
SIMPLIFYING THE ACCELERATION F A C T O R .................
37
6.
THE ACTION OF THE TRANSFORMATION Q4 s .... ...........
41
7.
RA DIA TIO N ............ ..................................
51
8.
ACCELERATED ROTATING SOLUTIONS......................
68
9.
SUMMARY AND CONCLUSIONS ........................ ......
73
I.
4 .
76
RE FE REN CES ............... . ....................... ..........
80
A P P EN DI CE S ........................ .............. . .......
85
Ap p e n d ix A - The Tetrad F o r m a l i s m ..................
A p p e n d i x B - An O r t h o n o r m a l T e t r a d for
Stationary, Axially Symmetric
Space-time s.......... ...............
Appendix C - Null Tetrads and the Notation
of the Newman— P e n r b se F o r m a l i s m .....
Appendix D - Null Tetrad for Stationary,
Axially Symmetric Space— times........
Appendix E - Prolate Spheroidal Coordinates.......
■
•
investigations......
Remaining problems and future
•
:
•
■-
86
;
,i
89
92
95
98
«
V
TABLE OF CONTENTS— continued
Page
Appendix F — Expansion Coefficients In
Equations 7.21a— 7. 21 c .........
Appendix G - Asymptotic Structure of Gravitational
Fields in terms of Linearized
M u l ti p ol e s ..............................
Appendix H - Transformation of the
Kerr solution..........................
101
103
105
LIST OF FIGURES
Figure I:
Schematic diagram of the physical
effect of Q 4 s on an
axially symmetric s ou rc e...........
Figure 2:
Trajectory of an observer in
uniform rectilinear accelerated
motion ('hyperbolic m o t i o n ' ) .......
Figure 3:
Preferred point and singularity
of acceleration factor........;....
Figure 4:
Source of the Schwarzchild solution
in Weyl canonical coordinates......
Figure 5:
Constant coordinate curves for
parabolic cylindrical coordinates..
Figure 6:
Zeroes of the functions in
Equations (6.29a) and (6.29b)......
Figure 7:
Infinite red-shift surface for
$ 4 «. acting on a Kerr solution
( q < < I ) ................................
8:
Constant coordinate curves for
prolate spheroidal co ordinates....
Figure
vii
ABSTRACT
By examining the effect of Cosgrove's transformation,
($4s » when acting on some simple space-times, the argument is
put f o r w a r d that the ne w s o l u t i o n s w h i c h are g e n e r a t e d
re p r e s e n t the space — times due to l i n e a r l y a c c e l e r a t e d
axially symmetric sources. The procedure used is to look at
the effect of the transformation on some simple space— times
and compare the results with space-times which are known to
represent
accelerated
sources.
T he
effect
of
the
t r a n s f o r m a t i o n on static s o l u t i o n s is c a r e f u l l y examined.
The author feels the transformation generates the metric of
linearly accelerated sources.
I
CHAPTER I
INTRODUCTION
In
all
t h e or ie s
p re di ct
the
gravitational
surroundings.
This
of
gravity,
is
affects the trajectories
vicini ty.
Newton' s
a scalar
mas s
field
an
often
V^(Hx)
in the
made
revealed
on
says
that
in its vicinity
of
its
in its
an object
described
containing
gravitational
by
a
p o te nt ia l
form
= 47iGe(x).
regions
to
by how
of test particles
gravity
d e n s i t y o (x ) , the N e w t o n i a n
is
object
f u n c t i o n (J(x)# In a r e g i o n of space
obeys Poisson's equation
For
of
of
most
the object
sets up a gravitational
attempt
influ en ce
influence
th e o r y
an
(1.1)
sp a c e
containing
no
mass,
d (x ) o b e y s
Laplace’s equation
= 0 .
V^(Hx)
(1 .2 )
The force of g r a v i t y on a test p a r t i c l e of m a s s m is
F = -mV<5>.
A particular
an integral
solution
(1-3)
to Poisson's
equation
is expressed
over the mass density .
d (£) = -G///er( x ') I£ - & ' I 1 d 3 x'.
For points
as
x outside
the
mass
density,
(1.4)
this
solution
obeys
2
Equation
(1.2).
Einstein's Gener al T h e o r y of R e l a t i v i t y says that the
gravitational
influe nce of an object
four-dimensional
space — time geometry
object. The mathematical
the
space — time
components
points
of the metric
separation
dx *
are
the
in the vicinity of the
function which gives information on
geometry
in space-time
is d e s c r i b e d by
is
tensor
are
which have
said
to
the
metric
tensor.
denoted by
have
a
k
g £j
an infinitesimal
Th e
Two
coordinate
space-time
interval
between them ds, where
(ds
Equation
= gjjdx^dx^
(1.5)
is the
.
(1.5)
most
common
components of the metric tensor.
time
said
interval
to
components
components
of
test
a metric
X i.
non-linear,
of
exhibiting
respect
to
a given
The E i n s t e i n field e q u a t i o n s
partial
of
with
the
differential
metric
of the metric
tensor
tensor
particles, th ro ug h
the
In most cases when a space-
of the form of Equation (1.5) is given,
have
coordinates
way
obey.
are known,
space-time
may
is
set
of
are a set of
equations
must
one
which
the
On c e
the
the trajectories
be
determined
(Oh an ia n ,I976,p .203).
Before continuing,
we need to explain some terminology.
When a metric tensor satisfying the Einstein field equations
is k n o w n
in cl ose d
solution'. Until
form
one
recently,
is
said to p o s s e s s
the n u m b e r
of
exact
an 'exact
solutions
3
was
small.
The
reason was
that the field equations
were so
complicated that straightforward solution was impossible for
all but
a few
cases.
Direct
s o l u t i o n often depe nd s
on the
ab i l i t y to make ce rt ai n s i m p l i f y i n g a s s u m p t i o n s about the
m e t r i c tensor or the form of the s p a c e - t i m e interval. This
work will consider a group of exact
solutions which describe
the s p a c e — time ex t e r i o r to an a x i a l l y s y m m e t r i c source of
gravity. There
phrase
is a second term which needs explanation. The
'solution
generation
c r e a t i o n of new exact
direct
solution
of
technique'
refers
solut io ns by m e t h o d s
the
differential
to
the
other than the
equations.
Often
an
initial 'seed' so l u t i o n is m a n i p u l a t e d to give a new exact
solution. The ad v a n t a g e in a c q u i r i n g s o lu ti on s in this w a y
is
that
one
avoids
solving
non-linear
partial
differential
e qua tions.
STATIONARY,
The
notation
introduction
detailed
to be
Kinnersley
considered
are
space-time
co or di na te,
is,
i n t ro d u c e d
here
f o l lo ws
the
(1977,pp.1529— 1530)
and
introduction of Kramer
stationary
that
of
AXIALLY SYMMETRIC SPACE-TIMES
stationary
is one
and a x i al ly
for which
there
the
more
space-times
sym me tr ic .
exists
A
a time
x 1 = t , such that all g .^ are i n d e p e n d e n t of t ,
g^j.^
derivative.
(1980,Ch.17) The
concise
= 0 wher e
the
comma
An axially symmetric
all c o m p o n e n t s
of the me t r i c
denotes
space— time
tensor are
the
partial
is one in which
in d e p e n d e n t
of an
4
azimuthal
angle
Therefore,
in a stationary,
components
of the metric
two remaining
S^j
=
c o o rd i n a t e
g.j(x^*x^)
called o r t h o g o n a l
interval,
(0 < 0 < 2 n),
B ijt0
= 0.
axially symmetric space— time the
are
functions
3
4
coordinates x and x .
spatial
The s p a c e - t i m e
=
tensor
i»j
only of the
(1 .6)
- 1 ,2,3,4
is also a s s u m e d to p o s s e s s
a property
tr an sit iv it y. This mea ns the s p a c e - t i m e
d S 2 , is u n c h a n g e d w h e n one ma k e s the c o o rd in at e
transformation
(0,t) — > (— 0,— t). Physically,
is unchanged when the
the
space— time
sources undergo motion reversal.
restricts the space— time
interval
This
to take the form
d0
ds 2 = Stt dt2 + 2 gt 0dtd0 +
3x2
+ S33Cdx")" + 2g34(dx3)(dx4)
(dx4)2
+ *44'
A t r a n s f o r m a t i o n to a new
al wa ys
be
Equation
found
which
(1.7)
set
of
allows
(x 3 ,x4 ) c o o r d i n a t e s
the
last
three
can
terms
of
(1.7) to be written in the form
f(x 3 ,x 4 )[(dx 3)2 + (dx4 )2 ] .
(Kramer,1980,p.195)
When
one
(1 .8 )
elects
to use
coordinates
such
that the last three ter ms of E q u a t i o n (1.7) can be w r i t t e n
in
the
form
isotropic
of
Equation
coordinates.
x3
(1.8),
Therefore,
stationary,
axially
symmetric
four metric
tensor components.
in
and
x4
isotropic
space— times
are
are
called
coordinates,
specified
by
The form of these four m e t r i c funct io ns used by Le wi s
5
(1932) allows the space-time
interval
to be written as
(ds )2 = e2 u (dt - o)d0)2 - e2 B- 2 u d 02
- e 2 (Q- u )[(dx 3)2 + (dx4 )2 ]
The
functions
u , to, B , Q
coordinates. Appendix B shows
Einstein
tensor
depend
(1.9)
only
E q u a t i o n (B. I 0 a ) in A p p e n d i x B. C o n s i d e r
interval
in this
Equat ion
(1.9)
The
of
the
subspace
(x ^ , x 4 )
in Equation(l .9) written with
re sp ect to an o r t h o n o r m a l tetrad. Notice
subspace
the
the non— zero components of the
for the metric
(x 3 f x 4)
on
metric
the V o p e r a t o r of
a two-dimensional
in E q u a t i o n
(1.9).
T he
is found by setting dt = d 0 = 0 in
(ds „)2 = e2 ( Q - u ) [(d x 3)2 + (dx4 )2 ].
2
gra di ent of a scalar f in this subspace
(1.10)
is a 2- ve c t o r
and has components
e 2 (8 - u ) j-f 3 ^ f 4 j ^
(1 .11 )
while the Laplacian has the form
2 (Q-u)
. - - - I f 33
In
Equations
function
f
+
(1.11)
indicate
and
(1.12),
partial
remainder
the
subscripts
derivatives.
indicate partial de ri vatives. It turns out to be
op e r a t o r on the t w o - d i m e n s i o n a l
2-vector,
work,
where
f u n ct io n s
19 7 7,
this
the
in
(Kinnersley
of
Except
on
necessary
advantageous
the
(1 .12 ).
f4 4 ],
p .I 52 9) to
su b s c r i p t s
de fine
on
a tilde
3 4
(x ,xs ) sub space. For each
A
[A3 ,A4 ]
.4,2
= (A3)2 + (*’ >
(1.13)
6
there
is a related vector,
X
Note
X
[A4 ,-A3 ]
=
(1.14)
that
V-X = O
where
implies
f is some
A = yf
(1.15)
scalar function.
VACUUM FIELD EQUATIONS
Einstein's
vacuum
field
equations
are
found
by
setting
the E i n s t e i n te nso r equal to zero. Fr o m E q u a t i o n s (1.9) and
(B. 6),
one
of these
O- b V V c b =
Because
field
equations
is
0.
(1.16)
of Equation
(1.15),
this
implies
Ve 8 = V f .
One
solution
(1.17)
to Equation (1.17)
is
eB = x 3
(1.18a)
f
(1.18b)
= x4
In fact,
without
accordance
called
these
loss
of generality one may choose
with Equation
Weyl canonical
coordinates,
x 3 ,x4 in
3
4
and then p = x , z = x
(1.18),
coordinates
(Kramer
1980,
are
p. 195).
In
the remaining vacuum field equations are
V
[e2 uV(pe- 2 u )l
V
[p- 1 e4u Vw]
= p- 1 e4u Vw Vw
= 0
V VQ = V Vu - Vu Vu + p 1 U
(1.19)
(1.20)
P
+ (1/4) p- 2 e4 u Vw Vw
(1.21)
O p - P [(U p)2 - < v 2 ]
- (1/4) P- 1 C4 u K w
P
)2 - (w )2 ]
%
(1.22)
7
“ B = 2 Pu p0 Z - ( 1 / 2 ) P le4U“ p«z
Equations
(1.19)
and
(1.20)
U.23)
involve
only
the
metric
functions, u and <o. Once these two funct io ns are found, Q is
found
by
simple
Equation
(1.21)
derived
(1.19)
from
and
integrations
gives
the
metric of stationary,
space-time
axis
the
new
other four
(1.20) are
The m e t r i c
no
of Equations
information
equations.
it
to
the
be
Equations
the
f u n c t i o n <o gives a m e a s u r e of how fast the
inertial
sources
which are not rotating,
with
can
for d e s c r i b i n g
frames
around
the
which
possess
axial
The s p a c e ­
symmetry,
are described by Equations
fu n c t i o n w set
equal
symmetry
for w /= 0 one expects
of the g r a v i t a t i o n to be rotating.
exterior
(1.23)
as
axially symmetric space-times.
(Bardeen 1970) . In other words,
time
and. (1.23).
Therefore,
the key eq u a t i o n s
is dragging
sources
(1.2 2)
to zero.
but
(1.19) to
Such space-
tim es are said to be static. These static s p a c e - t i m e s were
first
for
studied by W e y I. The key equation (Lewis 1932,
static
space — times
p. 182).
is Equation (1.19) with e> = 0
V I e 2 uV( pe- 2 u )I = 0 ,
(1.24)
u PP + P- l u P + " 2Z - O-
(1-25)
or
Equation (1.25)
c y l i nd r ic al
function
is of the
coordinates
u(p,z).
From
same
form
as Laplace's
(z,p, 0 ) for
Equation
(1.2),
equation
an a x i a l l y
any
axially
in
symmetric
symmetric
N e w t o n i a n 0 can be used as a s o l u t i o n to the E i n s t e i n v a c u u m
8
field
Equation
symmetric
(1.25).
This
implies
mass distribution a one has,
for
an y
axially
via Equation (1.4)
a
s o l u t i o n to the static, a x i a l l y s y m m e t r i c E i n s t e i n v a c u u m
fiel d equation.
Re l a t i v i t y ,
How ev er ,
the Weyl
one must
ca no ni ca l
be cautious.
coordinates
In General
p and z do not
n e c e s s a r i l y re t a i n their E u c l i d e a n m e a n i n g as c y l i n d r i c a l
coordinates.
symmetric
is not
So
mass
able
to
determine
immediately
able
the
general
space — time
That
re p r e s e n t
what
is
for
dis tr i bu ti o n.
necessarily
one
distributions
u (p ,z) represents
ma ss
while
mass
free
use
the
axially
(1.4),
one
to that particular
for a given a ,
is,
and
in Equation
external
same
0 and u do not
ph ysical
distribution
zeroes
select
to say the c o r r e s p o n d i n g
is
relativistic solution u(p,z),
u (p,z) exhibits
to
situation.
To
giving
a particular
one must
examine where
singularities.
SL(2,R) TENSOR
NOTATION
The S L (2 ,R) tensor f o r m u l a t i o n of the v a c u u m stationary,
axially
symmetric
(Kinnersley 1977).
field
Equations
Stationary
axially
is
now
symmetric
introduced
space-times
with orthogonal transitivity can be written
ds 2 - £A B d,A d,B
- e2 (fi" u ) t(dx 3)2
+ (dx4 )2 ]
(1.26)
The vacuum field equations can be written
V
tp- 1 fAX VfX B ] = 0
P2 = ~ de t (fAB).
.
U.27)
(1.28)
9
The
indices
(A,B = 1,2)
are raised and lowered according to
X
eAXh
eAB = s
(1.29)
8XAh
r 0
I I
L-I
0 J
(1.15),
the
-
Using Equation
(1.30)
field Equation
(1.27)
implies
the
-j
existence
Vg
of a set of functions
AB - -P
g^
such that
-If X ^f
fA
XB
(1.31)
which can be inverted to yield
V f AB =
(1.32)
P lfAX Vg XB
If one defines a tensor
H
(1.33)
fAB + igAB
Equations
(1.31)
and
(1.32)
can
be
written
In
the
single
complex equation
(1.34)
This
complex
equation
is e q u i v a l e n t
to
the
vacuum
field
e qua tions.
Kinnersley
properties
generation
exact
of
and Chitre
the
techniques
solutions.
in ter ms
field
of
(Kinnersley
Their
two
and
(1978),
eq ua tions,
which
yield
techniques
ge n e r a t i n g
Chitre
by e x a m i n i n g
discovered
asymptotically
are
most
functions
1978b).
These
easily
®^g(s,t)
generating
the group
solution
flat
new
expressed
and F^g(s)
functions
are defined such that
idG A B (s,t)/dt
i9 FAB<‘>'8 ‘
|t=Q
It-O
Hi
(1.35)
10
In
other
words,
Kinnersley
these
and
functions
Working
discovered
p r es er ve
these
the
Chitre
over
a set
a
of
the for m
finite
functions.
can be
involve
al g e b r a i c
different
of
manipulation
of
Cosgrove
(1979)
transformations
of the v a c u u m field equations.
Cosgrove
and
techniques
functions f .
AB*
route,
infinitesimal
transformations
transformations
expressed
Kinnersley
generating
rather than the metric
infinitesimal
give
so l u t i o n
has
also
in term s
Chitre
of the
(Cosgrove
yield
how
his
new
of
metric
transformations
generating
1980,
Some
were exponentiated to
which
shown
wh i c h
f u nc ti on s
of
pp.2422 — 2424).
PURPOSE
Sp e c i f i c a l l y ,
suggest
a
phy sic al
transformations.
solutions
static
the
what this w o r k will a t t e m p t to do is to
First
generated
axially
physical
interpretation
to
be
to
one
examined
will
when
Cosgrove's
technique
symmetric
solutions.
It will
effect
of his
transformation
of
Cosgrove's
be
the
new
is applied
be
is
argued
to
to
that
accelerate
the initial solution. The effect of this t r a n s f o r m a t i o n is
illustrated
in Fi gu re
I. Finally,
the
a c ti on
of
the
same
t r a n s f o r m a t i o n on 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 solutions
will be examined.
11
Accelerating
s ou r c e
yields
+
Initial
source
Figure I: Schematic diagram of the physical
effect of
^ on an axially symmetric
solution
12
CHAPTER 2
FLAT SPACE-TIME FOR ACCELERATED OBSERVERS
The
simplest
describing
flat
metric
tenso r
space— time.
The
is the m e t r i c
space— time
tensor
interval
appears
ds 2 = R ij d x idxj
where
is
x
Cartesian
a
coordi nate
time
spatial
co ord in at es .
and
are
x 2 ,x 3 ,x 4
Several
v
systems will prove useful. For example,
R co s0,
(2 .1 )
(dx 1)2 - (dx 2)2 - (dx 3 )2 - (dx4 )2 .
=
x
other
1
three
c o o rd in at e
2
= T, x
3
= Z, x =
x^ = R sin0 changes Equa tion (2.1) to th e form
ds 2 = dT 2 - dZ 2 - dR 2 - R 2 d02 .
(2.2)
All that has be en done is in E q u a t i o n (2.2) is to t r a n s f o r m
fr om C a r t e s i a n spatial c o o r d i n a t e s to c y l i n d r i c a l
coordinates.
metric
Even
tensor
c o o r di n at e
=
X
this
components
1
+
2
X
simple
change
transformations
interpretation.
u
under
may
With
, V
=
X
1
-
2
X
3
4
O
3 .
4
q = X 4 + ix
» q = X - ix
Equation
(2.1) becomes
coordinate
their
not
change*
functional
yie ld
spatial
such
form.
the
All
a direct
13
O
o
ds -6 = dudv - dT|dT).
The
point
using
is that
non-Cartesian
s imply
looking
tensor g ..,
metric
at
tensor
is
a
gravitational
the
impossible
of
the
flat
space-time
effects. However,
components
of
is flat.
metric
to di sguise
The re fo re ,
a particular
to tell
representing
curved
is easy
systems.
components
if the space-time
the
space-time
coordinate
it may be
represents
tell
flat
if that
when
metric
particular
space-time
with
by
or
if
it
corresponding
there is an effective way to
With the functional
tensor,
the
form
components
of
of
the
R i e m a n n c u r va t ur e tensor R ijkl m a y be computed. If all the
components of this curvature tensor are zero,
is
flat.
The
components
drawback
of R .
is
th a t
the
the space-time,
computation
is g e n e r a l l y long and tedi ou s
of
the
(Misner
IJ & I
et. a I .1 973 ,Ch.l 4 ) . T h i s
chapter
shows
flat
space-time
displayed, in ways which occur frequently in the remainder of
this
work.
A co o r d i n a t e
observer moving
notation
here
sy ste m
of
with uniform,
follows
1973,
C h . 6)
first
an observer
To
get
at
the
is
rectilinear
presentation
this
at rest
interest
co o r d i n a t e
in flat
s p e ci fi es an event by giving
one
used
acceleration.
in
(Misner
system,
space-time.
its c o o r d i n a t e s
{e
} ( i= l ,2,3,4 ) . The ob se rv er
an
The
et.al.
co ns id er
This observer
(x
Z, x 3 = X, x 4 = Y } w i t h respect to an o r t h o n o r m a l
basis v e ct o rs
by
=T,
x
=
tetrad of
speci fi es the
14
position four vector,
+ Z &2 + Xag +
~ =
This
0f the event as a four— vector
is all a mathematical
way of saying how
an observer
at
rest w o u l d m e a s u r e events. A ph ys i c a l ob se r v e r w o u l d use a
set of clocks to m e a s u r e the time T of an event and a set a
meter
sticks
event.
One of the reasons for putting these physical
mathematical
to m e a s u r e
language
for some observers
they
would
would
Assume
x=
what
another
such
l o c a t i o n X,Y ,Z of an
basis
vectors)
acts
is
in
that
measuring
but
Mathematics
about how
a given observer
the
time
same
it
a ll ow s
the observer would measure.
observer
that
at
things.
he
is
feels
moving
in the
a co nstant
positive
e^—
acceleration
of
A. The position four-vector of this observer is:
[A ^sinhArle 1
+
(Misn er
actually
things,
predictions as to
magnitude
(coordinates,
to sidestep questions
measure
direction
spatial
it is not easy to intuitively imagine how
go about
allows one
the
[A- 1 CoshAt
et.al.
+ Z q - A- 1 Ie2 ,
Ch. 6 ) The
symbol
(2.3)
t is the p r o p e r time along
the trajectory of the accelerated observer. It may be viewed
as
representing
time
intervals
measured
along w i t h the a c c e l e r a t e d observer.
by
a clock riding
The s y m b o l
Z q 18 the
position of the accelerated, observer at time t = 0. Equation
(2.3) cl a i m s
[Z (t ) - zQ - A- 1 I 2 “
[T(t) I 2 = A -2 = constant
for all va lue s of t . On a T -Z graph,
the t r a j e c t o r y of the
15
a c c e l e r a t e d o b s e r v e r is a hyperbola. For this reason,
dimensional
hyperbolic
u n i f o r m l y .a c c e l e r a t e d
motion.
2
Fi gu r e
shows
motion
the
is
trajectory
one­
termed
of
an
observer in hyperbolic motion.
The r e a s o n
motion
for i n t r o d u c i n g
is to look
observer's
at
point
flat
of
space — time
view.
One
observer to use his own set
to
the
orthonormal
tetrad
(Fermi-Walker
P .1 7 3)
can
it
coordinates
be
are
sh ow n
from
expe ct s
of basis
accelerated
the
accelerated
{t» z,x, y} adopted
vectors
Ce.)
Jie carries
the n o t i o n of 'carrying basis
tr an sport,
that
related
in h y p e r b o l i c
this
of coordinates
w i t h him. By m a k i n g pr ec is e
vectors',
an o b s e r v e r
to
Misner
et.al.
the
accelerated
the
stationary
1973,
observer's
observer's
coordinates by
T =
(z + A ^JsinhAt
Z - Z q - A*=.
(z + A *) c o sh At
(2.4b)
IM
Y = y
(2.4d)
It
(2.4c)
M
In
(2.4a)
these
interval
accelerated
of Equation
coordinates.
the
flat
space-time
(2 .1 ) becomes
ds 2 = (I + Az) 2 dt 2 - dz 2 - dx 2 - d y 2
Equation
(2.5) di s p l a y s
the
metric
of
(2.5)
flat
s p a c e — time
in
te rm s of c o o r d i n a t e s
used by an a c c e l e r a t e d observer. One
p r o p e r t y of e q u a t i o n
(2.5)
coordinate
is of p r i m a r y
transformation i = pcos^,
interest.
y = psin®
If the
is performed
16
T
Trajectory
Figure 2 : Trajectory of an observer in uniform rectilinear
accelerated motion ('hyperbolic moti on *)
17
in Equation (2.5),
the
space-time
interval
becomes
ds 2 = (I + Az) 2 dt 2 - di2- dp 2 - p 2 d02.
(2 .6 )
Comparison
of
0),
Equation
shows
symmetric
this e q u a t i o n w i t h E q u a t i o n
(2.6)
space-time.
As
is not in the canonical
represents
it
stands,
form of Lewis
a
(1.9) (with <o =
static,
however.
axially
Equation
(2.6)
(1932):
ds 2 = e2 u dt 2 - e“ 2 u p 2 d 02
- e2 (Q"u)[dp 2 +
The n e c e s s a r y
c o o r di na t e
dz2 ].
(2.7)
transformation
is (Kra me r 1980,
p. 202 )
F = 0
t = t,
(I + Az) = A1^2 Cr + z )1 '2
; = A-i n G
- z)1/2.
where
%2
P2 + % 2
(2 .8)
z = z + I / (2A),
puts Equation
(2.6)
in the
form
ds 2 = A(r + z)dt2 - (2Ar)- 1 [dp 2 + dz2 ]
- A- 1 Cr +
Comparison
of Equation
(2.9)
z)"1 p 2 d 0 2 .
(2.9)
with
metric functions of Lewis to be
Equation
(2.7)
allows
identified.
u= (I / 2 ) I n [A (r + z )]
Q = (l/2) I n H r
The
re as on
(2.10)
+ l)/2rl.
for w a n t i n g
the
(2J1)
the
metric
functions
in Weyl
18
canoni ca l
coordinates
te c h n i q u e s
is
that
to be d i s c u s s e d
are
action on Weyl coordinates.
(in Weyl
Z - C
an
The
of
the
2
space-time
remarks
to
sp ac e- ti me .
Equation
g e n e ra ti ng
in ter ms
of their
Whenever the metric function g^ ^
at
the
guess
(2.9),
fundamental
space-time
condition
(due
g tt
to
the
The
flat
g
space-time
ph ys i c a l
a localized
un it y
of Weyl
co o r d i n a t e
are
value
For any space-time,
chapter.
has
g^t
= +**
large
[(p )
A
source)
canonical
g
flat
(2.9) v i o l a t e s
the
physical
obeys
v a lu es
the
of
coordinates
the
this
+ ( z ) I — >«. That
asymptotically
of
is.
is a function of the
g r a vi ty
for
It
represents
s p a c e — time.
condition
space-times
asymptotic
(2.9)
0f Equation
a
case
the
function g ^
a p p ro a ch es
In the
phy si ca l
infinity.
for
of
Equation
the metric
test
coordinates.
w o u l d be
that
The
will be said to
of Equation (2.9) is an example
beginning
Normally,
g
form
factor'.
interval
(p ,z) co ord in a te s.
times.
defi ne d
2 1/2
+ p )
(C = constant),
'acceleration
di f f i c u l t
is,
solution
co ord ina te s) has a term w i t h the fu n c t i o n a l
+ (( Z - C )
contain
the
flat
in E q u a t i o n
space(2.9)
is
the surface defined by g^t = 0
is an in finite re d - s h i f t surface (Ohanian 197 6 , p.308). The
equation
of
(2.9)
p =
is
represents
the
infinite
0,
flat
red-shift
z < 1/2A.
space-time.
Equation
surface
(2.9)
for
Equation
nonetheless,
19
CHAPTER 3
NEW SOLUTIONS VIA ACCELERATION FACTORS
As
first
mentioned
interval for stationary,
in
Chapter
I,
the
space-time
axially symmetric space-time is
ds^ = e^u (dt — o)d0 )^ — e ^Tlp^d 0 ^
- e 2 (n" u)[dp 2 + dz2 ].
Equation
of
(3.1)
Lewis
is
(1932)
written
and
using
(3.1)
the
assuming
metric
(p,z)
as
parametrization
Weyl
canonical
coordinates. The Einstein vacuum field equations are written
out
in
Equations
parametrization
( 1 . 1 9 ) — ( 1 . 23 ).
and
the
selection
of
The
an
Lewis
(1932)
orthonormal
tetrad
(Appendix B) a l l o w s the field e q u a t i o n s to a s s u m e a fa irly
compact
form.
We yl
interpretation
axis
and
of
of
coordinates
z as a distance
p as
s y m m e t r y axis.
can on i c a l
parallel
a perpendicular
This
all ow
to the
di st an ce
interpretation
the
aw a y
loose
symmetry
from
is strict Iy true
the
only
w h e n U = M = Q = O. For this case. E q u a t i o n (3.1) reduces to
the flat
space-time
coordinates.
offset
tim e
by
a
the
The
interval
advantage
fact
Weyl
complicated
written
in Weyl
coordinates
coordinates
fu n c ti on al
in cylindrical
can give
form.
The
is
spatial
somewhat
simple
space-
vacuum
field
20
e q u at io n s
for this
metric
are E q u a t i o n s (1.19)-(1.23) w i t h
M=O
(3.2)
V [pV u ] = 0
(3.3)
(3.4)
Equations
follows
(3.3)
and
(3.4)
are
integrable,
since
Q
= Q
pz
zp
from Equation (3.2) Once a function u is found which
obeys E q u a t i o n (3.2), the m e t r i c f u n c t i o n Q m a y be found by
integration
b ec au se
of
Equations
Equation
is
solutions
are
to
will be
than
space-times
their complete
approach
to
static,
a x i al ly
symmetric
single
metric
is one r ea so n w h y static s p a c e - t i m e s are
analyze
stationary
how
completely characterized by the
f u n c t i o n u. This
easier
Notice
integrability
po ss ib le
E q u a t i o n s (3.3)-(3.4). A s o l u t i o n set to these eq ua t i o n s
<u,Q >.
the
is
on
by
is
This
condition
d en ote d
(3.2)
(3.3)— (3.4).
stationary
requ ir e
specification.
generating
the
space-times.
extra
function w
In this chapter,
static,
axially
T he
for
a very direct
symmetric
solution?
introduced.
In General
e q ua ti on s
one
g o v e r n i n g the m e t r i c
Therefore,
surprise.
Relativity,
the
One
form
adv an ta g e
of
generally
expects the field
fu n c t i o n s
to be nonlinear.
Equation
of this
(3.2)
equation
is
a pleasant
is d i s c u s s e d
In
Ch ap ter I f o l l o w i n g E q u a t i o n (1.25). So lu t i o n s to E q u a t i o n
(3.2) can be w r i t t e n as a P o i s s o n integral
(Equation (1.4))
21
of an axially symmetric mass distribution.
is
that
Equation
equation.
Equation
(3.2).
If
two
(3.2),
This
(3.2)
a li near
functions
the
is
is
partial
u ^ an(j ^
generation
differential
separately
(u1+ u 2 ) also
fu n c t i o n
so l u t i o n
Another advantage
at
obey
obe^ s E q u a t i o n
its
simplest:
the
linear s u p e r p o s i t i o n of two k n o w n m e t r i c f u n c t i o n s Uj a n ^
u 2 . if the
functions
gravitational
u are
However,
to con clu de
that
due
to
the
superposition
suppose one solution
(u^+Uj)
metric
is
anoth er
function
function A
the m e t r i c
of
is
not
not
Equations
corresponding
of
those
sources
sources.
is due to one
source.
solution,
$) is
<(u i + U 2 ),(fli + Q 2 ) >
nonlinearity
by
that this te ch n i q u e yields the
,Q2 * is due to an o th er
and
described
assumed to be produced by localized
it is t e m p t i n g
space-time
fields
equal
a
Equation
but
to
the
(3.3) — (3.4).
to (n^+*2),
(3.2)
"
Tb a *
because
To
says
corresponding
Qj+flj
solution
source
is*
of
the
determine
the
assume
Q = Q 1 + Q2 + u2 - P*
(3.5)
This auxiliary function p obeys an equation of the form
Vp =
(I
- 2 u V p )Vu 2 + 2 pu 1 ,zVn2 .
Equation (3.6) is the price
solutions,
of
integrating
the
simple
and the ease of the technique
Suppose
by
for this
(3.6)
metric
Equation
now
that
superposition of
depends on the ease
(3.6).
ong of the solutions
functions
of Equations
(2.10) — (2.11)
is given
22
U1 = (l/2)ln[A(r
=
where
r
+
(I/ 2 )I n [(r +
and
expressions
z
are
*>].
z)/2 r ],
(3 .8 )
defined
in E q u a t i o n s
field equa tio ns ,
(3.7)
as
in
Equation
(2.8).
(3.7)-(3. 8 ) do s a t is fy the v a c u u m
as they s i m p l y r e pr es en t flat s p a c e - t i m e
from the point of view of an accelerated observer.
s o l u t i o n <u 1 „ni>
^u2 #®2 ^ =
The
is
superposed
with
When this
another
Equa t i on (3.6) for the auxiliary
so lu t i o n
function
P
is
Vp = r 1 EzVu + p^u],
The space-time
(.3.9)
interval for this new
static
solution has the
f orm
ds 2 = A(r + z)e2 u dt 2 - A- 1 Er + z)- 1 e- 2 u p 2 d0 2
- (2A r)- 1 e2(fi" P) [ d p 2 + dz2 ] .
The s ol u ti on g e n e r a t i o n p r o c e d u r e
Equations
(3.2)-(3.4).
function p. Finally,
Second,
(3.10). W h i l e the p r o c e d u r e
see a p h ys ic al
this
is simple.
solve
the new metric
(3.10)
for
the
is displayed
is simple,
solve
auxiliary
in Equation
it is not so easy to
i n t e r p r e t a t i o n for the m e t r i c
g e n e r a t e d by
t e chnique .
First,
expressions
recall
(in
that
the
symbol
particular
(Equations
the
acceleration
is associated with an observer
moving
through
magnitude
’A'
represents
is
First,
flat
of
space-time.
an
in
the
(3.7)-(3.8))
acceleration.
This
above
(Chapter
observer
This,
2) who
describes
23
flat s p a c e - t i m e by me a n s of the s o l u t i o n <u ,O 1 > ." Wha t the
above s o lu ti on g e n e r a t i o n te ch n i q u e does is to add to this
a c c e l e r a t e d o b s e r v e r ’s v i e w of flat s p a c e - t i m e a so lu ti on
<u,0>.
be
If <u,Q> is asymptotically flat,
the
result
of
localized
sources
symmetric
around the z axis. Therefore,
technique
takes
observer)
and ’a d d s ’ the
sources.
time
The
around
flat
space-time
metric
sources
the new
time
localized
to
rectilinear
metric
seen by
field
acceleration
The
are
<u,Q>
for
the
the
argument
be viewed
which
magnitude
sour ce s
above
functions
c o n f i r m the as su mp ti o n.
assumes
for
curved
spaceor
as the spacein
uniform,
'A'.
To
find
the
simply find the
and
'add’ an
flat
that
Equations
space-time
charged C-metric.
from
the view
there is a way to
a p a r t i c u l a r m e t r i c called the
Among many properties they show how the C-
describes
the
ch ar ged
r e pr es en t
(3.7)— (3.8)
K i n n e r s l e y and W a l k e r (1970) in a
t h or ou gh a n a l y s i s , d i s c u s s
pape r
of lo ca l i z e d
are
at rest
of an a c c e l e r a t e d observer. F o r t u n a t e l y ,
accelerating
accelerated
factor.
the metric
metric
an
<u,Q>
metric of an accelerating source of gravity,
gravity
axially
accelerated observer:
might
of
are
the above generation
represents
sources
acceleration
which
field
as seen by the
equivalently,
due
(as
gravity
generated
it may be assumed to
the
space-time
around
a
mass.
The
symbols
e and
charge
and
mass
the
of
uniformly
m
in their
accelerating
24
particle.
The C-metric has the space-time
ds 2 = A~2(x
interval:
+ y)~ 2 [F(y)dt2 - F " 1 (y)dy2 - G™ 1 (x)dx2
— G(x)dz^],
(3.11)
where
G(x) = I - x 2 - 2 m Ax 3 - e2 A 2 x 4 ,
F (y ) = -G(-y).
The
flat
space
limit
(in this limit)
is w h e n e = m = 0 .
is c h an g e d to the Weyl
W h e n the C - m e t r i c
ca n o n i c a l
form,
the
m e t r i c f u n c t i o n u is of the form
u = (1 / 2 ) ln[(z + l/ 2 A )2 + (( z + l/2A) 2 +p 2)1/2] ,
whe re
(p»z)
Equation
Equation
are
Weyl
c an on ic al
coordinates.
(3.12)
Note
how
(3.12)
is of the same form as Equation (3.7). Since
0
(3.12) re p r e s e n t s a c c e l e r a t e d flat sp ac e- ti me .
E q u a t i o n (3.7) does also.
Equation
(3.9)
for
the
auxiliary
function
p
is
of
interest for a couple of reasons. The e q u a t i o n is in terms
of V0,
w h i c h m e a n s the e q u a t i o n is reall y tw o e q ua ti on s
since V i s a
two-dimensional
In
find
order
to
P given
u;
gradient operator: V = ^ 3 p ,SzI.
Equation
(3.9)
integrability condition V ^P — 0. Taking V
must
obey
the
of both sides of
Equat ion(3 .9 ) gives
V Vp =
But,
since
r-1V [p Vu ].
u obeys
E q u a t i o n (3.13)
function
P is
Equation
(3.13)
(3.2),
the
right hand
side
of
is zero and the e q u a t i o n for the au x i l i a r y
integrable.
25
The
that
discussion
solutions
to
following
Equation
Equation(3„2)
can
(1.25)
be
po ints
written
out
as
an
integral:
r 1 d 3 x'.
u (p » z) = -/.//oCp'.z') I
o(p,z)
can
density
be
thought
in (p,z) Weyl
outside
of
the
of
as
an
canonical
mass
density,
axially
coordinate
cr,
Equation
(3.14)
symmetric
space.
Equation
s ol u t i o n to E q u a t i o n (3.2). For point s
d e n s i t y o(p,z).
(3.14)
mass
For points
(3.14)
is
a
inside of the mass
is a s o l u t i o n of Poisson's
equation
p ^V[ pVu]
= 4jtcr(p»z).
(3.15)
If the u in E q u a t i o n (3.9) sa ti s f i e s E q u a t i o n (3.15) at all
points
in (p,z) space,
the c o r r e s p o n d i n g
Poisson equation
for the auxiliary function p is
p-^V [pV(r***p)]
= 4n r
z cr(p , z) ,
(3.16)
with a solution
P (p ,z) — — x ( p , z ) J f J x
x
I
^(p',z')(z'+l/ 2 A>o(p',z') x
j" 3^d3 X'.
(3.17)
P is determined by a modified mass
density:
r ^zo(p,z).
The
im p o r t a n t point
(3.18)
is that P ma y be w r i t t e n
as a P o i ss on
integral.
Recall, how ev er ,
that se le c t i n g a mass d e n s i t y o(p,z)
for an a x i a l l y s y m m e t r i c
c o n f i g u r a t i o n of m a t t e r does not
m e a n that going t h r o u g h E q u a t i o n s
(3.14)-(3.1 7 ) will yie ld
26
the
space-time
rectilinear
the
when
they
interpretation
in
coordinates
A
and
which
all
of
the
r
is
point
preferred
point
origin
the
preferred
of
solution
below
is
due
it
is
any
the
schemes
1980).
S (s ) d e f i n e d
=
In
flat
sources.
in
are
space-time
The
symbols
as
p ,z
cylindrical
Equation
Weyl
up
between
the
the
point
,z ) .
a
appears
other
their
system
troublesome
a localized
or ( p
symmetry
%o n e g a t i v e
out
As
mass- d e n s i t y
(0,— 1/2A)
preferred
point
the
more
is
,
the
toward
the
A—
>0
This
on
along
the
the
elaborate
motion
by
the
z— a x i s
of
an
z-axis
solution
( d i n n e r s ley
replaced
, the
o (p ,z) . For
appearance
investigators
r
>+<»
z- a x i s
unaccelerated
lies
the
work,
for
the
However,
in
A—
infinity..
if
( 0 , - 1 / 2 A)
the
axis
and
the
generation
P q • along
A.
is
coordinate
point
solution
point,
the
(3.13)
canonical
simple
coordinate
of
about
acceleration
selects
which
their
is
coordinates
interpreted
In
preferred
assumed
which
generation
S (s )
This
slides
to
r.
distance
slides
sources
something
Cosgrove
thing
is p o t e n t i a l l y
expression
is
a
(p,z)
point
jn r
moment,
the
on
canonical
the
not b e
configuration
values.
(p,z).
depends
of
from
factor
selects
Weyl
mass
retain
interesting
space ,
technique
away
cr (p , z ) c a n
second
the
The
only
far
for
appearance
(p,z)
particular
acceleration.
problem:
appearing
that
1977,
a function
as
[(I-Zsz)Z
+
4s2 p2 ]1 / 2 .
(3.19)
27
This
ubiquitous
t h eo re ti c
expression
ana lys is
differential
first
of K i n n e r s l e y
equations
derived.
for
the
and Ch itre
metric
gives
in
the
(1977b)
generating
the
first
group wh e n
functions
F a b (S)
are
simple
solution generation technique here may be related to
the techniques
This
appe ar s
clue
that
the
of other workers.
Another clue to a connection to other techniques, is the
appearance
p.
This
of the Equation
equation
resul ts
(3.9)
for the
primarily
auxiliary
from
using
function
Eq u a t i o n s
(3.3)-(3.4) to d e t e r m i n e a m e t r i c f u n c t i o n Q w h i c h results
from
adding
space-time
a general
solution.
with Equation
differences
of A). The
(2.17)
static solution to an accelerated flat
Equation
(3.9) corresponds
of Hoenselaers
et.al.
are a sign and a different
very closely
(1979b).
parameter
The
(t
only
instead
equation governing their P is
Vp = S- 1 (t) [ (l-2tz) Vu - 2tpVu]
(3.20)
The ir P is used in solving for the g e n e r a t i n g f u nc ti on s of
static,
as
a
axially
result
functions.
of
symmetric
their
There
is
space-times,
Their
definitions
no
superposition of solutions.
reference
of
to
The advantage
P comes
the
about
generating
acceleration
or
in their P is that
the limit t-->0 is s i m p l e r in m a n y cases. F ig ur e 3 shows the
preferred
point
and
singularity
expressed
in Weyl canonical
of
the
coordinates.
acceleration
factor
28
Z
Preferred
p o in t
_ S in g u l a r
re g io n
Figure 3: Preferred point and singularity of
acceleration factor
29
CHAPTER 4
SIMPLE EXAMPLES
Too
elaborate
often,,
mathematical
by no specific
of
reasons
described
al l o w s
for
this.
as the
fo u r
functions.
Eve n
these
displaying
g
notion
of
an
accompanied
are
a couple
g r a v i t y being
of a four-dimensional
space-time
, represents
® j,j k I ' c a n
is
components
is too compact. The
coordinates.
though
tensors
the
the notation
independent components,
of
introduced
or example s. There
First,
space-time
tensor,
is
R el at iv it y,
n o t a t i o n of R i e m a n n i a n g e o m e t r y to be
tensor,
curvature
of General
formalism
curvature
Sometimes
for the metric
areas
calculations
the c o mp a ct
employed^
the
in ma n y
sixteen functions
The
symbol
for
of
the
2 5 6 different
represent
symmetries
symbol
reduce
the
number
of
explicitly displaying the components
tedious
and
is necessary
time
consuming.
to examine
the
Yet,
structure
of a given space— time.
A second re as on for not
occurs
of
not
in
solution
having
equations,
to
including sp ec i f i c e x a m p l e s
generating
techniques.
explicitly
often the m e t h o d
solve
involves
the
Despite
promises
Einstein
solving
field
a d i ff ic ul t
30
differential
Once various
problem
of
e q u a t i o n or a c q u i r i n g an a u x i l i a r y function.
functions
giving
are found,
a phy si ca l
there
is the
interpretation
perpetual
to
all
the
mathematics.
This
chapter
in earli er
there
shows
specific
chapters. Not
examples of points raised
every algebra
step
is shown,
but
is enough so that any omitted steps are trivial.
One of the simplest metrics
is the Schwarzchild m e t r i c ,
(1916) which describes the space-time
outside
a point source
of mas s M
ds 2 = (I-ZMR- 1 IdT 2 - R2 Sin2OdO2
- (1-2MR - 1)""1 dR 2 - R 2 d 0 2 .
A spherically symmetric
so
it
is p o s s i b l e
Equation
space-time possesses
to wr i t e
(3.1). The
(4.1)
(T,0)
axial
Equation
(4;1)
in the
coordinates
here
are
symmetry
form
of
equa te d
to
the (t,0) c o o r d i n a t e s of E q u a t i o n (3.1). Next, the terms in
Equation (4.1)
form
involving dR and dO are
of Equation
cast
in the
isotropic
(1.8)
2
(l-2MR- 1)- 1dR 2 + R 2 d 6
= R 2 [(R2 -2MR)- 1 dR 2 + dO2 ].
This part of the s p a c e - t i m e interval is i s ot ro pi c
coordinate
if a n ew
a is introduced and defined as
do = (R 2 - 2 M R )-17 2dR
a = cosh 1 [(R— M )/M ]
R
MIcosha + I].
(4.2)
31
With the coordinate
a.
Equation (4.1) becomes
ds ^ = [c o she — I ]/[c o sha + Ildt^
[co sha + 11 ^ [sin^ 6 d 0^
—
— M^Icosha + I ]^ [da^ + d B^
»
(4.3)
and by a comparison with Equation (3.1) allows
functions
and coordinates to be
e ^u =
the following
identified:
[c o sha — 1] / [cosha + 11
(4.4)
(4.5)
Ms inhasin©
e2G _ e^ u [dp^ .+ dz^l- ^M^ [cosha + 11 ^
x [d a 2 + d © 2 ].
As
the
Weyl
both
(p,z) coordinates
isotropic,
transformation.
of
one
in two
Cauchy-Riemann
c o nj ug at e
they
and
and
ar e
Hence, p+iz
an ot her
equation
(4.6)
related
(a,©)
di m e n s i o n s ,
to p. In this
(a,©) coordinates
by
and a+i© are
(p,z),
equations.
the
and
each
are
conformal
analytic
functions
sati sf y
Laplace's
r e l at ed
Ther ef or e,
case,
a
are
z
is
by
a set
the
z = Mcoshacos©
of
fu nc t i o n
and E q u a t i o n
(4.6) r e d uc e s to
[c o sh2 a — 11/[c o sh 2 a — cos2©]
e2fl _
Writing
the
coordinates
metric
is made
functions
terms
of
easier by defining x = cosha,
p = M(X 2- I ) 2 (1-y2)
.2u
in
(4.7)
Weyl
y = cos©:
2 , z = Mxy
(4.8)
, .x
.,x _
(x— I) / (x+ 1) - g%t
(4.9)
(4.10)
(x 2 - l ) / ( x 2 - y 2 )
Equation
(4.8)
is
easily
(p,z)
recognized
as
de fi n i n g
prolate
32
spheroidal
coordinates
This
(Appendix
calculation
shows
familiar
metrics
familiar
form.
Equations
re c o g n i z e
as
representing
point
mass.
In
in Weyl
fact,
over
a
(3.16)) the mass
form
Putting
destroys
their
(4.9) and (4.10) are di f f i c u l t to
the
there
mass
several, things.
c an on ic al
s o m e t h i n g quite different.
integral
E).
space-time
is
a
o u t si de
temptationto
of
a
conclude
W h e n u is w r i t t e n as a P o i s s o n
density
in
(p ,z)
de ns it y w h i c h w o u l d
space
give the
(Equation
function
in
E q u a t i o n (4.9) is that of a thin rod of linear mass d e n si ty
1/2
and length 2M
lying
along the z-axis,
= 0, z = 0. The point
source
(4 .1 )
line
appears
as
coo rdinates.
The
a
source
of the
source
of
the
and centered on p
space-time
in
Weyl
of E q u a t i o n
ca nonical
Schwarzchild
is
(p,z)
sh ow n
in
F igure 4.
Equation
the
density
(4.9) has an interesting
constant,
if
the
top
limit.
of
the
While keeping
line
source
is
placed at p = 0 , z = 0 , and the lower end is allowed to tend
t o w a r d mi nu s
(4.9) goes
is
fu nc t i o n
u in E q u a t i o n
(l/ 2)l n [( 4M ) - 1 [z + (p2+z2)1/ 2]
the
same
as
(2.10) ). This
sug ges ts
fu n c t i o n
be
source
the m e t r i c
to
u -->
which
infinity,
can
with
an
an a c c e l e r a t i o n
viewed
linear
acceleration
mass
as
produced
by
(4.11)
factor
factor
an
d e n s i t y 1 / 2 . On one
(Equation
in a m e t r i c
infinite
line
hand this
is
33
desirable
thought
to
a
be ca use
this
of as providing
general
infinite
source
of
mass
acceleration
when
added
solution.
That
is,
static
can
be
(Chapter
3)
adding
an
a c c e l e r a t i o n factor is p h y s i c a l l y e q u i v a l e n t to adding.an
infinite line source
an
extended
phy si ca l
being
line
sources
of finite
to the sp ac e- t i m e . On the other hand,
source
does
of gr a v i t y
cause
are most
uneasiness
bec au se
often v i s u a l i z e d as
extent.
The metric function, u, for the Schwarzschild solution
is
a
special
case
of
called Zipoy-Voorhees
a
class
solutions
The metric functions for these
The
of
exact
static
(Zipoy 1966,
solutions
Voorhees 1970).
solutions are
e2u = [<x - I) /(x + I )]8
(4.12)
e2Q = (S 2 /2) [ ( x 2 - I) /(x 2 - y 2)]
(4.13)
S c h w a r z c h i l d s o l u t i o n is the 6 = 1
Voorhees
case
of the Zi poy-
solution.
A c c o r d i n g to Ch ap te r 3, if a new s o l u t i o n is found by
adding
an acceleration
factor
to a Zipoy-Voorhees
solution,
it will be necessary to solve for a function P (see Equation
(3 .1 1 )) to c o m p l e t e l y sp e ci fy the ne w s p a c e - t i m e
From
Equation
coordinates
(3.11),
the
two
e q ua ti on s
for
interval.
P
in
(x,y)
are
P x = ; ™ 1 [ z u x + p(i-y 2 )1 / 2 (x2-l)" 1/ 2 uy ]
(4.14)
Py = ; - l [ z u y - p<x 2 -l> 1 / 2 (l-y2 >"1/2*y ]
(4.15)
Using Equations
(4.8) and (4.12),
the
function
P is found
to
34
be
I n [(x + 2MAy - 2Ar)/(x 2 - 1)1/ 2]
P = S
(4.16)
Another simple static axially symmetric metric
is the Curzon
so l u t i o n (1924)'. The m e t r i c f u n c t i o n s for this s p a c e - t i m e
are
u = -m/(p^ + z2) ^ 2
-(m p )2 /2(p 2 + z2)2
fi =
Recall
in
(4.17)
(4.18)
that the field equation for u
(p ,z)
coo rdi n at es .
Laplace's
metric
The
equation outside
is
function
of a point
f u n c t i o n of E q u a t i o n (4.17)
earlier.
A
novice
Laplace's
is
the
raised
cl a i m
that the s i m p l e s t n o n - t r i v i a l
so lu t i o n
source. Therefore,
stresses
point
equation
theoretical
of
the
even b et te r
physicist
a
would
s o l u t i o n to Laplace's
equation
u
is
+ p-1u
+u
=O
H
p
zz
pp
given
monopole
it
is
by
source
the
(4 .9 )) — ■
leads
to
point
the
the
mass:
(4.17)
of strength m. Yet,
of
the
more
a thin
general
interpretation
authors
Equation
considerably
that
(4.19)
rod
source
relativistic
suggesting
the
Cu rz oh
compound
the
problem
of
solution
which
space-time
solution.
point
is
a
eventually
The
a
physical
su btle
canonical
giving
(Equation
surrounding
the space-time
a disk of radius 2m. At any rate, Weyl
truly
--
so lu t i o n
it represents
a
as previously discussed,
complicated
Schwarzschild
of
representing
--
some
exterior
to
coordinates
clear
ph ysical
35
phy si ca l
interpretation
to
so lu t i o n s
expressed
in
(p,z)
coordinates .
Finally,
the
function
P (Equation
(4.14)) rela te d
to
the Curzon solution obeys the equations
= mp [2z + I/ 2 A] / [r<p2 + z2)3/2]
P
Pz = m[z(z + 1/2A) - p 2 l/[?(p 2 + z2)3/2l
P
(4.20a)
(4.20b)
with solution
P = -2Amr/(p 2 + z2 )1 / 2 .
(4.20c)
36
Z
l i n e a r d e n s ity = 1
2
A
2M
»
Figure
4
f
: Source of the Schwarzchild solution
in Weyl canonical coordinates
37
CHAPTER 5
SIMPLIFYING THE ACCELERATION FACTOR
Chapter
2 points
by an accelerated
out
that
observer
flat
can
space-time,
when viewed
be written (Equations
(2.9)-
(2 .11 )):
ds 2 = e2 u dt 2 - e~ 2 ” p 2 d 02
- e2 (Q™ u > [dp 2 + dz2 ],
e^U = A(r + z),
r2 = p 2 + Z 2 ,
These
fu n c t i o n s
transformations
e^^
= (r + z)/(2r)
(5.1b)
z = z + 1/2A
were
on
(5.1a)
derived
flat
(5,1c)
by
performing
space— time,
so they
c o o rd in at e
are- guaranteed
to satisfy the Einstein vacuum field equations:
V [pVu]
= 0
(5.2a)
Q p = P t < V 2- U z)2]
The metric
function-.g
-1/2 A
extending
and
°z
= 2p upu z
(5.2b)
has a line singularity at p = 0, z =
along
the
negative
z-axis
to
min us
infinity.
One
of
the
reas ons
that the c o o r d i n a t e s
why E q u a t i o n s
(5.1)
look m e s s y
is
(p,z) are c e n t e r e d on the point p = 0 ,
z = 0 , wh i l e the s i n g u l a r i t y in the m e t r i c f u n c t i o n g 11 is
displaced away
from the origin down the z-axis. This chapter
38
will
show
some
unexpected
results.
The m e t r i c
f u n c t i o n g ^ ^ £n E q u a t i o n (5.1b) depends on
the
two
how
an attempt
coordinates
to clean up this problem
p and
z . It w o u l d
be
leads to
simpler
if g ^ ^
de p e n d e d on
a single coordinate. Call this new co o r d i n a t e
%.
coordinate
The
new
c o o r d in at e s:
on
is
some
function
§ = £(p,z). Ho we ve r,
if g
of
the
Weyl
£S to only depend
then
§ = €(y),
Assuming
£
y = z + r.
is
going
to
be
(5.3)
part
of
a set
of
isotropic
coordinates:
«„ + *„
Using Equation
$yy(y2 y)
«•
(5.3),
<5-4>
this
equation
becomes:
(5.5)
+ Sy = 0
with solution:
tty)
wh er e
= c 1 (y)1/2 + C 2
and
Equation
(5.1)
C2
are
(5.6)
co n s t a n t s
the s i m p l e s t
of
integration.
a pp ea ra nc e,
select
C^
To
give
I-1 / 2
and C 2 = °- Ther efo re :
S =
The
A-I/ 2 (r + z)1/2.
conjugate
(5.7)
coordinate q is found by means
of the Cauchy-
Riemann conditions
6P
=
" 11Z
The q coordinate
Gz
=
1Ip
( 5 *8)
is:
q = A- 1 / 2 (r - z )1^2
(5 .9 )
39
E q u a t i o n s (5,7) and (5,9) can be in ve r t e d to yield:
2A- 1 Z = §2 - Ti2
With
this
change
Equation
(5.1)
A- 1 p =
(5.10)
of coordinates,
the
space— time
interval
in
becomes:
ds 2 = A 2 § 2 dt 2 - ri2d0 2
- U g 2 + d q 2 ].
The
that
coordinate
transformation
in Equation
(§, Tj) are a set of p a r a b o l i c
cylindrical
centered
on
coordinate
5
The y
the
of
coordinates
However,
in
only
one
case
by
coordinate
are
shown
forcing
an d
shows
coordinates
z = — I /2 A . T he
coordinates
this
(5.10)
constant
in Figure
g
forcing
to
the
be
a
new
to m a i n t a i n the i s o t r o p i c form of the metric.
suggests
metric
over
for these
about
this
!Symmetric
P = O,
point
curves
come
function
(5.2)
(5.11)
to
changing
of Equation
the
the
(5.1a)
general
and
static
axially
the field equations
( £,q ) c o o r d i n a t e s .
Equation
(5.1a)
be comes:
ds 2 = e2 u dt 2 - e- 2 uA ~ 2 (§q)2 d0 2
- e2 (Q-u) a 2 ( 2 + ^2) [dq 2 + d % 2 ].
(5.12)
and the field equations become:
%q
+ T lu5 = 0.
(5.13a)
= (q 2 + S 2 )~ 2 q S [ S < ( n ^)2 - (n^)2 > + 2 q u ^ ^ ]
(5.13b)
+V
luV, +
0, = (q 2 + ? 2 )_ 1 q ? [ - q < ( u ^)2 - (u?)2 > + l ^ u ^ ]
In
a
(5.13c)
C h a p t e r 7, E q u a t i o n s (5.12)-(5il3) will prove useful
detailed
a n a ly si s
of
radiation
from
new
solutions.
in
40
Z
I \ \ \
Figure
5
\
: Constant coordinate curves for
parabolic cylindrical coordinates
41
CHAPTER 6
THE ACTION OF b
It
is
well
standard
functions
generating
H n (x),
known
ca n
generating
be
in m a t h e m a t i c a l
have,
functions.
For
p h y si cs
that
many
representations
in
ex am pl e,
polynomials,
completely
function,
4s
F (x,t),
Hermite
defined
in
terms
terms
of
of
their
as
CD
F(x,t) =
exp(2xt-t^)
= 5
L.
H (x)tn / n !
n
(6.1)
n =?0
(Mathews
and
Walker
(1970)).
At
first
that the generating function buries
about
the Hermite
However,
a short
polynomials
calculation
about H^(x ) £$ easily
glance,
it appears
all specific
information
inside
shows
an infinite
how
specific
summation.
information
obtained.
F (x ,t ) = exp( 2 xt-t2) = exp(x 2 )exp(-(t-x)2 )
CO
= ex p (x2 )
^
n =0
dn {exp(-(t-x) 2 ))}/dtn tn / n !
42
= exp(s2 )
(-l)n dn {exp(-x2 ) } / dx 11 tn / n !
^
(6 .2 )
n =0
When
Equation
functional
(6.2)
is
form of H^(%)
compared
is found
to
(6 .1 ) ,
Equation
to be
® n (x ) = exp(x^) (~l)n (d / dx)n (exp( — X 2))
The
main point
essential
In
is
that
a generating
(6.3)
function
contains
most
information about a set of functions.
a
series
of
papers
Kinnersley
( 19 78b )
Hoenselaers,
(1979a),(197 9b)
stationary,
axially
equations.
One
investigations
most
Kinnersley,
discovered
symmetry
symmetric
of
the
and
coworkers
and C h i t r e - ( I 977), (I 97 8 a ) ,
(K i n n e r s I e y - (I 977) , K i n n e r s l e y
are
the
and . X a n t h o p o u l o s -
groups
admitted
Einstein— Maxwell
many
consequences
of
by
the
field
their
is the fact that the symmetry transformations
easily
expressed
in
te r m s
of
suitably
defi ne d
g e n e r a t i n g functions. The g e n e r a t i n g f u n c t i o n s are SL(2, R)
tenso rs
and
are
following manner.
expressed
in
Equation
(1.26).
construct
H
tensors.
from
A stationary,
form
of
the
Equations
the complex tensor,
the notation
tensor
the
constructed
of Kinnersley
is used as the
t en so r
axially symmetric
S L (2,R)
tensor,
f
('1.31) — (1.33)- are
. (For a concise
(1977),
first
a metric
see
Co s grove
of an infinite
in the
metric
is
, as
in
used
to
summary of
(1980).) The
hierarchy
of
43
t(0 ,n)
Il
3
He
CO
Il
(n)
H (0)
H
CT
AB
AB
AB
AB'
(m +1 , n)
__(m. n + 1 )
(m
H
NAB
NAB
" iNAX
(6.4 a ,b » c )
B
(m,n) ^ (-1 ,0 ),( 0- 1 )
(6.5)
(m,'n) ^ (0 ,0 )
(6 .6 )
(6.7)
which
are,
in
turn,
employed
in
the
definition
of
two
generating functions
L
(6.8)
=A b ’ ^
n= 0
g a =<«-«>
-\
(6.9)
' Ir'
n =0
The
two
ge n e r a t i n g
coordinates
fun ct i o n s
being used
only the dependence
above equations.
contain
to describe
the
on the parameters,
If Weyl
coordinates
of E q u a t i o n
(6 .8 ) is w r i t t e n
(Hoenselaers
et.al. 1977)
that
governed by the differential
a dependence
space-time,
s,t,
are used,
functional
although
is shown
.It
the
on the
in the
the left
can be
form
side
sho wn
of F
AB
is
equation
V(Fa b ) = i t s " 2 (t)[(l- 2 t z)V(HA x ) _ 2 tpV(HA x )]FXB.
(6 .10 a)
S2 (t)
G
AJd
=
(1-2 1 z )2 +
(21p )2
can al wa ys be e x p r e s s e d in te rm s of F
(6 .10b)
AB
, so once F ln is
AB
found,
the
sol ut io n
function
generation
cl e a n l y
expressed
functions
metric
automatically
above
techniques
by
formulas
instead
tensor
f^g.
of
to
be
examined
invol vi ng
formulas
Roughly
d e t e r mi ne d.
the
direct Iy
speaking,
the
are
The
most
ge n e r a t i n g
involving
the
formulas
are
expressions of the form
(Ne. Fa b ) =
(Algebraic
D es pi te
involving
the fact that the g e n e r a t i n g
inf inite
number
ac t u a l l y
describes
which
expression
must
produced.
of
tensors,
the
is
s p a c e — time
eventually
This
it
be
old F ^ )
f u nc ti on s
the
and
recovered
(6 .11 )
involve an
tensor
it
is
from
f
wh i c h
this
a ny
tensor
new
is done using the definition of F^g
S^ve
(6.12)
Re tdtFA B (t)1 t= 0
Cosgrove's m e t h o d (1980) of so lu ti on g e n e r a t i o n stems
from
seeking
infinitesimal
change
the
form
of the vacuum
able
to
exponentiate
transformations
field equations.
some
of
his
which
do not
Cosgrove
was
infinitesimal
t r a n s f o r m a t i o n s to finite t r a n s f o r m a t i o n s . The p a r t i c u l a r
transformation
to be
Q . . The
4S
of this
effect
looked
at
is denoted
in Cosgrove
transformation acting
by
on a function
is written
I' = 8 4 s (I)
(6.13)
The symbol s is a parameter which may vary continuously from
minus
infinity
to plus
infinity.
When s = 0, Equation (6.13)
45
becomes
The effect of 8
._
. ...4s ° ?
space-times is summarized by
the i d e n t i t y t r a n s f o r m a t i o n .
stationary,
axially ,symmetric
the expression
(6.14)
fAB - .®4s(fA B ) = " S 1 (s)fX Y F X A (s)FYB (s)l
The metric tensor f
axially
symmetric
generating
vacuum
field
function constructed
t-*16 new
^AB
is a known, solution to the
Until
exact
now,
all
equations
from
the
stationary,
and
metric
£ s the
tensor
f^g'
solution.
expressions
are
written
without
an
exp li ci t c o o r d i n a t e de p e n d e n c e displayed. As it turns out,
when
^ acts
produced,
on
a function,
but
coordinates
also
not
produces
only
is
a new
fu nc ti on
new
co or dinates.
are changed according to
tp’.z’j = 5 4s
. {p
f o ,z
.z }
t = S ^(p,z,s){p,z-2.s(p^ + z^)}
The
co o r d i n a t e
(6.14)
should
d e p en d en c e
of
the
first
part
of
(6.15)
Equation
read
fA B * p,'z’'s) = ®4 s *f A B (p 'z) * *
In order to c o m p a r e
desirable
Weyl
old so l u t i o n s
to get rid of changes
coordinates.
This
^AB (p »z * s ) =
(6.16)
to new
solutions,
induced by Q4 s
it is
acting
on
is accomplished by writing Equation (6.14)
~ S 1 (p'#z,,s)fXY (p» ,z')F A (p ‘,z ‘ ,s )
% f Y b (p ',z ',s)
(6.17)
where,
{P ' ,z '}
Q
-4 s {P <z)
(6.18)
46
In
the
Lewis
parametrization
of
stationary
axi a l l y
s y m m e t r i c m e t r i c s (Equ a t i o n (1.19)), the m e t r i c tensor.
has components
2u
,
2u
»
fIl = e . ' f12 = f 2 1 = — me
-2u
- P2e
f22 = * 2 e 2 "
The
( A = I 1B=I)
and
( A = I 1B=Z)
(6.19)
components
of E q u a t i o n
(6.17)
can be w r i t t e n in the form
(6 .20 )
fIl - (S2fIl)- l u p U T ,)2 - uiI1T2)2'
(6 .21 )
f12 ' (s2 fll," 1 'p llp 1 2 <Tl )2 " «lla 1 2 <T2 )2'
with
(6.22)
fAB " PAB + iQAB
Z(T1 )2 = (I-Zsz)
2 ^ 2 )^ " -(I-Zsz)
(6,18),
metric
f^
(6.23b)
+ S ( P 1Z 1S)
The a u x i l i a r y f u n c t i o n s
original
(6.23a)
+ Stp.z.s)
S 1 T jl and T% do not d e pend on the
so the coordinate
may be performed
immediately.
substitution. Equation
When this
is done
we
get
(6.24)
S(P11Zr1S) = S * (p ,z ,- s )
T 1 ( P r 1 Z 1 1 S)
(6.25a)
= T 1 ( p 1z 1- s ) S ™ 1 (p»z.-s)
(6.25b)
T 2 (pr,zr,s) = T j ( P 1Z1-S)S *(p »z,-s)
The
explicit
(6.21)
coordinate
dependence
of
Equations
(6.20
is
^ j j ( P 1Z 1S).= [ s ^ f j j ( p r1Zr)S^(P'Z»~s)]
x T j ( P 1Z 1- S ) ) 2 -
[ ( P j j ( P r1Zr1S)
( Q j j ( P r1Zr1S)T2 (P1Z1-S))2 I
(6.26)
47
f % 2 (p,z,s) =
^'11 ( P ' » z ' »s )P 2 2 ( P ' » z ' *s)(Tj(p,z,-s))^
( p ' ,z ' ,s)Qj 2 ^P* # z ' ,s)(T 2 (p,z,-s))^]
where
the primed
Equations
fAB
is acted
the
three
coordinates
(6.26)
quantities
defined by Equation
(6.18).
(6 ;2 7) show that w h a t e v e r m e t r i c
and
on by Q45.,
are
(6.27)
the new
S.
T1.
metric
and
T2-
is going to involve
Notice
that
these
functions can be written
= 2 s[(z + l / 2 s )2 + p 2]1/2
S(p.z.-s)
(6.28)
2 ^x 1 (PtZf- S ))2 = 2 s [ ( z + 1/2 s) + (p2 + (z + l/ 2 s)2 )1/2l
(6.29a)
2(T2 ( p, z ,-s ))2 = 2 s [-(z + 1/2s) + (p2 + (z + l/ 2 s )2)1/21
(6.29b)
The expression for S in Equation
the quantity r defined
examining
symbol
flat
The expressions
much
like
introduced
an^ T^
(£ ,q)
observer.
The
is here replaced by the symbol
Equations
parabolic
in E q u a t i o n s
the
(5.7)
coordinate
of various quantities,
the
(2.8)
an accelerated
and
(6.29)
cy l i n d r i c a l
(5.9).
The
s.
look very
coordinates
zeroes
of T 1
are shown in Figure 6 .
Despite
in
from
for T 1 and T 2 in
the
like
in Equation (2.8) which came about by
space-time
A in Equation
(6.28) looks very much
ana ly si s
of
there
transformations
definitions
is one property that
Cha pt er s
a c c e l e r a t i o n or m e t r i c s
and
2
and
3.
a s s o c i a t e d wi t h
When
stands out
considering
ac ce l e r a t i o n ,
in
48
Weyl
coordinates
the
point
p = 0,
z = -1/2A
becomes
a
special point. Eve n th ou g h the poi nt has p e c u l i a r and even
inconvenient
properties,
it seems very much
associated with
accelerations
of m a g n i t u d e A. E q u a t i o n s (6.26) - (6.29) show
w h e n Q 4s
on a metric,
acts
p r e f e r r e d point
the point p = 0 , % = - l / 2 s is a
in the new
that Q 4 s has the physical
metric.
effect
an acceleration with magnitude
T h e r ef or e,
it
appears
of giving an initial metric
s. The
advantage
with Q 45 is
that it is not co n f i n e d to act on static so l u t i o n s but can
also
act
on
stationary
solutions.
Since
are often set up by rot at in g sources,
such
s p a c e — times
raises
the
and accelerating
effect
possibility
of
ex te r i o r to a localized,
source.
the c o n n e c t i o n b e t w e e n Q 45 and ac ce l e r a t i o n ,
To show
the
metrics
the effect of Q 4s on
tantalizing
ea s i l y ge n e r a t i n g the s p a c e - t i m e
rotating,
stationary
of
the
transformation
on
a
general
static.
axially symmetric metric is examined. The form of the metric
tensor,fAg
15
AB
As
sh ow n
by
F
e
L
0
0
2u
I
(6.30)
-p 2 e~2u J
Hoenselaers
et.al.
(1979a),
the
ge n e r a t i n g
functions are of the form
F.U
-
. F 12 - i.-1.'-'
(6.31a»b)
where
Vp = S- 1 [(1-2sz)Vu - 2sp^u]
(6.32)
49
Z
Figure 6:
Zeroes of the functions in
Equations (6.29a) and (6.29b)
50
Using
these
in E q u a t i o n s
s on a general
u' (p.z.s)
(6.26) - (6.27) gives the effect of
static solution,
= 3 4 s (u (p,z) )
= (1 / 2 )ln[s(z+l/ 2 s) + (p 2 +(z+l/ 2 s)2 )1/ 2J
+
(p,z,s)
Equation
the
(6.34)
metric.
says that Q 4 ^ preserves
factor
(6.33)
similar
This acceleration factor
the
P
d e fi n ed
(6.34)
= Q 4 s Cf12 )
Equation
acceleration
(6.33)
P (P ' , Z ' ,S )
in
shows
to
function
to P wh i l e
Equation
containing
in C h ap t er
that
one
static nature of
® 4s
produces
in Equation
an
(2.10).
is added to a function P similar to
b e t w e e n Q 4 ^ and the ana ly s i s
metric
the
the
an
(3.11).
The
only
difference
in C h a pt er 3 is that here the
acceleration
3 the m e t r i c
factor
is
added
f u n c t i o n c o n t a i n i n g an
acceleration factor is added directly to the original metric
function,
u.
51
CHAPTER 7
RADIATION
Previous chapters have shown that the transformation of
Cosgrove,
acts
on an exact
vacuum field equation and produces
the
ori ginal
solution.
The
s o l u t i o n of
the E i n s t e i n
an accelerated version of
argument
for
Q
producing
a c c e l e r a t i o n thus far hinge s on being able to c o m p a r e
newly generated
solution with already known
s p a c e - t i m e from an a c c e l e r a t e d observer,
H o we ve r,
since
the
transformation
ea sil y produced,
Q
4s
solutions:
the
flat
or the C - me trie.
allows
exact
of new
exact
s o l ut io ns
to be
sol ut io ns
q u i c k l y ove rr u n s the n u m b e r of k n o w n solutions.
For this reason,
whether
^ 4s
the
really
exact
do
of
the
electromagnetic
radiates
solutions
describe
an accelerating
One
more general
mos t
criteria
are n e e d e d to test
generated by the transformation
space-time
gravitational
geometry
external
to
source.
im p o r t a n t
predictions
of
accelerated
point
charge
(Jackson,1975,Chapter
14).
Since
theory
energy
electromagnetism
the
the n u m b e r
new
that
an
is a linear theory,
any finite
M a xw el l' s
size charged
b o d y ( s u p e r p o s i t i o n of point charges) will radi at e e n er gy
52
when
the
bo d y
that a distant
accelerating
is accele ra te d.
observer
body
the charged body.
the
magnitude
General
correspond
gravitational
which
perturbations
Relativity
without
haying
In particular,
a
source
the
claims
contrast
energy
as
electric
in
of the
interact
with
identify
the
energy
around
a source.
In
to
fields
~ (distance
the
form
carried
magnetic
theory,
by
General
radiation
in the s p a c e — time
of
theory
away
fields.
gravitational
electromagnetic
and magnetic
in
which
electromagnetic
being
in
solutions
energ y
and
away by p e r t u r b a t i o n s
the source
directly
exact
radiating
In
in the
(fields)
the attributes
the radiation will
allows
c ar ri ed
electric
to
is useful
and direction of the acceleration.
waves.
views
property
can determine
Relativity
to
This
is
geometry
when
the
around a source decrease as:
from
source)- *
is said to be radiating and somewhere
its charges
are ac ce ler ati ng. In General R e l a t i v i t y , the first partial
derivatives
e le ct ri c
of the metric
and m a g n e t i c
tensor,
fields.
g^j*
The
are analogous
temptation
to the
is to cla im
that if the first partial d e r i v a t i v e s of the m e t r i c tensor
dec re as e
source
of
as the first
gravitation
radiation
surrounding
to
the
inverse p o w e r of distance,
sho ws
a
up
source
second partial
is
as
radiating.
However,
perturbations
and
the
tidal
derivatives
in the
forces
then the
gravitational
tidal
are
of the metric
forces
proportional
tensor.
The
53
quantity
in vol vin g
metric
tensor
is
roughly
speaking,
the
the
se cond p a r ti al
Riemann
a source
derivatives
tensor,
of
the
Therefore,
of gravitation is radiating
if:
(Distance)
The purpose
holds
of this chapter is to see if the above condition
when
applied
to
the
solutions
generated
by
the
transformation 9„
4s"
Radiation
the
is
space — time
Appendices
Newman
with
conveniently. analyzed
respect
C and D intro du ce
and
Penrose.
co e f f i c i e n t s ,
In
are
symmetric
reason
can be
space-time.
seen by
a
complex
the null
by
appendices,
out
looking
null
tetrad
a n ^ the e m p t y
written
for using c o m p l e x null
to
these
f (a)(b)(c)'
components,
axially
most
tetrad.
n o t a t i o n of
the
space
rotation
Weyl
for a general
There
tensor
stationary,
are a couple
of reasons
tetrads to de sc ri be radiation.
considering
at
flat, space— time
One
(Equation
(2.2)) ,
ds 2 = dT 2 - dZ 2 - dR 2 - R 2 d 0 2
and
a
pulse
contravariant
of light
of
light
moving
(7.1)
down
the
+Z-axis.
components of the position vector of the pulse
are,
X i = [T,T,0,0].
The norm
of this vector
(7.2)
is
= X iX 1 = 0,
showing
The
that
the
position vector
(7.3)
of the
light
pulse
is
a
54
null vector. Null v e ct o rs point
in the d i r e c t i o n in w h i c h
radiation will
radiation
travel.
Examining
in curved
time
may be handled by using
null
tetrads. A second reason for using complex null
is
tha t
since
conjugates
complex
of
and
examining
components
Appendix
fr am e
(see
vectors
Equations
component
are
of
tetrads
complex
(D .I c ,d )),
projected
is e q u i v a l e n t
(4). The number
to
the
onto
the
interchanging
independent
tetrad
in half.
Dnti
the
(1962)
these
investigated
asymptotic
when projected
D,
tetrad
a tensor
(3) and
is cut
Ne wm an
by
of
the null directions defined by
the
other
tetrad
indices
components
of
each
conjugate
c o m p l e x null
tetrad
two
space-
behavior
onto
components
empty
are
of
Weyl
a c o m p l e x null
given
space-times
special
tensor
tetrad.
symbols
In
and
are d ef in ed in E q u a t i o n s (C.3e). N e w m a n and Unti (1962) and
Janis
and Ne wma n
(1965)
show
have Taylor series expansions
a
suitably
defined
that
these
tensor
components
in powers of the reciprocal of
distance
coordinate
along
a
nu l l
direction,
00
^ - 5
a
z.
,.(distance
(a)
coordinate) n * .
.-0
The
coefficients
whether
shape
of
or
not
the
in these
expansions
a space-time
gravitational
give
is r a d i a t i n g
radiation
i n f o r m a t i o n on
energy
pattern.
and
In
the
this
55
chapter
the
expansion coefficients
for the new
C ha pt er 3 and the t r a n s f o r m a t i o n 9 ^
for
some
cases
and
could represent
In
examined
a radiating
Chapter
3,
it
to
see
solutions
of
will now be c o m p u t e d
if the
ne w
solutions
source.
was
shown
that
w h en
the
metric
function u (g^^ = e^U ) is a superposition of an acceleration
factor
(Equation
equation
written
defined
(3.8)),
and
( E q u a t i on(3 .2) ) ,
as
Equation
by Equation
the
(3.10)
(3.9).
a
solution
space-t ime
with
Chapter
an
to
Laplace's
interval
auxiliary
5 showed how
can be
fu nc ti on
P
to simplify
s o lu ti on s w i t h an a c c e l e r a t i o n factor by d e f i n i n g two new
iso tr op ic
coordinates
(^,q). W h e n w r i t t e n
(£, H ) c o o r d i n a t e s de f i n e d by E q u a t i o n s
space— time
interval
of Equation
in te r m s
of the
(5.7) and (5.9), the
(3.10) takes
the
form
ds 2 = A 2 £2 e2 u dt 2 - e~ 2 n q 2 d 02
- g2(Q-p)[d %2 + dq2 ] .
With
respect
(Equa tio ns
Weyl
to
the
null
te trad
(7.4)
defined
in
Appendix
D
(D.la)-(D.ld) wi t h <o = 0), the c o m p o n e n t s of the
tensor
for the metric displayed
in Equation
(7.4) have
the form
= 2~ 1 e ^ [ ( n ^ - u ^ - 2 A g U g + 2 A q U q + S - l * ; - q
\ - S
\
-TT1A
+ i( 2 U $ T T 2 A £U T r 2Aq U S+ ?~ luq +Tl-lu5+T,~ l A S"~^~lATl)] <7,5a)
= 2 - 1 e-2A [-(u^ )2- (u^ )2-$ 1U ^ q 1^q]
(7.5b)
2
>
4 = -(V*
(7.5c)
56
= Yg = 0 ,
(7.5 d )
where
A = fi— p .
Later it becomes necessary to consider the we a k - 1 im it limit,
or linearization with respect
ex pre ssi ons .
function
( 3 .9 )
Equations
(3.3)
SI is q u a d r a t i c
indicates
that
to the function u of the above
and
in the
the
(3.4)
indicate
that
function u while
function
p
is
the
Equation
linear
in
the
fu n c t i o n u. The linear a p p r o x i m a t i o n to E q u a t i o n s (7.5) is
(going back to Weyl coordinates)
r
~ A[(- z(upP" u zz ) + 2 Pu pz + 3 u z
(7.6a)
+ h p u PP- u ZZ j + 2~*U pz + 3 u P jl
(7.6b)
~ A[~ uz + P 1^ u pI
(7.6c)
= 0
(7.6 d )
As expected. E q u a t i o n s (7.5) and (7.6) v a n i s h w h e n u = G = p
= 0. For this case,
ds 2 = A 2 ^2 d t2 - q 2 d0 2 - d ^2 - dq 2
and
the
co o r d i n a t e
Equations
transformation
(2.4) ,(5.7) , and
(7.7)
found
by
(5.9),
(7.8a)
T = ^sinhAt
z" zo - A
puts
combining
— ^ — ^coshAt
(7.8b)
R = q
(7.8c)
0 = 0
(7. 8 d)
Equation
(7.7)
into
cylindrical
spatial
coordinates
57
(Equation
(7.1))»
ds 2 = dT 2 - R 2 d0 2 - dZ 2 - dR2 .
With the further coordinate
transformation,
(7.9a)
U + r
Z - Z 0 - A -1
R=
(7.9b)
rcosO
(7.9c)
rsinO
(7.9 d )
this flat space-time interval becomes
2
ds 2 = dU 2 + ZdUdr - r 2 (d0 )
. — r2 sin 2 0d0 2
(7.10)
The coordinate U represents
( r , 0-, Of )
coordinates
coordinates..
A
pulse
0 = constant
constant,
given by U = constant.
the
flat
space-time
represent
of
light
in Equation
the three
spherical
spatial
traveling
trajectory has
Therefore,
time while
along
a 0
=
an equation of motion
a complex null
(7.10)
tetrad for
is
Il
h31
(7.11a)
d/dr
N = d/dU - Z-1SZdr
(7.11b)
M = 2 ~ 1/2 r- l [d /d0 + icscOd/d 01
(7.11c)
B = 2_ 1/ 2 r- 1 [d/d 0 - icsc 0 d/ d 0 ]
(7. lid)
where
the
tetrad
of Equation
a
a retarded
null
spatial
^ jg} = (E(a)} Js expressed
(A.2) . The
tetrad vector
direction
and
d i st a nc e
along
the
L points
coordinate
this
null
r
is
direction.
in the form
outward
a
along
measure
of
Ac c o r d i n g
to
N e w m a n and Unt i, this is ex a c t l y the sort of null tetrad and
58
coordinates
behavior
to
of
represents
be
the
used
Weyl
when
examining
tensor
gravitational
to
see
radiation.
found the leading terms of the Weyl
the
asymptotic
a
space-time
if
In pa rt ic u l a r ,
tensor components
they
should
behave as
in
I
U
O
Il
»0
(7.12a)
(7.12b)
T 1 = 0(r~4 >
W
I
H
O
Il
*2
(7.12c)
(7.1 2 d )
V 3 - 0<r'2 )
The
*4 - O U " 1 )
Weyl c o m p o n e n t s
are
the
components
(7.1 2 e )
expressed
in E q u a t i o n s
w i t h respect
(7.5) or (7.6)
to a c o m p l e x null
tetrad
which in the flat space-time limit reduces to
I = 2"1 / 2 [A- 1 r 1a/9t + r1" 1d/d0]
(7.13a)
n = 2-1 /2[A_ 1 ^- 1 3/3t - n ~ 1d/d0]
(7.13b)
m = 2- 1 / 2 [d/d5 + id/dn]
(7.13c)
O
m = 2- 1 / 2 [d/d§ - id/dq)
(7.13d)
w h i c h is, of course,
a null
a tetrad is very convenient
coefficients and Weyl
A summary
possible
at
for explicit
calculation of spin
tensor components.
this
radiation
tetra d for E q u a t i o n (7.7). Such
in
point
is
in order.
space-time
a c c e l e r a t i o n of the sources),
(and
To e x a m i n e
the
therefore
the
N e w m a n and Dnt i express the
asymptotic limit of the Weyl tensor components,*?^
in a set of c o o r d i n a t e s
expressed
(D, r,e,0) and defi ne d w i t h respect
59
to a c o m p l e x null
(7.11)
in
available
^ (a ) ,
the
tetrad
flat
space-time
in Equations
expressed
<E(a)) w h ich reduces
in
(7.5) are
a
set
defined
with
respect
to a
reduces
to Equations
(7.13)
Therefore,
what
must
*. - V
and
to
However,
the Weyl
of
coordinates
(t,0,g,T|)
tetrad
in the
of flat
limit
what
is
tensor components,
complex null
be done
is to determine relations
limit.
to E q u a t i o n
(e^^)
and
which
space-time.
to consider possible
radiation
of the form,
V
express
the
relations
in
terms
of
(U,r,9-,0)
coordi na tes . W h e n this is done, the Weyl c o m p o n e n t s ma y be
expanded
in powers of r- ^ to compare the asymptotic behavior
w i t h the p r e d i c t e d b e h a v i o r s
of .Newman and Un t i for these
components.
The
metric
calculation
functions
approximation
accelerating
may
is s i m p l i f i e d
are
be
sources
weak
used.
are
by a s s u m i n g
enough
Physically
creating
weak
so
th a t
this
that
the
means
the
linear
that
gravitational
the
fields
w h i c h are 'painted' on to a b a c k g r o u n d of flat space- ti me .
From
a computational
problem
of relating
the two
e x p r e s s i n g the null
the
nu l l
tetrad
transformations
relations to be
point
of view,
this
Equations
in Equations
that
the
sets of Weyl tensors reduces to
tetrad of E q u a t i o n s
of
means
(7.8)
(7.11)
(7.13).
and
(7.9)
The
give
in terms of
coordinate
the
tetrad
60
21/2L = (_g-lUcose)(l + s)
+ (— ^ ^ (U+rsin20 ) )(m + ^ )
+ (-isinOMm _
(7.14a)
23/2N = (£- 1 cos9(U+2r)) (I + s )
+ (—5 ^(U + r + rcos2@ ) ) (m + g )
+ isinO(m _ ^)
(7.14b)
2M = ( ^"1 SinO(UHr) ) (I + fl)
+ (-5~^rsinOcosO)(m + *)
(7.14c)
+ ^^2 - s) - icosO(jg — j®)
where
(from Equations
(7.8a)
and
(7.8b)),
(7.15)
S2 = (Z - z Q -A "1 )2 - T2
Us in g
Weyl
Equations
(7.14)
and
the
general
definition
of
the
tensor components with respect to a complex null tetrad
(Equat ions(C.3)) a long and
tedious
calculation
shows
that
= [2,In2S f 0]
+ U U c o s O f 1] r-1
+ [U2 CO s0csc2O (co s0fo_4<f1+3Cp s©f2) I r 2
+ [U3 cos©csc^©(— 2cos0f0+(7-3sin2U)f^— dcosOfj
+Cos 2 O f 3 ) ] T ~ 3
+
[U4 8- 1 csc60( ( 7sin40-42sin2 0 + 43)f (())
+ cos©(7 2s in20-104)f 1
+(24sin40-80sin20+ 68)f2
+ 2 4 c o s 3 0f 3 + (l-2sin26 + sin4 ©)f4)]r~ 4
(7.16a)
61
? I = [21/2sine(cos0<i)o_<5)1) ]
+ [Ucosd(Ziz 2 Sine)"1 ) (-fo+4CO SeY1-St2) Ir-1
+ [U2Cose(Ziz2Sine)-3 ((Z^Zcos2O)tP0
— 1Z c o s 0<?^4.(5 + 5C o s ^^^<^2— 3cos0<fg)]r
+ [U3 cose(Z1 / 2 sine)~5 ((Ilsin2O-IS) V0
+!Zcose(4-sin2e)?^+l2(3sin^e-4)?2
*
+4(cose)(4—sin2e)? —cos2eY )]r 3
3
- [co s^
q -Zco
4
(7.16b)
s d Y 2^
+ [Usecdosc2e (— Zcos d f q
+ (Z + Zco s2e) f^-Sco se?2+2<^3^r
+ [U2 cose4-1csc ^ e (c o s d (Z e o s2e + 5 )
— Z(Z+9cos2e)T^+6cose(cos^8+3)T2
— 4 (I + co s2e )T 2 + eo se*f^ ) ]r ^
(7.16c)
T3 = [(2lZ2sine)-1cos3eT0_3Cos2e<5>1
+ScoseT2-Y3]
+ [Ucose(Ziz2Sine)- 3 (-3coS2OT q
+Z c o s O(Zco s 20+3)T 1_3(1+3Co s 2®^2
+6coseT3-Y4)]r-1
(7.16d)
T 4 - [2~1csc209 (c o s ^6Tq -4 co s3^ 1?'^
+ 6 cos 2 eT 2 - 4 coseT 3 +t4)]
Tbe
expressions
above
are
written
(7.16e)
as
expansions
in
the
inverse p o w e r of the r co ordinate. The r e a s o n for carrying
62
out
the
later.
expansion
The
to
expressions
Equations.6d),
expressions
different
^2
=
j?3
in Equations
u— a s o l n t i o n to the
compute
above
the functional
orders
simplify
=
0 . To
(7.16),
will
somewhat,
further
a specific
field e q u a t i o n
become
apparent
since
from
analyze
the
metric
(3.2)—
function
is needed,
form of the remaining 9
to
via Equation
(7.6a)-(7.6c).
A s o l u t i o n to the field e q u a t i o n (3.2) can be w r i t t e n
as a multipole
expansion:
OO
u(p,z) = 5mn (r)"n-1Pn (x).
(7.17)
n=0
r^ =
+ z2 #
The symbol
Legendre
(7.17)
x = z/r
m^ r e p r e s e n t s the 2 n-pole m o m e n t and p n (x) is a
p o l y no mi a l.
The
metric
is ax i a l l y s y m m e t r i c
coordinates
function
and
is put
is
a x ia l l y
function
u
in
Equation
and w r i t t e n in We y l ca nonical
symmetric.
into E q u a t i o n s
When
this
metric
(7.6 a )-(7.6b ) , the results
are
(7.18)
^a =
n=0
where
63
?<*>
= A m ^ n (r )-n-2csc2©
[-e“
[Pn (cose) [ 8cos4® (n + 2 > +2cos2e('"5n''ll) + (2n+5 ) I
+cos@P%(cos@)[cos2@(-4n-6)+(4a+7)]]
-i2e“ ^®sin0[cos©Pa (cos©)[4cos^8(a+2)+(-3n-7)]
+p
(cos0)[c o s (—2n— 3)+(n+2)]]
n-1
- i 3 e iesinePn_ 1 (cose)
+i3sin0ei26P (cos©)]
n.
+(3/2)mn e l26Pn (cos9)r
— (I/2)m^nr n 2 c s c2©
[P^(CosO)E8cos*8(ii+2 )+2 cos2©(-5a-ll)+(2ii+5)]
+cos©P
j(cos©)[cos2©(-4a-6)+(4a+7)l
+ i2sia©[cos©P^(cos©)[4cos26(a+2)+(-3a-7) I
+p
a— I
(cos©) [cos2© (— 2 a-r 3) + (a+2)]]
(7.19a)
<S>U> = n (2 n + 3 sin2 6)-l
2
a'
Ip n (cos©)[-sia2© +2acos2 ©-a]-acos0Pn_ 1 (cose)
+2Aar[cos©Pn (cos0)-Pn_ 1 <cose>1]
<5>(a ) _ _(<p(a) ) »
4
0
W h e a E q u a t i o a s (7.19)
relatioas
betweea
coordiaates
(7.19c)
are
Weyl
( c o m b iaiag
followiag expaasioas
(7.19b)
combiaed
with
coordiaates
Equatioas
the
a a d the
(7.8)
aa d
co o r d i a a t e
( U, r» ©, 0)
(7.9))
the
are fouad:
p ~ (-iAsia2©)r2 + (- iAU)r+(2- 1 iAU2 eta2©)
Z ~ (-Asia2©)r2+(-AU)r+2"1 (-AU2-I/A)
(7.20a)
(7.20b)
64
have the following behavior with respect to the
r coordinate.:
%
(7.21a)
<?2 “
5 A2n ) ( D 'e,0)r" 3
(7.21b)
V4 =
^A4 n ) (U,e,0)r~1
(7.21c)
where
the
=
the
expansion
retarded
time
coordinate
A p p e n d i x F lists
summations
coefficients
these
su m m a t i o n .
The
is
V and
are
(7.21) . The
only
the angular
coefficients
in E q u a t i o n s
these e x p r e s s i o n s
A^^
out
functions
coordinates.
to n = 2 for
important
for
these
the
point about
the r-d e p e n d e n t factors outside
r behavior
of
expressions
the
inay be
summerized as.
T
-5 + 8)
0 (r
a
a = 0 ,2,4
a= 1 ,3
= 0
Equations
out
(7.22) e x p l a i n w h y E q u a t i o n s (7.16) were
to di f f e r e n t
are put
powers
in Equations
? a = 0(r“ 5+ a)
or more
(7.22)
carried
of r ~ *. W h e n the e q u a t i o n s
(7.16) the results
a = 0 ,I ,2,3,4
(7.22)
are
(7.23)
specifically,
r ^ contribution to *? •
0•
[Isie2#]?
+[SU2Ctn2S]?
2r
-2
+[S-1U4Ctn4Gcsc2S]?
4r
“4
(7.23a)
65
r ^ contribution to ¥ .
[-3U2-1/2ctn0]1,2r" 1
+ t-2-5/2D3 ctn30c sc 20]<li> r- 3
T
(7.23b)
'
r-3 contribution to f ^ ;
1P2 + E4- 1U 2Ctn2Sc s c 2OI 1F4r" 2
(7.23c)
r-2 contribution to *P .
[-2- 3/2UctUOcsc 2O l f 4r- 1
(7.23d)
r ^ contribution to f .
E2- 1CSC2Olf
Eq u a t i o n s
static,
4
(7.23a-e)
axially
(7.23 e )
are
the
symmetric
Weyl
tensor
space — time
components
which
is
adding an acceleration factor to a multipole
for
generated
a
by
solution of the
field e q u a t i o n (E qua tio n (3.2)). W i t h respect to the radial
dependence of these components,
fashion predicted
e)),
(also
Equations
(7.23)
by Ne wm an
see Appendix
(7.21)
and
G)
and Unti
for
(7.23),
the components behave
(see
radiating
the
Equations
systems.
expressions
in the
(7.12a-
Combining
in E q u a t i o n s
can be written:
r 3 contribution to f .
2sin2©^AQn>
+ 3U2 ctn20 ^ n>
+ 8 1U^ ctn^Ocsc 2O^A4n ^
(7.24a)
66
contribution to
- 3U2~1/2ctn@^A <n >
- 2“ 5/2u3 c t n3
s c 20^ A^ n )
(7.2#)
.-3 contribution to
^Ag**
(7.24c)
+ 4~lu2 ctn2@csc2e^Aj*)
.-2 contribution to
— 2
2Uctn0csc20 ^ A^ ^
r ■*" contribution to
(7.2 4 d )
4*
2“ 1 c s c 20^ a | 1i)
In order
tensor
(7.24)
in
to argue
components
containing
Appendix
a £0)
that
one would
expressed
G.
(7.24)
the leading
in Equations
radiation,
can be
Equations
(7.2 4 e )
If
in the
onlyone
is used,
= -Smgd
te r m s
of the We y l
(7.24) represent
a space-time
like
form
term
to show
that Equations
of Equations
in
the
(G.l) -(G.5)
summations
in
from Appendix F one has:
+ A2 U 2csc20 ) 2 /(8A7D 5)
(7.25 a)
A2 0> = ”y < 2 A 3 U3 )
(7.25b)
a |0)
(7.25c)
= 3m0 sin4©/(2A3U 5)
and putting these expressions
r 3 contribution to
(3V
in Equations
(7.24)
gives
.
0*
2A3 U)[-sin2©(l + A2 U2 CSC2G)2 / (2A4 U4 )
+ Ctn2G (I + cos2© / 8)
(7.26a)
67
r- ^ contribution to
— ^m QCtn@(l + c
.
o
)/(2^^
^
)
(7.26b)
r- ^ contribution to *P ;
m 0 (4 + 3 c o s 2®)/<8A3U3 )
(7.26 c )
r- ^ contribution to *P .
3'
—
s in©co s©/ (2^ ^ ^ A3
)
(7.26id)
r ^ contribution to *f .
4'
Sm0 sin2© / (4A3 D5 )
Comparing
Equations
(7.26)
(7.26e)
with
the
expressions
in Appendix
6 shows that it is not po s s i b l e to select a set of m u l t i p o l e
moments
(a^ in Appendix 6), which will yield the expressions
in Equations
(7.26).
this last issue.
Chapter
9 will
give
more
discussion
on
68
CHAPTER 8
ACCELERATED ROTATING SOLUTIONS
C h a p t e r 7 p e r f o r m s a r a d i a t i o n an al y s i s of the .spacetime
produced
when
the
transformation H
acts
on a static
sp ac e- ti me . This c h ap t er p o in ts out some d i f f e r e n c e s that
result
when
the
transformation
^ acts
on
a stationary
space-time.
As
mentioned
following
Equations
(1.19)-(1.21) ,
stationary space-times are described by two metric
.
u , and (i>. Inste ad of the static case
is
only
one
stationary
equations
need
and
(Equations
could
be
(3.9)).
general
metric
Chapter
by
3
an
how
new
acceleration
the price being
function
(Equation
Chapter
(1.19)),
3 showed
adding
f u n c t i o n appears
of
(Equa ti on
the s o l u t i o n to t wo field
function —
an a u x i l i a r y
solution
result
(w = 0) in w h i c h there
solve
requ ir e
ge n e r a t e d
A similar
static
key
for
to
(1.19) - (1.20.
to an initial
to solve
A
equation
space-times
s o lu ti on s
factor
field
functions
'
the
p (Equa ti on (3.5)
when
acts on a
(6.33)).
was
the
fa c t
that
the
a u x i l i a r y fu n c t i o n p can be w r i t t e n as a P o i s s o n integral.
Since
the
static
metric
function u obeys Laplace's equation.
69
it
can
be
written
symmetric
integral
the
the Poisson integral
over
an
function or. The auxiliary
as a similar Poisson
However,
a’ =
a Poisson
distribution
can be written
(3.17).
as
distribution
integral
as
axial Iy
function P
in Equation
function which
appears
in
of P is a modified distribution function
r~^zCT(p,z)
(8.1)
z = z + 1 / (2 A)
r 2 = p 2 + z2 .
This
is a pleasing
means
new
that
result for a couple of reasons.
in 'adding'
solutions
an a c c e l e r a t i o n factor
(Chapter
3) or
in letting
s o lu ti on (Chapter 6), the new
direct
manner
C ha pt er s
3
acceleration,
the
g ra vi ty
one
are
to
on a static
s o l u t i o n depends
in a fairly
giving
would expect
fields
acc el er a ti o n.
extension
6
to
Secondly,
the
ca se
to generate
act
on the ori ginal
and
First,, it
appear
solution.
a
If the m e t h o d s of
static
solution
the distributions
modified
when
a
linear
setting up
undergoing
E q u a t i o n (8.1) su gg es ts a po ss i b l e
of
acting
on
a
stationary
solution.
Ideally,
acts
on
wh at
one w o u l d like
a stationary
sp a c e - t i m e ,
to see
the
is that w h e n d
new
space-time
4s
is
determined by two auxiliary functions P^,
and P2 ,. which may,
in turn be w r i t t e n as P o i s s o n integ ra ls
over
two
separate
modified distribution functions. These modified distribution
70
functions would be related to distribution functions
CT2 » which determine
the original
Unfortunately,
an
such
field equations
c ou pl ed
write
over
set
alone
of n o n l i n e a r
sep arate
hope
separate
optimistic
that
new
The
It
is
impossible
to
in terms of Poisson integrals
distribution
the
tempered.
functions u and w are a
equations.
functions
and (0.
functions u»
is quickly
governing the metric
these original
two
metric
an<j
f u nc ti on s
s o lu ti on s
is
cr^ an(j c
governed
j et
by
two
modified mass densities.
The ac ti on of
co n f u s e d
by
the
^ on stationary
fact
that
the
new
so l u t i o n s
s o lu ti on s
is further
will
al ways
c o n t a i n a d e p e n d e n c e on two f u n c t i o n s T^ and T 2 as d e f in ed
in E q u a t i o n s
functional
(6.28a) and (6.28b). The f u n c t i o n Tj *s
form
of
the
an acceleration factor— an infinite
line
s i n g u l a r i t y at p = 0» z < - I /(2A) . The f u n c t i o n T^ j s also a
line
singularity
factor
compounds
at
P = 0,
the
trouble
z > - 1/(2 A ) . This
in analyzing
add it io na l
the new
solutions
generated by tfc ,
4s
Nonetheless,
stationary
the
solutions.
transformation
If the
can
initial
cr ea ted by r o t at in g sources,
be
applied
space-time
to
is being
^ should give the re su l t i n g
space-time when the sources are in uniform acceleration with
a c c eIe r at io n
represents
a ng ul ar
the
magnitude
s.
spacb-time
momentum.
The
Th e
Kerr
ex te r n a l
result
of
solution
to a po i n t
Q
acting
( 1 96 3 )
mass
with
on
Kerr
a
71
s o l u t i o n is p r e s e n t e d in A p p e n d i x H (Equations (H.3a,b,c)) .
The
of
parameter
rotation
q in those
in the original
no angular mom en tu m
= I re p r e s e n t s
th is
expressions
Kerr*.
Kerr
the
amount
solution— q = 0 represents
(Schwarzchild
'ex tre me
accelerated
Kerr
represents
solution
Any
solution
(1916)),
radiation
(as
in
while
analysis
Chapter
7)
q
of
is
i m m e d i a t e l y h i n d e r e d by the fact that the n ew so lu t i o n is
convenient Iy expressed
and
in prolate
However,
shift
in both parabolic
spheroidal
it
surface
0),
(T^.Tg)
(x» y) coordinates.
is possible
(g ^ ^ =
coordinates
to determine
for
the
the
infinite red-
so lu t i o n
in
Equation
(H.3 a) , in the limi t of small r o t a t i o n (q < < I). The e q u a t i o n
for this surface
x = l +
wh er e
q ^ [(1-y^)/2(1+2smy)^ ]
(x ,y ) are
canonical
(p,z)
prol ate
space
spheroidal
z = m(l-2sm)~*
p=0,
z=
7
value
in
Weyl
-m(l+2sm)~*
illustrates
a finite
coordinates
centered on the two points
p = 0,
Figure
for
is
this
of
red— shift
s. Notice
surface
how
the
for
s = 0,
surface
and
changes
shape for di f f e r e n t value s of the p a r a m e t e r s: a p a r a m e t e r
which
is
directly
acceleration.
proportional
to
the
magnitude
of
the
72
Z
q = .1
sm=0
sm = .48
Figure 7: Infinite red-shift surface
on a Kerr solution (q{<I )
for Q 4 g acting
73
CHAPTER 9
SUMMARY AND CONCLUSIONS
In this wo r k we have a t t e m p t e d to d e t e r m i n e
the
solution
stationary,
generating
techniques
to
show,
symmetric
in
has the physical
symmetry
the
the
space-time,
symmetric
axis.
the
and
The
ori gi nal
time
with
(Chapters
result
is
This
acceleration
This factor appears
sol ut io ns
outlined
sophisticated
ph ys ic al
observer
of
(3),(4),(5),(6)).
factor
of
in
Weyl
appears
axia ll y
Cosgrove,
static,
is ah accelerated
H
t
axially
in
from
(Chapter
d ^ ^
In
the v i e w
(2))
0n
both
was
of a
comparing
flat
spacewhen
coor di na te s,
metric
of
and
cases,
canonical
the
version
interpretation
flat s p a c e — time
action
expressed
static,
it an a c c e l e r a t i o n along the
metric
metric.
the
of
transformation
giving
new
accelerated
case
of taking an initial
s u r m i s e d by e x a m i n i n g
this
new
interpretation. The attempt has
sp ec if ic
effect
metric
uniformly
creating
axially symmetric space— times has an interesting
and straightforward physical
been
for
if one of
(Equation
the
an
(3.7)).
in the simplistic approach to generating
in
treatments
Chapter
(3)
as
of Kinnersley,
well
as
the
more
Cos grove, and other
74
workers.
The
hypothesis
is
made
that
if
(in
We y l
canonical
.
space) an acceleration factor represents
an o b se rv er
in linear
factor plus any static
th a t
static
acceleration.
to
radiate
to see
the
solution
Since
accelerated
of
S
sp ace .
4s
static
of
rectilinear
system
Chapter
is
expected
7 makes
an effort
space-times
radiation
solution to which
acts),
(7.17). This
generated by
and
therefore
seed
If
of
space-times,
®4S
acting
and
the
solution
on
radiating
field
when
an acceleration factor
is the m u l t i p o l e
enough to cover a wide
linearity
uniform,
static
represent
the space— time
sources.
accelerating
space,
energy.
the new
added (or on w h i c h
general
in
such an accelerated
in fact,
action
Equation
when
for
then an a c c e l e r a t i o n
solution represents
gravitational
if,
The
acceleration,
flat space-time
this
of sources
solution
space-time,
equation
so lu t i o n in
is used b e c a u s e
range
(Equation
it
produces
(3.2))
is
in Weyl p— z
then
^ acts on mo r e general
is
from,
an
the
for
static
sources
in p-z
the result should be radiation. The c a l c u l a t i o n s
in
C ha pt er 7 are s i m p l i f i e d by a s s u m i n g the l in ea r limit:
in
both
Weyl
coordinates
(t,0, p, z)
and
the
radiation
c o o r d i n a t e s of N e w m a n and Unti (U,r,6,0) the a s s u m p t i o n is
that
the observer
A comparison
is far away from the. accelerating source.
of Eq u a t i o n s
(7.23)
and
(7.12)
su ggests
75
cause for initial
concerning
optimism
acceleration.
w i t h respect
The
Weyl
tensor
to the h y p o t h e s i s
components
of
the
ne w g e n e r a t e d so l u t i o n s e x h ib it correct a s y m p t o t i c radial
co o r d i n a t e
'peeling'
fields
behavior.
and
of
new
behavior
a n a lo g o u s
to
electromagnetism
dependence
At
is
This
the
is
sometimes,
el ec tr ic
having
a
and
termed
magnetic
predictable
r
^
w h e n the sources of the field are acc el er at in g.
any rate,
static
there
is a certain
space-times
radial dependence as
gratification
produced by
space-times
at seeing
demonstrate
containing
the
the same
gravitational
radiation.
H o w ev er ,
to
accelerating
dependence
the
sources,
of
W h e n this
say
the
Weyl
is done,
new
the
space-times
other
tensor
those
coordinate
components
the a g r e e m e n t
are
must
be
of
(U,0-,(?)
examined.
between Equations
(7.23)
and wha t the d e p e n d e n c e shoul d be for r a d i a t i n g sources is
impossible
the
to find. The
(U,O,0)
coordinates)
calculated behavior (with respect to
of
the
Weyl
accelerated solutions produced by
the
predicted
or Janis
behavior
and N e w m a n
suggested
(1965).
tensor
components
demonstrate
some
singularities
as
of
very
the
the
assumed
does not coincide vith
by
This
Newma n
new
serious
demonstrated
and
disagreement
e x a m i n i n g A p p e n d i c e s F and G. As a m a t t e r
tensor
for
in
(1962)
is seen by
of fact the Weyl
accelerated
a n g ul ar
Unti
solutions
in f i n i t i e s
Equations
and
(7.26).
76
Therefore,
the
accelerated
initial
optimism
for
producing
solutions may have to be tempered.
DIFFICULTIES AND FUTURE INVESTIGATIONS
i) T h r o u g h o u t this work,
coordinates.
relativity
One
of
is. the
the
there
is a l w a y s the issue of
elegant
a b il i t y
to
coordinate
free fashion.
Yet,
different
s p a c e — times,
a set
aspects
formulate
singularity
showed how
as
in We yl
of c o o r d i n a t e s
a finite
rod
laws
canon ic al
a
aspects of
results.
coordinates
in Weyl
Ch a p t e r 7 issue b e c o m e s
in
is ne eded
and
of a Schwarzchild particle
source
general
to
Chapter 2
can ap pear as in infinite
canon ic al
the metric
its
to examine physical
perform calculations and express physical
s h o w e d how flat s p a c e - t i m e
of
line
Chap te r
4
can appear
coordinates.
i m p o r t a n t and more difficult.
In
The
problem boils down to having to abandon Weyl p— z coordinates
in
favor
of
coordinates
in
radiation
are
stationary,
coordinates
useful
from
accelerating
the
relation
(Equations
in
axially
are useful
coordinates
(U,r,G,@).
simplifying
symmetric
between
The
these
algebra
space— time.
in considering
sources.
the
Weyl
involved
radiation
the possible radiation
argument
two
(7.8),(7.9), and (7.15))
The
The
for
sets
the validity
of
of
coordinates
is not as c o m p e l l i n g
as
it mi g h t be. For one thing, the 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
assume
used
a flat
to
space-time
des cr ib e
background
accelerating
and yet
with
no
the results
constraint
are
on the
77
magnitude
of the acceleration.
i i) The Weyl tensor components demonstrate very awkward
behavior
in the limit
’A') going
the
One direction for future
a more
Weyl
convincing
coordinates
expressions
equations
at
vs',
or
boil
down
great
between
and the radiation coordinates.
The key
to
the
for
investigation would be to
connection
made
assuming
from
the
appea r
in
that
accelerated
distance
the
this p r o b l e m
assumptions
accelerating
a
argument
summarizing
(7.2 0 a ,b ) . The
the
acceleration magnitude
to zero.
iii)
make
if the
in E q u a t i o n s
arriving
an
at
observer
source
is
source.
watching
in
The
these
some
way
coordinate
t r a n s f o r m a t i o n s are base d on the a s s u m p t i o n that w h e n the
radiation
observer
coordinate
is
also
r. goes
a great
certainly valid when the
and
the
spatial
accelerations
region.
in finity
di st a n c e
sources
are
However,
to
away.
Such
of radiation
confined
the
this
to
some
accelerations
means
the
thinking
are
localized
well
here
is
defined
are
uniform
and linear and are by no me a n s c o n f i n e d to a s m a l l regio n of
space-time.
hyperbola
source
The sources here
in s p a c e — time.
which
then
Th er ef or e,
The
deaccelerates
i n f i n i t y (along
an d
follow
the
symmetry
accelerates
one
mi g h t
back
expect
a world— line
spatial
as
it
comes
axis),
out
a
motion
stops
to
mo r e
which
is a
is that
of a
in
from
mi n u s
for an instant,
spatial
strict
infinity.
criterion
is
78
needed to define what
is meant by an asymptotically distant
observer with respect to such a linearly accelerated source.
iv) Even when such a good transformation of coordinates
is developed,
show
the
there will remain the problem
space-time has a multipole
structure
of trying to
as suggested
by Janis and N e w m a n (see Appendix G). This would amount to
showing
a relationship between
the expansion constants
and a^ which appear in Appendices F and G.
v)
Initially,
physical
effect
symmetric
exact
this
of
work
began
^ acting
solution.
The
on
by
considering
a stationary,
stationary
case
the
axially
requires
the
consideration of an additional metric function a>. As Chapter
6 points out, when
of
two
mirror
solutions
^ acts on stationary space — times, a set
acceleration
(see
Equations
factors
(6.25a,b)).
appear
The
in
problems
the
new
concerned
with a double set of line singularities are quickly seen.
However,
stationary,
axially
symmetric
space — times, include!
rotating sources such as the Kerr solution. The action of
® 4 S on
such
possibility
solutions
of
cou l d
producing
an
produce
the
accelerated
tantalizing
and
spinning
gravitational source. The action of
^ acting on the Kerr
solution
8
is
Nonetheless,
included
in
Chapter
and
Appendix
G.
the algebraic difficulties and problems with
interpretations
of the. coordinates
not be underestimated.
for
theses
cases
should
v) In all the solution generation schemes,
forms of new
different
solutions
limits.
field limits,
factors,
should be
Particular
large acceleration factors,
slides
values
parameter
of
the
represent
source,
some
of
the
particles
It
in
the
the
If
the
z— axis
should
may
^ af fe cts
to what
's'
is
imparted
is
going
to
to the
on when
around.
test
of
be
the
for different
parameter
of the acceleration
trajectories
of a s p a c e - t i m e
point
test
interesting
geodesics
of
is the
particles
to
both
see
how
finite
or
the
mass
and photons.
be better
aid
displaying
's'.
slides
vii) The effect
an
acceleration
this work has glossed over
down
in t e r e s t i n g
motions.
of
and
consideration
v i) A n o t h e r
g eo de si c
up
the magnitude
this preferred point
as
small
on weak
of the p r e f e r r e d point in the a c c e l e r a t i o n factor
(p = 0 ,z=— I / (2s))
perhaps
should be placed
in
and the b e h a v i o r of the s o l u t i o n for p oi nt s along
the fact
ac ti on
calculated and examined
care
the symmetry axis. In particular,
shape
the explicit
in
of
^ o n d i f f e r e n t s p a c e - t i m e s could
studied
the
through
tedious
singularities,
newly
generated
computer assistance
of expertise
in this
the use
algebra
red-shift
as
area.
well
surfaces,
space-times.
in this work
of computer:
Part
of
both
as
visually
and
geodesics
the
lack
of
is due to the author’s lack
REFERENCES
81
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C.W.,
T h o r n e , K.S.,
Gravitation. San Francisco:
and W h e e l e r ,
J. A. 1 9 7 3 .
W.H. Freeman and Company.
Newman,
E.T.,
and P e n r o s e ,
R . 1962.
An a p p r o a c h to
gravitational
radiation
by
a method
of s p i n
c oe ff i ci en t s. J. Math. Phy s. 3;, 566.
Newman,
E.T.,
and Unti,
T.W.J.
1962.
Behavior
of
a s y m p t o t i c a l l y flat e m p t y spaces.
J. Math. Phy s.
3, 891-901.
Ohanian, H.C. 1976. G r a v i t a t i o n and
W.W. Norton and Company Inc.
s p a ce ti me .
New
York:
S c h w a r z c h i l d, K. 1 9 1 6. Uber das G r a v i t a t i o n s f e l d eines
M a s s e n p u n k t e s nach der E i n s t e i n s c h e n The o r ie. Sitz.
Pre u s s . Akad. Wiss. p.I 8 9 .
Voorhees, B.H. 1970. Static axially symmetric
fields. P h y s. Rev. D 2, 2119-2122.
Z i p o y , . D.M. 1966. T o p o l o g y of some
Math. Phy s. 7, 1137-1143.*
gravitational
s p h er oi da l
metrics.. J.
84
o
In the above references, the following sources,
Cosgrove (1980.)
Mi s n e r , Thorne,
and Wheeler (1973)
Kramer, Stephani, MacCallum, and Herlt (1980)
co n t a i n e x c el l en t and very c o m p r e h e n s i v e b i b l i o g r a p h i e s
concerning exact solutions, solution generation techniques,
and original papers in the field of general relativity.
85
APPENDICES
86
APPENDIX A
THE TETRAD FORMALISM
87
The
tetrad
places.
formalism
Treatments
is
which
in t r o d u c e d
are
in
a number
particularly
useful
of
are
C h a n d r a s e k h a r (1979), Mi s n e r et.al. (1973, C h a pt er 14), and
Kramer
(19 80,
Chapter
I). Four — d i m e n s i o n a l
space — time
is
u s u a l l y d e s c r i b e d by co or d i n a t e s , x 1, and a m e t r i c tensor,
g ij.(xk ). At each point
is
selected.
The
of space-time
contravariant
a basis
components
of four vectors
of
these
vectors
are written
(a )
The
index
indicates
(a , i
in
(A.I)
parenthesis
which
of
The
index
considered.
tensor
I , 2 , 3 ,4 )
the
is
called
four
without
a
tetrad
te trad
vectors
parenthesis
is
index
is
being
called
index and i n d ic at e s the p a r t i c u l a r c o m p o n e n t
tetrad
vector
differential
with
respect
g e o me tr y ,
the
to
the
tetra d
coordinates
vectors
and
the
of a
x *.
In
are v i e w e d
as
directional derivatives and are written
e , . = e, .iO Z d x i)
~(a)
(a)
Tensor
(A.2)
indices are ra ise d and l o w e r e d
with the metric
tensor g
in the usual m a n n e r
and its inverse g lj
(A. 3)
(a) i - ® ij 6 (a)
(a)
The
S iJe
(A. 4)
(a )j
c o n t r a v a r iant
co m p o n e n t s ,
matrix with an inverse,
e
(b )
(a)
(a)
J
(a)
e (a ) ' can
be
viewed
as
a
s0 that
(A .5)
(A.6)
88
The p o w e r in the tetrad f o r m a l i s m
the tetrad vector components
come s i n
selecting
to obey
(A.7)
C (a)i e (b)i = M a ) ( b )
where
is a constant
symmetric
The d e f i n i t i o n of n
raise and lower tetrad
The
quantities
signature
(a ) (b ) mea ns it can be used to
indices
(b) i
n
(b)i = e
je
n (a)(b)e
® (a )
(b)je
i
"
matrix with
e (,>j 8 j
(A.8)
e(a)
q . ., . a n d
are
u s e d t o raise
and
lower tetrad indices in the same manner as the metric tensor
g^
and its inverse,
indices .
Tensor
g*^>
are used to raise and lower tensor
,
components
are projected
on to the
tetrad basis
to get the tetrad components
A
(a)(b)
_
Tetrad vectors
®(a)
1e
commutation
Ricci
rotation
JA
(A.9)
ij
in general have non-zero commutators
Te
, _
~(a)'£(b)] - C
The
(b)
(A. 1 0)
(a)(b)&(c)'
coefficients,
coefficients,
^(a)(b)(c)
f.
are
related
to
the
x
m
a
r (a)(b)(c) " e (a) e (b) m :n® (c )
(A.11)
by the formula
2crU H b M c ) ^
= C (b )( a )(c ) + C (c)(a)(b)
- C (a)(b)(c)
(A.12)
89
APPENDIX
B
ORTHONORMAL TETRAD FOR STATIONARY
AXIALLY SYMMTERIC SPACE-TIMES
90
Stationary,
by
the
time
four
axially symmetric space-times are described
coordinates
(x* = t,x^ = 0,
) and
by
the
space-
interval:
(ds)^
=
e^u (dt-6)d0) ^ - e ^ ® - u ^d0^
- e2( Q- u) [dz32 + dx42].
An o r t h o r n o r m a l
(B. I)
tetrad (Misner et.al. 1973, p.354) is a set.
of four v e c t o r s such that the quantity,
in Eq.(A.7)
has the form:
tlU H b )
= + 1 0 0 0
0 - 1 0 0
An o r t h o n o r m a l
the
metric
• m 1
° m l
= (S)1
6 (4)1
0
0 - 1 0
0
0
0 -1
(B.2)
te tra d w i t h r e s pe ct to the c o o r d i n a t e s and
defined
by Eq.(B.l)
is:
(B.3 a)
=
=
r u 6 I1
O-s t l U e 1 U e 2 1I.-
(B.3b)
=
e— SH u5 ^i
(B. 3 c )
=
e-SH-u5 i
4
(B.3d)
W h e n the
components
of the E i n s t e i n tensor,
proj ected o n to the o r t h o n o r m a l
are
teteraj of Eq s.(B c3) using
Eq, (A.9), the n o n - z e r o c o m p o n e n t s of G(a )(|,) can be w r i t t e n
in the form:
.e2 ( 0 - u ) [G(2)
_ G <1>
(2 /
j = _ e" BV[e2nV e B" 2 u ]
(I)
+ e-2B+4uV())V(|)
2 e 2$2[G(2)(i)]
= V [e"B+4uV<o]
(B .4)
(B. 5)
91
_ e2 (il-u) [q (3 )
+ G
_ e2 (£i-u) [q (2)
+
BVVe B
(4)
(4)
„(1)
G
(i)]
-
(B.6)
2-1 C 20+4uVtoVto
+ e~ BVVeB
+2 VuVu-B — 2 VV Ti
(B .7)
+2 VVQ
- 2( n- ) le(3,m
- = U , (4)
2
- I _-2B+4u [(<o ) -(w
)2 )]
e
+ b 44+ <b 4 )2_B33- ( B 3 )
+2[B3Q3-B4 Q4
(B.8)
- ( U 3 ) 2 H-(U4 ) 2 ]
_ e ($2-n)
-E34-B3 S4 - 2U 3U4 +2
(4)
— I ■— 2 B+4 u,
3“4
(B.9)
+Q 3 B 4 +Q 4 B 3
For
convenience,
dimensional
VMVN
gradient
V,
is
used
as
a
operator:
I3 N3 + M 4 N 4
VVM = M 33 + M 44
a symbol
(B.lOa)
(BolOb)
92
APPENDIX C
NULL TETRADS
AND THE NEW MAN-PENROSE FORMALISM
93
A null
tetrad
quantity, ^ ( a )(b)<
1 ( a ) (b) =
0
of
four v e c to rs
such that
1
0
0
0
0
0
0 -1
0 0 - 1 0
The
Ricci
and
for
Penrose
rotation
a
null
the
*n E q 0(A .7) has the form:
1 0
0
is a set
(C.I)
coefficients
tetrad
are
are
gi v e n
defined
the
as
names
in Eq.(A.ll)
(Newman
and
1 962):
= k
r o ) (i) (4)
r (3) (I) (3) = a
T (2)(4)(3)
r (3)(I)(I)
r (2)(4)(4)
T (2)(4)(2)
= X
=
V
T ( S ) ( I ) (2)
T (2)(4)(I)
(1/2)<r(2)(I)(l)/r(3)(4)(1)) = c
(1/2>(r(2)(I)(2)+r (3)(4)(2)) = Y
U / 2 ) (,"(2) (I) (4)+r (3) (4) (4)) = °
(1/2)(r(2)(I)(S)+T ( S ) (4)(3) > = P
(C.2)
94
The
five
in d e p e n d e n t
components
tens or are (Newman and Penrose
of the v a c u u m
1962):
(C .3 a )
\
= " R (l>(3)(1)(3)
a)
1= " R (l)(2)(I)(3)
(j)
2 = - R (1)(3)(4)(2)
where
space Weyl
(C.3b)
(C .3 c )
V - w
3 - - R (I ) (2)(4)(2)
(C .3d)
V _ n
4 - - R (2)(4)(2)(4)
(C .3 e )
R(a)(b) (c) (d) ar®
curvature
tensor .
tetrad
components
of
the Riemann
95
APPENDIX
D
NULL TETRAD FOR STATIONARY
AXIALLY SYMMETRIC SPACE-TIMES
96
A null tetrad w i t h respect
metric
defined, by
Eq.(B.l)
(2)1/2
•
<
i
>
1
"
(2)1/2
to the c o o r d i n a t e s and the
is:
(e-u + <i>e""B+ u)5 1 +
I
(e-B+*)6 I
2
(D.l a)
(e™u -toe~B+u)61 i +
(-e~B+u)82 i
(D.lb)
(2)1 Z 2
e ~ fi+ u (83 i+ i64 i) = 2
i = e-Q + n (8 i_.8 i)
(2)1 Z 2 C
3
4
(4)
Fo r
th is
null
tetrad,
(E qn (D .2) -Eqn(D.4)
n o n - z ero
the
(D. lc )
(D.ld)
sp i n
coefficients
have the form:
T
=
Jt =
where
(D.2 a)
tB»
Kj
I
H
=
K
V
I
k = (X - Y )~ 1 m (x )
(D.2b)
(D.2c)
-2 1^(B)
*
-T
(D.2d)
a
2 " 2 (£ + v ) - 2 (- 3 Z 2 ) e - 2( n" u ) [(e£1~u )3_ i(e£2- u )4 ] (D.2e)
P
2 " 2 (k + v)
+ 2 (" 3Z 2 ) e7'2(£2~ u ) [(efi™ u )3 + i(eQ-u )4 ] (D.2 f)
the auxiliary functions:
X = to + eB~ 2u
(D.3)
Y _ to - eB-2u
(D.4)
have been n s e d .
component s of the Weyl tensor are:
vO
rI
Il
I
m
P
The
+ 2k t . + k(3p + u^
(D.5 a)
= 0
(D.5b)
= rr + kv
(D.5 c)
T3 = 0
O
= %(v)
(D.5 d )
- 2vr + v(3o+^)
In the case of static,
axially symmetric
(D.5 e)
space-times
97
(I) = 0
the
(D.6)
spin coefficients
k
= 2™ 1!!!(x )
V
=
T
=
rednce
to:
(D.7 a)
- t
(D.7b)
m
BI
I
s—4
I
n
(D.7c )
c
It
(D.7d)
-T
a
(D.7 e)
-a
P
(D.7 f)
where:
X
-
The Weyl
0
i
2
)
4
(D.8)
eB-2u
tensor components
rednce
to:
- Q (k ) + 2 k (t - o ^
(D.9 a)
TT -
(D.9b)
-(^o>
3
(D.9c)
(D.9d)
98
APPENDIX E
PROLATE SPHEROIDAL COORDINATES
99
Prolate
cylindrical
sphereidaL coordinates
coordinates
(p»z) by
(z ,y) are related to the
the relations
(E.l)
= k(x2- l ) l / 2 (l-y2 )^/2
P
(E.2)
z = kxy+c
where
(p,z) plane
preferred
s e pa ra te d
by
(E.3)
(E.4)
+
r+ and r_ are
in the
The
(2k)-l(
+ r+ )
H
y =
M
I
x = (2k)-1 ( r_
I
Geometrically,
from
the distances
an arbitrary point P
to two preferred points
points
are
a dis ta nc e
for
centered
2k.
F ig ur e
y.
point
8 shows
From
the
the
Z = C
curves
(Equations
(E.3) and <E„4)), the range of the x and y
relations
I < x < 00
»
The square
of the di st a n c e b e t w e e n two p oi nt s
(p,z) plane
< y < +1 .
with an infintesimal
and
constant
coordinate
-I
x and
on the
along the z-axis.
above
(E.5)
in the
separation is
(dp)2 + (dz)2 =
k 2 (x2-y2) [(dx)2 / (x2-!)
+ (dy) 2 / (1 - y 2 ) I .
The relations between differential
operators are
9
= z 2-l)1/ 2 (l-y2 )1/2/(k(x2-y2 )) [ zd
- y3 I,
p
x
x
.y
d z = l/(k(x2- y 2))r (X2- D y d x + (l-y2)xay] .
These
allow
be expressed
equations
in Weyl canonical
in prolate
spheroidal
(E.6)
(E.7)
■
(E,8)
(p,z) coordinates to
coordinates.
100
2k
z
MZ
con stant
x - coordinate
constant
y - c o o r d in a te
Figure 8
Constant coordinate curves for
prolate spheroidal coordinates
101
APPENDIX F
EXPANSION COEFFICIENTS IN
EQUATIONS (7„21a)-(7.21c)
102
The
E q u a tions
expansion
coeficcients
(7.21a— c) are
in
the
summations
of
(to order n =2 ),
A (0, = -3m0 [i+ (AUcsc6)2 ]2 [8A7U5 ]-1
2
= -3
^ +(
sc © ) 2 1
(F .I )
4 1J
x [2(AU) 2-sin2© ] [SA8U 7 ]™ 1
(F.2)
= -Sm2 SiB2O[! + (AUcscO)2 1
x [3sin2©-4(AU)2 ] [IdA9U9 ]-1
a
(F .3 )
(F .4 )
'»> = = V 2a3u31" 1
A^n
= m 1 [2(AU)2 - sih20][2A4 U 5 ]-1
(F .5)
4 2’ = -Sja^ sin2© [-5 s in20+4 (AU) 2 ]
(F.6)
>
O
x [4A 5U 7 ]-1
= 3m()sin4©[2A3U 5]"1
Ad)
4
= 3 m iSin4©[2(AU)2+5sin2 ©]
(F.7)
x [2A4 U7 ]_1
a !2’
(F.8)
= 3ni2sin6©[35sin2©-20(AU)2 ]
x [SA5U 9 ]"1
The
symbols
and m 2 are
summation of Equation
(7.17).
(F .9 )
the multipole
moments
in the
APPENDIX G
ASYMPTOTIC STRUCTURE OF
GRAVITATIONAL FIELDS IN TERMS OF
LINEARIZED MULTIPOLES
104
In an a t t e m p t to def ine a m u l t i p o l e s t r u c t u r e for the
sources of a g r a v i t a t i o n a l
field,
Janis and N e w m a n
have t h e
show the Weyl tensor components,
%
=
H a
(1965)
general
form:
2 S i n 2 6 I *"5
+ terms of order r ^
(G.l)
= U 1 S i n e - S m iz2a2 1 * s i n 6 c o s 6 1 r ~ 4
+ terms
of order
r"5
(G.2)
^ 2 = [ao_ ( 2 )1^2aj1 ^cosG+a^2 ^ (Scos2O l ) ]
+ terms of order r"^
(G.3)
Yg = [(2)lZ2a2 3 *sin6c os6I*- 2
+ terms of order r
(G .4)
Y. = [2-lal4)sin2e]r-1
4
. 2
+ terms of order r~2
These
expressions
are
(G .5)
written
in terms
of
the
coordinates used in Chapter 7 and with respect
d ef in ed
their
in E q u a t i o n s
work
to
proportional
of
the
represent
to a^« The
coordinate
superscript
U.
The
symbol
When
an
is
symbol
2 N-pole
a
a^ *8»
a the
(n), this represents
°f ajj with respect
function
(7.11).
symbol
a^
(U,r,,O,0)
to the tetrad
*8 defined
whose
general,
moment
is
a function
a^ appears
the n-th partial
in
with
a
derivative
to the coordinate D. The real part of the
id e n t i f i e d
as
an
'electric?
type
of
pole
c o r r e s p o n d i n g to m a s s e s at rest. The i m a g i n a r y part of the
function
a jj i8
corresponding
to
identified
as
a 'magnetic'
time-independent
mass
type
'currents'.
of
In
pole
their
105
analysis the following physical
assumptions
are us e d :
First,
p h y s i c a l Iy
'magnetic'
states
type
the
conservation
monopoles.
The
of
'electric'
and
condition:
a<2> - 0
physically
states
there
is
the
absence
of
gravitational
dipole radiation. Bo th these c o n d i t i o n s
came
detailed
gravity
mathematical
analysis
of
the
p a r t i c u l a r by c o n s i d e r i n g the initial
about
in the
field:
in
data on the surface
U=Const ant .
The
main
point
should be possible
about
the
to write
the
above
form ul J
expressions
is
derived
that
for
it
the
f's of an a c c e l e r a t e d object (E qu ations (7.23)) in the form
of the e x p r e s s i o n s ap p e a r i n g in E q u a t i o n s (G. I) ~ (G. 5 ). This
would make
it possible to examine the multipole
the r a d i a t i o n from the a c c e l e r a t e d solut io ns
S
4 s*
structure of
ge ne r a t e d by
106
APPENDIX H
TRANSFORMATION OF THE KERR SOLUTION
107
In the p a r a m e t r i z a t i o n of L e w i s , ( E q u a t i o n (1.9)) the
metric
fu n c t i o n s
for
the Kerr
so lu ti on
(1963)
are
of the
for m,
e2u
= ( ( p X )2 + (q
(a =
2mq (1-y^ )(px + l ) ( (pz )2 + (qy)2 - l)- ^,
(H.lb)
e2Q
= [(px)2 +(qy)2 - 1 ] [p- 2 (x2- y 2 )I~ 1 ,
(H.lc)
with x-y
Weyl
y
)2 -l)((pX + l)2 + (q
being prolate
spheroidal
y
)2)
- I f
(H.I a )
coordinates related to the
coordinates by,
o = mp,
(H.2 a )
p = o((x2- l ) ( l - y 2 ))1 / 2 ,
z= o x y ,
q2 =1 — p 2 .
(H.2b)
(H .2 c )
W h e n the t r a n s f o r m a t i o n
Kerr solution metric
^ 0 f Co sgrove (1979) act s oh the
functions,
the results are,
Q a (e2 u ) = (1-4 s 2 o 2)x
4s'
[[(pX+'S) (px-l)+q2Yy] (T1 )2
- q 2 [(pX+'S)y-(p3E-l)Y] (T2 )2 Ix
[(pX)2 + (qY)2-(‘S)2 ]-1 [(pX+'S)2 +(qY)2 ]"1 ,
(H. 3 a )
«4,(.)
= qs- 1 <l-4s2,2)-l«
[-[(pX+'S) (p x- l) +q2 Yy ] [(p2+l)Y-(pX+3)y] ( T x 2
I
+ [ (pX+'S) y- (px-l)Y) [q2yY+ (pX+'S) (px + 1 ) ] (T^)2 1 x
[[ (pX+'S) (px-1) +Q2Y yl 2 (Ti )2
-q2 [(pX+‘S)y - (PX-I)Yl2 (T2 )2 I"1 ,
(H.3b)
Q4 s(e2(fi_u)) = (I-.4 s2 o2 ) E (pX+15) 2 + (qY)2 ]x
p " 2 ('S)-1 [x
2- y 2 ]-1 j
(B .3 c )
108
where,
X = x+2soy,
Y = y + 2 scrx,
(H.4 a »b )
CS)2 = I + 4s<rxy + 4 s 2 o 2 (x2 + y 2 -l),
with
(H.4c)
2(T^)2 = I + 2 scrxy + “S,
(H.4d)
2(T^)2 = -I - 2 axy + 3,
(H.4e)
x and
y being
a set
of
prolate
spheroidal
coordinates
centered on the two points.
P =
0,
z = +o (I-2 s<y)- ^.
(H .5 a )
P =
o,
Z
— a (1+2 so)
(H.5b)
=
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