Locally Inertial Reference Frames in Lorentzian
and Riemann-Cartan Spacetimes
(2)
J. F. T. Giglio(1) and W. A. Rodrigues Jr.(2)
(1)
FEA-CEUNSP. 13320-902 - Salto, SP Brazil.
Institute of Mathematics Statistics and Scienti…c Computation
IMECC-UNICAMP
13083950 Campinas, SP Brazil
walrod@ime.unicamp.br or walrod@mpc.com.br
February 8, 2012
Abstract
In this paper the concept of locally inertial reference frames (LIRFs)
in Lorentzian and Riemann-Cartan spacetime structures is scrutinized. A
rigorous mathematical de…nition of a LIRF in both structures is given,
something that needs preliminary a clear mathematical distinction between the concepts of observers, reference frames, naturally adapted coordinate functions to a given reference frame and which properties may
characterize an inertial reference frame (if any) in the Lorentzian and
Riemann-Cartan structures. Hopefully, the paper clari…es some obscure
issues associated to the concept of a LIRF appearing in the literature, in
particular the relationship between LIRFs in Lorentzian and RiemannCartan spacetimes and Einstein’s most happy thought, i.e., the equivalence principle.
1
Introduction
In this note we investigate if it is possible to have in a general Riemann-Cartan
spacetime a locally inertial reference frame in an analogous sense in which this
concept is de…ned in a Lorentzian spacetime that models possible gravitational
…elds in General Relativity.
To answer the above question which is a¢ rmative in a well de…ned sense we
are going to recall the precise de…nitions of the following fundamental concepts:
(i) de…nition of a general reference frame in Lorentzian and Riemann-Cartan
spacetimes;
(ii) de…nition of observers in a Lorentzian or Riemannian spacetime;
1
(iii) classi…cation of reference frames in Lorentzian spacetimes1 ;
(iv) de…nition of an inertial reference frame (IRF) in Minkowski spacetime;
(v) de…nition of a locally inertial reference frame (LIRF) in Lorentzian and
Riemann-Cartan spacetimes.
However, in order for it to be possible to present precise de…nitions of the
concepts just mentioned we need to recall some basic facts of di¤erential geometry and …x some notation. This will be done in Section 2.
2
Lorentzian and Riemann-Cartan Spacetimes
To start we introduce a Lorentzian manifold as a pair hM; gi where M is a
4-dimensional manifold and g 2 sec T20 M is a Lorentz metric of signature (1; 3).
V4
We suppose that hM; gi is orientable by a global volume form g 2 sec T M
and also time orientable by an equivalence relation here denoted ". We next
introduce on M two metric compatible connections, namely D the Levi-Civita
connection of g and D a general Riemann-Cartan connection.
De…nition 1 We call the pentuple hM; g; D; g ; "i a Lorentzian spacetime and
the pentuple hM; g; D; g ; "i a Riemann-Cartan spacetime.
m
Remark 2 Minkowski spacetime structure is denoted by hM ' R4 ; ; D;
; "i.
Let U; V; W M with U \ V \ W 6= ? and introduce the local charts ('; U )
and ( ; V ) and ( ; W ) with coordinate functions h i, hx i,hx0 i respectively.
Recall to …x notation that, e.g., given p 2 M and V R4 we have : V ! V;
(p) = (x0 (p); x1 (p); x2 (p); x3 (p)): The coordinate chart
determines a socalled coordinate basis for T V denoted by he = @=@x i. We denoted by h# =
dx i a basis for T V dual to fe = @=@x g, this meaning that # (e ) = .
We also write
g=g #
# =g
#
# ; g
:= g(e ; e ); g
:= g(# ; # );
were we denoted by he i and h# i the reciprocal bases of he i and h# i, i.e.,
we have g(e ; e ) = ; g(# ; # ) = . Moreover, we denote by g = g e
e =g e
e 2 sec T02 M the metric on the cotangent bundle and of course
g g = .
A curve in M is a mapping c : R I ! M;
7! c( ). As usual the tangent
vector …eld to the curve c is denoted by c or by as more convenient. In the
coordinate basis he = @=@x i we write c = dd v ( ) @=@x jc( ) . In particular
we write when c(0) = po , c j =o = v e jpo 2 Tpo M .
The metric structure permits one to classify curves as timelike (g(c ; c ) >
0; 8 2 I)) or spacelike (g(c ; c ) < 0; 8 2 I)) or lightlike (g(c ; c ) = 0; 8 2
I)).
1 The classi…cation of reference frames will not be presented in this paper.
Lorentzian spacetime case, see [18].
2
For the
For timelike curve c : 7! c( ), such that g(c ; c ) = 1 the parameter is
called the proper time.
Given U; V
M and coordinate functions h i; hx i covering U and V for
the structure hM; g; D; g ; "i and coordinate functions h& i; hx i covering U
and V for the structure hM; g; D; g ; "i we …x here the following notation
De # :=
# ;
D@=@& d& :=
d& ;
De # :=
D@=@ d
De e :=
D@=@& @=@& :=
# ;
:=
d
e ;
De e :=
;
@=@& ;
e ;
D@=@ @=@
=
@=@
:
(1)
For the connection coe¢ cients in coordinate basis h@=@x0 i; hdx0 i we use
. Finally for an arbitrary basis he i for T (U \ V \ W ) and dual basis h i
for T (U \ V \ W ) such that [e ; e ] = c e we write
0
2.1
De
:=
; De e :=
e ;
De
:=
; De e :=
e :
(2)
Torsion and Curvature Tensors
V1
Let u; v; w 2 sec T M and 2 sec
T M . We recall that the torsion and curV1
vature tensors of an arbitrary connection r are the mappings T : sec( T M
V
1
TM
T M ) ! C 1 (M; R) and R : sec(T M
T M
TM
TM) !
1
1
1
C (M; R) (C (M; R), the set of C real functions on M ) given by T ( ; u; v) =
( (u; v)) and R(w; ; u; v) = ( (u; v)w), where : sec(T M T M ) ! sec T M
(torsion operator) and : sec(T M T M ) ! End(sec T M ) (curvature operator)
are given by (u; v) = ru v rv u [u; v] and (u; v) = ru rv rv ru r[u;v] .
In an arbitrary basis he i for T (U \V \W ) and dual basis h i for T (U \V \W )
we have
T(
R
:= e (
; e ; e ) := T
)
e (
=
c
=
)
(3a)
T
,
c
with an analogous formula for R
.
2.2
and
Relation between
(3b)
We have that (see, e.g.,[13]):
=
where
K
:=
1
(T
2
+S
+K
)=
3
1
(T
2
;
(4)
T
+T
)
and where we introduce
S
=
g
(g
T
+g
T
)=S
:
(5)
The tensor …eld K = K @=@x
dx
dxv is known as the contorsion
tensor of the connection D and S = S @=@ dx
dxv has been called in [18]
(using certain analogies from the theory of continuous media) the strain tensor
of the connection D.
2.2.1
Relation Between the Curvature Tensors of D and D
The components of the curvature tensors relative to the coordinate basis associated to the coordinates hx i covering V are:
R
where J
:= D K
We need also the
K
=R
K
+J
and J
[
[
];
]
=J
(6)
J
.
Proposition 3 Let Z 2 sec T V be a normalized timelike vector …eld such that
g(Z; Z) = 1. Then, there exist, in a coordinate neighborhood V , three spacelike
vector …elds ei which together with Z form an orthogonal moving frame for
x2V
M [3].
Proof. Suppose that the metric of the manifold in a chart ( ; V ) with coordinate functions hx i is g = g dx
dx . Let Z = (Z @=@x ) 2 sec T V be
as above and Z = g(Z; ) = Z dx ; Z = g Z Then, g Z Z = 1. Now,
de…ne 0 = ( Z ) dx = Z dx ,
= g
Z Z . Then the metric g can
be written due to the hyperbolicity of the manifold as g =
, with
P3
i
i
=
(x)dx
dx
.
Now,
call
e
=
Z
and
take
e
such
that
0
i
i=1
i
i
(ej ) = j . It follows immediately that g(e ; e ) =
, ; = 0; 1; 2; 3:
3
Observers and Reference Frames
De…nition 4 An observer in a Lorentzian structure hM; gi is a timelike curve
pointing to the future such that g( ; ) = 1.
De…nition 5 A reference frame in U \ V \ W M in a Lorentzian structure
hM; gi is a timelike vector …eld Z (g(Z; Z) = 1) such that each one of its integral
lines is an observer.
So if is an integral line of Z its parametric equations are d(x
Z (x ( )).
( ))=d =
De…nition 6 A naturally adapted coordinate system hx i to a reference frame
Z 2 sec T V (denoted hnacsjZi) is one where the spacelike components of Z
vanish. Note that such a chart always exist [2].
4
Remark 7 The de…nition of a reference frame in a Lorentzian spacetime or in
a Riemann-Cartan spacetime is the same as above since that de…nition does not
depends on the additional objects entering these structures.
3.1
References Frames in hM; g; D;
g ; "i
and hM; g; D;
g ; "i
Given a reference frame Z in U \V \W M , consider the physically equivalent
1-form …eld = g(Z; ).Then we have (see, e.g.,[22]):
1
+ Eh;
(7)
3
where h = g
is the projection tensor, a is the (form) acceleration of
Z, ! is the rotation tensor (or vorticity) of Z, is the shear of Z and E is the
expansion ratio of Z. In a coordinate chart ( ; V ) with coordinates x covering
(V ), writing Z = Z @=@x and h = (g
Z Z )dx
dx we have (with
Z ; = g(D@=@x Z; @=@x ) and Z[ ; ] and Z( ; ) denoting the antisymmetric
and symmetric parts of Z ; )
D
=a
+!+
a = g(DZ Z; ) = DZ ;
!
= Z[
; ]h
h ;
= [Z(
; )
1
Eh ]h h ;
3
E=D Z :
(8)
Remark 8 We observe that the vorticity tensor has the same components as
the object g(?( ^ d ); ), where ? is the Hodge star operator. Indeed, a trivial calculation gives g(?( ^ d ); ) = c023 e1 + c031 e2 + c012 e3 which are the
components of ! in an orthonormal basis.
Remark 9 Eq.(8) is the basis for the classi…cation of reference frames in a
Lorentzian spacetime structure2 [15, 18, 22] and in order to be possible to talk
about the classi…cation of reference frames in a Riemann-Cartan spacetime
structure we need the
Proposition 10
D
=a
1 0
T
2 ij
1
+ Eh;
3
a = DZ ;
1
= + (E
3
! = ! + T 0;
T0 =
+!+
i
^
j
;
S0 =
(9)
(10a)
E)h + S 0 ;
1 0
S
2 ij
i
j
:
Proof. It is a simple exercise using an orthonormal basis where
2 For
(10b)
(10c)
=
0
.
the classi…cation of reference frames in a Newtonian spacetime structure, see [16].
5
Remark 11 We observe that in a Riemann-Cartan spacetime the interpretation of ! (in the decomposition of D given by Eq.(9)) is the same as ! in a
Lorentzian spacetime [18], i.e., it measures the rotation that one of the in…nitesimally nearby curves to an integral curve (an ‘observer’) of Z had in an
in…nitesimal lapse of proper time with relation to an orthonormal basis Fermitransported by the ‘observer’ . The interpretation of the terms
and E are
also analogous to the corresponding terms in a Lorentzian spacetime. Thus, a
reference frame is non-rotating if ! = 0, i.e., ! = T 0 and Eq.(10c) shows
that torsion is indeed related to rotation from the point of view of a Lorentzian
spacetime structure.
m
3.2
Inertial Reference Frames in hM ' R4 ; ; D;
; "i
m
Now, let hM; g; D; g ; "i = hM ' R4 ; ; D; ; "i and let hx i be coordinates
in the Einstein-Lorentz-Poincaré gauge for M . Then if the matrix with entries
is the diagonal matrix diag(1; 1; 1; 1), we have =
dx
dx . If
we put I = @=@x0 we see immediately that that hx i is a hnacsjIi. We have
m
trivially that D I = 0, which means that for the reference frame I = @=@x0
we have a = 0, ! = 0, = 0, E = 0:
m
De…nition 12 A inertial reference frame (IRF) in hM ' R4 ; ; D;
m
reference frame I such that D
I
= 0.
; "i is a
So, inertial reference frames in Special Relativity are not accelerating, not
m
rotating, have no shear and no deformation. Of course, since D@=@x0 @=@x0 =
0, each one of the integral lines of the vector …eld I = @=@x0 is a timelike
autoparallel (in this case, also a geodesic) of Minkowski spacetime (a straight
line).
3.3
Are there IRFs in Lorentzian and Riemann-Cartan
Spacetimes?
The answer is yes for the Lorentzian case only if we can …nd a reference frame
I such that D I = 0. In general this equation has no solution in a general
hM; g; D; g ; "i structure and indeed we have the
Proposition 13 [22] An IRF exists in the Lorentzian structure hM; g; D;
only if the Ricci tensor satis…es Ricci(I; Y ) = 0 for any Y 2 sec T M .
g ; "i
Remark 14 This excludes, e.g., Friedmann universe spacetimes, Einstein-de
Sitter spacetime. So, no IRF exist in many models of GRT considered to be of
interest by one reason or another by ‘professional relativists’.
Remark 15 The situation in a Riemann-Cartan spacetime is more complicated
and will be analyzed elsewhere, but we observe that in an arbitrary teleparallel
6
e
spacetime structure 3 hM; g; r; g ; "i the teleparallel basis he i satis…es re e =
0. Then the reference frame e0 is a IRF.
3.4
Pseudo Inertial Reference Frames
De…nition 16 A reference frame I 2 sec T U; U
M is said to be a pseudo
inertial reference frame (PIRF) if DI I = 0, I ^ d I = 0 and I = g(I; ).
This de…nition means that a PIRF is in free fall and it is non rotating. It
means also that it is at least locally synchronizable, but we are not going to
discuss synchronizability here (details may be found, e.g., in ([18]).
4
4.1
What is a LIRF in hM; g; r;
g ; "i
Normal Coordinate Functions at po 2 M
In what follows r denotes D or D. We will specialize our discourse at appropriate places.
Let ('; U ) be a local chart around po 2 U with coordinate functions h i.
Let : R I ! M , 7! ( ) an autoparallel4 in M according to an arbitrary
connection r, i.e.,
r
= 0:
(11)
Take arbitrary points po ; q 2 M . Put
(0) = po ,
q
:= d=d jpo =
(po ) =
po
= 0;
q
@=@
jpo =
(q) =
q
q
e 2 Tpo M;
6= 0
(12)
(13)
Although the notation may looks strange it will become clear in a while.
Now, any autoparallel emanating from po is speci…ed by a given q 2 Tpo M .
Indeed, take q ‘near’ po , this statement simply meaning here that the coordinates di¤erence 4 = q
po = q << 1. In general there may be many
autoparallels that connect po to q. However, there exists a unique autoparallel
I ! M , 7! q ( ) such that q (0) = po ;
q :R
q (1) = q.
So, under the above conditions we see that if q << 1, then q uniquely
speci…es a vector q = q e 2 Tpo M: It is evident that ' : q 7! f q g serves as a
good coordinate system in a neighborhood of po . We have
De…nition 17 ' : q 7! ( q0 ; q1 ; q2 ; q3 ) := f q g ('(po ) = (0; 0; 0; 0)) is called a
normal coordinate chart based on po [(nccbjpo )] with basis e = @=@ jpo .
With respect to the (nccbjpo ) an autoparallel ( ) with (0) = po and (1) =
q is clearly represented by
'( ( )) = f q g:
(14)
3 A teleparallel spacetime is one equipped with a metric compatible connection for which
its Riemann curvature tensor vanishes, but its torsion tensor is non vanishing.
4 Autoparallels in a Lorentzian spacetime structure coincide with geodesics.
7
4.2
Autoparallels Passing Trough po in hM; g; D;
g ; "i
If ( ) is an autoparallel in the structure hM; g; D; g ; "i it satis…es the equation
D
= 0, which gives recalling that
=
and Eq.(14) that the connection coe¢ cients
vanishes at po , i.e.,
('( (0)) = 0. Moreover, in what
follows for simplicity of notation and when no confusion arises we eventually
use the sloppy notation
(po ) :=
('( (0))
@g =@ ) we
Since it is well known that
= 12 g (@g =@ +@g =@
can take at po = (0), g (@=@ ; @=@ )jp0 =
and of course, @g =@ jp0 =
0.
We can show through a simple computation that for any q 0 2 U , q 0 2
= po we
have
1
(R
( )+R
( )) 0 ;
(15)
( ) 0 = @=@
( ) 0=
;
3
q
q
q
and also that for the chart ( ; V ) with coordinate functions hx i we have for q 0
near po
=x +
1
2
(po )x x ;
1
2
x =
(po )
;
where
(po ) here denotes the values of the connection coe¢ cients at po in
the coordinates hx i, i.e., D@=@x @=@x =
@=@x .
Remark 18 Let
U
M be the world line of an observer in autoparallel
motion in spacetime, i.e., D
= 0. Then the above developments show that we
can introduce in U normal coordinate functions h i such that for every p 2
we have @=@
g
g(@=@
0
p2
=
; D@=@ @=@
R
jp ,
)
g(@=@
; @=@
)jp2
=
and
(
)
=
p2
= 0. Also,
p2
=@
=@
@
=@
(16)
which is non vanishing if the curvature tensor is non vanishing in U .
4.3
LIRFs in hM; g; D;
g ; "i
De…nition 19 Given a timelike autoparallel line
U
M and coordinates
h i covering U we say that a reference frame L = @=@ 0 2 sec T U is a local inertial Lorentz reference frame associated to
jp ,
L
^d
L jp2
= 0, g(@=@
; @=@
(LIRF )5 i¤ Ljp2 =
)jp2 =
and @g
=@
@
@ 0
p2
=
jp2 = 0.
Moreover, we say also that the normal coordinate functions (also called in
Physics textbooks local Lorentz coordinate functions) h i are associated with
the LIRF :
5 When
no confusion arises and
is clear from the context we simply write LIRF.
8
Remark 20 It is very important to have in mind that for a LIRF L, in
6= 0 (i.e., only the integral line of L is in free fall in general),
general DL L
p2
=
and also eventually L ^ d L jp2= 6= 0, which may be a surprising result. In
contrast, a PIRF I such that Ij = Lj has all its integral lines in free fall and
the rotation of the frame is always null in all points where the frame is de…ned.
Finally its is worth to recall that both I and L may eventually have shear and
expansion even at the points of the autoparallel line that they have in common.
More details in [18].
De…nition 21 Let be an autoparallel line as in de…nition 19. A section s
of the orthogonal frame bundle F U; U
M is called an inertial moving frame
along (IMF ) when the set s = f(e0 (p); e1 (p); e2 (p); e3 (p)); p 2 \ U g s is
such that 8p 2 we have e0 (p) = jp and g(e ; e )jp2 =
, with
(p) =
g
g(e (p); De
(p) e
(p)) = 0.
Remark 22 The existence of s 2 sec F U satisfying the above conditions can
be easily proved. Introduce coordinate functions <
> for U such that at
po 2 ; e0 (po ) = @=@ 0 po = jpo , and ei (po ) = @=@ i po , i = 1; 2; 3 (three
orthonormal vectors) satisfying g(e ; e )jp0 =
and
p0
= 0 and parallel
transport the set e (po ) along . The set fe (po )g will then also be Fermi
transported since is a geodesic and as such they de…ne the standard of no
rotation along . See details in [18].
Remark 23 Let I 2 sec T V be a PIRF and
U \ V one of its integral lines
and let <
>; U M be a normal coordinate system through all the points of
the world line such that
= Ij . Then, in general <
> is not a (nacsjI)
in U , i.e., Ijp2= 6= @=@ 0 p2= even if Ijp2 = @=@ 0 p2 .
Remark 24 It is very much important to recall that a reference frame …eld as
introduced above is a mathematical instrument. It does not necessarily need to
have a material substratum (i.e., to be realized as a material physical system) in
the points of the spacetime manifold where it is de…ned. More properly, we state
that the integral lines of the vector …eld representing a given reference frame do
not need to correspond to worldlines of real particles. If this crucial aspect is
not taken into account we may incur in serious misunderstandings.
Remark 25 Physics textbooks and even most of the professional articles in
GR do not distinguish between the very di¤ erent concepts of reference frames,
coordinate systems, sections of the frame bundle and does not leave clear what
is meant by the word local. In general what authors mean by a local inertial
reference system is the concept of normal coordinates associated to a timelike
autoparallel curve as described above. Moreover, keep in mind that of course,
= d=d = @=@ 0 .
9
4.4
LIRFs in hM; g; D;
g ; "i
We have seen above that we can always introduce around a point po 2 U M in
a Lorentzian hM; g; D; g ; "i or in a Riemann-Cartan hM; g; D; g ; "i structure
a chart ('; U ) with normal coordinate functions.
However, it is not correct a priory to assume that the normal coordinate
functions of the two structures coincide. So, we denote by h i the RiemannCartan normal coordinate functions around po in what follows. In the case of a
Lorentzian structure we found that at po the connection coe¢ cients
(po ) =
(D@=@ @=@ ) (g @=@ ) = 0. However,we are not going to suppose that this
is generally the case in a Riemann-Cartan structure. So, let us investigate which
conditions
(po ) (given by a formula analogous to the last one just above)
must satisfy in normal coordinates h i. A Riemann-Cartan autoparallel
passing through po and neighboring points q 0 (in the sense mentioned above),
satisfy D
= 0, and we have
d2
d
2
+
d
d
d
[
d
('( ( ))] = 0:
(17)
If the auto-parallel equation is for points from po to q given by
(recall Eq.(??)) then since d2 =d 2 =0 = 0, at po we must have
( ) = q0
(po ) =
(
)
1
2
(po ) +
(po ) = 0, i.e.,
(po ) =
(po ).
Now, if we recall Eq.(4) and Eq.(5), which give the components of the torsion
tensor and the strain tensor of the connection D, we see that in the case of
normal coordinates h i we must have
T
(po ) = 2
(po ); S
(po ) =
2
(po ):
which are the conditions that select the normal coordinate functions h
po in a Riemann-Cartan spacetime.
(18a)
i near
Remark 26 We did not suppose, of course, that the autoparallels of the LeviCivita and Riemann-Cartan connections coincide (since this is trivially false)6 .
So, we have the question: when does the two kinds of autoparallels coincide? If
they do coincide then the Lorentzian and Riemann-Cartan normal coordinate
functions around po must coincide and since for a autoparallel from po to q, it
(po ) =
(po ). But now
is d2 =d 2 =0 = 0 we must have again that
since
(po ) = 0 we arrive at the conclusion that
T
=2
(po ); S
(po ) = 0:
(19a)
S (po ) = 0 implies moreover that T (po ) = T (po ) = T (po ), i.e.,
the torsion tensor must be completely anti-symmetric at all manifold points
(since po is arbitrary):
T (po ) = T[ ] (po ):
(20)
6 E.g., the geodesics of the Levi-Civita and the teleparallel connection on the punctured
sphere S are very di¤erent, the latter one are the so-called loxodromic spirals and the former
are the maximum circles [18].
10
Eq.(20) is then the condition for the two kinds of autoparallel to coincide. It is
a very particular condition and contrary to what is stated in [4, 5, 12, 21, 23] it
is not satis…ed by a general Riemann-Cartan connection and thus cannot serve
the purpose of …xing coordinate functions that could model LIRF analogous
to the ones used in the Lorentzian case 7 . We recall moreover that the connection coe¢ cients of the Riemann-Cartan connection although anti-symmetric
using the normal coordinate functions will have no particular symmetry if arbi@
@
+
trary coordinate functions hx i are used, since we have
= @x
@
@x @x
@x
@
@2
@x @x
.The symmetric part is, of course, the same one that appears also in
the transformation law for the Levi-Civita connection coe¢ cients. We arrive at
the conclusion that only for very particular spacetimes, the ones in which the
strain tensor of the connection vanishes, we can build around a point po normal
coordinate functions for which the formulas in Eq.(19a) hold and it is clear that
. However, for the case of non vanishing
in this case g (@=@ ; @=@ )jpo =
of the strain tensor of the connection D, when the formulas in Eq.(18a) hold,
and @g =@ jpo = 0 for otherwise
we cannot have g (@=@ ; @=@ )jpo =
(po ) would be null.
So, de…nitively normal coordinate functions are not useful to model a LIRF
in Riemann-Cartan spacetimes. So, what can we do to model such a LIRF in
this case?
To answer that question we need the following result:
Proposition 27 Along any timelike autoparallel line
M in a RiemannCartan spacetime structure there exists a section s of the orthogonal subframe
bundle F U
F M; U
M called an inertial moving frame along (IMF )
s = f(e0 (p); e1 (p); e2 (p); e3 (p)); p 2 U M g s, such that 8p 2 : e0 jp2 =
p1
g00
jp2 , g(e ; e )jp2 =
where the c
, [e ; e ]jp2 = c
e
p2
and
p2
= 0,
are not all vanishing (i.e., the he i is not a coordinate basis).
Proof. Indeed, at that any point p 2 U
M given ('; U ), a (nccbj p), with
bases h@=@ i and hd i for T U and T U we can …nd given an arbitrary vector
…eld X, a non coordinate basis he i and h i for T U and T U by …nding a
solution to the matrix equation8 ,
0=
0
X
1
=
X
+
1
X( );
(21)
satisfying the conditions [e ; e ] = c e ; c =
e ( )
the c
are not all vanishing and where the matrix function
is de…ned by
DX e := (
e0 =
X)
DX e0 := (
e ;
e ;
0
=(
1
)
0
X)
:
e ( ), where
with entries
e0 ;
(22)
7 Also, [21] who cites [4, 5] did not realize that total antisymmetry of the components of
the torsion tensor is no more than the conditon for two kinds of autoparallels (the Lorentzians
and the Riemann-Cartan ones) to coincide.
8 X( ) is the matrix with entries X(
) = X @=@ ( ).
11
To accomplish our enterprise we choose at an arbitrary po 2 and take e0 (po ) =
g00 1=2 po = g00 1=2 @=@ 0 po and recall that from Proposition 3 there exists
ei (po ), i = 1; 2; 3 that together with e0 (po ) satisfy g(e ; e )jpo 2 =
. So, our
task is simply reduced to …nding solutions for Eq.(21). Now, taking X = @=@
we have that ( X ) :=
and Eq.(21) is the system of di¤erential equations
@=@
(
)=
:
(23)
whose solution with given boundary conditions is well known [9]. Once that
solution is known we have that De e jp0 = 0 and thus we construct the IMF
by simply parallel transporting the basis fe (po )g of Tpo M along , getting for
any p 2 , De e jp2 = 0.
Taking into account the previous proposition we …nally propose the following:
De…nition 28 Given a timelike autoparallel line
U
M in a RiemannCartan spacetime structure and coordinate functions h i covering U
M we
say that a reference frame L 2 sec T U is a local inertial reference frame associated to (LIRF ) i¤ for all p 2 there exists exists a section s of the orthogonal frame bundle F U F M; U M such that s = f(e0 (p); e1 (p); e2 (p); e3 (p)); p 2
U
Mg
s, where e0 jp2 = Ljp2 = g00 1=2 @=@ 0 p2 = g00 1=2 p2 ,
!jp2 =
T0
p2
and
coordinates.
p2
= 0. Moreover we say that h
i are inertial
Remark 29 Di¤ erently from the case of the LIRF in a Lorentzian spacetime,
in a general Riemann-Cartan spacetime we do not have g(@=@ ; @=@ )j =
for all
;
= 0; 1; 2; 3, for otherwise we get
= 0 which, as we
saw above, implies a completely antisymmetric torsion and we cannot have
(p) = 0. Moreover, we observe that in the basis fe g the components of
the torsion tensor are according to Eq.(3a) T
=
nents of the Riemann curvature tensor are R
=e (
Keep in mind that although
tensor is non null in all p 2
5
and the compo-
c
)
e (
) .
= 0 we have that the Riemann curvature
since e (
)
6= 0.
Equivalence Principle and Einstein’s Most
Happy Thought
At last we want to comment that, as is well known, in Einstein’s GR one can
easily distinguish (despite some claims on the contrary) in any real physical
laboratory, (i.e., not one modelled by a timelike worldline) a true gravitational
…eld from an acceleration …eld of a given reference frame in Minkowski spacetime [20, 17]. This is because in GR the mark of a real gravitational …eld is the
12
non null Riemann curvature tensor of D, and the Riemann curvature tensor of
m
D (present in the de…nition of Minkowski spacetime) is null. In [6, 19] we interpreted a gravitational …eld as deformation of Minkowski spacetime and showed
that it can also be alternatively described as the torsion 2-forms of an e¤ ective
e
teleparallel spacetime (M; ; r; ; "). We show now that one can also interpret
the acceleration …eld of an accelerated reference frame in Minkowski spacetime
e
as generating an e¤ective teleparallel spacetime (M; ; r; ; ") structure. This
can be done as follows. Let Z 2 sec T U , U
M with (Z; Z) = 1 be an
accelerated reference frame on Minkowski spacetime. This means as we know
m
from Section 3.1 that a = DZ Z 6= 0. Put e0 = Z and de…ne an accelerated reference frame as non trivial if 0 = (e0 ; ) is not an exact di¤erential.
Next recall that in V
M there always exist three other -orthonormal vector …elds ei , i = 1; 2; 3 such that he i is an -orthonormal basis for T U ,
i.e.,
=
, where h i is the dual basis9 of he i. We then have,
m
m
De e =
e , De
=
.
What remains in order to be possible to interpret an acceleration …eld as
a kind of ‘gravitational …eld’is to introduce on M a -metric compatible cone
e
nection r such that the fe g is teleparallel according to it, i.e., re e
e
4
=
e
0; re
= 0. Indeed, with this connection the structure hM ' R ; ; r; ; "i
has null Riemann curvature tensor but a non null torsion tensor, whose components are related10 with the components of the acceleration a and with the other
m
coe¢ cients
of the connection D, which describe the motion on Minkowski
spacetime of a grid represented by the orthonormal frame he i. Schücking
[23] thinks that such a description of the gravitational …eld makes Einstein
most happy thought, i.e., the equivalence principle (understood as equivalence
between acceleration and gravitational …eld) a legitimate mathematical idea.
However, a true gravitational …eld must satisfy (at least with good approximation) Einstein equation or the equivalent equation for the tetrad …elds he i
[6, 19], whereas there is no single reason for an acceleration …eld to satisfy that
equation.
6
Conclusions
In this paper we have recalled the de…nitions of observers, reference frames
and naturally adapted coordinate chart to a given reference frame. Equipped
with these de…nitions and some basic results such as the proper meaning of an
inertial reference frame in Minkowski spacetime and the notion of pseudo-inertial
reference frames and locally inertial reference frames in a Lorentzian spacetime,
general we will also have that d i 6= 0, i = 1; 2; 3.
explict formulas can be easily derived using the equations of section 4.5.8 of [18]
which generalizes for connections with non vanishing nonmetricty tensors the second formula
in Eq.(5).
9 In
1 0 The
13
we showed how to de…ne consistently locally inertial reference systems in a
general Riemann-Cartan spacetime structure hM; g; D; g ; "i. We prove that
a set of normal coordinate functions covering a timelike autoparallel do not
de…ne a LIRF in hM; g; D; g ; "i as it is the case in a Lorentzian spacetime,
but can be used to de…ne a LIRF (De…nition 28) once we take into account
Proposition 27. We also brie‡y recalled how the concepts of LIRF in Lorentzian
and Riemann-Cartan spacetimes are linked to “Einstein’s most happy thought”,
i.e., the equivalence principle. Summing up we think that our paper complement
and help to clarify presentations of related issues appearing in [1, 7, 8, 9, 10, 11,
14, 24], besides clarifying some misconceptions like the ones in [4, 5, 12, 21, 23]
as exposed in Remark 26.
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