On source coupling and the teleparallel equivalent to GR
J.M. Nester and L.L. So
a
a ,b
b
Department of Physics, National Central University, Chungli, Taiwan 320
Alternatives to the usual general relativity (GR) Riemannian
framework include Riemann-Cartan and teleparallel geometry.
The “teleparallel equivalent of GR” (TEGR, aka GR//) has certain
virtues; however there have been some differences of opinion over
source coupling limitations. Now it is quite straightforward to
show that the coupled dynamical field equations of Einstein’s GR
with any source can be accurately represented in terms of any other
connection, in particular teleparallel geometry. Using an argument
similar to one used long ago to show the “effective equivalence”
between GR and the Einstein-Cartan theory, we construct the
teleparallel action which is equivalent to a given Riemanian one;
thereby finding the “effectively equivalent” coupling principle for
all sources, including spinors. No auxiliary field is required. Can
one decide which is the real “physical” geometry? Invoking the
minimal coupling principle may give a unique answer.
1
1 Introduction
Geometry concerns about the length and angle. The world has
geometry, but what is the geometry of the world? How can one
determine it? How to compare the direction at different points?
How to say two lines are parallel? The answer is simple. One may
just simply makes an assumption, determine the necessary
equations and make the experimental measurements.
However, one can consider alternative geometries. In general,
geometry study the metric (determines the length and angle) and
connection (determines the parallelism). From these two, one can
obtain the non-metricity, torsion and curvature. These three
properties come from the same metric, but with different ideas of
parallelism.
The alternative to the standard Riemannian geometry include (i)
Metric Affine, (ii) Poincare gauge theory and Einstein-Cartan
(vanishing non-metricity) and (iii) Teleparallel (vanishing
curvature). For simplicity, this paper just consider the vanishing
non-metricity only.
GR field equations can be represented in terms of any other
connection. This introduces an extra field and involves nonminimal coupling in general. In particular we can use teleparallel
geometry. For complete equivalence we want not only the
corresponding field equations but also all of the conserved
quantities. Hence we should find the equivalent action. In general
the effectively equivalent action needs extra fields and nonminimal coupling. We can find the effectively equivalent action
using an argument similar to that used to establish the effective
equivalence of GR and the Einstein-Cartan theory. Similar to what
happened in that case, we find that the GR// effective equivalent
2
field equations and action do not require an extra field. We obtain
a complete equivalence for all sources, that is, including spinors.
Can one decide which is the real “physical” geometry? Invoking
the minimal coupling principle hopefully gives a unique answer.
3
2 Virtues of Teleparalel gravity
As Friedrich Hehl said: “I would call general relativity theory GR
the best available alternative gravitational theory and the next best
one its teleparallel equivalent GR//[1].
GR// has a metric compatible connection. While the curvature
vanishes the torsion does not; it acts like a gravitational force. The
simplest description is in terms of orthonormal-teleparallel frames.
This representation is easily constructed: given the metric for GR,
simply choose any orthonormal frame and declare it to be parallel.
Thus each Riemannian geometry is represented by not just one but
rather a whole gauge equivalence class of teleparallel geometries.
In classical GR, the presence of a gravitational field is expressed
by the torsionless Levi-Civita metric-connection. The curvature
determines the intensity of the gravitational field. However,
contrast in GR//, the presence of a gravitational field is expressed
by the flat Weitzenbock connection. The torsion gives the
intensity of the gravitational field. Thus gravitational interaction
can be described by curved or torsioned spacetime.
The virtues of teleparallel gravity are: it is the gauge theory of
spacetime translations [2] and it is generally believed that have the
advantage for the energy-momentum localization[3].
4
3 Source Coupling Controversy
There have been some differences of opinion over alleged GR//
source coupling limitations. For example: Gronwald said that only
scalar matter fields or gauge fields, such as the electromagnetic
field, are allowed as materials sources, whereas matter carrying
spin cannot be consistently coupled in such a framework.” [4]
Also “any post-Riemannian geometry, that is, any spacetime
geometry which has more geometric field variable (gravitational
fields) than the metric, is irrelevant to the Maxwell equations. In
particular neither torsion nor non-metricity couple to it…” [5]
On the other hand, “it is shown that the electromagnetic field is
able to not only couple to torsion, but also, through its energy
momentum tensor, to produce torsion. Furthermore, it is shown
that the coupling of the electromagnetic field with torsion
preserves the local gauge invariance of Maxwell’s theory.” [6]
It should be keep in mind that, as Pereira has emphasized, spaces
do not have curvature and torsion, rather it is connections that have
curvature and torsion. Since any two connections differ by a
tensor it is straightforward to make a change from one connection
to any other. Then, in this way we can take the field equations of,
say the Poincare Gauge theory, or the Einstein Cartan theory and
represent them in terms of the Riemannian connection. The price
is that one needs to add to the Riemannian geometry an extra
tensor field (e.g., the contortion). And, moreover, the extra field
couples in a characteristic “non-minimal” fashion. This example is
typical. As long as one allows some extra tensor field with its
associated special type of non-minimal coupling, one can
transcribe any physical equations from Riemann-Cartan to
Riemannian to teleparallel geometry and vice versa. However, are
these different “isomorphic” representations physically equivalent?
5
In addition to field equations we also have conserved currents like
energy-momentum and angular momentum. They too should be
considered for a real “equivalence”. In fact one should find an
equivalent action. It turns out that we can show how this can be
done, but the general argument is a longer story.
There are two special cases of particular interest where things
simplify considerably. (1) The “equivalence” of the EinsteinCartan theory and Einstein’s GR, which was established long ago
[7]. (2) And, via a similar argument, GR and TEGR, aka GR//.
6
4 Riemannian equations re-expressed in teleparallel
form
All the dynamics of standard GR can be represented in terms of
any other connection (e.g., Riemann-Cartan with torsion) simply
by a transformation involving an extra field     K  , where

is the contorsion tensor.
K 
The Einstein field equation is the most minimal coupling.
Therefore, mininal coupled Riemannian equations will become
non-minimally coupled to the contorsion. In particular we can

force the new connection  to be teleparallel, so that R
 0 . (In
fact there are then an infinite number of choices for  )
In more detail, one can get the Riemannian GR equations for any
source from an action of the form:
ab
L  eea eb R
()  Ta  a  L(, e,    ) leading to
 :
 ea :
0
L L {} L

  {}
 
 
Ga   Ta
ab :
 a :
Ta  0
matter source equation
Einstein equation
determine Lagrangian multipler  a
Riemannian torsion vanishes
Within these equations we make the above substitution. In this
way, all of these Riemannian equations of motion can be
transcribed into teleparallel form. But the form of the coupling has
become non-minimal.
However, but what about the conserved quantities? Can we find
an action that gives the new equations?
7
5 Equivalent Teleparallel Action
For complete equivalence we need an equivalent teleparallel action.
It can be obtained by the same substitution ba  ba  K ba and
forcing Rab  0 with a Lagrange multiplier field. The result is of the
form:

ab
L//    14 T  T  12 T T  T 
T     R 
ab   L( , e,  , K )
Varying L// gives the typical teleparallel form of equations of
motions.
G   R  (   K )  12 g  R(   K )
 D K  KK  T 
0
L//

  K

L//
L
 (  K ) //

 
T    K    K  
GR// Einstein equation
GR// matter source equation
GR// torsion non-vanishing
In short:
LGR

Riemannian form of equations
K
L//
K

Teleparallel form of equations
The established result is similar to what we found many years ago
in connection with the effective equivalence of the Einstein-Cartan
and GR theories. Remarkably, in these cases (unlike the
corresponding situation for the Poincare Gauge theory or MetricAffine gravity) we do not need any extra field. We obtain a
complete equivalence for all sources --- including spinors!
8
Conclusion
There is a one-to-one correspondence between the two
representations. There is no physical experiment that can
distinguish between them, no way to decide whether spacetime has
curvature without torsion or torsion with vanishing curvature. We
have established a complete equivalence of GR and GR// for all
kind of sources including spinors. In other words, GR and GR//
are fully equivalent for all sources.
So, how we determine the true “physical” geometry? Is it
Riemannian or teleparallel? This may be possible with the aid of
the minimal coupling principle. Sources which are minimally
coupled with respect to the Riemannian geometry will not
generally be minimally coupled with respect to the teleparallel
geometry, and vice versa. We could hope that observations would
yield results which are consistent with one of these geometries and
the associated minimal coupling.
9
References
[1] Alternative Gravitational Theories in Four Dimension (gr-qc/9712096)].
[2] gr-qc/9602013
[3] C. Moller, Ann Phys. 12 (1961) 118-133; J.W. Maluf, J. Math, Phys. 35
(1994) 335-343; V.C. de Andrade, L.C.T. Guillen & J.G. Pereira, Phys,
Rev. Lett. 84 (2000) 4533-36.
[4] Gronwald: Int J Mod Phys D6(1997) 263 (MAG I)
[5] Puntigam, Lammerzhal & Hehl, CQG 14 (1997) 1347 “E &
M”
[6] de Andrade & Pereira: Int J Mod Phys A “Torsion and EM
field”
[7 ] J. M. Nester, Phys Rev D16 (1977) 2395.
[*] V.C. de Andrade, L.C.T. Guillen & J.G. Pereira, Phys. Rev. D 64 (2001)
027502; gr-qc/0104102.
[**] F. Gronwald, Int. J. Mod. Phys. D 6 (1997) 263; V.C. de Andrade &
J.G. Periera, Phys. Rev. D56 (1997) 5689, Int. J. Mod. Phys. D8 (1999)
141; F.W. Hehl & Y.N. Obukhov, Springer Lecture Notes in Physics 562
(2001) 479, gr-qc/0001010.
10