In Search of the Spacetime Torsion
7 IN SEARCH OF THE SPACETIME TORSION
J. G. PEREIRA
Instituto de F́ısica Teórica, Universidade Estadual Paulista
Rua Pamplona 145, 01405-900 São Paulo, Brazil
Whether torsion plays or not a role in the description of the gravitational interaction is a
problem that can only be solved by experiment. This is, however, a difficult task: since there
are different possible interpretations for torsion, there is no a model–independent way to look
for it. In these notes, two different possibilities will be reviewed, their consistency analyzed,
and the corresponding experimental outputs briefly discussed.
1 Gravitation and Universality
Gravitation has a quite peculiar property: particles with different masses and different compo-
sitions feel it in such a way that all of them acquire the same acceleration and, given the same
initial conditions, follow the same path. Such universality of response — usually referred to as
universality of free fall— is the most fundamental characteristic of the gravitational interaction.1
It is unique, peculiar to gravitation: no other basic interaction of nature has it. Universality of
free fall is usually identified with the weak equivalence principle, which establishes the equality
between inertial and gravitational masses. In fact, in order to move along the same trajectory,
the motion of different particles must be independent of their masses, which have then to be
canceled out from the equations of motion. Since this cancellation can only be made when the
inertial and gravitational masses coincide, this coincidence naturally implies universality.
Einstein’s general relativity is a theory fundamentally based on the weak equivalence prin-
ciple. In fact, to comply with universality, the presence of a gravitational field is supposed to
produce curvature in spacetime, the gravitational interaction being achieved by letting (spin-
less) particles to follow the geodesics of the curved spacetime. In general relativity, therefore,
geometry replaces the concept of gravitational force, and the trajectories are determined, not
by force equations, but by geodesics. It is important to emphasize that only a universal inter-
action can be described by such a geometrization, in which the responsibility for describing the
interaction is transferred to spacetime. It is also important to remark that, in the eventual lack
of universality, the geometrical description of general relativity would break down.2
The fundamental connection of general relativity is the Christoffel connection, a Lorentz-
valued connection written in a coordinate basis.3 In terms of the spacetimea metric gµν , it is
written as
gρλ(∂µgλν + ∂νgλµ − ∂λgµν). (1)
It is a connection with vanishing torsion,
νµ = 0, but non- vanishing curvature,
λνµ 6= 0.
In terms of this connection, the equation of motion of a test particle is given by the geodesic
We use the Greek alphabet µ, ν, ρ, . . . = 0, 1, 2, 3 to denote spacetime indices.
http://arxiv.org/abs/0704.1141v1
equation
µν uθ u
ν = 0, (2)
which says that the particle four-acceleration vanishes identically. This property reveals the
absence of the concept of gravitational force, a basic characteristic of the geometric description.
2 What About Torsion?
A general Lorentz connection has two fundamental properties: curvature and torsion.4 Why
should then matter produce only curvature? Was Einstein wrong when he made this assumption
by choosing the Christoffel connection? Does torsion play any role in gravitation? Cartan was
the first to ask these questions, soon after the advent of general relativity. As a possible answer, a
new theory was formulated, called Einstein–Cartan theory,5 in which the Christoffel connection
was replaced by a general connection presenting both curvature and torsion.
The main idea behind the Einstein–Cartan construction is the fact that, at the microscopic
level, matter is represented by elementary particles, which in turn are characterized by mass
(that is, energy and momentum) and spin. If one adopts the same geometrical spirit of general
relativity, not only mass but also spin should be source of gravitation at this level. Like in
general relativity, energy and momentum should appear as source of curvature, whereas spin
should appear as source of torsion. This means essentially that, in this theory, curvature and
torsion represent independent degrees of freedom of gravity. As a consequence, there might
exist new physics associated to torsion. Of course, at the macroscopic level, where spins vanish,
the Einstein–Cartan theory coincides with general relativity. At the microscopic level, however,
where spins are relevant, it shows different results from general relativity. If this is interpretation
is correct, Einstein made a mistake when he did not include torsion in general relativity.
Because of the Einstein–Cartan theory, there is a widespread belief that torsion is intimately
related to spin, and consequently important only at the microscopic level. This belief, however, is
not fully justified: in addition to lack experimental support, it is based on a very particular model
for gravitation, which is well known to present several consistency problems. For example, it is
not consistent with the strong equivalence principle.6 Another relevant problem is that, when
used to describe the interaction of the electromagnetic field with gravitation, the Einstein–
Cartan coupling prescription violates the U(1) gauge invariance of Maxwell’s theory.7 This last
problem is usually circumvented by postulating that the electromagnetic field does not couple
to torsion.8 This “solution”, however, is quite unreasonable.9 The purpose of these notes is
to call the attention to another possible interpretation for torsion,6 which is consistent and
does not present the above mentioned problems. This solution is based on a different model
for gravitation, known as the teleparallel gravity equivalent of general relativity,10 or simply
teleparallel gravity.
3 A Glimpse on Teleparallel Gravity
Teleparallel gravity corresponds to a gauge theory for the translation group, with the field
strength given by the torsion tensor. Its main difference in relation to general relativity is the
connection field: instead of Christoffel, the fundamental connection of teleparallel gravity is the
so called Weitzenböck connection. In terms of the tetrad b haµ, it is written as
µν = ha
ρ ∂νh
µ. (3)
We use the Latin alphabet a, b, c, . . . = 0, 1, 2, 3 to denote algebraic indices related to the tangent Minkowski
spaces. These indices are raised and lowered with the Minkowski metric ηab, whereas the spacetime indices are
raised and lowered with the metric gµν = ηab h
In contrast to Christoffel, it is a connection with non-vanishing torsion,
νµ 6= 0, but vanishing
curvature,
νµ = 0. The Weitzenböck and the Christoffel connections are related by
µν , (4)
where
µν) (5)
is the contortion tensor.
In teleparallel gravity, the equation of motion of a test particle is given by the force equa-
tion 11
µν uθ u
µν uθ u
, (6)
with torsion playing the role of gravitational force. It is similar to the Lorentz force equation of
electrodynamics, a property related to the fact that, like Maxwell’s theory, teleparallel gravity
is also a gauge theory. Using expression (4), the force equation (6) can be rewritten in terms of
the Christoffel connection, in which case it reduces to the geodesic equation of general relativity:
µν uθ u
ν = 0. (7)
The force equation (6) of teleparallel gravity and the geodesic equation (7) of general relativity,
therefore, describe the same physical trajectory. This means that the gravitational interaction
has two equivalent descriptions: one in terms of curvature, and another in terms of torsion.12
Although equivalent, however, there are conceptual differences between these two descriptions.
In general relativity, curvature is used to geometrize the gravitational interaction. In teleparallel
gravity, on the other hand, torsion accounts for gravitation, not by geometrizing the interaction,
but by acting as a force. As a consequence, there are no geodesics in teleparallel gravity, but
only force equations, quite analogous to the Lorentz force equation of electrodynamics.
One may wonder why gravitation has two equivalent descriptions. This duplicity is related
precisely to that peculiarity: universality. Like the other fundamental interactions of nature,
gravitation can be described in terms of a gauge theory – - just teleparallel gravity. Universality
of free fall, on the other hand, makes it possible a second, geometrized description, based on
the weak equivalence principle — just general relativity. As the sole universal interaction, it is
the only one to allow also a geometrical interpretation, and hence two alternative descriptions.
From this point of view, curvature and torsion are simply alternative ways of describing the
gravitational field,13 and consequently related to the same degrees of freedom of gravity. If this
interpretation is correct, Einstein was right when he did not include torsion in general relativity.
4 Conclusions
According to the Einstein–Cartan theory, as well as to other generalizations of general relativity,
torsion represents additional degrees of freedom of gravity. As a consequence, new physical
phenomena associated to its presence are predicted to exist. With this point of view in mind,
there has been recently a proposal to look for these new phenomena using the Gravity Probe B
data.14 The idea is that, if torsion is able to couple to spin, consistency arguments require that
it might also be able to couple to rotation — that is, to orbital angular momentum. The data of
Gravity Probe B could then be used to look for signs of this coupling. On the other hand, from
the point of view of teleparallel gravity, torsion does not represent additional degrees of freedom,
but simply an alternative to curvature in the description of gravitation. In this case, there are no
new physical effects associated with torsion. According to teleparallel gravity, therefore, torsion
has already been detected: it is the responsible for all known gravitational effects, including the
physics of the solar system, which can be reinterpreted in terms of a force equation, with torsion
playing the role of gravitational force.
Which of these interpretations is the correct one? From the theoretical point of view, we
can say that the teleparallel interpretation presents several conceptual advantages in relation to
the Einstein–Cartan theory: it is consistent with the strong equivalence principle,6 and when
applied to describe the interaction of the electromagnetic field with gravitation, it does not
violate the U(1) gauge invariance of Maxwell’s theory.9 From the experimental point of view, on
the other hand, at least up to now, there are no evidences for new physics associated with torsion.
We could then say that the existing experimental data favor the teleparallel point of view, and
consequently general relativity. However, due to the weakness of the gravitational interaction, no
experimental data exist on the coupling of the spin of the fundamental particles to gravitation.
For this reason, in spite of the conceptual soundness of the teleparallel interpretation, we prefer
to say that a definitive answer can only be achieved by further experiments.
Acknowledgments
The author would like to thank R. Aldrovandi and H. I. Arcos for useful discussions. He would
like to thank also FAPESP, CNPq and CAPES for partial financial support.
References
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http://arxiv.org/abs/gr-qc/0304106
http://arxiv.org/abs/gr-qc/0408096
http://arxiv.org/abs/gr-qc/9708051
http://arxiv.org/abs/gr-qc/0501017
http://arxiv.org/abs/gr-qc/9703059
http://arxiv.org/abs/gr-qc/0104102
http://arxiv.org/abs/gr-qc/0403074
http://arxiv.org/abs/gr-qc/0608121
	Gravitation and Universality
	What About Torsion?
	A Glimpse on Teleparallel Gravity
	Conclusions
