IN CONNECTION-DYNAMIC THEORIES OF GRAVITY ∗
Sean M. Carroll(1) and George B. Field(2)
(1)Center for Theoretical Physics, Laboratory for Nuclear Science
and Department of Physics
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
email:
(2)Harvard-Smithsonian Center for Astrophysics
Cambridge, Massachusetts 02138
email:
Abstract
"We discuss the possibility of constraining theories of gravity in which the connection is
a fundamental variable by searching for observational consequences of the torsion degrees
of freedom. In a wide class of models, the only modes of the torsion tensor which interact
with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar
version, we study constraints on the two-dimensional parameter space characterizing the
theory. For reasonable choices of these parameters the torsion decays quickly into matter
fields, and no long-range fields are generated which could be discovered by ground-based
or astrophysical experiments.
...
Thus, the curvature and torsion have a similar status as tensors which characterize a
specified connection. Special relativity posits a spacetime connection for which both tensors
vanish; the transition from special to general relativity may be thought of as allowing for
the dynamics of a nonzero curvature, while constraining the torsion to vanish. From
a point of view which takes the connection as an independent variable, this restriction
seems somewhat arbitrary (although it is nevertheless possible, by judicious choice of
Lagrangian, to make the torsion nonpropagating or even vanishing). We are therefore led
to consider theories in which both the curvature and torsion are determined dynamically
by the response of the metric and connection to matter fields."
Even in special relativity, physics in an accelerating frame has a non-vanishing connection.
The local value of connection describes covariant accelerations of the (LNIF) detectors not that of the test particle that is detected. A non-vanishing covariant curl between neighboring values of the connection will describe the intrinsic curvature of the spacetime detected by pairs of neighboring non-accelerating local inertial frame detectors (LIF).
"The introduction of additional propagating degrees of freedom opens the possibility
that such a theory could lead to observable deviations from general relativity. Experiments
in the solar system and in binary pulsar 1913+16 offer strong evidence that the metric
must not deviate too far from the form specified by Einstein’s equations [7]. The situation
with respect to torsion is less clear, as the literature contains various different proposals
for what the dynamics of torsion could be. ...
Our goal in this paper is to determine whether there are any observational consequences
of propagating torsion which are relatively independent of any specific gravitational
model. To that end, we discuss possible actions for torsion and its interaction with
matter fields such as those in the standard model of particle physics. In these theories we
construct a free Lagrangian from powers and derivatives of the torsion, and couple “minimally”
to matter through the covariant derivative. We find that there is only a small range
of models possible without placing arbitrary restrictions on the dynamics. In these models
only a single mode interacts with matter, either a massive scalar or a massive spin-1 field,
and each model is parameterized by two constants with the dimensions of mass. In this
paper we concentrate on the scalar theory, which is related to several different proposals
found in the literature. We discuss what regions of parameter space are excluded by laboratory
and astrophysical data. A reasonable expectation, however, would be for each of
the two mass parameters to be of order the Planck scale; such a choice is a safe distance
away from the regions excluded by experiment. We conclude that, while there are reasons
to expect that the torsion degrees of freedom exist as propagating fields, there is no reason
to expect any observable signature from torsion.
...
The picture of torsion as an extremely short-range field runs somewhat counter to the
intuitive conception of torsion as a part of spacetime geometry. More concretely, we are
used to gauge theories giving rise to massless, long-range fields, and the status of torsion
as part of the connection on the tangent bundle might lead us to expect the same in
this case. This conflict with intuition may be resolved by noticing that the torsion is a
tensor which is linear in the connection. It therefore becomes possible to construct gauge
invariant interactions which give a mass to some of the connection degrees of freedom.
This is in contrast with the pure metric theory, or with gauge theories on internal vector
bundles, where all gauge invariant terms involve the curvature tensor, constructed from
derivatives of the fundamental fields. Thus, despite its origin as part of the geometry of
spacetime, the physical manifestation of torsion can be significantly different from that of
other “geometrical” fields.
The possible existence of torsion is of interest both in the construction of quantum
theories of gravity and in the experimental search for deviations from general relativity.
The important lesson of this paper is that the absence of effects of torsion in experiments
should not lead us to discount the possibility of torsion playing a role in the ultimate
theory of gravity."
This paper http://arxiv.org/abs/gr-qc/9403058 has a lot of interesting details.