CONSEQUENCES OF PROPAGATING TORSION

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: carroll@marie.mit.edu

(2)Harvard-Smithsonian Center for Astrophysics

Cambridge, Massachusetts 02138

email: field@cfa.harvard.edu

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.