A crucial concept of gravitation is that the metric tensor itself defines neither curvature nor torsion. As a matter of fact, curvature and torsion require a connection to be defined, and many different connections with different curvature and torsion tensors can be defined on the very same metric spacetime. A general Lorentz connection has 24 independent components. However, any gravitational theory in which the source is the 10 components symmetric energy-momentum tensor, will not be able to determine uniquely the connection. There are only two possible options.
The first is to choose the Levi-Civita (or Christoffel) connection, which is a torsionless connection completely specified by the 10 components of the metric tensor. The second option is to choose a Lorentz connection not related to gravitation, but to inertial effects only. This is the so-called Weitzenböck connection. In this case, the gravitational field turns out to be fully represented by the non-trivial part of the tetrad field, that is, the part that has nothing to do with inertial effects. In the first case, which corresponds to Einstein's choice, we get general relativity, a theory that assumes torsion to vanish from the very beginning. In the second case, we get teleparallel gravity, a gauge theory for the translation group, in which curvature is assumed to vanish from the very beginning.
Einstein's choice, it must be said, is the most intuitive from the point of view of universality. Gravitation can be easily understood by incorporating the Levi-Civita curvature in the definition of spacetime, in such a way that all (spinless) particles, independently of their masses and constitutions, will follow a geodesic of the (supposed) curved spacetime. Universality of free fall is, in this way, naturally incorporated into gravitation. Geometry replaces the concept of force and the trajectories are solutions, not of a force equation, but of a geodesic equation. Nevertheless, because such a geometrization can only be done for a universal interaction, in the absence of universality (or of the weak equivalence principle), the general-relativistic description of gravitation would break down.
Although conceptually different, general relativity and teleparallel gravity provide completely equivalent descriptions of the gravitational interaction. An immediate implication of this equivalence is that curvature and torsion turn out to be simply alternative ways of describing the gravitational field, and are consequently related to the same degrees of freedom of gravity. This is corroborated by the fact that the same matter energy-momentum tensor appears as source in both theories: of curvature in general relativity, of torsion in teleparallel gravity. There are, however, conceptual differences between these two theories. In general relativity, curvature is used to geometrize
the gravitational interaction: geometry replaces the concept of gravitational force, and the trajectories are determined, not by force equations, but by geodesics. Teleparallel gravity, on the other hand, attributes gravitation to torsion, but not through a geometrization: it acts as a gravitational force
. In consequence, there are no geodesics in teleparallel gravity, only force equations quite analogous to the Lorentz force equation of electrodynamics. This is, by the way, what one should expect from a gauge theory for gravitation.
The reason for gravitation to present two equivalent descriptions lies in its most peculiar property: universality
. Like the other fundamental interactions of Nature, gravitation can be described in terms of a gauge theory. In fact, teleparallel gravity is just a gauge theory for the translation group. Universality of free fall, on the other hand, makes it possible a second, geometrized description, based on the 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. This means that all known gravitational phenomena, including the physics of the solar system, can be consistently reinterpreted in terms of teleparallel torsion. Though unnoticed by many, therefore, torsion has already been detected.
Among many peculiar properties of teleparallel gravity, it is worth mentioning here a quite interesting one. All fundamental source fields of nature have a well-defined local energy-momentum density. It would then be natural to expect that the same should happen to the gravitational field. However, as is well known, no tensorial expression for the gravitational energy-momentum density can be defined in the context of general relativity. The basic reason for this impossibility is that both gravitational and inertial effects are mixed in the spin connection of the theory, and cannot be separated. Even though some quantities, like for example curvature and torsion, are not affected by inertia, some others turn out to depend on it. For example, the energy-momentum density of gravitation will necessarily include both the energy-momentum density of gravity and the energy-momentum density of inertia. Since the inertial effects are essentially non-tensorial (they depend on the frame), it is not surprising that in this theory the complex defining the energy-momentum density of the gravitational field shows up as a non-tensorial object. On the other hand, although equivalent to general relativity, teleparallel gravity naturally separates gravitation from inertia. The reason behind such possibility is that in this theory the spin connection represents inertia only, whereas gravitation is represented by the non-trivial part of the tetrad (the part that preclude it to be a closed 1-form). As a consequence, in teleparallel gravity it turns out possible to write down an energy-momentum density for gravitation only, excluding the contribution from inertia. This object is a true tensor, which means that gravitation alone, like any other field of nature, does have a tensorial energy-momentum definition
We are currently exploring deeper the foundations of teleparallel gravity. At the same time, we are using this theory to re-analyze different problems in the hope to unveil some new aspects of gravitational physics, which like the existence of a tensorial energy-momentum density for the gravitational field, cannot be seen in the usual context of general relativity.
|de Sitter Special Relativity
Cosmological observations and quantum gravity arguments seem to indicate the necessity of an invariant length-parameter on both scales. At the cosmological level, this parameter is related to the accelerated expansion rate of the universe, which could be explained by a non-vanishing cosmological term Λ. At the quantum level, this parameter is related to the Planck length, which shows up as the threshold of quantum gravity. Considering that Poincaré, the kinematic group of ordinary special relativity, does not allow the existence of an invariant length-parameter, it turns out necessary to look for another group to replace Poincaré as the group ruling the kinematics at those scales.
For the case of physics at the Planck scale, several deformed special relativities have been proposed, as for example those based on a κ-deformed Poincaré group. These models presuppose a violation of the Lorentz symmetry to allow the existence of an invariant length-parameter. However, there some drawbacks. For example, in these models the causal structure of spacetime is broken down, and consequently they spoil one of the most important principle of physics. Not to mention the apparent inconsistency of assuming simultaneously the invariance of the speed of light and the breakdown of Lorentz symmetry. Such Lorentz violation changes also the notions of angular momentum and spin, posing some problems for the interpretation of fields and particles as representations of the Poincaré group. Furthermore, because the energy-momentum relations (also called dispersion relations) are not invariant under change of scale of mass, energy and momentum, these models are applicable only near the Planck scale, giving rise to a kind of patchwork special relativity, with different theories governing the kinematics at different scales.
For the case of physics at the cosmological scale, a cosmological term Λ can be considered the simplest and most economical explanation for the observed acceleration in the universe expansion rate. On the other hand, the most simple and natural way to incorporate a cosmological term into spacetime is arguably to assume that, instead of Poincaré, spacetime kinematics is ruled by the de Sitter group. As is well known, the de Sitter group has Lorentz as subgroup, and consequently preserves the causal structure of spacetime. Furthermore, it involves an invariant length-parameter related to the cosmological term. Accordingly, we can say that the most simple and natural spacetime kinematics that preserves the Lorentz symmetry and involves an invariant length-parameter (in addition to the speed of light), is that provided by the de Sitter special relativity. This theory can then be considered a natural alternative relativity to deal with the existence of invariant length-parameters in physics. It should be emphasized that the existence of an invariant length-parameter does not necessarily requires violation of the Lorentz symmetry.
An important property of the de Sitter special relativity is that its dispersion relation is invariant under change of scale of mass, energy and momentum. As a consequence, it can be applied at any energy scale, from cosmology to quantum gravity, being in this sense universal. For this reason, it is a natural alternative to deal with the existence of invariant length-parameters at any scale of energy. Notice furthermore that, if kinematics is to be ruled by the de Sitter group, the underlying de Sitter spacetime is no longer to be viewed as a dynamical solution of Einstein equation. This is quite natural because, as a quotient space, it is more fundamental than Einstein equation in the sense that, like Minkowski, it is known a priori, independently of any gravitational theory.
Another important property of the de Sitter special relativity is that it introduces the conformal transformation in the spacetime kinematics, and consequently the conformal current into the source of spacetime curvature. In fact, whereas Minkowski spacetime in transitive under spacetime translations, the de Sitter spacetime is transitive under a combination of translations and proper conformal transformations. Instead of violating the Lorentz symmetry, the de Sitter special relativity presupposes a violation of the spacetime translations. This property has concomitant consequences for general relativity: whereas the energy-momentum current keeps its general relativity role of dynamical source of ordinary curvature, the conformal current appears as the algebraic source of the de Sitter curvature, that is to say, of the cosmological constant. The de Sitter special relativity, therefore, allows a new and consistent interpretation of the dark energy as an entity which is encoded in the structure of the kinematic group of spacetime, rather than produced by the energy-momentum tensor of an exotic dynamical entity that fills uniformly the whole Universe. Its source in de Sitter relativity is the conformal current of ordinary matter.
We are currently developing further the conceptual basis of the theory, as well as trying to apply it to both cosmological and quantum gravity problems.