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.