On conserved quantities for the Schwarzschild black hole in teleparallel gravity

Abstract

We examine various methods of constructing conserved quantities in the Teleparallel Equivalent of General Relativity (TEGR). We demonstrate that in the covariant formulation the preferred method are the Noether charges that are true invariant quantities. The Noether charges depend on the vector field $$\xi $$ ξ and we consider two different options where $$\xi $$ ξ is chosen as either a Killing vector or a four-velocity of the observer. We discuss the physical meaning of each choice on the example of the Schwarzschild solution in different frames: static, freely falling Lemaitre frame, and a newly obtained generalised freely falling frame with an arbitrary initial velocity. We also demonstrate how to determine an inertial spin connection for various tetrads used in our calculations, and find a certain ambiguity in the “switching-off” gravity method where different tetrads can share the same inertial spin connection.

The Geometrical Trinity of Gravity

“…However, the proposal is at odds with the usual considerations of energy in metric-teleparallel gravity [12][13][14][15]. Though the gravitational energy-momentum introduced in the metric-teleparallel theory is generally covariant, it is not Lorentz covariant.…”

Energy and entropy in the Geometrical Trinity of gravity

“…The Lemaître frame has been considered previously in metric teleparallelism [100] and the components of the superpotential have been computed for the regularised spin connection [101],…”

“…According to the Ref. [101], the gravitational energymomentum charges are vanishing in the freely falling frame. Our interpretation is quite different as we maintain that the gravitational energy momentum current should always vanish in an inertial frame and it is only in such a frame that we may compute the physically meaningful result for the gravitational charges.…”

“…which could have been guessed since it is (up to a constant prefactor) the only available non-derivative tensor compatible with the symmetry of l α νµβ . Note, in particular, that since 2∇ α q [αβ] α = ∇ α h αβ α , the kinetic excitation cannot be given by the Noether potential in (101). For the antisymmetrised tensor in the Noether potential we obtain from (112a)…”

Noether currents in theories with higher derivatives and theories with differential field transformation in action (DFTA)

“…In this article we investigate several important issues that can accompany Noether currents in the general case of theories with high order field derivatives. Here we are focused on the original definition, that have been proposed in the famous work of Noether [4] but sill widely used (see, for example, [6][7][8][9][10] and also [11]). It is also much simpler than some others (see, for example, [12]) and is better suited for generalizations.…”