Mass and angular momentum for the Kerr black hole in TEGR and STEGR

Abstract We study the energy–momentum characteristics of the rotating black hole–Kerr solution of general relativity in the teleparallel equivalent of general relativity (TEGR) and the symmetric teleparallel equivalent of general relativity (STEGR). The previously constructed spacetime-covariant and Lorentz-invariant expressions for conserved Noether currents, superpotentials, and charges are used. The Noether charges describe the total energy, momentum, or angular momentum of a gravitational system depending on the choice of displacement vector $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> . To define the covariant and invariant conserved quantities in both TEGR and STEGR, one needs to use external fields which are flat teleparallel connections. To determine the non-dynamical connections in TEGR and STEGR, we use the unified “turning-off” gravity principle. In addition, to analyze the Noether conserved quantities in these theories, we use the concept of “gauges.” Changes in the gauge can affect the Noether conserved quantities. We highlight two ways to turn off gravity—by $$M \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and by $$M \rightarrow 0, ~ a \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>a</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> —which give us different gauges in TEGR and STEGR. In both kinds of gauges, we obtain the expected values of black hole mass and angular momentum. Our attempts to find gauges which could lead to a correspondence to Einstein’s equivalence principle for the Kerr solution were unsuccessful in both TEGR and STEGR. However, these exercises helped us to find a related gauge for the Schwarzschild solution in STEGR that is a novel finding.