⚛️ general relativity

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

This paper calculates the mass and angular momentum of the Kerr black hole within the Teleparallel Equivalent (TEGR) and Symmetric Teleparallel Equivalent (STEGR) of General Relativity by applying covariant Noether charge formalisms with specific "turning off" gravity gauges, successfully recovering expected values while revealing limitations in satisfying Einstein's equivalence principle for rotating solutions.

Original authors: E. D. Emtsova, A. N. Petrov, A. V. Toporensky

Published 2026-01-26

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CC BY 4.0

Original authors: E. D. Emtsova, A. N. Petrov, A. V. Toporensky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Measuring a Spinning Black Hole

Imagine a black hole not just as a dark hole, but as a massive, spinning top in space. This specific type of black hole is called the Kerr black hole. It has two main "weights" we want to measure:

Mass: How heavy it is (like the weight of the top).
Angular Momentum: How fast and hard it is spinning (like the spin of the top).

In standard physics (General Relativity), calculating these numbers is tricky because gravity isn't a force you can just put on a scale; it's the shape of space itself. This paper explores two alternative ways of describing gravity, called TEGR and STEGR. Think of these as two different "languages" or "maps" used to describe the same terrain. The authors wanted to see if these new maps could accurately measure the weight and spin of the Kerr black hole.

The Problem: The "Empty" Background

In these new theories (TEGR and STEGR), gravity is described using a "flat" background, like a perfectly smooth, empty sheet of paper. However, a real black hole bends that paper. To do the math, the authors have to decide: What does the paper look like if we magically turn off the black hole's gravity?

This is where they use a concept called "Turning Off Gravity."

Analogy: Imagine you are trying to measure the weight of a heavy backpack. To do it, you need to know what the scale reads when the backpack is empty.
The Twist: In these theories, there isn't just one way to "empty" the backpack. You can take out the heavy books (mass) but leave the spinning wheels (spin), or you can take out everything.
The "Gauge" Concept: The authors call these different ways of emptying the backpack "gauges." It's like choosing different reference points. If you choose the wrong reference point, your measurement might be wrong.

The Journey: Testing Different Maps

The authors tested several different "gauges" (different ways of defining the empty background) to see which one gave the correct answer for the black hole's mass and spin.

1. The First Attempt (Gauge I & I*)

They tried a simple way to turn off gravity.

Result for Mass: They got the correct weight ( $M$ ).
Result for Spin: They got the wrong answer. It was like measuring a spinning top and saying it's spinning at 1/3rd of its actual speed.
Verdict: This gauge failed to capture the full spin.

2. The Second Attempt (Gauge II & II*)

They tried a more complex way to set up the "empty" background.

Result for Mass: Correct! ( $M$ )
Result for Spin: Correct! ($aM$)
Verdict: Success! By choosing the right "gauge," they could measure both the weight and the spin perfectly in both TEGR and STEGR.

Key Finding: The paper proves that even though these theories are complex, if you pick the right "reference frame" (gauge), you can get the exact same correct answers for a spinning black hole that standard physics predicts.

The Equivalence Principle Test: The "Free Fall" Experiment

The authors also tried to test a famous rule called the Equivalence Principle.

The Rule: If you are falling freely in space (like an astronaut in orbit), you shouldn't feel any gravity. You should feel weightless.
The Test: They tried to calculate the "force" felt by an observer falling into the black hole. If the theory is perfect, the math should show zero force (weightlessness).
The Result: It failed. Even when they used special coordinates designed for falling observers (called Doran coordinates), the math showed a non-zero force.
Why? The force they calculated was directly linked to the black hole's spin (the parameter $a$ $a$ ).
- Analogy: Imagine falling into a whirlpool. Even if you are falling freely, the spinning water pushes you sideways. The authors found that in these theories, the "spin" of the black hole creates a leftover "push" that shouldn't be there if the Equivalence Principle held perfectly for this specific setup.
The Silver Lining: When they turned off the spin (making the black hole non-spinning, like a Schwarzschild black hole), the force disappeared, and the rule worked. This helped them discover a new, correct way to describe a non-spinning black hole in STEGR that they hadn't found before.

Summary of Conclusions

Success: The authors successfully calculated the correct mass and angular momentum (spin) for a spinning black hole using two alternative gravity theories (TEGR and STEGR), provided they chose the correct "gauge" (reference frame).
Failure: They could not make these theories perfectly match the "Equivalence Principle" (feeling weightless while falling) for a spinning black hole. The spin itself seemed to break the rule in their calculations.
New Discovery: While trying to fix the spinning problem, they accidentally found a new, correct way to describe a non-spinning black hole in STEGR that satisfies the Equivalence Principle.

In short: They found the right "ruler" to measure a spinning black hole's weight and spin in these new theories, but they also found that the spin makes the "weightless falling" rule a bit wobbly in these specific mathematical frameworks.

Technical Summary: Mass and Angular Momentum for the Kerr Black Hole in TEGR and STEGR

Problem Statement
The paper addresses the challenge of constructing covariant and Lorentz-invariant conserved quantities (specifically mass and angular momentum) for the Kerr black hole solution within the frameworks of the Teleparallel Equivalent of General Relativity (TEGR) and the Symmetric Teleparallel Equivalent of General Relativity (STEGR). While these theories are dynamically equivalent to General Relativity (GR), they utilize non-dynamical connections (inertial spin connections in TEGR and symmetric teleparallel connections in STEGR) that are not determined by the field equations themselves. A significant gap in the literature exists regarding the application of fully covariant formalisms to the Kerr solution; previous works often relied on non-covariant formulations or failed to simultaneously satisfy requirements for coordinate covariance, local Lorentz invariance, and well-defined global charges. Furthermore, attempts to verify the Einstein equivalence principle (where the energy-momentum current vanishes for a freely falling observer) in these teleparallel frameworks for rotating black holes had yielded inconclusive or unsuccessful results.

Methodology
The authors apply a recently developed formalism based on Noether's theorem, utilizing spacetime-covariant and Lorentz-invariant expressions for conserved currents, superpotentials, and charges. The methodology involves several key steps:

Formalism Application: The authors employ the Noether procedure with an arbitrary displacement vector $\xi$ . By choosing $\xi$ as a timelike Killing vector, they calculate total energy (mass); by choosing a rotating spacelike Killing vector, they calculate angular momentum.
Defining Connections via "Turning Off" Gravity: Since the teleparallel connections (Inertial Spin Connection, ISC, in TEGR; Symmetric Teleparallel Connection, STC, in STEGR) are external structures, the authors utilize the "turning off" gravity principle. This involves:
- Selecting a specific tetrad (for TEGR) or coordinate system (for STEGR).
- Calculating the associated Levi-Civita connection (or curvature).
- Solving for the parameters of the solution (mass $M$ and spin parameter $a$ ) such that the curvature vanishes (representing "no gravity").
- Using these parameter values to define the flat teleparallel connection for the full solution.
Gauge Analysis: The authors investigate the concept of "gauges," defined as equivalence classes of pairs (tetrad/connection or coordinates/connection). They test two distinct methods of "turning off" gravity:
- Limit $M \to 0$ (retaining rotation $a$ ).
- Limit $M \to 0$ and $a \to 0$ (reducing to the Schwarzschild limit).
  They apply these methods to different tetrads in TEGR and different coordinate systems in STEGR to determine if specific gauges yield physically meaningful results.
Equivalence Principle Test: The authors test the equivalence principle by calculating the Noether current for a freely falling observer (using proper vectors) in both Boyer-Lindquist and Doran coordinates. A vanishing current is required to satisfy the principle.

Key Contributions and Results

Conserved Quantities in TEGR:
- The authors identified four distinct gauges (combinations of tetrads and ISCs).
- Two gauges (Gauge I and I*) yielded the correct black hole mass ( $M$ ) but failed to reproduce the correct angular momentum, resulting in $aM/3$.
- Two other gauges (Gauge II and II*), utilizing a more complex tetrad structure, successfully yielded both the correct mass ( $M$ ) and the correct angular momentum ($aM$).
- This demonstrates that the choice of gauge is critical in TEGR for recovering the correct angular momentum of the Kerr black hole.
Conserved Quantities in STEGR:
- Two different gauges (Gauge SI and SI*) were constructed using different methods of turning off gravity.
- Both gauges successfully yielded the correct mass ( $M$ ) and angular momentum ($aM$).
- The authors note that the total superpotential in STEGR consists of the Komar superpotential (which alone gives half the mass, known as "Komar's anomaly") and a divergence term. In their calculation, the divergence term contributes an equal amount to the Komar term, correcting the mass to the expected value $M$ .
Equivalence Principle Analysis:
- The authors attempted to find a gauge where the Noether current vanishes for a freely falling observer in the Kerr solution.
- Using both Boyer-Lindquist and Doran coordinates (the latter chosen because the proper time of a free-falling observer coincides with coordinate time), the calculated currents were non-zero in both TEGR and STEGR.
- The non-zero current components were found to be proportional to the spin parameter $a$ .
- Schwarzschild Limit: When taking the limit $a \to 0$ , the current vanishes in both theories, recovering the equivalence principle for the non-rotating Schwarzschild solution.
- Novelty in STEGR: While the authors previously failed to find a well-defined gauge for the Schwarzschild solution in STEGR using Lemaitre coordinates, the current study successfully identified a well-defined gauge (Gauge SII**) derived from the Kerr solution's limit that satisfies the equivalence principle for the Schwarzschild case.

Significance and Claims
The paper claims to successfully resolve the calculation of mass and angular momentum for the Kerr black hole in both TEGR and STEGR using a fully covariant and invariant formalism. The primary significance lies in demonstrating that while the field equations of these theories are equivalent to GR, the construction of conserved quantities requires a careful selection of "gauges" (specifically the non-dynamical connections). The authors highlight that:

Correct angular momentum in TEGR is not guaranteed by all valid gauges; specific tetrad/connection combinations are required.
STEGR naturally resolves the "Komar anomaly" for mass through the inclusion of the divergence term in the superpotential.
The failure to satisfy the equivalence principle for the Kerr solution (non-vanishing current for free fall) is attributed to the presence of angular momentum ( $a$ ), as the principle is restored when $a \to 0$ .
A new, well-defined gauge for the Schwarzschild solution in STEGR was discovered as a byproduct of the Kerr analysis, which had previously eluded the authors.

The authors explicitly state that their attempts to find a gauge corresponding to the equivalence principle for the rotating Kerr solution were unsuccessful in both theories, suggesting that the rotation parameter introduces complexities that prevent the current from vanishing for freely falling observers in these specific teleparallel formulations.