On black holes in new general relativity

The paper demonstrates that all physically viable models of New General Relativity, including TEGR and the Hayashi-Shirafuji model, inevitably exhibit divergences in torsion scalars at local horizons, thereby obstructing their interpretation as valid black hole configurations.

The Gist

The Big Picture: A New Map for Gravity

Imagine gravity as a landscape. For nearly a century, we've used General Relativity (GR) as our map. In this old map, gravity is like the curvature of a trampoline; heavy objects bend the fabric, and things roll toward them. This map works great for planets and stars, and it predicts Black Holes—places where the trampoline is bent so sharply that nothing, not even light, can escape.

But physicists have been wondering: Is there another way to draw this map?

Enter New General Relativity (NGR). This is a "teleparallel" theory. Instead of bending the fabric (curvature), imagine gravity as twisting the fabric (torsion). Think of it like a rubber band that is stretched and twisted rather than just bent.

The authors of this paper asked a simple but crucial question: If we use this "twisting" map (NGR) to draw a Black Hole, does the map break down?

The Experiment: Drawing the Black Hole

To test this, the authors tried to draw a static, spherical black hole (like the classic Schwarzschild black hole) using the rules of NGR. They looked for a specific spot called the Event Horizon.

• The Analogy: Imagine the Event Horizon as the "point of no return" on a waterfall. Once you cross it, you can't swim back up. In physics, this is where light stops being able to escape.

In the old map (General Relativity), the waterfall is smooth right up to the edge. You can cross it without the water suddenly turning into fire.

In the new map (NGR), the authors tried to calculate what happens to the "twist" (torsion) as you approach this edge.

The Discovery: The Map Explodes

The results were shocking. The authors found that in almost every version of this new theory that makes physical sense, the "twist" becomes infinite right at the edge of the black hole.

• The Metaphor: Imagine you are driving toward a bridge. In General Relativity, the bridge is solid all the way to the other side. In New General Relativity, as you approach the bridge, the road suddenly turns into a vertical wall of infinite height. The math "blows up."

In physics, when a number becomes infinite (diverges), it usually means the theory has broken down. It's like a calculator saying "Error" because you tried to divide by zero.

The Specific Culprits

The paper looked at several specific versions of this new theory:

1. The "Standard" Twist (TEGR): This is the version that is mathematically identical to Einstein's General Relativity in most ways.
   • Result: Even here, the "twist" explodes at the horizon. It's like having a perfect map that suddenly turns into static noise right at the destination.
2. The "One-Parameter" Twist (Hayashi & Shirafuji): A popular, slightly modified version.
   • Result: Same problem. The math breaks at the horizon.
3. The "Safe" Twists: There were a few weird, highly specific versions of the theory where the math didn't explode at the horizon.
   • Result: However, these versions had other problems. They were "ghosts" (unstable, imaginary particles that shouldn't exist) or they couldn't explain how gravity works in our solar system (no Newtonian limit). They were mathematically safe at the horizon but physically useless everywhere else.

The Conclusion: No Black Holes in This Theory

The authors conclude that New General Relativity cannot describe a black hole.

Here is the logic in plain English:

• A black hole is defined by having a horizon (the point of no return) and a region inside it.
• In NGR, if you try to draw this, the "twist" of space becomes infinite at the horizon.
• Because the math breaks there, that part of the universe (the horizon and the inside) effectively ceases to exist in the theory's description.
• If the theory says the black hole doesn't exist because the math explodes, then the theory cannot be a valid description of a black hole.

Why Does This Matter?

Think of it like testing a new engine design. You build a prototype (New General Relativity) and run it on a test track.

• It drives fine on the highway (cosmology/universe expansion).
• But the moment you try to drive it into a tunnel (a black hole), the engine explodes.

The paper tells us that while this new theory might be interesting for other things, it fails the "Black Hole Test." It cannot consistently describe the most extreme objects in the universe without breaking its own rules.

Summary in One Sentence

The paper proves that in "New General Relativity," the mathematical description of a black hole's edge causes the theory to explode into infinity, meaning this specific theory cannot actually describe black holes as real, physical objects.

Technical Summary

1. Problem Statement

The paper addresses a fundamental issue in Teleparallel Gravity (TG) theories, specifically New General Relativity (NGR): the inability to consistently describe static, spherically symmetric black hole configurations.

• Context: In General Relativity (GR), the Schwarzschild solution describes a black hole with a smooth event horizon and a singularity only at the origin ($r=0$). In TG, where gravity is described by torsion rather than curvature, previous studies (including on the Teleparallel Equivalent of General Relativity, TEGR, and $F(T)$ gravity) have shown that torsion invariants often diverge not just at the origin, but also at the local horizon (the apparent horizon).
• The Conflict: If torsion scalars diverge at the horizon, the horizon and the region inside it cannot be part of the spacetime manifold. This implies that standard black hole solutions do not exist in these theories, or the theories are physically ill-defined in the strong-field regime.
• Objective: The authors investigate whether this "horizon singularity" is an unavoidable feature of New General Relativity (NGR), a broad class of TG theories defined by three free parameters ($c_v, c_a, c_t$) acting on irreducible torsion scalars ($V, A, T$). They aim to determine if any physically viable NGR model can support a black hole geometry without singular torsion invariants at the horizon.

2. Methodology

The authors employ a rigorous perturbative analysis within the framework of static, spherically symmetric teleparallel geometries.

• Geometric Framework:
  • They utilize the most general diagonal tetrad and spin connection compatible with static spherical symmetry, defined by arbitrary functions $A_1(r), A_2(r), A_3(r)$ and spin connection functions $\chi(r), \psi(r)$.
  • They define the local horizon (Apparent Horizon) not by a coordinate-dependent metric function (like $g_{tt}=0$), but by the geometric condition that the expansion of outgoing null geodesics vanishes: $\theta(\ell) = 0$.
• Field Equations:
  • They derive the Field Equations (FE) for NGR, separating them into Antisymmetric (AFE) and Symmetric (SFE) parts. The AFE imposes constraints on the spin connection functions $\chi$ and $\psi$, while the SFE governs the metric functions $A_1$ and $A_2$.
  • They introduce a reparameterization ($b_1, b_2, b_3$) to simplify the algebraic structure of the FE.
• Perturbative Approach:
  • To analyze the behavior near the horizon ($r = r_h$), they introduce a small parameter $\epsilon$ such that $r = r_h + \epsilon$.
  • They assume power-law expansions (ansatz) for the arbitrary functions $A_1, A_2, \chi,$ and $\psi$ near the horizon (e.g., $A_2 \sim \epsilon^{-p}$).
  • They solve the AFE and SFE order-by-order in $\epsilon$ to identify consistent solution branches.
• Scalar Analysis:
  • For each valid solution branch, they evaluate the behavior of the torsion scalars ($V, A, T$) and the torsion scalar $T$ as $\epsilon \to 0$.
  • They check for divergences. If any scalar diverges at the horizon, the geometry is deemed singular and the black hole configuration is considered non-viable.

3. Key Contributions

• Systematic Classification of NGR Models: The paper provides a comprehensive classification of all possible static, spherically symmetric vacuum solutions in NGR, categorized by the values of the free parameters ($b_1, b_2, b_3$) and the behavior of the spin connection functions.
• Resolution of the Horizon Singularity Problem: The authors demonstrate that for the most physically relevant and viable models (including TEGR and the Hayashi-Shirafuji model), the torsion scalars inevitably diverge at the local horizon.
• Identification of "Pathological" Viable Models: They identify specific sub-classes of NGR (Cases 3, 4, and 5) where the geometry remains finite at the horizon. However, they prove that these models suffer from severe physical defects: they either lack a Newtonian limit, fail to propagate spin-2 gravitational waves, or contain ghost instabilities.
• Refinement of Horizon Definition: The work reinforces the use of the expansion scalar $\theta(\ell)$ as the robust, coordinate-independent definition of a local horizon in TG, showing how it interacts with the torsion invariants.

4. Results

The analysis yields five distinct solution branches (Cases 1–5) based on the perturbative expansion. The results are summarized as follows:

• Case 1 (TEGR): Corresponds to the Teleparallel Equivalent of General Relativity.
  • Result: The torsion invariants $V$ and $T$ diverge at the horizon for generic choices of the spin connection. Even in specific sub-cases where $T$ might remain finite, $V$ diverges.
  • Conclusion: TEGR cannot describe a black hole; the horizon is a geometric singularity.
• Case 2 (1P-H&S Model): The one-parameter Hayashi and Shirafuji model.
  • Result: Effectively reduces to TEGR due to the constraints on the spin connection ($\chi = n\pi$).
  • Conclusion: Like TEGR, it exhibits divergent torsion at the horizon and cannot support black holes.
• Cases 3, 4, and 5 (Other NGR Models):
  • Result: These models allow for finite torsion scalars at the horizon ($r=r_h$). The geometry is well-behaved, and the interior region is part of the manifold.
  • Physical Viability: Despite the geometric regularity, these models are physically unacceptable:
    • Case 3 & 5: They correspond to models that do not propagate spin-2 fields (no gravitational waves) or contain ghost instabilities (violating ghost-free conditions).
    • Case 4: Corresponds to a model with no Newtonian limit.
  • Conclusion: While geometrically possible, these "black holes" exist only in theories that are physically inconsistent with observation or fundamental stability.

Table 4 Summary: The paper explicitly tabulates that for all ghost-free, physically viable models (Types 2 and 6 in the NGR classification), the scalars $V$ and/or $T$ diverge as $\epsilon \to 0$.

5. Significance

• Negative Result for Black Holes in TG: The paper provides strong evidence that no physically viable teleparallel theory (specifically within the NGR class) can consistently describe a static black hole. The "horizon singularity" is not an artifact of a specific choice of frame but a fundamental feature of the theory's interaction with local horizons.
• Implications for Modified Gravity: This challenges the viability of NGR and related TG theories as alternatives to GR for describing strong-field astrophysical objects. If a theory cannot describe a black hole (a key prediction of GR), its status as a viable gravitational theory is severely compromised.
• Distinction between Geometry and Physics: The work highlights a crucial distinction: a theory can be mathematically capable of producing a regular horizon geometry (Cases 3–5), but if it fails to propagate gravitational waves or introduces ghosts, it is physically irrelevant. Conversely, the most physically robust theories (TEGR, 1P-H&S) fail geometrically at the horizon.
• Future Directions: The authors suggest that the only way to resolve this might be to find a specific "proper" tetrad or spin connection that avoids these divergences, or to reconsider the definition of horizons in TG. However, their analysis suggests such a solution is unlikely within the current NGR framework.

In conclusion, the paper demonstrates that New General Relativity, including its most popular sub-cases, inevitably leads to singular torsion invariants at the local horizon for physically viable models, thereby precluding the existence of well-defined black hole configurations in these theories.