On non-vacuum black holes in new general relativity

This paper demonstrates that static, spherically symmetric black hole solutions in New General Relativity containing matter or electromagnetic fields are only mathematically possible in parameter regimes that suffer from fundamental theoretical pathologies, thereby implying the theory admits no physically meaningful black holes distinct from those in General Relativity.

The Gist

Imagine the universe as a giant, stretchy trampoline. For decades, physicists have used General Relativity (GR) to describe how heavy objects (like stars and black holes) warp this trampoline, creating gravity. In this old view, gravity is caused by the curvature of the fabric.

However, a newer group of physicists has proposed a different way to look at the trampoline. They call it New General Relativity (NGR). Instead of looking at how the fabric curves, they look at how it twists. Think of it like this: if General Relativity is about bending a rubber sheet, New General Relativity is about twisting the threads of the sheet.

This paper is a detective story where the authors try to see if this "twisting" theory can explain black holes—the most extreme objects in the universe.

The Mystery: Can "Twisting" Gravity Make Black Holes?

The authors wanted to know: If we use this new "twisting" theory, can we build a black hole that looks and behaves like the ones we see in the sky, but without breaking the laws of physics?

To solve this, they had to check three strict "rules of the road" that any good theory of gravity must follow:

1. No Ghosts: The theory shouldn't predict "ghost particles" (unstable energy that makes the universe explode).
2. Gravitational Waves: The theory must allow ripples in gravity (like the ones detected by LIGO) to travel through space.
3. Newtonian Limit: The theory must match our everyday experience of gravity (like apples falling) when things aren't moving too fast or are too heavy.

The Investigation: The "Black Hole" Test

The authors set up a mathematical experiment. They tried to build a black hole using the "twisting" rules of NGR. They checked two scenarios:

• The Empty Room (Vacuum): A black hole in a completely empty universe.
• The Busy Room (Non-Vacuum): A black hole surrounded by gas, dust, or electric fields (like a real star collapsing).

They used a special mathematical tool called perturbation theory. Imagine trying to balance a pencil on its tip. You start with the pencil perfectly upright (the "leading order"), and then you wiggle it slightly to see if it stays balanced or falls over. They wiggled their equations near the edge of the black hole (the "horizon") to see if the theory held up.

The Findings: A Dead End

Here is the twist in the story: The theory failed.

Every time they tried to build a black hole that wasn't just a copy of the old General Relativity, the math forced the theory into a corner where it broke the rules.

• The "Ghost" Trap: To make the black hole work, the theory had to choose parameters that created "ghosts." It's like trying to build a house, but the only way to make the roof stay up is to use bricks that are actually made of smoke. The house might look okay for a second, but it's fundamentally unstable.
• The "Silent" Trap: In other cases, the theory worked mathematically, but it stopped allowing gravitational waves to travel. It's like building a radio that can receive signals but can't broadcast them. Since we know the universe does broadcast gravitational waves, this version of the theory is wrong.
• The "Broken Scale" Trap: In other scenarios, the theory couldn't explain why an apple falls to the ground the way Newton described. It was like a scale that works for heavy rocks but says a feather weighs a ton.

The Conclusion: Back to the Drawing Board

The authors concluded that New General Relativity cannot create "new" black holes.

If you try to use this twisting theory to describe a black hole, you are forced to turn the "twist" off completely. When you turn the twist off, the theory just becomes the old, standard General Relativity we already know.

The Analogy:
Imagine you are trying to invent a new type of car engine that runs on "twisting" air instead of burning fuel. You build a prototype, but every time you try to drive it, the engine either explodes (ghosts), makes no sound (no waves), or refuses to move (no Newtonian limit). The only way to make the car drive is to remove the "twisting" part entirely and put in a standard engine.

Why This Matters

This paper is important because it tells us that while "twisting" gravity is an interesting idea, it likely cannot explain the black holes we see in the universe unless it is exactly the same as the old theory.

It doesn't mean the idea is useless for everything (it might still work for the early universe or cosmology), but for black holes, the "twist" doesn't seem to add anything new. The universe, it seems, prefers the classic "curved" explanation for its darkest secrets.

In short: The authors tried to find a new, cooler way to describe black holes using a "twisting" theory of gravity, but they found that the math forces the theory to collapse into the old, standard version. No new black holes allowed!

Technical Summary

1. Problem Statement

The paper investigates whether New General Relativity (NGR), a torsion-based modification of General Relativity (GR), admits physically meaningful non-vacuum black hole solutions that are distinct from the Teleparallel Equivalent of General Relativity (TEGR).

• Context: NGR is defined by a Lagrangian depending on three free parameters ($c_a, c_v, c_t$) associated with irreducible torsion scalars. While TEGR is dynamically equivalent to GR, NGR allows for deviations.
• Previous Findings: A prior study by the authors [5] demonstrated that vacuum static, spherically symmetric (SSS) black hole solutions in NGR only exist when the theory parameters are fixed to values corresponding to pathological models (those containing ghost instabilities, lacking spin-2 propagation, or failing to recover the Newtonian limit).
• The Open Question: The authors hypothesize that the presence of matter (a perfect fluid and/or electromagnetic field) might relax the algebraic constraints imposed by the vacuum field equations, potentially allowing for physically viable black hole solutions within the "healthy" region of the NGR parameter space.

2. Methodology

The authors employ a rigorous perturbative analysis within a fully covariant teleparallel framework.

A. Geometric Framework

• Geometry: They utilize the tetrad formalism with a flat, metric-compatible spin connection.
• Symmetry: They assume Static, Spherically Symmetric (SSS) configurations. The geometry is defined by four arbitrary functions: $A_1(r), A_2(r), \chi(r), \psi(r)$.
• Horizon Definition: Instead of relying on coordinate-dependent conditions (like $g_{tt}=0$), they define a Local Horizon (LH) via the vanishing of the expansion scalar of outgoing null rays ($\theta(\ell) = 0$). This is a coordinate-independent condition essential for teleparallel theories.

B. Perturbative Approach

• Expansion: To analyze the behavior near the horizon ($r = r_h$), they introduce a perturbative parameter $\epsilon$ such that $r = r_h + \epsilon$.
• Ansatz: The geometric functions and matter fields (pressure $P$, density $\rho$) are expanded in powers of $\epsilon$.
  • $A_2 \sim \epsilon^{-p}$, $A_1 \sim \epsilon^q$, $\chi \sim \epsilon^u$, $\psi \sim \epsilon^v$.
  • Matter fields are expanded as $P \sim \epsilon^y$ and $\rho \sim \epsilon^z$.
• Conservation: The energy-momentum tensor conservation ($\nabla_\mu \Theta^{\mu\nu} = 0$) is enforced, linking the exponents of the matter fields ($y=z$) to the geometric exponents.

C. Field Equations

The analysis solves the Antisymmetric Field Equations (AFE) and Symmetric Field Equations (SFE) order by order in $\epsilon$:

1. AFE: Constraints on the geometry independent of matter.
2. SFE: Coupled equations involving the energy-momentum tensor ($\Theta_{\mu\nu}$) of a perfect fluid and an electromagnetic field.

3. Key Contributions

1. Extension to Non-Vacuum: This is the first comprehensive analysis extending the vacuum black hole constraints of NGR to non-vacuum scenarios (perfect fluids and electrovacuum).
2. Systematic Branch Classification: The authors systematically classify solution branches based on the leading-order behavior of the perturbative expansion. They identify 18 inequivalent branches satisfying the leading-order equations.
3. Parameter Space Filtering: They rigorously test these branches against the physical viability criteria for NGR:
   • Ghost-freedom: Absence of unstable modes.
   • Spin-2 Propagation: Ability to support gravitational waves.
   • Newtonian Limit: Consistency with solar-system tests (recovering GR at low energies).
4. Higher-Order Consistency Check: Crucially, the paper does not stop at leading order. It analyzes next-to-leading-order terms to check for internal consistency, revealing that many branches that appear valid at leading order fail at higher orders.

4. Results

The analysis yields a definitive negative result regarding the existence of novel black holes in NGR.

• Vacuum Confirmation: The study confirms previous findings: vacuum solutions only exist in unphysical parameter regions (Type I: ghosts; Type II: no spin-2 mode).
• Non-Vacuum Analysis:
  • Starting with 18 candidate branches satisfying the leading-order SFE and conservation laws, the authors apply physical and analytical filters.
  • Discarded Branches: 12 branches were discarded immediately due to geometric inconsistencies (e.g., $\alpha_1=0$ implying the horizon escapes the perturbative description) or equivalence to TEGR.
  • Remaining Candidates: 6 branches (A.2, A.3, A.4, F.2, H.2) survived the leading-order filter.
  • Higher-Order Failure: Upon analyzing next-to-leading-order terms:
    • Branch A.2, A.3, A.4, F.2, H.2: All required the parameter $b_1$ (a reparametrization of the NGR constants) to take values outside the physically admissible region ($-2/3 < b_1 \leq 0$).
    • Specifically, solutions required $b_1 = 2$ (no Newtonian limit), $b_1 = -2/3$ (Type II, no spin-2 mode), or $b_1 < -2/3$ (ghosts).
    • In the electrovacuum (Reissner-Nordström analogue) limit, the constraints were even stricter, forcing unphysical parameter values.
• Conclusion: No physically consistent black hole solutions exist in NGR that are distinct from TEGR. The mere existence of a horizon in these configurations forces the theory into pathological regimes.

5. Significance

• Theoretical Constraints: The paper establishes a strong "no-go" theorem for NGR black holes. It demonstrates that the presence of a horizon acts as a rigid constraint that eliminates the possibility of viable modified gravity black holes within this specific class of theories.
• Distinction from TEGR: It reinforces that while NGR is a generalization of TEGR, it does not offer a viable alternative for describing black hole spacetimes without sacrificing fundamental physical requirements (stability, wave propagation, or classical limits).
• Methodological Rigor: The work highlights the importance of checking higher-order perturbative terms. Many branches that looked promising at the leading order collapsed when subleading terms were considered, a nuance often missed in simpler analyses.
• Future Directions: The authors note that their conclusion is specific to static, spherically symmetric configurations with minimal coupling. They suggest that relaxing these assumptions (e.g., axial symmetry, time-dependence, or non-minimal couplings) might still yield viable solutions, though the current results cast doubt on the viability of NGR as a black-hole-forming theory in its standard form.

Final Conclusion: Within the class of static, spherically symmetric models with minimal matter coupling, New General Relativity does not admit physically meaningful non-trivial black holes. Any solution admitting a horizon inevitably reduces to TEGR or corresponds to a pathological theory with ghosts, no gravitational waves, or no Newtonian limit.