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Pathology of the Unified Dark Sectors in Modified General Relativity

Mohsen Khodadi

TL;DR

This work analyzes the linear stability of a black hole in Modified General Relativity (MGR), a theory positing a unified geometric origin for dark matter and dark energy via a line element field. Using gauge-invariant perturbation theory (including the Kodama-Ishibashi formalism) and a Zerilli-type master equation, the polar sector is shown to be driven by a sourced coupling to the line element perturbation, with a source term $S_{MGR} = \mathcal{C}(r) \partial_t^2 (\delta f)$ that diverges as $\mathcal{C}(r) \sim r^{5/2}$ in the far field, signaling a strong infrared instability on a non-asymptotically flat background; in contrast, the axial sector remains governed by a homogeneous, positive, well-behaved potential, indicating stability. The analysis demonstrates that the instability arises from the specific polar coupling to the line element field, not from the background metric itself, thereby challenging MGR's viability unless asymptotic flatness or another stabilizing mechanism is introduced. The results motivate exploring modifications such as adding a potential for the line element field or implementing screening to suppress the problematic coupling, as well as studying cosmological backgrounds to assess the theory's broader viability.

Abstract

This paper presents a comprehensive stability analysis of the black hole solution within Modified General Relativity (MGR), a theory proposing a unified geometric description of dark matter (DM) and dark energy (DE). A rigorous gauge-invariant formalism is employed to analyze gravitational perturbations of the extended Schwarzschild metric. The central finding is a critical pathology within the polar perturbation sector, where metric fluctuations couple to the theory's fundamental line element field. This coupling is governed by a factor that, while well-behaved at the horizon, diverges powerfully in the far-field limit as a direct consequence of the theory's non-asymptotically flat nature. This indicates a strong infrared instability that overwhelms perturbations at large distances. In stark contrast, the axial perturbation sector is found to be completely stable. This dichotomy proves that the instability is not inherent to the background metric but is specifically generated by the novel coupling mechanism encoding MGR's unified dark sectors. The results reveal a fundamental strong-coupling problem within the MGR framework, challenging its physical viability as an alternative to Einstein's General Relativity (EGR).

Pathology of the Unified Dark Sectors in Modified General Relativity

TL;DR

This work analyzes the linear stability of a black hole in Modified General Relativity (MGR), a theory positing a unified geometric origin for dark matter and dark energy via a line element field. Using gauge-invariant perturbation theory (including the Kodama-Ishibashi formalism) and a Zerilli-type master equation, the polar sector is shown to be driven by a sourced coupling to the line element perturbation, with a source term that diverges as in the far field, signaling a strong infrared instability on a non-asymptotically flat background; in contrast, the axial sector remains governed by a homogeneous, positive, well-behaved potential, indicating stability. The analysis demonstrates that the instability arises from the specific polar coupling to the line element field, not from the background metric itself, thereby challenging MGR's viability unless asymptotic flatness or another stabilizing mechanism is introduced. The results motivate exploring modifications such as adding a potential for the line element field or implementing screening to suppress the problematic coupling, as well as studying cosmological backgrounds to assess the theory's broader viability.

Abstract

This paper presents a comprehensive stability analysis of the black hole solution within Modified General Relativity (MGR), a theory proposing a unified geometric description of dark matter (DM) and dark energy (DE). A rigorous gauge-invariant formalism is employed to analyze gravitational perturbations of the extended Schwarzschild metric. The central finding is a critical pathology within the polar perturbation sector, where metric fluctuations couple to the theory's fundamental line element field. This coupling is governed by a factor that, while well-behaved at the horizon, diverges powerfully in the far-field limit as a direct consequence of the theory's non-asymptotically flat nature. This indicates a strong infrared instability that overwhelms perturbations at large distances. In stark contrast, the axial perturbation sector is found to be completely stable. This dichotomy proves that the instability is not inherent to the background metric but is specifically generated by the novel coupling mechanism encoding MGR's unified dark sectors. The results reveal a fundamental strong-coupling problem within the MGR framework, challenging its physical viability as an alternative to Einstein's General Relativity (EGR).
Paper Structure (14 sections, 151 equations, 1 figure)

This paper contains 14 sections, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of Black Hole Solution in MGR
  3. Polar Perturbation Analysis
  4. Perturb the Einstein-MGR field equations $\delta G_{\mu\nu} + \delta \Phi_{\mu\nu} = 0$
  5. Master equation using the Zerilli function
  6. Master equation using the gauge-invariant formalism
  7. Toward determining the function $f$
  8. Stability analysis and Numerical Demonstration
  9. Axial Perturbations
  10. discussion and conclusion
  11. Algebraic manipulation of Eq. (\ref{['Zer']})
  12. Gauge-invariant perturbation approach
  13. Treatment of the unit vector field perturbation and constraint structure
  14. A list of some MG theories hosting the strong coupling problem at low-energy limit

Figures (1)

  • Figure 1: Log-log plot of the magnitude of the dominant source term, $| \mathcal{C}(r) \delta f(r) |$, versus radial coordinate $r$. The solid (blue) curve comes from approximation (\ref{['C']}), the dashed (ref) curve from Eq. (\ref{['far']}), $\delta f(r) \sim r^{m_+}$, and the asymptotic form of the coupling factor $\mathcal{C}(r) \sim r^{5/2}$.