Covariant Tolman-Oppenheimer-Volkoff equations in Energy-Momentum Squared Gravity

Technical Summary: Covariant Tolman-Oppenheimer-Volkoff Equations in Energy-Momentum Squared Gravity

Problem Statement
General Relativity (GR) faces challenges regarding dark matter, dark energy, and cosmological singularities, motivating the study of modified gravity theories. Energy-Momentum Squared Gravity (EMSG), defined by supplementing the Einstein-Hilbert action with a function of the scalar $T \equiv T_{\mu\nu}T^{\mu\nu}$, offers a framework where vacuum solutions match GR, but matter sources acquire nonlinear self-interactions at high densities. While EMSG has been studied cosmologically and via standard metric approaches for stellar configurations, a covariant formulation for static, spherically symmetric stars within the $1+1+2$ semi-tetrad formalism has been lacking. The authors aim to construct such a formulation to determine if the "effective fluid" picture, successful in cosmology, extends consistently to relativistic stellar structure, and to analyze the resulting phase-space dynamics.

Methodology
The authors employ the $1+1+2$ covariant formalism, which decomposes spacetime relative to a timelike unit vector $u^a$ (fluid four-velocity) and a spacelike unit vector $e^a$ (preferred radial direction). For static, spherically symmetric (LRS-II) configurations, the dynamics are encoded entirely in scalar variables: radial acceleration ($A$), sheet expansion ($\phi$), and the electric part of the Weyl tensor ($E$).

1. Effective Fluid Formulation: The field equations of EMSG are recast into a GR-like form, $G_{\mu\nu} = T^{(eff)}_{\mu\nu}$, where the effective energy-momentum tensor $T^{(eff)}_{\mu\nu}$ retains the perfect fluid form ($\rho_{eff}u_\mu u_\nu + p_{eff}h_{\mu\nu}$). The effective density $\rho_{eff}$ and pressure $p_{eff}$ are algebraic functions of the physical density $\rho$, pressure $p$, and the coupling constants $\eta$ and $n$. Two matter Lagrangian choices are considered: $L_m = p$ (standard) and $L_m = -\rho$ (dust-safe).
2. Covariant TOV Equations: The authors derive the propagation equations for the covariant scalars in EMSG. These equations are formally identical to the GR TOV equations but utilize the effective variables $\rho_{eff}$ and $p_{eff}$ as the gravitational sources.
3. Dimensionless Dynamical System: The system is transformed into a dimensionless autonomous planar dynamical system using normalized variables ($\tilde{K}, \tilde{p}$) derived from the Gaussian curvature and effective pressure. This reduction is possible only if an "effective closure relation" $\tilde{\mu} = \tilde{\mu}(\tilde{p})$ exists, linking effective density to effective pressure.
4. Qualitative Analysis: The authors perform a dynamical systems analysis, identifying nullclines, invariant sets, and critical points (finite and at infinity) to describe the global structure of the stellar phase space. They specialize to linear physical equations of state (EoS), $p = \lambda \rho$, to compare with the GR benchmark.

Key Contributions and Results

• Unified Effective Fluid Framework: The paper demonstrates that for perfect physical fluids, the nonlinear corrections in EMSG can be fully reinterpreted as an effective perfect fluid. Consequently, the covariant stellar equilibrium equations retain the standard TOV structure when expressed in terms of effective variables. This unifies the description of EMSG in both cosmological and stellar sectors.
• Phase-Space Reduction and Closure: The authors establish that the stellar equations reduce to a two-dimensional autonomous system on the $(\tilde{K}, \tilde{p})$-plane if a closure relation exists.
  • Linear EoS Cases: For specific cases (e.g., $n=1/2$ or $n=1$ with $L_m=p$), the system exhibits exact or piecewise linear effective closures. In these sectors, the phase portrait is topologically equivalent to the GR benchmark, though with modified effective slopes ($\lambda_{eff}$).
  • Dust Configurations: A critical finding is that for dust ($p=0$), the effective pressure $p_{eff}$ is generally non-zero (induced by EMSG corrections). Consequently, the planar reduction fails because $\tilde{\mu}$ cannot be written solely as a function of $\tilde{p}$. In these cases, the full three-dimensional covariant flow must be analyzed. The authors show that static dust configurations in EMSG do not possess interior fixed points but are organized as orbits connecting a central asymptotic regime to a vacuum surface.
• Metric Reconstruction and Matching Conditions: The authors map the covariant variables back to standard metric functions ($m_{eff}, \Phi$). They derive the modified TOV equation in terms of effective variables.
  • Surface Definition: A significant result is the identification of the stellar surface. Since EMSG coincides with GR in vacuum ($T=0$), the exterior is Schwarzschild. However, the natural matching condition is $p_{eff}(R) = 0$, not necessarily $p(R) = 0$. For self-bound matter, the physical radius ($p=0$) and the gravitational matching radius ($p_{eff}=0$) may differ, a distinction the authors argue is crucial for interpreting stellar models in EMSG.
• Specific Sector Recoveries:
  • For radiation ($p=\rho/3$) with $n=1$ and $L_m=p$, the theory is exactly equivalent to GR in terms of effective variables, recovering the standard GR phase portrait.
  • For $n=1/2$, the system is piecewise linear, with different effective slopes for positive and negative pressure branches.

Significance and Claims
The paper claims to provide a "global qualitative description" of the stellar phase space in EMSG, moving beyond numerical integration of specific equations of state to a structural understanding of the theory's behavior.

• Unification: It establishes a unified framework where cosmological and stellar tests of EMSG can be interpreted using the same effective-fluid description.
• Limitations of Planar Reduction: The work highlights that while many sectors of EMSG mimic GR's dynamical structure, others (specifically dust) require a higher-dimensional analysis, challenging the assumption that all modified gravity stellar problems can be reduced to planar systems.
• Physical Interpretation of Boundaries: The paper emphasizes that the distinction between the physical surface ($p=0$) and the gravitational surface ($p_{eff}=0$) is a genuine physical consequence of the effective-fluid interpretation, particularly relevant for self-bound matter. This distinction affects mass-radius relations and the definition of the stellar radius in observational comparisons.
• Benchmarking: The study recovers the standard GR benchmark in the limit $\eta \to 0$ and identifies specific sectors (like the radiation sector with $n=1$) where EMSG is exactly equivalent to GR, providing clear benchmarks for future observational constraints.

The authors conclude that while the formalism recovers known results and extends them to general $n$, the primary value lies in the qualitative dynamical systems approach, which reveals the conditions under which EMSG behaves like GR and where it necessitates a fundamentally different treatment (e.g., dust). They note that applying this formalism to realistic equations of state and calculating specific observables (mass-radius relations, tidal deformabilities) constitutes the necessary next step.