Buchdahl bound, photon ring, ISCO and radial acceleration in Einstein-æther theory

TL;DR

This work analyzes spherically symmetric configurations in the minimal Einstein–æther theory with a Maxwell-like kinetic term, focusing on vacuum exterior solutions and interior matter. The authors derive the Eling–Jacobson exterior solution, revealing that finite $K_{b}$ enlarges central radii such as the ISCO and photon ring while preserving a Schwarzschild-like Newtonian limit; they also show a gauge freedom that allows a diagonal metric and a purely time-like æther in vacuum, with a two-variable reduction to $X$ and $Y$ capturing the physics. Extending to matter, they formulate the Einstein–æther TOV equations, introduce an æther energy density, and derive a Buchdahl bound $\frac{M}{r_{b}} \le \frac{4(1-K_{b})}{(3-2K_{b})^{2}}$ for $0\le K_{b}\le 1/2$, with a saturated analytic solution supporting the bound. Numerical results for uniform density illustrate that the bound tightens with $K_{b}$ and that the radial acceleration relation scales as $g_{\text{EA}} \approx \frac{2}{2-K_{b}}\, g_{N}$ at low pressures, indicating a parallel but amplified Newtonian behavior. Overall, the paper provides exact exterior solutions, a gauge-invariant interior framework, and physically interpretable bounds that inform potential observational tests and connections to ÆST.

Abstract

Spherically symmetric Einstein-æther (EÆ) theory with a Maxwell-like kinetic term is revisited. We consider a general choice of the metric and the æther field, finding that:~(i) there is a gauge freedom allowing one always to use a diagonal metric; and~(ii) the nature of the Maxwell equation forces the æther field to be time-like in the coordinate basis. We derive the vacuum solution and confirm that the innermost stable circular orbit (ISCO) and photon ring are enlarged relative to general relativity (GR). Buchdahl's theorem in EÆ theory is derived. For a uniform physical density, we find that the upper bound on compactness is always lower than in GR. Additionally, we observe that the Newtonian and EÆ radial acceleration relations run parallel in the low pressure limit. Our analysis of EÆ theory may offer novel insights into its interesting phenomenological generalization: Æther--scalar--tensor theory (ÆST).

Figures (4)

• Figure 1: Upper panel: the $\TimeFunc$ and $\SpaceFunc$ functions in \ref{['Schwarzschild']} for various $\Kb$ and fixed $\Rg$, according to the exact EÆ exterior solution in \ref{['TIntegral', 'InverseFunction', 'HorizonFunction']}. Lower panel: the central structures comprise the throat $\Rh$ in \ref{['HorizonFunction']}, the ISCO-like radii $\Risco$ in \ref{['ISCO']}, and the photon ring $\Rl$ in \ref{['Luminal']}. The dimensionless numbers $\zeta_{\pm}$ correspond (in units of $\Rg$) to the upper bounds of the ISCO-like radii in \ref{['ISCORange']}. These radii all connect with their counterparts for the Schwarzschild black hole of GR in the $\Kb\mapsto 0$ limit.
• Figure 2: The change of $h(\Kb,M/r_{\text{b}})$, which is the final inequality of Buchdahl theorem \ref{['eq:final_ineq']} with $0$ being the limit, with different $\Kb$ and $M/r_{\text{b}}$. The dotted line is the analytical solution from \ref{['buchdahl']}. It can be seen that there is a white region near $h(\Kb,M/r_{\text{b}})=0$ between $0<\Kb\leq 1/2$, which corresponds to \ref{['buchdahl']}. For the remaining region, there is no additional bound from \ref{['eq:final_ineq']}.
• Figure 3: The variation of $M_{\text{Sz}}/r_{\text{b}}$ with different values of $\Kb$. The red curve represents the theoretical prediction derived from \ref{['buchdahl']}, while the blue curve corresponds to the result from numerical integration, where a uniform physical density is assumed. The horizontal grey line marks the GR limit of $4/9$ for comparison. The theoretical prediction indicates that the bound slightly exceeds the GR limit for small $\Kb$, while it falls below the GR limit as $\Kb$ increases. In the case of uniform density, the bound is consistently stricter than the GR limit.
• Figure 4: Radial acceleration relation (RAR), where $g_{\text{Dyn}}$ and $g_{\text{Bar}}$ are dynamical acceleration and Newtonian acceleration with baryons only, respectively. The blue line shows a numerical evaluation of the EÆ case with $\Kb=5/3$, while the black dotted line shows the Newtonian case which runs straight to the origin with a slope near unity. The red dashed line shows a prediction from MOND --- see e.g. Famaey:2011kh. The EÆ profile is roughly six times larger than the Newtonian one and parallel to it, which verifies the scaling relation \ref{['eq:scaling']} for this particular $\Kb$.