The Relativistic Gravitational Field of a Spherically Symmetric Extended Body

Abstract: We investigate the gravitational field of an extended spherically symmetric body within the framework of Extended Relativity (ER), a Lorentz-covariant formulation of relativistic gravity on a Minkowski background. Using a relativistic superposition principle for retarded gravitational fields, we derive an explicit metric for an extended body by integrating the contributions of its mass elements. The resulting metric reproduces the standard gravitational time dilation of a point source and agrees with the classical tests of General Relativity in the appropriate limits. However, unlike the exact Newtonian shell theorem and the Schwarzschild exterior solution, the external field depends weakly on the internal mass distribution through higher-order corrections. These corrections decay rapidly with distance but become significant near compact objects. We analyze the corresponding admissible-velocity geometry, derive the motion equations for test particles, and compare the predictions of extended-body and point-source models. For neutron stars, the corrections noticeably modify the local light-velocity structure near the surface. For the Earth, the corrections are small but produce measurable differences in round-trip light travel times to the International Space Station. The formalism provides a transparent relativistic description of extended gravitational sources and offers a framework for studying relativistic corrections due to internal structure in strong-field and precision-measurement regimes.

• The paper presents a novel Lorentz-covariant metric for extended spherical bodies by integrating retarded field contributions of individual mass elements.
• It details corrections to the Schwarzschild solution, including modified time dilation and alterations of local light cone geometry.
• It highlights practical implications for probing astrophysical interiors and refining precision satellite timing experiments.

Relativistic Gravitational Field of a Spherically Symmetric Extended Body in Extended Relativity

Introduction

This work presents a detailed analysis of the gravitational field generated by an extended, spherically symmetric mass distribution within the framework of Extended Relativity (ER), a Lorentz-covariant formulation of relativistic gravity on a Minkowski background (2605.21551). Unlike General Relativity (GR), which employs general covariance and describes curvature directly via Einstein’s field equations, ER maintains underlying flat geometry and utilizes a Lorentz-covariant description, allowing explicit linear superposition of gravitational fields. This property is leveraged to derive an explicit, non-perturbative metric for extended spherically symmetric bodies by integrating retarded field contributions from mass elements.

Extended Relativity Foundations and Metric Construction

Extended Relativity substitutes the general covariance principle of GR with Lorentz covariance and treats gravitational sources as producing metric deviations $h_{\mu\nu}$ from the Minkowski background $\eta_{\mu\nu}$, yielding a physical metric $g_{\mu\nu} = \eta_{\mu\nu} - h_{\mu\nu}$. The metric deviation of a single point-like source is modeled with a Kerr-Schild-type degenerate symmetric tensor linear in the mass, facilitating a relativistic superposition principle. For a point source at rest, the explicit deviation tensor becomes

$h_{\mu\nu}(x) = 2M l_\mu(x) l_\nu(x)$

where $l_\mu(x)$ constructs the gravitational four-potential from the geometry determined by the retarded position and velocity of the source.

For extended bodies, the total metric deviation is obtained via direct integration (superposition) of all mass contributions:

$h_{\mu\nu}(x) = \int\! 2 m' l_\mu(x) l_\nu(x)\,,$

where $m'$ refers to the distributed mass elements. This construction is analogous to the Liénard–Wiechert potential superposition in electrodynamics.

Explicit Metric of a Spherically Symmetric Extended Body

The explicit deviation tensor outside a spherically symmetric body decomposes into the standard point-mass term and rapidly decaying correction terms parameterized by the geometry (radius $R$) and mass moments of the source. The leading term in the deviation tensor, identical to the Newtonian potential $(\propto 1/p)$, ensures the classical shell theorem holds at leading order and reproduces standard Schwarzschild-like time dilation. Higher-order corrections ($\propto 1/p^3$, $\eta_{\mu\nu}$0, etc.) encode dependence on the internal mass distribution, in contrast to the strict point-mass behavior predicted by the Schwarzschild solution and Newtonian theory.

At distance $\eta_{\mu\nu}$1 from the center of a uniform sphere,

• The time dilation term $\eta_{\mu\nu}$2 matches that of the point mass.
• The spatial metric components acquire additional terms involving the ratios $\eta_{\mu\nu}$3, decaying rapidly with distance.

Consequently, in the far-field limit ($\eta_{\mu\nu}$4), the external metric asymptotically approaches the Schwarzschild (point source) solution. However, near the source—especially for compact objects—the corrections are non-negligible.

Breakdown of the Shell Theorem and Physical Consequences

A central finding is the violation of the Newtonian shell theorem in the relativistic regime: the ER metric for an extended spherically symmetric mass does not preserve the shell theorem or Birkhoff's theorem exactly. The external metric is sensitive to the internal mass distribution through higher-order correction terms, although these effects diminish rapidly with distance.

For neutron stars, with $\eta_{\mu\nu}$5 a few Schwarzschild radii, the corrections to the admissible velocity ellipsoid at the surface are significant. In the outward radial direction, the predicted speed of light is greater in the extended model than in the point-mass case, while in the transverse direction it is lower. For movement toward the center, the deviation induces a small reduction in the speed of light compared to vacuum, a feature absent in the point-mass model. These modifications to the local light cone geometry could be relevant for near-surface propagation of photons in strongly gravitating astrophysical environments.

For weak-field sources such as Earth, extended-body corrections are minute yet measurable. For instance, in the context of time-of-flight experiments between the Earth's surface and the International Space Station (ISS), the round-trip gravitational time delay differs by $\eta_{\mu\nu}$6 picoseconds between the point-mass and extended-mass models. The extended correction is of the same order as the standard GR correction itself, indicating practical relevance for future precision-measurement experiments.

Test Particle Dynamics and Acceleration in the Extended Field

Equations of motion for test particles are rigorously derived using the ER geodesic equation, reformulated in terms of the physical coordinate time parameter. The influence of the extended mass distribution is incorporated into the Christoffel symbols through explicit dependence on the higher-order mass moments, yielding corrections of quadratic and higher order in spatial velocities. The corrections to acceleration are subdominant for non-relativistic velocities but become relevant in strong-field contexts.

Theoretical and Practical Implications

This approach offers several theoretical and practical implications:

• Unification and computation: The linear superposition property of the ER deviation tensor enables analytically tractable construction of the gravitational fields for complex mass distributions, in stark contrast to the highly nonlinear structure of Einstein’s equations in GR.
• Sensitivity to internal structure: The external metric’s sensitivity to internal mass moments, though weak, opens the possibility for probing the internal structure of astrophysical bodies via high-precision external field measurements.
• Satellite navigation and relativistic geodesy: Even for weak sources (Earth), extended-model corrections are potentially observable in high-precision space-based ranging and timing experiments, necessitating their inclusion for next-generation geodetic techniques or deep-space navigation.
• Astrophysical relevance: For compact objects such as neutron stars, the magnitude of corrections may influence photon propagation, emission signatures, and timing observations.

Conclusions

This work extends the geometric description of relativistic gravity for spherically symmetric bodies beyond both Newtonian gravity and standard Schwarzschild geometry. Using Extended Relativity, the explicit linear superposition of retarded field contributions enables derivation of a fully Lorentz-covariant metric for extended bodies. External fields of such sources contain weak but nonzero imprints of the internal mass distribution, violating the shell theorem at higher order and potentially enabling new observational probes of astronomical interiors. The results are directly relevant for strong-field problems (neutron stars) and for precision timing and ranging within weak fields (Earth–satellite systems). Future directions include extension of the formalism to rotating bodies and nontrivial internal anisotropy, further enhancing its applicability to celestial mechanics, astrophysics, and gravitational wave science.