[Paper Review] Gravity Inside a Nonrotating, Homogeneous, Spherical Body

Homer G. Ellis|arXiv (Cornell University)|Mar 20, 2012

Cosmology and Gravitation Theories5 references4 citations

TL;DR

This paper proposes a new interior metric for a nonrotating, homogeneous, spherical body by deriving improved Einstein field equations from a variational principle that correctly identifies active gravitational mass density as the source term, avoiding Einstein's unjustified identification of passive and active mass. The resulting metric, matched smoothly to the Schwarzschild exterior solution, predicts modified photon and neutrino flight times through the body, offering a logically consistent alternative to the standard Schwarzschild interior solution.

ABSTRACT

Schwarzschild's 'interior solution' is a space-time metric that satisfies Einstein's gravitational field equations with a source term that Einstein created on the basis of an unjustified identification of the conceptually distinct notions of the passive gravitational mass of matter and the active gravitational mass of matter. Giving up that assumption allows deriving from a variational principle new and better (because logically obtained) field equations that more faithfully extend the Poisson equation for Newton's gravity than do Einstein's, with an active gravitational mass density providing the source term. Solving these equations for a nonrotating spherical ball of matter with uniform mass density produces a new, improved interior metric matched at the surface of the ball to Schwarzschild's 'exterior' solution metric. This new metric can then be used to address questions about the flight times of photons and neutrinos through such a ball of matter.

Motivation & Objective

• To correct a foundational error in Einstein's field equations where passive gravitational mass was unjustifiably identified with active gravitational mass.
• To derive a logically consistent set of field equations from a variational principle that properly incorporates active gravitational mass density as the source term.
• To construct a new interior metric for a nonrotating, homogeneous, spherical body that smoothly matches the Schwarzschild exterior solution at the surface.
• To apply the new metric to compute flight times of photons and neutrinos through such a body, particularly relevant to experimental neutrino timing measurements.

Proposed method

• Derive new field equations from the variational principle $\delta\!\int({\bm{R}}-\frac{8\pi\kappa}{c^{2}}\mu+2\,\phi^{.\gamma}\phi_{.\gamma})\,|g|^{\frac{1}{2}}\,d^{4}\!x=0$, introducing an auxiliary scalar field $\phi$ to generalize solutions.
• Solve the resulting field equations $ {\bm{R}}_{\alpha\beta}=\frac{4\pi\kappa}{c^{2}}\mu\,g_{\alpha\beta}-2\,\phi_{.\alpha}\phi_{.\beta} $ and $ \square\phi=0 $ for a static, spherically symmetric, homogeneous sphere.
• Construct a new interior metric in the form $ d\tau^{2}=\frac{1-2m/R}{1+\lambda m/R}\left[1+\frac{\lambda m}{R}\frac{r^{2}(\rho)}{R^{2}}\right]\,dt^{2}-\cdots $, parameterized by $ \lambda $, with $ r(\rho)=\lambda(\rho-\rho_0) $ and $ \rho_0=\frac{\lambda-1}{\lambda}R $.
• Match the interior metric to the Schwarzschild exterior solution at $ r=R $ by solving $ \left(1-\frac{2m}{R}\right)\lambda^{2}-\frac{m}{R}\lambda-1=0 $ to determine $ \lambda $.
• Express the metric in 'ether-flow' form to clarify geodesic dynamics and radial acceleration behavior.
• Compute radial geodesic equations for test particles and analyze the gravitational and centrifugal components of radial acceleration, showing continuity of gravity but discontinuity of centrifugal effects at the boundary.

Experimental results

Research questions

• RQ1 How can Einstein's field equations be corrected to properly identify active gravitational mass density as the source term, avoiding the unjustified identification of passive and active mass?
• RQ2 What is the resulting interior metric for a nonrotating, homogeneous, spherical body that smoothly matches the Schwarzschild exterior solution?
• RQ3 How do the flight times of photons and neutrinos through such a body differ under the new metric compared to the standard Schwarzschild interior solution?
• RQ4 What is the behavior of radial acceleration for test particles in the new interior metric, particularly at the center and boundary?
• RQ5 Does the new metric preserve continuity of gravitational force while introducing discontinuity in centrifugal effects at the surface?

Key findings

• The new interior metric is derived from a corrected variational principle that properly identifies active gravitational mass density $ \mu $ as the source term, avoiding Einstein’s unjustified identification of passive and active mass.
• The solution yields $ \lambda = \frac{m+\sqrt{m^{2}+4R(R-2m)}}{2(R-2m)} $, which determines the radial scaling $ r(\rho)=\lambda(\rho-\rho_0) $ and ensures smooth matching to the exterior Schwarzschild solution at $ r=R $.
• The metric shows that at the center of the sphere, both gravitational and centrifugal radial accelerations vanish, consistent with Newtonian expectations.
• At the boundary, the gravitational component of radial acceleration is continuous between the new interior and exterior metrics, but the centrifugal component is discontinuous due to $ R/\lambda \neq R-3m $.
• The new metric enables precise computation of photon and neutrino flight times through the body, with potential implications for interpreting results from experiments like OPERA and ICARUS.
• The 'ether-flow' form of the metric reveals a radial flow of the gravitational field, with the radial coordinate $ \rho $ increasing faster than the areal radius $ r(\rho) $, reflecting a non-Euclidean geometry in the interior.