[Paper Review] A Locally Anisotropic Metric for Matter in an Expanding Universe: I. The Ansatz and the Modified Newton Law

P. Castelo Ferreira|ArXiv.org|Jul 5, 2009

Cosmology and Gravitation Theories42 references5 citations

TL;DR

This paper proposes a locally anisotropic metric that interpolates between the Schwarzschild metric near a central mass and the FLRW metric at large distances, using a radial-dependent exponent $α(r_1) = \alpha_0 + \alpha_1 \frac{2GM}{c^2 r_1}$ to eliminate singularities at the event horizon and ensure finite, positive mass-energy density. The key result is a modified Newtonian law that induces galaxy-scale deviations from standard gravity, potentially explaining flat rotation curves without dark matter.

ABSTRACT

It is suggested a metric ansatz to describe local matter in an expanding universe, hence interpolating between the Schwarzschild metric at small spatial scales and the FLRW metric at large spatial scales. This is acomplished maintaining space-time free of singularities except for the Schwarzschild mass pole at the origin as opposed to metrics already considered in the literature with the same purpose, namely the McVittie metric. The modified Newton law is analyzed and the static orbit solutions computed. It is concluded that the effects of expantion in the solar system are negligible, however depending on the metric parameter value, at galactic scales there is a significant deviation from the General Relativity Newton law which may contribute to dark matter effects allowing for a flattening of galaxy rotation curves.

Motivation & Objective

• To resolve the longstanding problem of describing localized matter in an expanding universe while avoiding singularities in existing metrics like McVittie and Cosmological-Schwarzschild.
• To construct a space-time metric that maintains local anisotropy near massive bodies yet preserves global spatial isotropy at infinity.
• To ensure the stress-energy tensor describes finite, positive-definite mass-energy density outside the event horizon.
• To derive a modified Newtonian law that may account for dark matter-like effects via cosmic expansion.

Proposed method

• Introduces a metric ansatz with a radial coordinate-dependent exponent $\alpha(r_1) = \alpha_0 + \alpha_1 \frac{2GM}{c^2 r_1}$ to remove essential singularities at the Schwarzschild radius.
• Analyzes curvature invariants and the stress-energy tensor to derive bounds on $\alpha_0$ and $\alpha_1$ ensuring singularity-free spacetime and positive mass-energy density.
• Derives the modified Newtonian gravitational acceleration by solving the geodesic equation in the weak-field, slow-motion limit.
• Computes orbital solutions for circular and elliptical orbits using perturbative methods, including corrections to precession, period, and radial evolution.
• Evaluates the effective potential and orbital dynamics in the presence of the Hubble expansion term, showing deviations from standard GR at large scales.
• Uses the Lagrangian formalism to derive the orbital differential equation, including $H_0^2$-order corrections due to cosmic expansion.

Experimental results

Research questions

• RQ1 Can a metric be constructed that smoothly interpolates between Schwarzschild and FLRW spacetimes while avoiding singularities at the event horizon?
• RQ2 What constraints on the parameters $\alpha_0$ and $\alpha_1$ ensure the spacetime is free of essential singularities and the mass-energy density remains positive definite?
• RQ3 How does the modified gravitational potential affect planetary and galactic-scale orbital dynamics compared to standard General Relativity?
• RQ4 To what extent do the corrections from the expanding background explain the observed flatness of galaxy rotation curves?
• RQ5 Are the predicted orbital precession and radial evolution consistent with solar system observations?

Key findings

• For the radial-dependent exponent $\alpha(r_1) = \alpha_0 + \alpha_1 \frac{2GM}{c^2 r_1}$ with $\alpha_1 < 0$, spacetime is singularity-free at the event horizon if $\alpha_0 - |\alpha_1| \geq 3$, and asymptotically flat if $\alpha_0 - |\alpha_1| > 5$.
• The modified Newtonian potential introduces a correction term proportional to $\left(\frac{H_0}{c}\right)^2 \left(1 - \frac{2GM}{c^2 r_1}\right)^{\alpha(r_1) - 1}$, which becomes significant at galaxy scales for large $\alpha_0$.
• For $\alpha_0 > \frac{c^2}{\sqrt{q_0 (GM H_0)^2}}$, the deviation from Newtonian gravity becomes substantial, potentially explaining flat rotation curves without dark matter.
• Orbital precession, period corrections, and radial evolution remain within observational bounds for the solar system, validating the model in the planetary regime.
• The stress-energy tensor derived from the metric confirms positive-definite mass-energy density outside the event horizon under the same parameter constraints.
• The model preserves global spatial isotropy at infinity while allowing local anisotropy near massive bodies, consistent with cosmological observations.