Title: Tensor Analysis of the Mass-Space Relationship in Einstein's Field Equations

Author: 
Date: 2025.5.1

Abstract

This study explores a novel physical relation Gm=c2sGm = c^2 s between mass and space, hypothesizing that mass itself causes the expansion of space. This relation is interpreted in the context of Einstein's field equations, the Friedmann equation, and the FLRW metric. Through a tensor formulation, we derive implications for energy density, spacetime curvature, and cosmic phenomena such as dark energy and cosmic inflation. The model not only reproduces the observed radius of the universe with 1.15% accuracy but also suggests that the universe is approaching a limiting physical configuration dictated by this relation.

1. Introduction

In classical general relativity, mass curves spacetime, resulting in gravitational attraction. However, observations such as cosmic expansion and dark energy imply a repulsive force. We propose a dual nature: while gravity attracts, mass can also induce expansion via a gravitomagnetic-like force. A key expression derived is: Gm=c2sGm = c^2 s where mm is mass, cc the speed of light, GG the gravitational constant, and ss a characteristic spatial distance.

2. FLRW Metric and Spatial Scaling

The FLRW metric describes a homogeneous, isotropic universe: ds2=−c2dt2+a(t)2[dr21−kr2+r2(dθ2+sin2θdϕ2)]ds^2 = -c^2 dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2) \right] Assuming s=a(t)rs = a(t) r, and using s=Gmc2s = \frac{Gm}{c^2}, we get: a(t)=Gmc2ra(t) = \frac{Gm}{c^2 r} This suggests that the scale factor may locally depend on mass, supporting the idea of mass-driven expansion.

3. Friedmann Equation with Substitution

The Friedmann equation is: H2=(a˙a)2=8πG3ρ−kc2a2+Λc23H^2 = \left( \frac{\dot{a}}{a} \right)^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} Using s=Gmc2s = \frac{Gm}{c^2}, and ρ=mV\rho = \frac{m}{V} with V=43πs3V = \frac{4}{3}\pi s^3, we derive: ρ=3c24πGs2\rho = \frac{3c^2}{4\pi G s^2} Substituting into the Friedmann equation yields: H2=2c2s2⇒s=2cHH^2 = \frac{2c^2}{s^2} \Rightarrow s = \frac{\sqrt{2}c}{H} This is consistent with the Hubble radius s=cHs = \frac{c}{H}, differing only by a factor of 2\sqrt{2}.

4. Einstein Field Equations in Tensor Form

The Einstein field equations are: Gμν+Λgμν=8πGc4TμνG_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} In the perfect fluid form: Tμν=(ρ+pc2)uμuν+pgμνT_{\mu\nu} = (\rho + \frac{p}{c^2}) u_\mu u_\nu + p g_{\mu\nu} Using the derived ρ\rho: ρ=3c24πGs2⇒T00=3c44πGs2\rho = \frac{3c^2}{4\pi G s^2} \Rightarrow T_{00} = \frac{3c^4}{4\pi G s^2} Then: G00+Λg00=6s2G_{00} + \Lambda g_{00} = \frac{6}{s^2} This implies spacetime curvature is inversely proportional to the square of the spatial scale ss.

5. Implications and Phenomena

  • Dark Energy: Expansion is an induced response to mass density.

  • Inflation: Close proximity of mass shortly after the Big Bang leads to massive expansion via induced gravitomagnetic fields.

  • CMB Anisotropy: Minor differences in spatial curvature from mass distributions yield anisotropies.

  • Galactic Rotation Curves: Gravitomagnetic feedback explains flat curves without dark matter.

6. Conclusion

The tensor formulation of Gm=c2sGm = c^2 s integrates well with general relativity, reproducing observed cosmic parameters and offering new explanations for dark phenomena. The universe may be asymptotically approaching a limit where this relation becomes exact, revealing a possible endpoint in cosmic evolution.

References

  • Einstein, A. (1915). The Field Equations of Gravitation.

  • Friedmann, A. (1922). On the Curvature of Space.

  • Carroll, S. (2004). Spacetime and Geometry: An Introduction to General Relativity.

  • Riess, A. et al. (1998). Observational Evidence from Supernovae.

  • Han, D. (2025). Private derivation notes on mass-space expansion.