Uniform density static fluid sphere in Einstein-Gauss-Bonnet gravity and its universality

In Newtonian theory, gravity inside a constant density static sphere is independent of spacetime dimension. Interestingly this general result is also carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet (Lovelock) gravity notwithstanding their nonlinearity. We prove that the necessary and sufficient condition for universality of Schwarzschild interior solution describing a uniform density sphere for all $n\geq4$ is that its density is constant.

💡 Research Summary

The paper investigates a striking dimensional universality that originates in Newtonian gravity and persists through Einstein’s General Relativity and further into the second‑order Lovelock theory, specifically Einstein‑Gauss‑Bonnet (EGB) gravity. In the Newtonian framework, the gravitational field inside a static sphere of constant density depends only on the radial coordinate and the density; it is completely independent of the number of spatial dimensions. The authors first recapitulate this elementary result and then turn to the relativistic case.

In General Relativity the interior solution for a static, spherically symmetric perfect fluid is the well‑known Schwarzschild interior metric. Although the explicit coefficients in the field equations change with the spacetime dimension (n\ge4), the functional form of the solution remains the same provided the fluid density is uniform. The paper shows that this “dimensional independence” is not an accident of the Einstein equations but a deeper property that survives the inclusion of higher‑order curvature corrections.

The authors consider the Einstein‑Gauss‑Bonnet action \