Deriving Schwarzschild and Kerr from the Einstein Equivalence Principle
Abstract
This manuscript demonstrates that the Schwarzschild and Kerr metrics can be derived fundamentally from the Einstein Equivalence Principle (EEP) without explicit reliance on the Einstein field equations. We show that the EEP implies a gravitational stretching of length in the direction of acceleration, a spatial counterpart to time dilation. In the static case, this effect yields the constraint g00g11=−1, from which the Schwarzschild solution is recovered by taking the Newtonian limit. For the stationary case, we employ oblate spheroidal coordinates and construct a physically motivated tetrad to implement the EP, introducing gravity through three unknown functions. By applying the EEP-induced constraints and showing that the principle requires the preservation of the spacetime volume determinant (g = g0), the system is reduced to a single unknown function that yields the Kerr metric. These results suggest that classical tests of General Relativity are, at their core, fundamental validations of the EEP. We further argue that the EEP leads naturally to Unimodular Gravity, implying that the existence of the cosmological constant is a direct consequence of this foundational postulate.
Versions
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BibTeX
@misc{alencar2026deriving,
title = {{Deriving Schwarzschild and Kerr from the Einstein Equivalence Principle}},
author = {G. Alencar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12708},
eprint = {2610.12708},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12708},
}
RIS
TY - UNPB
TI - Deriving Schwarzschild and Kerr from the Einstein Equivalence Principle
AU - G. Alencar
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12708
UR - https://arxiv.org/abs/2610.12708
AB - This manuscript demonstrates that the Schwarzschild and Kerr metrics can be derived fundamentally from the Einstein Equivalence Principle (EEP) without explicit reliance on the Einstein field equations. We show that the EEP implies a gravitational stretching of length in the direction of acceleration, a spatial counterpart to time dilation. In the static case, this effect yields the constraint g_00g_11=−1, from which the Schwarzschild solution is recovered by taking the Newtonian limit. For the stationary case, we employ oblate spheroidal coordinates and construct a physically motivated tetrad to implement the EP, introducing gravity through three unknown functions. By applying the EEP-induced constraints and showing that the principle requires the preservation of the spacetime volume determinant (g = g_0), the system is reduced to a single unknown function that yields the Kerr metric. These results suggest that classical tests of General Relativity are, at their core, fundamental validations of the EEP. We further argue that the EEP leads naturally to Unimodular Gravity, implying that the existence of the cosmological constant is a direct consequence of this foundational postulate.
ER -