Is Gravity Truly Balanced? A Historical-Critical Journey Through the Equivalence Principle and the Genesis of Spacetime Geometry
Jaume de Haro, Emilio Elizalde
TL;DR
This work addresses deriving the spacetime metric from the Equivalence Principle without invoking Einstein's field equations, focusing on weak-field gravity relevant to the Solar System. It first obtains a static conformastatic metric with $ds^2=(1+2\Phi)dt^2-\frac{1}{1+2\Phi}d\mathbf{x}\cdot d\mathbf{x}$ and shows geodesics reproduce a relativistic form of Newton's second law for free fall, then extends to moving sources via Lorentz boosts, yielding a nonstatic metric with gravitomagnetic potential $\mathbf{N}$ that matches the linearized GR metric in the Lorenz gauge. The framework recasts the Equivalence Principle as a dynamic balance between a real inertial force and gravity, enabling a tractable Cauchy problem for evolving $\Phi$, $\rho$, and $\mathbf{v}$, with $\Box\Phi=-4\pi G\rho$ and $\Box\mathbf{N}=-4\pi G\rho\mathbf{v}$, and a Lorenz gauge condition $\partial_t\Phi+\nabla\cdot\mathbf{N}=0$. By demonstrating agreement with the weak-field limit of GR and providing a physically transparent derivation that does not require solving the full Einstein equations, the paper offers both conceptual clarity and a practical pedagogical bridge between Newtonian inertia and relativistic gravity.
Abstract
We present a novel derivation of the spacetime metric generated by matter, without invoking Einstein's field equations. For static sources, the metric arises from a relativistic formulation of D'Alembert's principle, where the inertial force is treated as a real dynamical entity that exactly compensates gravity. This leads to a conformastatic metric whose geodesic equation, parametrized by proper time, reproduces the relativistic version of Newton's second law for free fall. To extend the description to moving matter, uniformly or otherwise, we apply a Lorentz transformation to the static metric. The resulting non static metric accounts for the motion of the sources and, remarkably, matches the weak field limit of general relativity as obtained from the linearized Einstein equations in the de Donder or Lorenz gauge. This approach, at least at Solar System scales, where gravitational fields are weak, is grounded in a new dynamical interpretation of the Equivalence Principle. It demonstrates how gravity can emerge from the relativistic structure of inertia, without postulating or solving Einstein's equations.