Einstein’s equations and the Newtonian limit

Keep the earlier Riemann convention and $(+,-,-,-)$ signature. Define $G_{\mu\nu}=R_{\mu\nu}-Rg_{\mu\nu}/2$. With zero cosmological constant, Einstein’s equation is $G_{\mu\nu}=8\pi G T_{\mu\nu}/c^4$. Curvature has units of inverse length squared, so the coupling converts energy density into the same units. The equation relates tensor components at one event, while its solutions depend on data across a region.

The contracted Bianchi identity gives $\nabla^\mu G_{\mu\nu}=0$, making $\nabla^\mu T_{\mu\nu}=0$ a consistency requirement. This is a local conservation statement, not an extra coordinate choice. A cosmological term proportional to the metric can also be included consistently because the metric’s covariant derivative vanishes. We set it to zero in this weak-field derivation; the cosmology lesson will state its positive-expansion convention separately to avoid signature-dependent ambiguity.

For a weak static field with negligible anisotropic stress, use $ds^2\simeq(1+2\Phi/c^2)c^2dt^2-(1-2\Phi/c^2)d\mathbf x^2$. Small $|\Phi|/c^2$ controls the metric expansion. A slowly moving test particle additionally has $v/c\ll1$, and a nearly stationary source lets us neglect time derivatives of the field. None of these assumptions follows automatically from either of the others.

The leading spatial geodesic equation becomes $d^2x^i/dt^2\simeq-c^2\Gamma^i_{00}=-\partial_i\Phi$. For a pressureless source, $T_{00}\simeq\rho c^2$, where rho is rest-mass density. The leading time-time field equation then recovers $\nabla^2\Phi=4\pi G\rho$. A relativistic fluid, fast-moving compact binary, or rapidly changing geometry requires terms discarded in this limit, so the recovery of Newtonian gravity is a consistency check with a stated domain.

The metric has ten independent symmetric components in four dimensions, but coordinate freedom and differential identities mean they are not ten independent unconstrained wave equations. A spacelike initial slice carries geometric data that must satisfy constraint equations. A coordinate or gauge prescription then organizes the evolution. Giving an arbitrary density field without matter velocities, consistent geometric data, and boundary conditions does not uniquely specify a relativistic spacetime.

Even in Newtonian gravity, Poisson’s equation needs boundary conditions: adding a harmonic function changes the solution without changing the local density. In general relativity, gravitational waves can enter through initial or boundary data even in vacuum. “Matter tells spacetime how to curve” is therefore a useful mnemonic, provided it does not conceal free gravitational degrees of freedom or the role of global conditions. The next lessons study exact exterior geometry and propagating perturbations separately.