The Einstein Field Equations

Every piece is now in place. The stress–energy tensor supplies a symmetric, conserved description of matter and fields; the Einstein tensor supplies a symmetric, divergence-free description of geometry; the geodesic equation fixed the Newtonian limit and identified $g_{00}$ with the potential. This lesson assembles the Einstein field equations by matching these objects, determines the coupling constant from Newtonian gravity, adds the cosmological constant, and rederives the whole equation from the Einstein–Hilbert action.

The source of curvature

Newtonian gravity is sourced by mass density $\rho$ through Poisson's equation ${\nabla }^2\Phi =4\pi G\rho$. Relativity forbids so narrow a source: mass is only one component of energy, energy and momentum mix under boosts, and pressure and stress carry energy too. The correct source is the full stress–energy tensor$T^{\mu \nu }$, a symmetric $(2,0)$ tensor whose components are the flux of the $\mu$-th momentum component across a surface of constant $x^{\nu }$:

• $T^{00}$ is the energy density.
• $T^{0i}=T^{i0}$ is the energy flux / momentum density.
• $T^{ij}$ is the momentum flux — pressure on the diagonal, shear off it.

For a perfect fluid with rest-frame energy density $\rho c^2$ and isotropic pressure $p$, moving with four-velocity $u^{\mu }$,

Local conservation of energy and momentum is the covariant statement

which in flat spacetime reduces to the continuity and Euler equations of fluid mechanics. Any geometric object set equal to $T^{\mu \nu }$ must therefore also be symmetric and divergence-free, or conservation would fail.

Building the equation from requirements

The field equation should read, schematically, curvature $=$ constant $\times$ matter, with the left side a tensor built from the metric. A short list of demands fixes it almost uniquely.

• Tensorial. Both sides are $(0,2)$ tensors, so the equation holds in every frame — the general-covariance requirement.
• Symmetric. The right side $T_{\mu \nu }$ is symmetric, so the left must be.
• Second order. To recover Poisson's equation (second-order in $\Phi$) and to match the Newtonian limit $g_{00}\sim \Phi$, the left side should involve at most second derivatives of the metric, and be linear in those.
• Divergence-free. Since ${\nabla }_{\mu }{T}^{\mu \nu }=0$ identically, the left side must be divergence-free identically, for consistency at all times.

The most general symmetric $(0,2)$ tensor built from the metric and its first two derivatives, linear in the second derivatives, and with vanishing divergence is a linear combination of the Einstein tensor and the metric,

where $\Lambda$ is a constant (the metric is divergence-free because ${\nabla }_{\mu }{g}^{\mu \nu }=0$ by metric compatibility). This is a theorem — Lovelock's theorem identifies $G_{\mu \nu }+\Lambda g_{\mu \nu }$ as the unique such tensor in four dimensions. Setting it proportional to $T_{\mu \nu }$ gives the Einstein field equations,

with a coupling constant $\kappa$ to be fixed and the cosmological constant $\Lambda$ deferred to the next section.

Fixing the coupling from the Newtonian limit

The constant $\kappa$ is determined by requiring that the equation reduce to Poisson's equation in the weak, slow, static limit, exactly the regime of the Newtonian-limit derivation. Take $\Lambda =0$ for this match. It is convenient to trace-reverse the equation: contracting $G_{\mu \nu }=\kappa T_{\mu \nu }$ with $g^{\mu \nu }$ gives $R-2R=\kappa T$, so $R=-\kappa T$, and substituting back,

In the weak static field with slow, low-pressure matter, the dominant stress–energy component is $T_{00}=\rho c^2$ and the trace is $T\approx -\rho c^2$, so the $00$ component reads $R_{00}\approx \frac{1}{2}\kappa \rho c^2$. The relevant curvature component in the weak-field limit is $R_{00}\approx {\nabla }^2\Phi /c^2$ — the time-time curvature is the Laplacian of the potential, inherited from $g_{00}=-(1+2\Phi /c^2)$ and the Riemann–tidal correspondence. Equating,

Poisson's equation ${\nabla }^2\Phi =4\pi G\rho$ then forces

The field equations in their standard form are

The factor $c^4$ in the denominator is enormous, so a large stress–energy produces a tiny curvature: spacetime is stiff, which is why gravity is the weakest interaction yet dominates on astronomical scales, where masses accumulate without cancellation. In geometrized units $c=G=1$ the coupling is simply $8\pi$.

The cosmological constant

The term $\Lambda g_{\mu \nu }$ is permitted by every requirement — it is symmetric, constructed from the metric, and divergence-free — so consistency alone does not forbid it. Einstein introduced it to allow a static universe and later regretted it; modern cosmology reinstated it as the leading model for the observed accelerating expansion. Moved to the right-hand side it looks like a source,

a perfect fluid with energy density ${\rho }_{\Lambda }{c}^2=\Lambda c^4/(8\pi G)$ and pressure $p_{\Lambda }=-{\rho }_{\Lambda }{c}^2$. This equation of state, pressure equal to minus the energy density, is the vacuum energy that drives accelerated expansion. Its magnitude is tiny, $\Lambda \sim 10^{-52}{\text{m}}^{-2}$, negligible on solar-system and galactic scales and dominant only over cosmological distances, which is why the Schwarzschild and orbit problems of the next module set $\Lambda =0$ without measurable error. The interpretation and consequences belong to the cosmology bridge; here it is the one term the derivation cannot rule out.

The Einstein–Hilbert action

The requirement-matching argument gives the equation but hides its economy. The modern derivation is variational: postulate an action and demand it be stationary. The Einstein–Hilbert action is the simplest coordinate-invariant scalar built from the metric,

with $R$ the Ricci scalar and $\sqrt{-g}{\mathbf{d}}^{4}x$ the invariant volume element. Adding a matter action $S_{\text{m}}$ whose variation defines the stress–energy tensor,

and requiring $\delta (S_{\text{EH}}+S_{\text{m}})/\delta g^{\mu \nu }=0$ reproduces the field equations with $\Lambda =0$. The variation of the $\sqrt{-g}R$ term yields exactly $R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R$; the boundary terms from varying the Ricci scalar cancel against the Gibbons–Hawking surface term. Including a constant term $-2\Lambda$ under the integral, $\int (R-2\Lambda )\sqrt{-g}{\mathbf{d}}^{4}x$, restores the cosmological constant. The action route shows that the field equation is not a guess but the Euler–Lagrange equation of the unique lowest-order invariant, and it is the starting point for every extension and quantization programme.

Matter and geometry, coupled

The field equations close the logic the module opened with. Geometry is not a fixed backdrop; it is a dynamical field $g_{\mu \nu }$ determined by its matter content, and matter moves along the geodesics of the geometry it produces. The two statements are coupled and nonlinear — the metric enters $T_{\mu \nu }$ (through $u^{\mu }$, $\sqrt{-g}$, index-raising), and $T_{\mu \nu }$ sources the metric — which is why exact solutions are rare and precious. Wheeler's summary compresses the content: matter tells spacetime how to curve, and curved spacetime tells matter how to move.

The first exact solution of these equations — the geometry outside a static spherical mass, found by Schwarzschild within months of the field equations — opens the next module, along with the orbits, light bending, and horizons that turn the field equation into predictions the solar system and telescopes test.