Einstein Field Equations

Having developed the mathematical language of differential geometry, we now arrive at the crown jewel of classical physics: Einstein's field equations. These ten coupled, nonlinear partial differential equations relate the curvature of spacetime to the distribution of matter and energy. They encode a revolutionary idea: gravity is not a force acting at a distance, but a manifestation of curved geometry.

The Equivalence Principle

Einstein's path to general relativity began with a deceptively simple observation: in a freely falling elevator, you feel weightless. Locally, gravity is indistinguishable from acceleration. This is the equivalence principle, and it comes in several strengths.

Weak Equivalence Principle

The trajectory of a freely falling body depends only on its initial position and velocity, not on its composition. This is the universality of free fall, tested to extraordinary precision (one part in $10^{15}$). In Newtonian terms, it says that gravitational mass equals inertial mass.

Einstein Equivalence Principle

The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the laboratory's velocity and position in a gravitational field. This means that in a small enough region, the laws of physics reduce to those of special relativity.

Strong Equivalence Principle

The same holds even for gravitational experiments. This is the most restrictive form, and it is satisfied by general relativity but not by most alternative theories of gravity.

The equivalence principle tells us that gravity can be "transformed away" locally by choosing a freely falling reference frame. But it cannot be transformed away globally if there is genuine curvature, tidal forces remain. This is the clue that gravity is geometry.

Gravity as Geometry

Consider two particles in free fall near the Earth. If they start at the same height but separated horizontally, they both fall toward the center of the Earth, so they converge. If they start at the same horizontal position but different heights, the lower one accelerates faster, so they diverge. These tidal effects are the signature of genuine gravity, they cannot be removed by any coordinate transformation.

In the geometric picture, the two particles follow geodesics. The convergence or divergence of nearby geodesics is governed by the Riemann curvature tensor through the geodesic deviation equation:

$$\frac{D^2 \xi^\mu}{d\tau^2} = -R^\mu{}_{\nu\alpha\beta} u^\nu \xi^\alpha u^\beta$$

where $\xi^\mu$ is the separation vector between nearby geodesics and $u^\mu$ is the 4-velocity. Tidal forces are curvature.

The Einstein Tensor

We need a tensor equation that relates curvature (geometry) to matter (physics). On the geometry side, we need a symmetric $(0,2)$ tensor built from the metric. The natural candidate is formed from the Ricci tensor and scalar:

The Bianchi identity is a purely geometric result: $\nabla_{[\lambda} R_{\rho\sigma]\mu\nu} = 0$. When contracted, it yields $\nabla^\mu G_{\mu\nu} = 0$. This is not a physical assumption, it is a mathematical theorem about the Riemann tensor. But it has profound physical consequences: it guarantees that the source of gravity (matter and energy) is automatically conserved.

The Stress-Energy Tensor

On the matter side of the equation, we need the stress-energy tensor $T_{\mu\nu}$, which encodes the density and flow of energy and momentum.

For a perfect fluid with energy density $\rho$, pressure $p$, and 4-velocity $u^\mu$:

$$T_{\mu\nu} = (\rho + p) u_\mu u_\nu + p \, g_{\mu\nu}$$

The components have clear physical meanings:

• $T_{00}$ = energy density
• $T_{0i}$ = energy flux = momentum density
• $T_{ij}$ = stress (pressure and shear)

Conservation of energy and momentum is expressed as $\nabla^\mu T_{\mu\nu} = 0$: which is consistent with $\nabla^\mu G_{\mu\nu} = 0$ precisely because the Einstein equations link the two tensors.

The Einstein Field Equations

The constant $\frac{8\pi G}{c^4}$ is fixed by requiring that the equations reduce to Newtonian gravity in the appropriate limit. The cosmological constant $\Lambda$ was introduced by Einstein in 1917 to allow a static universe; he later called it his "greatest blunder." We now know it describes the accelerating expansion of the universe and is associated with dark energy.

These equations are far more than a relativistic upgrade of Newton's law. They are nonlinear: gravity gravitates. The energy of the gravitational field itself contributes to the curvature. This nonlinearity makes exact solutions rare and precious.

The Cosmological Constant

The term $\Lambda g_{\mu\nu}$ can be moved to the right-hand side and interpreted as a contribution to the stress-energy tensor:

$$T^{\Lambda}_{\mu\nu} = -\frac{\Lambda c^4}{8\pi G} g_{\mu\nu}$$

This corresponds to a perfect fluid with equation of state $p = -\rho c^2$: negative pressure! A positive $\Lambda$ acts as a repulsive "antigravity" on large scales. Observationally, $\Lambda \approx 1.1 \times 10^{-52}$ m$^{-2}$: tiny, but enough to dominate the energy budget of the universe today.

The Newtonian Limit

For Einstein's theory to be correct, it must reduce to Newton's in the regime of weak gravity and slow motion. Consider a weak, static gravitational field: $g_{\mu\nu} \approx \eta_{\mu\nu} + h_{\mu\nu}$ with $|h_{\mu\nu}| \ll 1$. For a slowly moving particle ($v \ll c$), the geodesic equation reduces to:

$$\frac{d^2 x^i}{dt^2} \approx -\frac{c^2}{2} \partial_i h_{00}$$

Comparing with Newton's $\ddot{x}^i = -\partial_i \Phi$, we identify $h_{00} = -2\Phi/c^2$. The Einstein equations then reduce to Poisson's equation:

$$\nabla^2 \Phi = 4\pi G \rho$$

Newton's gravity is the weak-field, slow-motion limit of Einstein's geometry.

The Einstein-Hilbert Action

Just as in all of physics, the field equations can be derived from a variational principle. The action for general relativity is:

The factor $\sqrt{-g}$ (where $g = \det(g_{\mu\nu})$) ensures the integration measure is coordinate-invariant. The Ricci scalar $R$ is the simplest scalar that can be built from the metric and its first two derivatives. This is the most economical gravitational action, the unique choice (up to topological terms) with second-order field equations.

Counting Degrees of Freedom

The metric $g_{\mu\nu}$ is a symmetric $4 \times 4$ matrix, so it has 10 independent components. The Einstein equations provide 10 equations. But 4 of these are constraint equations (due to $\nabla^\mu G_{\mu\nu} = 0$), and we have 4 gauge degrees of freedom (coordinate choice). So the true dynamical content is $10 - 4 - 4 = 2$ degrees of freedom. These correspond to the two polarizations of gravitational waves, a prediction we will explore in lesson 6.5.

Key Insights

• The equivalence principle, the universality of free fall, implies that gravity is a property of spacetime itself, not a force
• The Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$ is the unique divergence-free, symmetric tensor built from the metric and its first two derivatives
• The field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$ are nonlinear: gravity itself gravitates
• The cosmological constant $\Lambda$ acts as a repulsive energy pervading all of space
• In the weak-field, slow-motion limit, Einstein's equations reduce to Newton's law of gravitation
• The equations follow from the Einstein-Hilbert action via the variational principle
• The true dynamical content is two degrees of freedom, the two polarizations of gravitational waves

Looking Ahead

With the field equations in hand, we can begin solving them. The first and most important exact solution was found by Karl Schwarzschild in 1916, just months after Einstein published his theory. The Schwarzschild solution describes the spacetime around a spherically symmetric mass and predicts gravitational redshift, orbital precession, and light bending, all confirmed by observation.