Einstein Field Equations

The Einstein field equations are the heart of General Relativity, describing how matter and energy curve spacetime.

The Equations

Einstein's field equations relate the geometry of spacetime to the distribution of matter and energy:

G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

Or equivalently, using the definition of the Einstein tensor:

Expanded Form

R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

The Terms Explained

Left Side: Geometry

$G_{\mu\nu}$ - Einstein tensor: describes spacetime curvature
$R_{\mu\nu}$ - Ricci tensor: contracted curvature
$R$ - Ricci scalar: total curvature at a point
$\Lambda$ - Cosmological constant: energy density of empty space
$g_{\mu\nu}$ - Metric tensor: describes spacetime geometry

Right Side: Matter

$T_{\mu\nu}$ - Stress-energy tensor: describes matter/energy distribution
$G$ - Newton's gravitational constant: $6.67 \times 10^{-11}$ N·m²/kg²
$c$ - Speed of light: $3 \times 10^8$ m/s

The Stress-Energy Tensor

The stress-energy tensor $T_{\mu\nu}$ encodes the density and flow of energy and momentum:

Stress-Energy Components

T_{\mu\nu} = \begin{pmatrix} \text{energy density} & \text{energy flux} \\ \text{momentum density} & \text{stress tensor} \end{pmatrix}

For a perfect fluid (like a star or the universe):

Perfect Fluid

T_{\mu\nu} = \left(\rho + \frac{p}{c^2}\right) u_\mu u_\nu + p \, g_{\mu\nu}

where $\rho$ is mass-energy density, $p$ is pressure, and $u^\mu$ is the 4-velocity of the fluid.

Conservation Laws

The contracted Bianchi identity $\nabla_\mu G^{\mu\nu} = 0$ implies:

Energy-Momentum Conservation

\nabla_\mu T^{\mu\nu} = 0

This is the relativistic generalization of conservation of energy and momentum. It's automatic in GR - not an additional assumption!

The Vacuum Equations

In empty space ( $T_{\mu\nu} = 0$ ), without cosmological constant:

Vacuum Einstein Equations

R_{\mu\nu} = 0

This doesn't mean spacetime is flat! The full Riemann tensor can still be non-zero. The Schwarzschild solution (black hole) satisfies these vacuum equations.

Notable Solutions

Schwarzschild (1916): Non-rotating black hole, spherical symmetry
Kerr (1963): Rotating black hole
FLRW: Cosmological solution - expanding universe
de Sitter/Anti-de Sitter: Vacuum solutions with cosmological constant
Gravitational waves: Ripples in spacetime (detected 2015!)

Matter Curving Spacetime

See how mass warps the fabric of spacetime according to Einstein's equations.

Mass: 1.0

Show orbiting particle (geodesic motion)

Gravitational Waves

Ripples in spacetime from a binary black hole merger. Notice the quadrupole (plus) polarization pattern.

Amplitude: 1.0

Frequency: 1.0

Show binary black hole system

Counting Equations

The field equations are 10 coupled, non-linear, second-order PDEs for the 10 components of the metric. However:

4 Bianchi identities reduce independent equations to 6
4 coordinate freedoms reduce independent metric components to 6
Perfect match: 6 equations for 6 unknowns!

The Newtonian Limit

In the weak-field, slow-motion limit, Einstein's equations reduce to Newton's:

Newtonian Limit

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

where $\Phi$ is the Newtonian gravitational potential and $g_{00} \approx -(1 + 2\Phi/c^2)$ .

Wheeler's Summary

"Spacetime tells matter how to move; matter tells spacetime how to curve."

— John Archibald Wheeler

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