General Relativity

Introduction

General Relativity, formulated by Albert Einstein in 1915, is a geometric theory of gravitation that describes gravity not as a force, but as a consequence of the curvature of spacetime caused by the presence of mass and energy.

1. Metric Tensor and Line Element

The metric tensor {`$g_{\\mu\\nu}$`} defines the geometry of spacetime. The infinitesimal proper time interval is:

$$ds^2 = g_{\mu\nu} dx^\mu dx^\nu$$

In Minkowski spacetime (flat space), the metric is:

$$\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$$
$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

2. Christoffel Symbols (Connection Coefficients)

Derivation from Metric Compatibility

The Christoffel symbols describe how the basis vectors change from point to point in curved spacetime. We derive them from the condition of metric compatibility:

$$\nabla_\lambda g_{\mu\nu} = 0$$

Expanding the covariant derivative:

$$\partial_\lambda g_{\mu\nu} - \Gamma^\sigma_{\lambda\mu} g_{\sigma\nu} - \Gamma^\sigma_{\lambda\nu} g_{\mu\sigma} = 0$$

Step 1: Write this equation for three cyclic permutations of indices:

$$\partial_\lambda g_{\mu\nu} = \Gamma^\sigma_{\lambda\mu} g_{\sigma\nu} + \Gamma^\sigma_{\lambda\nu} g_{\mu\sigma} \quad (1)$$$$\partial_\mu g_{\nu\lambda} = \Gamma^\sigma_{\mu\nu} g_{\sigma\lambda} + \Gamma^\sigma_{\mu\lambda} g_{\nu\sigma} \quad (2)$$$$\partial_\nu g_{\lambda\mu} = \Gamma^\sigma_{\nu\lambda} g_{\sigma\mu} + \Gamma^\sigma_{\nu\mu} g_{\lambda\sigma} \quad (3)$$

Step 2: Add equations (2) and (3), then subtract (1):

$$\partial_\mu g_{\nu\lambda} + \partial_\nu g_{\lambda\mu} - \partial_\lambda g_{\mu\nu} = 2\Gamma^\sigma_{\mu\nu} g_{\sigma\lambda}$$

Step 3: Multiply both sides by $g^{\lambda\rho}$ (inverse metric) to isolate $\Gamma$:

$$\Gamma^\rho_{\mu\nu} = \frac{1}{2}g^{\rho\lambda}\left(\partial_\mu g_{\nu\lambda} + \partial_\nu g_{\lambda\mu} - \partial_\lambda g_{\mu\nu}\right)$$

Final result: Using standard notation $\frac{\partial}{\partial x^\mu} = \partial_\mu$:

$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}\left(\frac{\partial g_{\sigma\nu}}{\partial x^\mu} + \frac{\partial g_{\sigma\mu}}{\partial x^\nu} - \frac{\partial g_{\mu\nu}}{\partial x^\sigma}\right)$$

Important properties:

  • • Symmetric in lower indices: $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$
  • • Not a tensor (transforms inhomogeneously)
  • • Vanishes in locally inertial coordinates
  • • In 4D, there are $4 \times 10 = 40$ independent components

3. Riemann Curvature Tensor

Derivation from Commutator of Covariant Derivatives

The Riemann tensor measures the curvature of spacetime. It arises when we compute the commutator of two covariant derivatives acting on a vector field $V^\rho$:

$$[\nabla_\mu, \nabla_\nu]V^\rho = R^\rho_{\sigma\mu\nu} V^\sigma$$

Step 1: Apply first covariant derivative:

$$\nabla_\nu V^\rho = \partial_\nu V^\rho + \Gamma^\rho_{\nu\sigma} V^\sigma$$

Step 2: Apply second covariant derivative $\nabla_\mu$ to this result:

$$\nabla_\mu\nabla_\nu V^\rho = \partial_\mu(\partial_\nu V^\rho) + \partial_\mu(\Gamma^\rho_{\nu\sigma} V^\sigma) + \Gamma^\rho_{\mu\lambda}(\partial_\nu V^\lambda + \Gamma^\lambda_{\nu\sigma} V^\sigma) - \Gamma^\lambda_{\mu\nu}(\partial_\lambda V^\rho + \Gamma^\rho_{\lambda\sigma} V^\sigma)$$

Step 3: Expand and collect terms:

$$\nabla_\mu\nabla_\nu V^\rho = \partial_\mu\partial_\nu V^\rho + (\partial_\mu\Gamma^\rho_{\nu\sigma})V^\sigma + \Gamma^\rho_{\nu\sigma}\partial_\mu V^\sigma + \Gamma^\rho_{\mu\lambda}\partial_\nu V^\lambda + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} V^\sigma - \Gamma^\lambda_{\mu\nu}\partial_\lambda V^\rho - \Gamma^\lambda_{\mu\nu}\Gamma^\rho_{\lambda\sigma} V^\sigma$$

Step 4: Compute the commutator $[\nabla_\mu, \nabla_\nu]V^\rho = \nabla_\mu\nabla_\nu V^\rho - \nabla_\nu\nabla_\mu V^\rho$. The second derivative $\partial_\mu\partial_\nu V^\rho$ cancels, leaving:

$$[\nabla_\mu, \nabla_\nu]V^\rho = \left(\partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\right)V^\sigma$$

Final Formula: The Riemann curvature tensor is defined as:

$$R^\rho_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$$

Symmetries of the Riemann Tensor

Lowering the first index with the metric: $R_{\rho\sigma\mu\nu} = g_{\rho\lambda}R^\lambda_{\sigma\mu\nu}$

  • • Antisymmetric in first pair: $R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$
  • • Antisymmetric in second pair: $R_{\rho\sigma\mu\nu} = -R_{\rho\sigma\nu\mu}$
  • • Symmetric under pair exchange: $R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$
  • • First Bianchi identity: $R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0$ (cyclic sum vanishes)
  • • Second Bianchi identity: $\nabla_\lambda R_{\rho\sigma\mu\nu} + \nabla_\mu R_{\rho\sigma\nu\lambda} + \nabla_\nu R_{\rho\sigma\lambda\mu} = 0$

In 4D spacetime, these symmetries reduce the number of independent components from $4^4 = 256$ to just 20 independent components.

4. Ricci Tensor and Scalar

The Ricci tensor is a contraction of the Riemann tensor:

$$R_{\mu\nu} = R^\lambda_{\mu\lambda\nu} = g^{\rho\sigma}R_{\rho\mu\sigma\nu}$$

The Ricci scalar (scalar curvature) is the trace of the Ricci tensor:

$$R = g^{\mu\nu}R_{\mu\nu}$$

5. Einstein Tensor

The Einstein tensor combines the Ricci tensor and scalar to satisfy the contracted Bianchi identities:

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$$

This tensor is automatically divergence-free: $\nabla^\mu G_{\mu\nu} = 0$, which ensures conservation of energy-momentum.

6. Einstein Field Equations

Derivation from the Einstein-Hilbert Action

The Einstein field equations can be derived from the variational principle. Start with the Einstein-Hilbert action:

$$S = S_{\text{EH}} + S_{\text{matter}} = \frac{c^4}{16\pi G}\int R\sqrt{-g}\,d^4x + \int \mathcal{L}_{\text{matter}}\sqrt{-g}\,d^4x$$

Step 1: Vary the action with respect to the metric $g^{\mu\nu}$:

$$\delta S = \frac{c^4}{16\pi G}\int \left(\delta R \sqrt{-g} + R \delta\sqrt{-g}\right)d^4x + \int \frac{\delta(\mathcal{L}_{\text{matter}}\sqrt{-g})}{\delta g^{\mu\nu}}\delta g^{\mu\nu}\,d^4x$$

Step 2: Use the variation formulas:

$$\delta\sqrt{-g} = -\frac{1}{2}\sqrt{-g} g_{\mu\nu}\delta g^{\mu\nu}$$$$\delta R = R_{\mu\nu}\delta g^{\mu\nu} + g^{\mu\nu}\delta R_{\mu\nu}$$

Step 3: The term $g^{\mu\nu}\delta R_{\mu\nu}$ integrates to a surface term (Gauss theorem):

$$\int g^{\mu\nu}\delta R_{\mu\nu}\sqrt{-g}\,d^4x = \int \nabla_\lambda K^\lambda\sqrt{-g}\,d^4x = \oint K^\lambda dS_\lambda = 0$$

Step 4: Define the stress-energy tensor from matter variation:

$$T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L}_{\text{matter}}\sqrt{-g})}{\delta g^{\mu\nu}} = 2\frac{\partial\mathcal{L}_{\text{matter}}}{\partial g^{\mu\nu}} - g_{\mu\nu}\mathcal{L}_{\text{matter}}$$

Step 5: Collect all terms and set $\delta S = 0$:

$$\frac{c^4}{16\pi G}\left(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\right) = \frac{1}{2}T_{\mu\nu}$$

The Einstein Field Equations

The fundamental equations of General Relativity relate spacetime curvature to matter-energy content:

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

Equivalently, using the Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$:

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

where:

  • • $G_{\mu\nu}$ is the Einstein tensor (describes spacetime curvature)
  • • $\Lambda$ is the cosmological constant (vacuum energy density)
  • • $T_{\mu\nu}$ is the stress-energy tensor (matter-energy content)
  • • $G$ is Newton's gravitational constant
  • • $c$ is the speed of light

Physical Interpretation: "Spacetime tells matter how to move; matter tells spacetime how to curve" (John Archibald Wheeler)

7. Stress-Energy Tensor

For a perfect fluid:

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

where $\rho$ is energy density, $p$ is pressure, and $u^\mu$ is the four-velocity.

For electromagnetic fields:

$$T_{\mu\nu} = \frac{1}{\mu_0}\left(F_{\mu\alpha}F_\nu^\alpha - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)$$

8. Geodesic Equation

Free particles follow geodesics, the straightest possible paths in curved spacetime:

$$\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$$

This can also be written using the covariant derivative along the path:

$$\frac{Du^\mu}{D\tau} = u^\nu\nabla_\nu u^\mu = 0$$

9. Schwarzschild Solution

Derivation of the Schwarzschild Metric

We seek a spherically symmetric, static, vacuum solution to Einstein's equations ($R_{\mu\nu} = 0$).

Step 1: Assume the most general spherically symmetric, static metric ansatz:

$$ds^2 = -e^{2\alpha(r)}c^2dt^2 + e^{2\beta(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

where $\alpha(r)$ and $\beta(r)$ are unknown functions, and $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the metric on the unit 2-sphere.

Step 2: The non-zero Christoffel symbols include:

$$\Gamma^t_{tr} = \alpha'(r), \quad \Gamma^r_{tt} = \alpha'e^{2(\alpha-\beta)}c^2, \quad \Gamma^r_{rr} = \beta'$$$$\Gamma^r_{\theta\theta} = -re^{-2\beta}, \quad \Gamma^r_{\phi\phi} = -re^{-2\beta}\sin^2\theta$$$$\Gamma^\theta_{r\theta} = \frac{1}{r}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta, \quad \Gamma^\phi_{r\phi} = \frac{1}{r}, \quad \Gamma^\phi_{\theta\phi} = \cot\theta$$

Step 3: Compute the non-zero components of Ricci tensor $R_{\mu\nu} = 0$:

$$R_{tt} = e^{2(\alpha-\beta)}\left(\alpha'' + \alpha'^2 - \alpha'\beta' + \frac{2\alpha'}{r}\right) = 0$$$$R_{rr} = -\alpha'' - \alpha'^2 + \alpha'\beta' + \frac{2\beta'}{r} = 0$$$$R_{\theta\theta} = e^{-2\beta}\left(r\beta' - r\alpha' - 1\right) + 1 = 0$$

Step 4: From $R_{tt} + R_{rr} = 0$:

$$\frac{2}{r}(\alpha' + \beta') = 0 \quad \Rightarrow \quad \alpha + \beta = \text{const}$$

Choose the constant to be zero (by rescaling time): $\beta = -\alpha$.

Step 5: Substituting $\beta = -\alpha$ into $R_{\theta\theta} = 0$:

$$e^{-2\beta}(1 + 2r\alpha') = 1$$$$e^{2\alpha}(1 + 2r\alpha') = 1$$$$\frac{d}{dr}(re^{2\alpha}) = 1$$

Step 6: Integrating:

$$re^{2\alpha} = r + C$$$$e^{2\alpha} = 1 + \frac{C}{r}$$

Step 7: Matching to Newtonian limit as $r \to \infty$ requires $\Phi = -GM/r$, giving:

$$e^{2\alpha} = 1 - \frac{2GM}{c^2r} = 1 - \frac{r_s}{r}$$

The Schwarzschild Metric

The unique spherically symmetric, static, vacuum solution to Einstein's equations:

$$ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

The Schwarzschild radius (event horizon for black holes):

$$r_s = \frac{2GM}{c^2} \approx 2.95 \text{ km} \left(\frac{M}{M_\odot}\right)$$

Key Properties:

  • • Asymptotically flat: $ds^2 \to \eta_{\mu\nu}dx^\mu dx^\nu$ as $r \to \infty$
  • • Coordinate singularity at $r = r_s$ (removable by coordinate transformation)
  • • Physical singularity at $r = 0$ (curvature invariants diverge)
  • • Describes spacetime outside any spherically symmetric mass distribution
  • • Valid for stars, planets, and black holes

10. Weak Field Limit and Gravitational Waves

For weak fields, we write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $|h_{\mu\nu}| \ll 1$. To first order, the field equations become:

$$\Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}$$

where $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$ is the trace-reversed metric perturbation.

In vacuum ($T_{\mu\nu} = 0$), this gives the wave equation:

$$\Box \bar{h}_{\mu\nu} = 0$$

describing gravitational waves propagating at the speed of light.

11. Newtonian Limit

In the weak field, slow motion limit, GR reduces to Newton's theory. For $h_{00} = -2\Phi/c^2$:

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

This is Poisson's equation for the Newtonian gravitational potential.

12. Einstein-Hilbert Action

The Einstein field equations can be derived from the action principle:

$$S = \frac{c^4}{16\pi G}\int R\sqrt{-g}d^4x + S_{\text{matter}}$$

Varying with respect to the metric gives:

$$\delta S = 0 \quad \Rightarrow \quad R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$$