General Relativity

A comprehensive graduate-level course on Einstein's theory of gravity—from tensor calculus and differential geometry through the Einstein field equations to black holes, cosmology, and gravitational waves.

Course Overview

General Relativity, formulated by Albert Einstein in 1915, is the modern theory of gravitation. It describes gravity not as a force, but as a manifestation of the curvature of spacetime caused by mass and energy. This course provides a rigorous mathematical treatment of GR, starting from differential geometry and culminating in applications to black holes, cosmology, and gravitational wave astronomy.

What You'll Learn

• Tensor analysis and differential geometry
• Riemann curvature tensor and Einstein equations
• Schwarzschild and Kerr solutions
• Black hole physics and thermodynamics
• Cosmological models (FLRW universe)
• Gravitational waves and LIGO detections
• Singularity theorems and quantum gravity
• Numerical relativity and modern applications

Prerequisites

• Special Relativity
• Advanced Mathematics (calculus, linear algebra)
• Vector calculus and differential equations
• Tensor Calculus (100 video lectures)
• Classical mechanics (Lagrangian formulation)
• Some exposure to Quantum Mechanics helpful

Course Structure

5 Parts covering 29 chapters • From differential geometry to quantum gravity • Includes detailed derivations, worked examples, and observational evidence • Suitable for advanced undergraduates and graduate students

Course Parts

📐

Part I: Differential Geometry

Mathematical foundations: manifolds, tensors, metric, connections, and covariant derivatives. Master the geometric language needed for General Relativity.

ManifoldsTensorsConnections6 Chapters

🌀

Part II: Curvature of Spacetime

The Riemann tensor, Ricci curvature, Bianchi identities, and geodesic deviation. Learn how curvature encodes gravity.

Riemann TensorGeodesicsKilling Vectors5 Chapters

⚛️

Part III: Einstein Field Equations

Derive Einstein's field equations from the Einstein-Hilbert action. Study the stress-energy tensor, weak field limit, and Newtonian approximation.

Field EquationsVariational PrincipleConservation Laws6 Chapters

🌌

Part IV: Classic Solutions

Exact solutions: Schwarzschild (spherical black holes), Kerr (rotating black holes), Reissner-Nordström (charged), FLRW cosmology, and gravitational waves.

SchwarzschildKerr MetricCosmology6 Chapters

🔬

Part V: Advanced Topics

Singularity theorems, black hole thermodynamics, ADM formalism, numerical relativity, gravitational lensing, and the path to quantum gravity.

SingularitiesBH ThermodynamicsQuantum Gravity6 Chapters

Key Equations — Step-by-Step Derivations

Below we develop the ten core equations of General Relativity from first principles. Each derivation walks through the key mathematical steps and ends with a boxed final result and a physical interpretation.

1. Metric Tensor & Line Element

In flat Minkowski spacetime the infinitesimal interval is

$$ds^{2}=\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}=-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}$$

To generalise to curved spacetime we replace the constant matrix $\eta_{\mu\nu}$ with a position-dependent symmetric tensor $g_{\mu\nu}(x)$, the metric tensor. The interval becomes

$$ds^{2}=g_{\mu\nu}(x)\,dx^{\mu}\,dx^{\nu}$$

Because the metric is symmetric ($g_{\mu\nu}=g_{\nu\mu}$), in four dimensions it has$\tfrac{4\times5}{2}=10$ independent components. Its inverse is defined by

$$g^{\mu\alpha}\,g_{\alpha\nu}=\delta^{\mu}{}_{\nu}$$

The metric raises and lowers indices: $V^{\mu}=g^{\mu\nu}V_{\nu}$ and$V_{\mu}=g_{\mu\nu}V^{\nu}$. It encodes all information about distances, angles, volumes, and causal structure.

Final Result

$$\boxed{ds^{2}=g_{\mu\nu}(x)\,dx^{\mu}\,dx^{\nu}}$$

Physical interpretation: The metric tensor determines the geometry of spacetime. It tells us how to compute proper time, proper distance, and the causal relationship (timelike, null, or spacelike) between neighbouring events.

2. Christoffel Symbols (Levi-Civita Connection)

We require a connection that is (i) torsion-free ($\Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu}$) and (ii) metric-compatible:

$$\nabla_{\alpha}\,g_{\mu\nu}=0$$

Writing out metric compatibility explicitly:

$$\partial_{\alpha}g_{\mu\nu}-\Gamma^{\lambda}{}_{\alpha\mu}\,g_{\lambda\nu}-\Gamma^{\lambda}{}_{\alpha\nu}\,g_{\mu\lambda}=0$$

Cyclic-permute the free indices $(\alpha,\mu,\nu)$ to obtain three equations. Add the first two and subtract the third; symmetry of $\Gamma$ in its lower indices causes four of the six connection terms to cancel, leaving:

$$2\,g_{\lambda\nu}\,\Gamma^{\lambda}{}_{\mu\alpha}=\partial_{\mu}g_{\nu\alpha}+\partial_{\alpha}g_{\mu\nu}-\partial_{\nu}g_{\mu\alpha}$$

Contract with $g^{\lambda\sigma}\!\cdot\! g_{\lambda\nu}=\delta^{\sigma}{}_{\nu}$(relabelling the free upper index) to isolate the connection:

Final Result

$$\boxed{\Gamma^{\lambda}{}_{\mu\nu}=\frac{1}{2}\,g^{\lambda\sigma}\!\left(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu}\right)}$$

Physical interpretation: The Christoffel symbols encode how coordinate basis vectors change from point to point. They are not tensors; they vanish in a local inertial frame. They play the role of "gravitational force" in the equation of motion.

3. Geodesic Equation

A free particle extremises its proper time between two events:

$$\delta\!\int d\tau=\delta\!\int\!\sqrt{-g_{\mu\nu}\,\frac{dx^{\mu}}{d\lambda}\,\frac{dx^{\nu}}{d\lambda}}\;d\lambda=0$$

Define the Lagrangian $\mathcal{L}=g_{\mu\nu}\,\dot{x}^{\mu}\dot{x}^{\nu}$ (dot = $d/d\tau$). The Euler-Lagrange equation gives:

$$\frac{d}{d\tau}\!\left(g_{\mu\alpha}\,\dot{x}^{\alpha}\right)-\frac{1}{2}\,\partial_{\mu}g_{\alpha\beta}\,\dot{x}^{\alpha}\dot{x}^{\beta}=0$$

Expanding the total derivative and contracting with $g^{\mu\lambda}$ yields:

$$\ddot{x}^{\mu}+\frac{1}{2}\,g^{\mu\lambda}\!\left(\partial_{\alpha}g_{\beta\lambda}+\partial_{\beta}g_{\alpha\lambda}-\partial_{\lambda}g_{\alpha\beta}\right)\dot{x}^{\alpha}\dot{x}^{\beta}=0$$

Recognising the Christoffel symbol from Derivation 2:

Final Result

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

Physical interpretation: Free-falling particles follow geodesics -- the straightest possible paths in curved spacetime. The Christoffel term acts as a "gravitational acceleration" that is purely geometric in origin.

4. Riemann Curvature Tensor

Curvature is detected by the failure of covariant derivatives to commute. For any vector$V^{\rho}$:

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

Evaluating the left side explicitly using$\nabla_{\mu}V^{\rho}=\partial_{\mu}V^{\rho}+\Gamma^{\rho}{}_{\mu\lambda}V^{\lambda}$:

$$\nabla_{\mu}\nabla_{\nu}V^{\rho}=\partial_{\mu}\!\left(\nabla_{\nu}V^{\rho}\right)+\Gamma^{\rho}{}_{\mu\lambda}\nabla_{\nu}V^{\lambda}-\Gamma^{\lambda}{}_{\mu\nu}\nabla_{\lambda}V^{\rho}$$

Antisymmetrising in $\mu\leftrightarrow\nu$, all terms involving$\partial V$ cancel (torsion-free), leaving only terms built from$\Gamma$ and $\partial\Gamma$:

Final Result

$$\boxed{R^{\rho}{}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma}+\Gamma^{\rho}{}_{\mu\lambda}\,\Gamma^{\lambda}{}_{\nu\sigma}-\Gamma^{\rho}{}_{\nu\lambda}\,\Gamma^{\lambda}{}_{\mu\sigma}}$$

Physical interpretation: The Riemann tensor measures the intrinsic curvature of spacetime. It determines geodesic deviation (tidal forces): nearby free-falling particles accelerate relative to each other whenever $R^{\rho}{}_{\sigma\mu\nu}\neq 0$.

5. Ricci Tensor & Ricci Scalar

The Riemann tensor has $20$ independent components in 4D. We extract physically relevant traces by contraction.

Step 1 — Ricci tensor. Contract the first and third indices of Riemann:

$$R_{\mu\nu}\;\equiv\;R^{\lambda}{}_{\mu\lambda\nu}=\partial_{\lambda}\Gamma^{\lambda}{}_{\nu\mu}-\partial_{\nu}\Gamma^{\lambda}{}_{\lambda\mu}+\Gamma^{\lambda}{}_{\lambda\sigma}\,\Gamma^{\sigma}{}_{\nu\mu}-\Gamma^{\lambda}{}_{\nu\sigma}\,\Gamma^{\sigma}{}_{\lambda\mu}$$

Step 2 — Ricci scalar. Contract the Ricci tensor with the inverse metric:

$$R\;\equiv\;g^{\mu\nu}\,R_{\mu\nu}$$

The contracted Bianchi identity$\nabla_{\mu}\!\left(R^{\mu\nu}-\tfrac{1}{2}g^{\mu\nu}R\right)=0$then defines the divergence-free Einstein tensor:

$$G_{\mu\nu}\equiv R_{\mu\nu}-\tfrac{1}{2}\,g_{\mu\nu}\,R,\qquad \nabla^{\mu}G_{\mu\nu}=0$$

Final Results

$$\boxed{R_{\mu\nu}=R^{\lambda}{}_{\mu\lambda\nu}},\qquad \boxed{R=g^{\mu\nu}R_{\mu\nu}},\qquad \boxed{G_{\mu\nu}=R_{\mu\nu}-\tfrac{1}{2}\,g_{\mu\nu}R}$$

Physical interpretation: The Ricci tensor encodes how volumes of geodesic balls change due to matter, while the Ricci scalar gives the overall scalar curvature. The Einstein tensor, being divergence-free, is the natural geometric quantity to equate to the stress-energy tensor.

6. Einstein Field Equations

Start from the Einstein-Hilbert action with matter and cosmological constant:

$$S=\int\!\left[\frac{1}{16\pi G}\!\left(R-2\Lambda\right)+\mathcal{L}_{\mathrm{m}}\right]\sqrt{-g}\;d^{4}x$$

Step 1. Vary with respect to $g^{\mu\nu}$. Using the Palatini identity$\delta R_{\mu\nu}=\nabla_{\lambda}\delta\Gamma^{\lambda}{}_{\nu\mu}-\nabla_{\nu}\delta\Gamma^{\lambda}{}_{\lambda\mu}$and the fact that the covariant-divergence terms integrate to a boundary (discarded):

$$\delta S_{\text{grav}}=\frac{1}{16\pi G}\int\!\left(R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R+\Lambda\,g_{\mu\nu}\right)\delta g^{\mu\nu}\sqrt{-g}\;d^{4}x$$

Step 2. Define the stress-energy tensor from the matter part:

$$T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\,\frac{\delta\!\left(\sqrt{-g}\,\mathcal{L}_{\mathrm{m}}\right)}{\delta g^{\mu\nu}}$$

Step 3. Setting $\delta S=0$ for arbitrary $\delta g^{\mu\nu}$yields (restoring $c$):

Final Result

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

Physical interpretation: The left side is pure geometry (curvature plus cosmological constant); the right side is the matter-energy content of spacetime. Energy-momentum conservation $\nabla^{\mu}T_{\mu\nu}=0$ follows automatically from the Bianchi identity.

7. Schwarzschild Metric

Ansatz. Seek a static, spherically symmetric vacuum ($T_{\mu\nu}=0$) solution. The most general such line element is:

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

Step 1. Compute the non-zero Christoffel symbols from this metric and substitute into the Ricci tensor. The vacuum field equations reduce to $R_{\mu\nu}=0$.

Step 2. The $R_{tt}=0$ and $R_{rr}=0$ components give$\alpha'+ \beta'=0$, so $\alpha=-\beta+\text{const}$. Absorbing the constant into the time coordinate: $\alpha=-\beta$.

Step 3. The $R_{\theta\theta}=0$ equation then yields:

$$e^{2\alpha}\!\left(1+2r\alpha'\right)=1 \quad\Longrightarrow\quad e^{2\alpha}=1-\frac{C}{r}$$

where $C$ is an integration constant. Matching to the Newtonian limit$g_{tt}\approx-(1-2GM/c^{2}r)$ fixes $C=2GM/c^{2}\equiv r_{s}$, the Schwarzschild radius.

Final Result

$$\boxed{ds^{2}=-\!\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}+\left(1-\frac{r_{s}}{r}\right)^{\!-1}dr^{2}+r^{2}d\Omega^{2},\qquad r_{s}=\frac{2GM}{c^{2}}}$$

Physical interpretation: This is the unique spherically symmetric vacuum solution (Birkhoff's theorem). It describes spacetime outside any non-rotating, uncharged spherical mass. At $r=r_{s}$ lies the event horizon; the coordinate singularity there is removable (e.g. Eddington-Finkelstein coordinates), while $r=0$ is a true curvature singularity.

8. Gravitational Redshift

Consider two stationary observers in the Schwarzschild geometry at radii$r_{1}$ (emitter) and $r_{2}\to\infty$ (distant receiver).

Step 1. Proper time for a stationary observer at radius $r$ is related to coordinate time by:

$$d\tau=\sqrt{-g_{tt}}\;dt=\sqrt{1-\frac{r_{s}}{r}}\;dt$$

Step 2. Light signals travel along null geodesics. For a static metric the coordinate time between successive crests is the same at emission and reception:$\Delta t_{\text{emit}}=\Delta t_{\text{rec}}$. But the proper-time intervals differ:

$$\frac{\Delta\tau_{\text{rec}}}{\Delta\tau_{\text{emit}}}=\frac{\sqrt{1-r_{s}/r_{2}}}{\sqrt{1-r_{s}/r_{1}}}$$

Step 3. Since frequency is inversely proportional to proper period ($\nu\propto1/\Delta\tau$), taking $r_{2}\to\infty$:

Final Result

$$\boxed{\frac{\nu_{\infty}}{\nu_{r}}=\sqrt{1-\frac{r_{s}}{r}}}$$

Physical interpretation: Light climbing out of a gravitational well loses energy and is red-shifted. Near the event horizon ($r\to r_{s}$) the redshift becomes infinite. This effect has been verified by the Pound-Rebka experiment and is essential for GPS satellite corrections.

9. Orbital Equation & Effective Potential

Step 1. The Schwarzschild metric has two Killing vectors ($\partial_{t}$ and $\partial_{\phi}$), yielding conserved energy and angular momentum per unit mass:

$$E=\left(1-\frac{r_{s}}{r}\right)c^{2}\frac{dt}{d\tau},\qquad L=r^{2}\frac{d\phi}{d\tau}$$

Step 2. Substituting into the normalisation condition$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-c^{2}$ (for massive particles) and restricting to the equatorial plane ($\theta=\pi/2$):

$$\frac{1}{2}\!\left(\frac{dr}{d\tau}\right)^{\!2}+V_{\mathrm{eff}}(r)=\frac{E^{2}-c^{4}}{2c^{2}}$$

Step 3. The effective potential is:

Final Result

$$\boxed{V_{\mathrm{eff}}(r)=-\frac{GM}{r}+\frac{L^{2}}{2r^{2}}-\frac{GML^{2}}{c^{2}r^{3}}}$$

The three terms are: (i) Newtonian gravity, (ii) centrifugal barrier (also present in Newtonian theory), and (iii) the relativistic correction unique to GR.

Physical interpretation: The GR correction term$-GML^{2}/(c^{2}r^{3})$ is attractive and dominates at small $r$, causing the innermost stable circular orbit (ISCO) at $r=3r_{s}$. It also produces the perihelion precession of Mercury ($\Delta\phi=6\pi GM/(c^{2}a(1-e^{2}))$ per orbit).

10. Gravitational Wave Equation

Step 1 — Linearisation. Write the metric as a small perturbation about flat spacetime:

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\qquad |h_{\mu\nu}|\ll 1$$

Step 2. To first order in $h$, the Einstein tensor becomes (after lengthy algebra):

$$G^{(1)}_{\mu\nu}=\frac{1}{2}\!\left(\partial_{\sigma}\partial_{\nu}h^{\sigma}{}_{\mu}+\partial_{\sigma}\partial_{\mu}h^{\sigma}{}_{\nu}-\Box h_{\mu\nu}-\partial_{\mu}\partial_{\nu}h-\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}+\eta_{\mu\nu}\Box h\right)$$

where $h=\eta^{\mu\nu}h_{\mu\nu}$ is the trace and$\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ is the flat d'Alembertian.

Step 3 — Trace-reversed perturbation. Define$\bar{h}_{\mu\nu}=h_{\mu\nu}-\tfrac{1}{2}\eta_{\mu\nu}h$.

Step 4 — Lorenz gauge. Impose the gauge condition$\partial^{\nu}\bar{h}_{\mu\nu}=0$, which eliminates all but the$\Box\bar{h}_{\mu\nu}$ term:

Final Result

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

In vacuum ($T_{\mu\nu}=0$) this reduces to the wave equation$\Box\,\bar{h}_{\mu\nu}=0$, predicting gravitational waves propagating at the speed of light. In the transverse-traceless (TT) gauge the two physical polarisations are $h_{+}$ and $h_{\times}$.

Physical interpretation: Gravitational waves are ripples in spacetime generated by accelerating masses. They were first directly detected by LIGO in 2015 (event GW150914), confirming a century-old prediction of General Relativity.

Quick-Reference Summary

#	Name	Equation
1	Line Element	$ds^{2}=g_{\mu\nu}\,dx^{\mu}dx^{\nu}$
2	Christoffel Symbols	$\Gamma^{\lambda}{}_{\mu\nu}=\tfrac{1}{2}g^{\lambda\sigma}(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu})$
3	Geodesic Equation	$\ddot{x}^{\mu}+\Gamma^{\mu}{}_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}=0$
4	Riemann Tensor	$R^{\rho}{}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma}+\Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma}-\Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}$
5	Ricci Tensor / Scalar	$R_{\mu\nu}=R^{\lambda}{}_{\mu\lambda\nu};\quad R=g^{\mu\nu}R_{\mu\nu}$
6	Einstein Field Equations	$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{\mu\nu}$
7	Schwarzschild Metric	$ds^{2}=-(1-r_{s}/r)\,c^{2}dt^{2}+(1-r_{s}/r)^{-1}dr^{2}+r^{2}d\Omega^{2}$
8	Gravitational Redshift	$\nu_{\infty}/\nu_{r}=\sqrt{1-r_{s}/r}$
9	Effective Potential	$V_{\text{eff}}=-GM/r+L^{2}/(2r^{2})-GML^{2}/(c^{2}r^{3})$
10	Gravitational Waves	$\Box\bar{h}_{\mu\nu}=-\frac{16\pi G}{c^{4}}T_{\mu\nu}$