PU-403 · General Relativity
The unit builds the geometric theory of gravitation from the equivalence principle, developing the differential geometry of curved spacetime until the Einstein field equations emerge as the unique low-derivative relation between matter and curvature. It then extracts the observational and structural consequences — light bending, perihelion precession, black holes, gravitational waves, and cosmological expansion — showing how a single geometric law reproduces Newtonian gravity and predicts phenomena beyond it.
Lectures
| L01 | Why Newtonian Gravity Is Incompatible with Relativity — |
| L02 | The Equivalence Principle and the Turn to Geometry |
| L03 | Manifolds, Tangent Spaces, and Tensors — |
| L04 | The Metric and Proper Time |
| L05 | Geodesics as Extremal Worldlines |
| L06 | The Covariant Derivative and the Connection |
| L07 | The Riemann Curvature Tensor |
| L08 | Geodesic Deviation and Tidal Forces |
| L09 | Ricci, the Bianchi Identity, and the Einstein Tensor |
| L10 | Stress-Energy and the Source of Gravity |
| L11 | The Einstein-Hilbert Action and the Field Equations |
| L12 | The Newtonian Limit and the Coupling Constant |
| L13 | The Schwarzschild Solution and Birkhoff's Theorem |
| L14 | Orbits in Schwarzschild and Mercury's Perihelion |
| L15 | Light Bending and Gravitational Redshift |
| L16 | The Classical Tests and Their Modern Precision |
| L17 | Inside the Horizon: Coordinate vs Physical Singularities |
| L18 | Kruskal Extension and Causal Structure |
| L19 | Rotating and Charged Black Holes: A Survey |
| L20 | Linearized Gravity and Gauge Freedom |
| L21 | Gravitational Waves and Their Polarizations |
| L22 | The Quadrupole Formula and Binary Inspiral |
| L23 | Detecting Gravitational Waves: LIGO and Beyond |
| L24 | Symmetry, Homogeneity, and the FLRW Metric |
| L25 | The Friedmann Equations and Cosmic Dynamics |
| L26 | Cosmological Redshift and the Hubble Law |
| L27 | The Cosmological Constant and Dark Energy |
| L28 | Energy Conditions and the Raychaudhuri Equation |
| L29 | Singularity Theorems: A Structural Overview |
| L30 | The Frontier: Quantum Gravity and Open Problems — |
Derivations homed in this unit
Equivalence Principle Forces a Curved Metric
Deriving that the universality of free fall implies gravity must be encoded in a curved spacetime metric rather than a force field on flat spacetime.
Geodesic Equation from Extremal Proper Time
Deriving the geodesic equation and Christoffel symbols by extremizing proper time along a worldline in a general metric.
Covariant Derivative and the Metric Connection
Constructing the unique torsion-free metric-compatible connection and its covariant derivative from the demand that tensor differentiation be tensorial.
Riemann Tensor from the Covariant Commutator
Deriving the Riemann curvature tensor as the commutator of covariant derivatives and establishing its algebraic and Bianchi symmetries.
Parallel Transport and Geodesic Deviation
Deriving the geodesic deviation equation showing that relative acceleration of nearby free-falling particles measures the Riemann tensor.
Contracted Bianchi Identity and the Einstein Tensor
Deriving the contracted Bianchi identity and showing the Einstein tensor is the unique divergence-free combination of Ricci curvature and metric.
Stress-Energy Tensor as the Conserved Source
Deriving the symmetric stress-energy tensor from translation invariance and showing its covariant conservation is required by the Bianchi identity.
Field Equations from the Einstein-Hilbert Action
Deriving the Einstein field equations by varying the Einstein-Hilbert action with respect to the metric and identifying the stress-energy tensor as its matter-source.
Newtonian Limit Fixes the Coupling Constant
Deriving the weak-field, slow-motion limit that reduces the field equations to Poisson's equation and fixes the 8 pi G over c to the fourth coupling.
Schwarzschild Solution from Spherical Vacuum
Deriving the unique static spherically symmetric vacuum solution and, via Birkhoff's theorem, its uniqueness even in the dynamic case.
Perihelion Precession from Schwarzschild Orbits
Deriving the anomalous perihelion advance of a bound orbit from the effective potential of Schwarzschild geodesics.
Light Deflection and Gravitational Redshift
Deriving the bending of light by the Sun and the gravitational shift of spectral lines from null geodesics in the Schwarzschild metric.
Event Horizon and Maximal Extension
Showing the Schwarzschild coordinate singularity at r equals two G M is a regular horizon by constructing Kruskal-Szekeres coordinates and the causal structure.
Linearized Gravity and the Wave Equation
Deriving the linearized field equations in transverse-traceless gauge and showing metric perturbations propagate as waves at the speed of light.
Quadrupole Formula for Radiated Power
Deriving the power radiated in gravitational waves by a source in terms of the third time derivative of its mass quadrupole moment.
Friedmann Equations from the FLRW Metric
Deriving the Friedmann equations for the scale factor by inserting the homogeneous isotropic FLRW metric and a perfect fluid into the field equations.
Cosmological Redshift and Hubble's Law
Deriving the redshift-scale-factor relation and the linear Hubble law for nearby sources from null geodesics in an expanding FLRW spacetime.
Raychaudhuri Equation and Geodesic Focusing
Deriving the Raychaudhuri equation for congruence expansion and showing the energy conditions force geodesic focusing, seeding the singularity theorems.