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General Relativity

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2025 · Graduate Refresher Course

General Relativity

General Relativity Visual

The following lectures were part of the two-week refresher course on General Relativity held at IUCAA, from 25 June to 4 July 2025. The course features two complementary tracks: Track A with a differential geometry approach to general relativity, including modern developments and conceptual connections with quantum theory, and Track B focusing on tensor calculus and the physical aspects of general relativity.

The course is ideal for university teachers, graduate students, and anyone seeking a structured, in-depth refresher in General Relativity.

Meet the speakers
Course Index ▼
Track A — Prof. T. P. Singh
  1. History of gravitation
  2. Modern perspectives on gravity and quantum theory
  3. Rindler coordinates
  4. Gravitation vs electromagnetism
  5. Differential forms
  6. Exterior calculus
  7. Connection 1-forms
  8. Computing curvature using differential forms
  9. Einstein's equations & Weyl curvature
  10. Gravitational collapse
  11. Naked singularities
Track B — Prof. Patrick Das Gupta
  1. Inertial frames, good clocks and the equivalence principle
  2. Differential geometry – I
  3. Differential geometry – II
  4. Differential geometry – III
  5. Tensor calculus and the Newtonian limit
  6. Physical aspects of general relativity
  7. Geometrical aspects & Energy-Momentum tensor
  8. Einstein-Hilbert action
  9. Deriving Einstein's equations
  10. Killing vectors & cosmology
Special Content

Video Lectures

Track A — Prof. T. P. Singh▼

Lecture A1: History of gravitation

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A historical overview of gravitational theory — from Ptolemy's geocentric model and Copernican heliocentrism, through Kepler's laws and Newtonian mechanics, to Maxwell's electromagnetism, Einstein's relativity, and the ongoing tension between general relativity and quantum theory.

Lecture A2: Modern perspectives on gravity and quantum theory

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Structural parallels between gauge theories in particle physics and general relativity. Reviews multiple formulations of GR — metric, tetrad (vierbein), Palatini, ADM, Ashtekar, and Cartan — and addresses quantum gravity on spacetime using division algebras and the Dirac operator.

Lecture A3: Uniformly accelerated observers – the Rindler coordinates

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Introduces uniformly accelerating reference frames in flat spacetime, Rindler coordinates for an accelerating observer, and the emergence of the Rindler horizon.

Lecture A4: Gravitation vs electromagnetism

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Comparison of gravitation and electrodynamics at non-relativistic and relativistic levels — Newton vs Coulomb, magneto-statics and gravito-magnetism, Maxwell vs special-relativistic Newtonian gravitation. Discusses gravity as spacetime curvature and gauge theories in both frameworks.

Lecture A5: General relativity in the language of differential forms

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Introduces differential geometry for GR: manifolds, vectors, one-forms, exterior derivative, gradient vector and one-form, dual basis, metric 2-form, Riemann tensor, and wedge products.

Lecture A6: Differential forms and exterior calculus

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Area 2-forms, dot and vector products, Maxwell's equations via differential forms, Hodge star dual, exterior derivative, closed and exact forms, the fundamental theorem of exterior calculus, and revisiting Green's, Stokes' and Gauss' theorems.

Lecture A7: Connection 1-forms and Cartan structural equations

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Curvilinear coordinates, parallel transport, covariant derivative in curvilinear coordinates, connection 1-forms, torsion and curvature 2-forms, Cartan structural equations, and the relation between connection 1-forms and connection coefficients.

Lecture A8: Computing curvature using differential forms

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Different definitions of the Riemann tensor, curvature 2-form as vector parallel transport around a closed loop, computing curvature via Cartan's second structure equation, and Bianchi identities of the curvature 2-form.

Lecture A9: Motivating Einstein's equations and Weyl curvature

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Concludes the differential-forms formulation of GR. Motivates the Einstein field equations from the Riemann curvature tensor, discusses the Weyl curvature tensor for matter-free spacetime, and outlines general properties of the Einstein equations in the language of differential geometry.

Lecture A10: Continuous gravitational collapse and black holes

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Centrally symmetric gravitational field in vacuum, gravitational collapse of a spherical body, Tolman-Bondi and Dutt-Oppenheimer-Snyder solutions for dust collapse in Newtonian gravity and GR, and the formation of a black hole.

Lecture A11: Formation of naked singularities

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Formation of naked singularities in the light of cosmic censorship conjectures. Introduces Horizonless Compact Objects (HCO) and their possible observational signatures.
Track B — Prof. Patrick Das Gupta▼

Lecture B1: Inertial frames, good clocks and the equivalence principle

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Foundations of special and early relativistic physics. Defines inertial and non-inertial reference frames, explains the role of ideal clocks in measuring proper time, and introduces the invariant spacetime interval and the Minkowski metric.

Lecture B2: Differential geometry – I

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Introduction to basic elements of differential geometry: open and closed sets, topology and topological space, covering of a topological space, functions and homeomorphisms, topological manifolds, and curves on manifolds.

Lecture B3: Differential geometry – II

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Differentiable atlases and manifolds, smooth functions between manifolds, smooth curves, tangent and cotangent spaces, vectors and one-forms, coordinate bases, tensor products, and wedge products.

Lecture B4: Differential geometry – III

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Coordinate bases, vector and 1-form fields, general tensor fields, p-forms, wedge products, exterior derivative, exact and closed forms, the metric tensor, Hodge star operation, Minkowski metric and Lorentz invariance, and proper time in accelerating frames.

Lecture B5: Tensor calculus and the Newtonian limit of general relativity

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Introduction to tensor calculus — general tensors, connection coefficients, covariant derivative, metric tensor, transformation properties of tensors. Ends with linearised gravity and deriving the Newtonian limit of GR.

Lecture B6: Physical aspects of general relativity

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Spherical polar coordinates, gravitational time dilation, gravitational frequency shift, 4-velocity in curvilinear coordinates, causality in curved spacetime, the Riemann curvature tensor, and the geodesic deviation equation.

Lecture B7: Geometrical aspects of general relativity and the energy-momentum tensor

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Spacetime metric, Christoffel symbols, Riemann tensor in terms of Christoffel symbols, parallel transport, geodesic equation, Levi-Civita connection, Bianchi identities, the Einstein tensor, and the energy-momentum tensor.

Lecture B8: Einstein–Hilbert action and derivation of the energy-momentum tensor

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Einstein equations, Weyl curvature tensor, Noether's theorem and conserved quantities in flat spacetime, derivation of the energy-momentum tensor via Hilbert's variational method, the Einstein-Hilbert action, and geometrodynamics.

Lecture B9: Deriving Einstein's equations and the Killing equation

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Einstein field equations derived by extremizing the Einstein-Hilbert action. Gauge invariance of GR under infinitesimal coordinate transformations and derivation of the Killing equation.

Lecture B10: Killing vectors, conservation laws and cosmology

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Killing vector fields and conserved quantities. Conservation of energy and angular momentum in Schwarzschild spacetime. Affine parameterisation of geodesics. Application of GR to cosmology via the FRW metric and Friedmann equations.
Special Lectures▼

Lecture 1: A refresher on quantum field theory

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Poisson brackets and commutation relations, canonical quantization, quantising the simple harmonic oscillator, real scalar field and the Klein-Gordon equation, quantising the real scalar field, and the complex scalar field.

Lecture 2: Gravitational waves and LIGO India

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Gravitational waves in the linearised approximation, the quadrupole formula, sources and properties, Compact Binary Coalescence, gravitational wave detection, the LIGO detectors, and details of the upcoming LIGO India detector.

Video Tutorials

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Tutorial 1

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Derivation of Maxwell's equations in covariant form from the action principle, deriving the continuity equation, and showing charge conservation in flat and curved spacetimes.

Tutorial 2

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Gravity as a central force, motion in polar coordinates, effective potentials, Kepler's problem, and perihelion precession of the planets using a perturbation term to the Newtonian gravitational potential.

Tutorial 3

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Use of the Mathematica package xAct and its extension xTensor in general relativity.

Tutorial 4

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The exterior derivative of the electromagnetic field 2-form is shown to be identically zero (homogeneous Maxwell's equations). The Hodge star dual of the 2-form is shown to equal the Hodge star dual of the current-density 1-form.

Tutorial 5

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Rindler spacetime for uniformly accelerating observers, the Unruh-Fulling effect, Rindler wedges, observer-dependent quantum states, inner products, field quantisation, and Bogoliubov coefficients.

Tutorial 6

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Classical tests of GR: gravitational frequency shift, the weak field limit, the Pound-Rebka experiment, Schild's argument, and gravitational frequency shift for observers in general spacetime.

Tutorial 7

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Geodesic equations for cylindrical surfaces, 2-spheres, and the Schwarzschild metric. The geodesic deviation equation and its relation to the Riemann tensor.

Tutorial 8

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Ricci tensor components in the FRW spacetime, the energy-momentum tensor of a perfect fluid in a homogeneous and isotropic universe, and the first-order trajectory equation in Schwarzschild metric using Killing vector fields.

Tutorial 9

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Continuation of the xAct and xTensor tutorial in general relativity.

Tutorial 10

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Demonstrates the Unruh effect using the Bogoliubov transformation.

Tutorial 11

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Applications of the xCoba extension of xAct in general relativity and beyond.

Tutorial 12

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The Kerr metric and black holes — ergosphere, frame dragging, Killing horizons, and the extraction of energy from a rotating black hole.

Tutorial 13

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Near-horizon geometries of Schwarzschild and Reissner-Nordström black holes. Field quantisation, observer-dependent vacuum states, and their relation to Hawking radiation.

Tutorial 14

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Continuation of the xCoba extension of xAct tutorial in general relativity and beyond.

Tutorial 15

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Continuation of the tutorial and discussion on Hawking radiation.

Tutorial 16

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Applications of the xTras extension of xAct in general relativity and beyond.

Assignments

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Assignment 1: Central forces, Keplerian mechanics and orbital precession

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Problems covering gravity as a central force, motion in polar coordinates, effective potentials, Kepler's problem, and perihelion precession using a perturbation term to the Newtonian gravitational potential.

Assignment 2: Covariant Maxwell's equations and charge conservation

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Derivation of Maxwell's equations in covariant form from the action principle, the continuity equation, and charge conservation in flat and curved spacetimes.

Assignment 3: Gauge invariance and exterior derivative of the EM field 2-form

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Properties of the electromagnetic 2-form tensor and showing that a mass term in the Proca Lagrangian breaks gauge invariance under U(1) transformation — concluding the photon must be massless.

Assignment 4: Rindler wedges and understanding the Unruh-Fulling effect

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Rindler spacetime for uniformly accelerating observers, Rindler wedges, observer-dependent quantum states, inner products, field quantisation, and Bogoliubov coefficients.

Assignment 5: Classical tests of general relativity

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Classical experimental tests: gravitational frequency shift, the Shapiro delay, bending of light, and precession of planetary orbits.

Assignment 6: Geodesic equation and geodesic deviation

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Geodesic equations for cylindrical surfaces, 2-spheres, and the Schwarzschild metric. The geodesic deviation equation and its relation to the Riemann tensor.

Assignment 7: Ricci tensor and energy-momentum tensor of a homogeneous universe

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Ricci tensor components in FRW spacetime, the energy-momentum tensor for a perfect fluid in a homogeneous and isotropic universe.

Assignment 8: Trajectory equations in Schwarzschild spacetime using Killing vectors

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Killing vector fields of the Schwarzschild metric, conserved quantities, and first-order differential equations for radial trajectories.

Assignment 9: Unruh effect via the Bogoliubov coefficients

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Demonstrating the Unruh effect using Bogoliubov transformations and field quantisation in non-inertial frames.

Assignment 10: Kerr black holes and energy extraction

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The Kerr metric — ergosphere, frame dragging, Killing horizons, and the Penrose process for extracting energy from rotating black holes.

Assignment 11: Near horizon geometries and Hawking radiation

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Near-horizon geometries of Schwarzschild and Reissner-Nordström black holes. Field quantisation and observer-dependent vacuum states in relation to Hawking radiation.

Tutorial Notebooks: Applications of xAct in General Relativity

The following notebooks demonstrate the use of the Mathematica package xAct and its extensions xTensor, xCoba, and xTras in general relativity.

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