General relativity and cosmology course

Define a smooth manifold using charts and atlases, and provide examples such as the n-sphere.

Define the tangent space at a point and construct vector fields as derivations.

Define the cotangent space (dual space) and construct one-forms.

Define a general (r,s) tensor as a multilinear map and explain the tensor product.

Define the metric tensor and its signature, and use it to raise and lower indices.

Define the Lie derivative of a tensor field with respect to a vector field.

Define differential forms and the exterior derivative operator.

Define integration on manifolds using volume forms.

Define an affine connection and the covariant derivative of a tensor field.

Derive the Levi-Civita connection from the metric compatibility and torsion-free conditions.

Define parallel transport along a curve and its relation to the covariant derivative.

Derive the geodesic equation as the curve of zero acceleration.

Define the Riemann curvature tensor in terms of the commutator of covariant derivatives.

Derive the algebraic symmetries of the Riemann tensor and the Bianchi identities.

Perform the Ricci decomposition of the Riemann tensor into the Ricci tensor, Ricci scalar, and Weyl tensor.

Interpret curvature physically via geodesic deviation.

Formulate Special Relativity in the language of Minkowski spacetime as a manifold.

State the Principle of Equivalence and derive its immediate consequences, such as gravitational time dilation.

Define the stress-energy tensor and construct it for a perfect fluid and the electromagnetic field.

Derive the covariant conservation law of the stress-energy tensor (∇μ Tμν = 0).

Formulate the Einstein-Hilbert action and derive the Einstein Field Equations (EFE) via the variational principle.

Analyze the Newtonian limit of the EFE to fix the value of the coupling constant κ.

Formulate the initial value problem of General Relativity using a spacelike hypersurface.

Define Killing vector fields and their connection to spacetime symmetries and conserved quantities via Noether's theorem.

Derive the Schwarzschild metric as the unique static, spherically symmetric vacuum solution (Birkhoff's theorem).

Analyze the causal structure of Schwarzschild spacetime, identifying the event horizon and physical singularity.

Analyze null and timelike geodesics in the Schwarzschild metric to describe particle orbits and light rays.

Calculate the perihelion precession of Mercury using the derived geodesic equations.

Calculate the deflection of light by a massive object (gravitational lensing).

Analyze the coordinate singularity at the event horizon and remove it using Kruskal-Szekeres coordinates.

Construct the Penrose diagram for the maximally extended Schwarzschild spacetime.

Derive the Reissner-Nordström metric for a charged, non-rotating black hole.

Analyze the Kerr metric for a rotating black hole, identifying its horizons and the static limit.

Define the ergosphere and derive the frame-dragging effect (Lense-Thirring effect).

Describe the Penrose process for extracting energy from a rotating black hole.

Construct the Penrose diagram for the maximal analytic extension of the Kerr spacetime.

Define surface gravity (κ) for a stationary black hole.

State and interpret the four laws of black hole mechanics.

Explain the physical analogy between black hole mechanics and the laws of thermodynamics.

Derive the Raychaudhuri equation and use it to prove the Penrose-Hawking singularity theorems under specific energy conditions.

Derive the linearized Einstein equations in the weak-field limit.

Perform a gauge transformation on the metric perturbation and fix the transverse-traceless (TT) gauge.

Solve the linearized vacuum equations to find plane wave solutions and identify the two polarization states (h+ and hx).

Analyze the effect of a passing gravitational wave on a ring of test particles.

Derive the quadrupole formula for the power radiated by a non-relativistic source.

Apply the quadrupole formula to calculate the energy loss from a binary star system.

Estimate the orbital decay and gravitational wave signal from a binary system like the Hulse-Taylor pulsar.

Describe the basic principles of gravitational wave detection by laser interferometers (LIGO/Virgo).

Postulate the Cosmological Principle and derive the Friedmann-Robertson-Walker (FRW) metric.

Apply the EFE to the FRW metric to derive the Friedmann equations.

Solve the Friedmann equations for universes dominated by matter, radiation, and vacuum energy (dark energy).

Define cosmological redshift and derive the Hubble-Lemaître law.

Define and calculate different distance measures in cosmology (luminosity, angular diameter).

Explain the horizon, flatness, and monopole problems of the standard Big Bang model.

Introduce a scalar field (the inflaton) and derive its equations of motion in an FRW background.

Derive the slow-roll conditions for inflation and show how inflation solves the Big Bang problems.

Outline the canonical quantization of a scalar field in Minkowski spacetime, defining creation and annihilation operators.

Explain the ambiguity in defining particles in a general curved spacetime due to the lack of a global timelike Killing vector.

Define Bogoliubov transformations relating different sets of creation/annihilation operators.

Derive the Unruh effect: the thermal particle spectrum observed by a uniformly accelerating observer in Minkowski vacuum.

Formulate quantum field theory near a black hole's event horizon.

Derive the Hawking temperature of a Schwarzschild black hole by analyzing the Bogoliubov transformation between past and future null infinity.

Relate Hawking radiation to black hole thermodynamics, identifying the Bekenstein-Hawking entropy.

Discuss the black hole information paradox and its implications for quantum mechanics.

Decompose metric and matter perturbations into scalar, vector, and tensor modes (SVT decomposition).

Explain the concept of gauge freedom in perturbation theory and construct gauge-invariant variables (e.g., Bardeen potentials).

Derive the linearized Einstein equations for scalar perturbations in an expanding background.

Model quantum fluctuations of the inflaton field during inflation as the origin of primordial perturbations.

Calculate the power spectrum of primordial curvature perturbations generated during inflation.

Analyze the evolution of perturbations after they re-enter the horizon during radiation- and matter-dominated eras.

Derive the Sachs-Wolfe effect, relating gravitational potential perturbations on the last scattering surface to CMB temperature anisotropies.

Perform the 3+1 ADM decomposition of spacetime into space and time (lapse, shift, and spatial metric).

Derive the extrinsic curvature of the spatial slice.

Derive the Hamiltonian and momentum constraint equations of General Relativity.

Formulate the Wheeler-DeWitt equation as a candidate for a quantum theory of gravity.

Introduce the basics of Loop Quantum Gravity: spin networks and the quantization of area and volume.

Provide a conceptual overview of String Theory, including extra dimensions and the role of strings as fundamental objects.

Discuss the AdS/CFT correspondence as a concrete realization of the holographic principle.

Explore a class of modified gravity theories (e.g., f(R) gravity) as an alternative to dark energy.