Physics / advanced
Gravitational field equations and exact geometries
Derive Einstein dynamics and analyze representative spacetime solutions using stress-energy conservation, symmetry, observable clocks, and global structure.
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Module 01 advanced
Gravitational field equations and exact geometries
Derive Einstein dynamics and analyze representative spacetime solutions using stress-energy conservation, symmetry, observable clocks, and global structure.
- 01
Einstein–Hilbert action and boundary terms
Derive the field equations with explicit curvature conventions and identify what a fixed boundary metric actually fixes.
30 min - 02
Stress-energy variation and covariant conservation
Connect metric variation to local conservation, conserved Killing currents, and the separate assumptions called energy conditions.
30 min - 03
Weak-field Newtonian limit
Recover acceleration and Poisson’s equation from a controlled weak metric, keeping the pressure and cosmological assumptions visible.
30 min - 04
Schwarzschild geometry and coordinate horizons
Separate exterior coordinates, invariant curvature, and operational clock comparisons in a spherical vacuum geometry.
30 min - 05
Timelike orbits and innermost stable circular orbits
Derive circular-orbit energy and stability in Schwarzschild geometry and distinguish an ISCO from the last possible circular orbit.
30 min - 06
Null geodesics and gravitational lensing
Relate the weak-deflection limit to lens geometry while distinguishing strong-field capture from a thin-lens approximation.
30 min - 07
Reissner–Nordström and charged horizons
Classify the roots of the spherical Einstein–Maxwell metric while distinguishing electrovacuum stress from vanishing matter curvature.
30 min - 08
Kerr geometry, frame dragging, and ergoregions
Distinguish a rotating horizon from its stationary-limit surface and identify the conserved energy relevant to extraction.
30 min - 09
de Sitter and anti-de Sitter geometries
Relate cosmological constant to curvature scale and distinguish observer horizons, cosmological patches, and timelike boundaries.
30 min - 10
Junction conditions and thin-shell spacetimes
Match induced geometry across a timelike surface and convert an oriented extrinsic-curvature jump into surface stress-energy.
30 min