Introduction à la relativité générale

Enseignant : Nick Kaiser
Chargé des TD : Erwan Allys
ECTS : 3
Langue d’enseignement : Anglais
Site web :

 Description

 Historical development :

Newtonian gravity - 1=r2 force - potential, gravity and tidal -elds - Pois-
son’s equation. Similar to electricity/magnetism - key di-erence (Galileo). 1905 : Special relativity -
4-vectors & tensors as the language of physics - conservation laws ; particle number, charge, energy
& momentum. 1910 : Einstein’s \happiest thought" - equivalence principle - gravitational redshift
- space-time is `warped’ - particles follow geodesics in a curved space-time. 1915 : Einstein’s -eld
equations.

 Special relativity (review).

Principles : no absolute velocity ; constancy of c - time dilation & length
contraction - simultaneity. Reference frames : clocks, rulers and events. Lorentz transformation : boosts
as 4-vector `rotations’ in non-Euclidean space-time. Time-like, space-like and null displacements.
Invariance of ds2. Twin & other paradoxes. Particle trajectories : proper-time - 4-velocity. Momentum
- why p =
mv - 4-momentum, E = mc2. Particle kinematics. Other 4-vectors : number & charge
density - the EM -eld - stress, energy and momentum conservation. Other invariants.

 Vector and tensor analysis in SR.

Formal de-nition of a vector - transformation laws. Vector
algebra - basis vectors - components. Scalar products. De-nition of tensors : 1-forms (gradient vectors)
- the basis for 1-forms - the metric tensor. General tensors - transformation laws - index raising and
lowering - di-erentiation of tensors. Tensor algebra and calculus in arbitrary coordinate systems -
generalisation of Lorentz matrix - curves and tangent vectors - (covariant) derivatives of vectors - the
connection/Christo-el symbols - relation to the metric - derivatives of 1-forms and general tensors.

 Space-time curvature.

Preface : relation of gravity to curvature - gravitational time dilation. Curved
manifolds - metric and local
atness. Covariant di-erentiation. Parallel transport, geodesics and
curvature. Geodesic deviation. The gravitational -eld : curvature tensor R - how we measure curvature
- relation to the Newtonian tidal -eld. Bianchi identities : Ricci and Einstein tensors.

 The Einstein’s theory of gravity.

Founding principles : locally Lorentian manifold - metric - equiv-
alence principle(s). Laws of physics in curved space-time - comma ! semi-colon. Einstein’s equations
- `geometrized’ units - relativistic generalisation of Poisson’s equation : G = 8-T - cosmological con-
stant -. Weak—eld gravity - the Newtonian limit - gauge transformations - wave equation. The RHS :
T = stress-energy tensor - dust - perfect
uids - collisionless matter - scalar -elds - conservation laws
- weak and strong energy conditions.

 Stars and black holes.

Coordinates for spherically symmetric space-times. The Schwarzschild
metric. The nature of the `horizon’ at r = 2m. Orbits in Schwarzschild geometry. Precession of the
perihelion of mercury. Other black-hole solutions. Equations of stellar structure for spherical stars
(static perfect
uid). The exterior geometry and the e-ective stress-tensor. Gravitational collapse -
Oppenheimer & Snyder solution - Raychaudhuri and the role of pressure - Hawking & Ellis singularity
theorem.

 Gravitational lensing

Lensing by refractive media ; index of refraction - Snell’s law - arrival time
delay - Fermat’s theorem. Refraction (& di-raction) by randomly disturbed medium. Lensing by
gravity ; geodesic equation - extremal paths - e-ective refractive index. Astrophysical applications ;
micro-lensing by stars - weak lensing and large-scale-structure. Lensing as a test of GR.

 Gravitational radiation.

The propagation of GWs. Detection of GWs. Generation of GWs. The
energy of GWs. Astrophysical sources of GWs. LIGO discoveries. Constraints on modi-ed (i.e.
non-GR) gravity theories. Future GW observatories.