Wave propagation in curved spacetimes

Basic idea of the course

I will present the general theory of wave propagation in curved spacetime in general relativity. The course will focus mainly on electromagnetic waves, and to a lesser degree the gravitational waves, as these two types of waves have important astrophysical applications. I will describe the theory of geometrical optics, its assumptions, applications and limitations, including the theory of gravitational lensing and the theory of light bundles. In the second part of the lecture I will focus on wave effects.

1. Wave equations in GR

- scalar wave equation

- electromagnetic wave equation

- gravitational wave equation

- waves in Minkowski, waves in curved spacetimes

2. Geometrical optics approximation

- eikonal approximation

- null geodesic equation, transport of polarization tensor

- limitations

- properties: time of arrival, multiple images etc.

- energy flux

- SR effects: relativistic light beaming, red/blueshift, aberration effect

- conformal invariance of null geodesics. Light in FRLW spacetime.

3. Theory of infinitesimal light bundles

- geodesic deviation equation of the 1st order: timelike, null

- resolvent, Jacobi matrix, etc.

- momentary measurements. Flux, angular diameter distance, luminosity distiance

- drifts, parallax etc.

- world-function formalism (perhaps)

- light bundles, Raychaudhuri equation

4. Lensing theory

- light rays in Newtonian approximation, gravitational light bending formula

- lensing in thin lens approximation: lensing equation

- Fermat principle, TOA

- lensing in FLRW spacetime

- conservation of flux, light intensity

- moving lens - perhaps?

- caustics

- beyond the thin lens approximation

- Schwarzschild BH: photonsphere, BH shadow

- Kerr: photonsphere, BH shadow

- multiple images

5. Focus on gravitational waves

- gauge, polarization

- energy flux

- source: quadrupole formula

- BBH signal

- detection: TOA of electromagntetic waves

- interferometers

- PTA's

- memory effect

6. Wave effects

- beyond the 0th order eikonal equation

- spin-Hall effect

- Fresnel formula?

- waves in Schwarzschild, quasinormal modes?

- superradiance?