Gravitational waves

Course goals

1. The student is able to work with contravariant and covariant tensors.
2. The student is able to discuss what are the physical degrees of freedom in the linearized Einstein
equations and how these relate to gravitational waves.
3. The student is able to discuss what are the properties of gravitational waves: polarizations, effect on
matter, energy carried by them.
4. The student knows where the quadrupole formula comes from, and is able to use it to derive the
gravitational radiation caused by accelerating bodies in example settings.
5. In particular, the student is able to calculate the gravitational waves emitted by two compact objects
orbiting each other.
6. The student is able to explain how interferometers like LIGO and Virgo detect gravitational waves.
7. The student is able to explain the basic principles of gravitational wave data analysis.
8. The student is able to discuss in broad strokes what has been the scientific pay-off of discoveries by
LIGO/Virgo in the past few years, and what can be expected from next-generation gravitational wave
observatories.

Content

The course starts with an introduction to the basic mathematical tools needed: tensors (in particular the metric
tensor), index notation, coordinate transformations. Special relativity is introduced, and a basic overview of
general relativity is given. The linearized Einstein equations are discussed and their physical degrees of freedom
are identified; it is shown how this leads to a wave equation and hence gravitational waves. The basic properties
of gravitational waves are studied: what polarizations they have, how they interact with matter, and the energy
they carry. Next we derive the quadrupole formula, which describes how gravitational waves are generated by the
motion of masses. An important example is the gravitational radiation emitted by two compact objects (neutron
stars and/or black holes) that orbit each other, and spiral towards each other until they merge together. We
discuss how these and other gravitational wave signals are detected with interferometers such as LIGO and Virgo,
including the basics of gravitational wave data analysis: how to identify and study weak signals in noisy detector
data. The final few lectures make a connection with discoveries made by LIGO and Virgo in the past few years,
and their impact on fundamental physics, astrophysics, and cosmology. We end with a discussion of future
gravitational wave observatories such as the underground Einstein Telescope and the space-based LISA,
together with the scientific output that can be expected from these.