Gravitational Wave Astronomy Dr. Giles Hammond Institute for Gravitational Research SUPA, University of Glasgow Universität Jena, August 2010

Spacetime and Matter This statement is embodied in the description of the Riemann curvature tensor and the geodesic equation of motion But what about The source of spacetime curvature is the Energy-Momentum tensor which is used to describe the presence of gravitating matter (and energy). Spacetime tells matter how to move Matter tells spacetime how to curve.

Perfect Fluids We will define the Energy-Momentum tensor for a perfect fluid In a fluid description we treat our physical system as a smooth continuum, and describe its behaviour in terms of locally averaged properties in each fluid element. Particles in a fluid element will contribute to: Pressure (molecules in an ideal gas) Heat conduction (energy exchange with neighbours) Viscous forces (shearing of fluid) The Energy-Momentum tensor has 16 components in 4-D (t, x, y, z):

Perfect Fluids Without the first row and the first column we have a 3x3 matrix known as the stress tensor. Assume directions 1, 2 and 3 are x, y and z respectively. If we take a small surface in the xy plane, T 13 is the stress acting in the x direction and T 33 is the stress acting in the z direction. The components in the diagonal of the matrix are normal stresses (i.e. like pressure) while the off-diagonal elements are shear stresses. For our perfect fluid all off-diagonal elements are zero. Units: x y pxpx pypy pzpz

Perfect Fluids The component T 00 is the energy flux or the energy per unit volume. The components T 10, T 20, T 30 are the momentum flux across a surface of constant time (i.e = 0 ). The components T 0 define the energy flux across a surface of constant (i.e = 1,2,3 : x, y or z). When the only energy is matter at rest all the components in the top row and the left column, exceptT 00, are zero. A perfect fluid (at rest) has an Energy-Momentum tensor: Units: x y T 03 z

Einstein Equations A particular combination of the Ricci tensor and scalar, called the Einstein tensor, embodies curvature The Energy-Momentum tensor describes how energy/mass curves spacetime Combining these gives the Einstein equation (the constant is obtained from Newtonian limit) Matter tells spacetime how to curve. Spacetime tells matter how to move

Gravitational Waves The following slides provide a brief derivation of how a wave equation comes from the spacetime of General Relativity I have tried to write the mathematics in a fairly complete way (something you dont always see in books/papers). But this means plenty of equations and indices

Gravitational Waves Assume weak gravity: is the flat spacetime metric and The Riemann tensor defined previously (slide 19) is Keeping terms which are linear in the metric gives

Gravitational Waves In the weak field limit the Christoffel symbols (slide 21) are The Riemann tensor is then which gives and the Ricci tensor

Gravitational Waves The Ricci scalar is and the Einstein tensor

Gravitational Waves By choice of a suitable coordinate system (or gauge) all terms in the Einstein tensor except for one can be set equal to zero. This results in: But this is just the DAlembertian operator Thus we find a wave equation in terms of the metric perturbation

Gravitational Waves Far from a source the Einstein equations in vacuum are The simplest solution is for plane waves with amplitude A ab and wave number k : h is dimensionless. It is often called the strain (h= L/L)

Gravitational Waves The 16 components of the amplitude can be reduced to 4 by suitable choice of gauge (Transverse-Traceless; metric perturbation is transverse to propagation direction and traceless) Consider a free particle initially at rest. The geodesic equation is as velocities are zero (dv/dt=0). Thus we find

Gravitational Waves From our matrix all 3 terms are zero => A particle initially at rest will remain at rest. In the TT gauge we have a coordinate system which remains attached to individual particles Now consider two particles, one at (0,0,0) and the other at (,0,0) Proper distance between them is and this DOES change with time