General Relativity Physics Honours 2005

1. General RelativityPhysics Honours 2005 Dr Geraint F. Lewis Rm 557, A29 gfl@physics.usyd.edu.au

2. OK – So what do I do? • Given your choice of coordinates (and hence metric) you • Calculate all the Christoffel symbols abc • Use geodesic equation to determine four equations of motion • Determine your initial conditions and ensure your 4-velocity is properly normalized (another constraint). • Solve your equations of motion

3. The Geodesic Equation In non-Minkowski coordinates, the terms of the geodesic equation can be though of as forces, i.e. as a straight line runs across a plane at constant cartesian velocity, its vr and v are not constant and it is the “force terms” due to non-zero abc that governs this. In fact, as abc depends upon the derivatives of the metric, the metric components gab can be thought of as a potential. Important: the metric does not describe only inertial frames, but can also represent non-inertial (accelerating or rotating) frames.

4. Equivalence (again) Einstein realized that there was an equivalence between accelerated reference frames and the “force” of gravity. Hence he determined that the action of gravity can be explained with the same mathematics of special relativity i.e. you use the geodesic equation to calculate the path of a beam of light etc. The difference is that the metric gab now represents curved rather than flat space-time.

5. Meaning of the Metric We will be working with the Schwarzschild metric, which represents the space-time outside of any spherically symmetric mass distribution of mass m. In polar coordinates, this is were G=c=1. What does this mean?

6. Embedding the Metric Consider t=constant and =/2. The line element reduces to This relates a coordinate distance, dr, to a proper (i.e. physical) distance, dr’, such that so a coordinate metre corresponds to physical distance of more than a metre. Embedding Diagram

7. Embedding the Metric We can do a similar thing with time and see that and a coordinate second is longer than a physical (or proper) second i.e. there is a time dilation. In general, for a time-like path, then

8. Meaning of the Metric As r!1this metric becomes the Minkowski metric of SR. Hence, at large distances the coordinate and physical quantities (ie. units of time and distance agree). Therefore, the quantities determined at small r can be thought of as being relative to those of a distant observer. Note: not all metrics become flat at larger distances!

9. OK – So what do I do? • Given your choice of coordinates (and hence metric) you • Calculate all the Christoffel symbols abc • Use geodesic equation to determine four equations of motion • Determine your initial conditions and ensure your 4-velocity is properly normalized • Solve your equations of motion

10. Is there any other way? The calculation of Christoffel symbols can be fiddley, but they are a generalization of the variational approach to mechanics (Ch 7.5). In a similar fashion, we can define a factor K such that This is equivalent to the classical Lagrangian and can be treated with in the same fashion.

11. Euler-Lagrange With the definition of K we can use the Euler-Lagrange equation to determine the equations of motion from the metric. With the definition of the Lagrangian, the Euler-Lagrange results in 5 equations of motion. The results of this and the Christoffel approach should be the same (first assignment)!

12. Where are we? • With the geodesic equation or the Euler-Lagrange approach, you are now armed with the mathematical tools necessary to calculate the vast majority of tests of General Relativity. • Perihelion shift of Mercury • Deflection of light • Redshift of light in a gravitational field • The Shapiro time-delay