General Relativity Physics Honours 2005

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

2. Prior Knowledge • Differential Equations • Special Relativity • Lagrangian Mechanics • Maxwell’s Equations This course is structured slightly differently to other GR courses. You will gain a working knowledge of GR before you are presented with a formal definition of tensors and the derivation of the field equations.

3. Textbooks Introducing Einstein’s Relativity à Course Book R. D’Inverno Gravity: An Introduction to Einstein’s GR J. B. Hartle Spacetime and Geometry S. M. Carroll A First Course in GR B. F. Schutz

4. Introducing Einstein’s Relativity • Special Relativity • 2.1*, 2.2-2.6, 2.7-2.8*, 2.9-2.13, 3.1-3.10, 4.1-4.2*, 4.3-4.5 • Formalism of Tensors • 5.1-5.2, 5.3*, 5.4-5.8, 5.9*, 6.1-6.12, 7.1-7.3, 7.5* • General Relativity • 8.1-8.5, 8.8, 9.1-9.7, 10.1-10.7, 12.1-12.6, 12.9-12.10 • 13.1-13.4, 14.1-14.3*, 14.4-14.6, 15.1-15.7, 15.8-15.9*, 15.10 • Black Holes • 16.1-16.11, 16.12*, 17.1-17.5, 18.1-18.3, 18.5, 19.2*, 19.3-19.5 • Gravitational Waves • 20.1-20.4, 20.9 • Cosmology • 22.1-22.12, 23.1-23.3, 23.17, 23.18 * means review & self-study

5. Additional Resources • http://www.physics.usyd.edu.au/~gfl/Lecture/ • http://spacetimeandgeometry.net/ • “A no-nonsense introduction to general relativity” by Sean Carroll (http://pancake.uchicago.edu/~carroll/notes) • “Living Reviews in Relativity” (http://www.livingreviews.org) • “The meaning of Einstein’s equations” by John Baez and Emory Bunn (arxiv.org/gr-qc/0103044) • Preprint archive (http://www.arxiv.org) • Type “general relativity” etc into google.

6. Assessment Final exam (80%) Three assignments (20%) Assignments are due at the start of the lecture on the specified date. Late assignments will be penalized 20% for each day they are late. Assignments more than one week late will not be accepted without a formal special consideration. Postgraduate students must achieve >70% on all assignments and will sit the exam in an open book environment.

7. The Newtonian Framework (2.3) • An Event occurs at a point in space and instant in time. • Events do not move. • The motion of a particle can be represented by a world-line, or collection of events specified by rand t (Fig 2.2). • An observer is equipped with a clock and measuring rod. • All observers agree on t: time is absolute. • Each observer can set up coordinates (t,x,y,z) to describe the location of events (with t the same for all observers).

8. Galilean Transformations Newtonian mechanics argues that there are preferred frames of reference, the unaccelerated inertial frames. Observers can transform between the coordinates of an event in different inertial frames with Galilean Transformations. If the frames S and S’ have collinear x-axes (Fig 2.5) then; where S’ moves along the x-axis with constant velocity v.

9. Newtonian Relativity • Newtonian theory we can only determine: • events relative to other events • velocity of a body relative to another body • Hence Newtonian theory makes the following postulate All inertial observers are equivalent as far as dynamical experiments are concerned. However, this is in conflict with Maxwell’s equations.

10. Special Relativity Einstein extended Newtonian relativity into The Principle of Special Relativity; All inertial observers are equivalent As Maxwell’s equations provide a single wave solution, with a velocity c, Einstein proposed the postulate of the constancy of the speed of light. The velocity of light in free space is the same for all inertial observers.

11. Special Relativity (Review Chapter 2) A new composition law for velocities (Eqn 2.6) where v in units of the speed of light (c). Note that if the velocities are «1 then this becomes the Newtonian formula. How does this behave if one of the velocities is c?

12. Special Relativity Einstein considered a simple thought experiment to examine the implications of the constancy of the speed of light (Fig 2.13). In classical physics both observers A & B would agree that the two events occurred simultaneously. However, as c is a constant to all observers, A & B now disagree on the simultaneous nature of the two events, removing the notion of an absolute time. We can consider the order of events in terms of a light cone or null cone (Fig 2.14).

13. Lorentz Transforms The Lorentz transformations are the SR versions of the Galilean transformations. Considering motion in the x-direction (Fig 2.17); It can be seen that the squared interval (pp.25-26); is invariant under Lorentz transformations.

14. 4-Dimensional Space-time In Newtonian physics we can use the Euclidian distance to measure the separation of two events, with time kept separate. With relativistic space-time, we use the interval squared This geometry is known as Minkowski space-time; intervals in this space-time can be negative. Note, different books have different sign conventions for ds2

15. Matrix Representation We can write the transformation between two frames S and S’ as where L is a 4x4 matrix (Ch 3). By considering a new variable T=i ct then the Lorentz transformations can be seen as rotations in the x-T plane (although this complex time formalism is frowned upon by some. See Fig 3.6 for a more accurate description).

16. Length Contraction Consider a rod fixed in S’ with ends at xA’and xB’a distance lo apart. The distance l as measured in S is given by; where  = ( 1 - v2/c2 )-1/2 (Note textbooks non-standard use of  rather than !). Hence the length in S is less than the length in S’. This effect is entirely reciprocal.

17. Time Dilation Suppose a clock fixed at x=x’Ain S’ records two events separated by an interval To. Hence, more time is measured to pass in S than in S’, a phenomenon known as time dilation. The time measured in the rest frame of the clock is known as the proper time. Again, the effect is reciprocal between frames.

18. Clock Hypothesis The clock hypothesis asserts that will measure a proper time of  even in non-inertial frames of where t is the time in some inertial system and v is the velocity measured in that system. This has been flying atomic clocks around the world. It is instructive to read about the twin paradox!

19. Velocity Transformations (again) Consider a particle in motion along x with velocity u in S, and u’ along x’ in S’ (Fig 3.5). Differentials of the Lorentz transforms give You should check the effects on transverse velocities yourself. Replacing u with u’ and v with –v transforms between S’ and S.

20. Accelerations We can calculate accelerations in a similar fashion, using the relations accelerations in S and S’ are

21. Accelerations We cannot assume du/dt = constant as velocities >c would occur. Instead we consider a instantaneous comoving frame in which the accelerating body is at rest (u’=0) with an acceleration a. The final form is a hyperbola on the x-t plane (Fig 3.8).

22. Accelerations For the case where xo – c2/a = to = 0 the null cone becomes an event horizon and a light ray sent out from O at times t>to will never be received by the accelerating particle. In fact the event horizon defines two regions of the universe where particles accelerating away from one another can never communicate (why?). More about event horizons when we examine black holes.

23. The Doppler Effect A source in a frame S’, moving at radial velocity ur, emits pulses separated by a time dt’. Due to the effects of time dilation, S infers that the pulses are separated by dt =  dt’. But how often does S receive the pulses? In a time dt the source has moved by ur dt =  ur dt’ and the pulses have to travel more distance back to the source. Hence, pulses are received at S with a time separation of

24. The Doppler Effect Using o = c dt’ and  = c dt then For purely radial motion

25. The Doppler Effect However, purely tangential motion results in the transverse Doppler shift. This purely relativistic effect (it is due to time dilation) has been experimentally verified to high accuracy.