1. General Relativity Physics Honours 2006 A/Prof. Geraint F. Lewis Rm 557, A29 gfl@physics.usyd.edu.au Lecture Notes 5

2. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Light Ray Orbits The equivalence principle tells us that light rays should be influenced as they pass through a gravitational field. We can use the same geodesic formulism to study this. Firstly, we still have two conserved quantities due to the symmetries of the Schwarzschild metric. Ch. 9.4 Note that the derivatives are with respect to the affine parameter, not the proper time. Secondly we have the normalization of the 4-velocity

3. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Light Ray Orbits So; again assuming =/2. Using the conserved quantities And so we can write;

4. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Light Ray Orbits This has the same form as the massive particle orbits if we assume an effective potential of the form; And treating b -2 as an energy term. What is the physical meaning of b? Consider orbits that start at r>>2M; Also for large r, then And so d=b and b is the impact parameter of the orbit!

5. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Light Ray Orbits The effective potential has a peak, and so unstable circular orbit, at Considering light rays starting from infinity, those with b -1 less than this scatter back to infinity, while those with more than this exceed the potential barrier and fall into the centre.

6. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Escaping to Infinity Consider a source at r<3M emitting light in all directions. Some light will escape to infinity, while some will fall into the black hole. What is the critical angle at which light barely escapes? Example 9.2 We need to consider the light ray as seen in the orthonormal basis. Again, working in a plane where =/2, then;

7. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Escaping to Infinity As the metric is diagonal, we can simply define the orthogonal basis vectors as; Hence, the photon 4-velocity in the orthonormal frame is (you should convince yourself that the 4-velocity of the photon in the orthonormal frame is null!)

8. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Escaping to Infinity Each angle corresponds to a different value of b -1 and an examination of the potential shows that rays which escape to infinity have “energies” greater than the potential barrier. Hence the critical angle occurs at b 2 = 27M 2 and

9. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Deflection of Light How much is light deflected by a massive, spherical object? From our conserved and geodesic equations we have; And so

10. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Deflection of Light From infinity, the photon travels to a radius r 1, before heading out again. This radius occurs at And the angle swept out is; The smaller the impact parameter, the larger the deflection angle, to the point where the photon enters a circular orbit or falls into the centre.

11. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Deflection of Light In solving for the deflection angle, introduce a new variable; Note, if M=0 then the resulting integral is , no deflection. Considering a light ray grazing the surface of the Sun; We can write the deflection angle as

12. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Deflection of Light We can expand this expression out in the lowest order terms of 2M/b and get Remember, w 1 is the root of the denominator. The result is that the deflection is given by

13. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Shapiro Time Delay The Shapiro time delay is apparent when photons are send on a return path near a massive object. In the Solar System, this involves “bouncing” radar off a reflector (space ship or planet) located on the other side of the Sun, and seeing how long the signal takes to return. The result is different to what you would expect in flat (special relativisitic) spacetime.

14. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Shapiro Time Delay As with the deflection of light, we can write And the total time taken for the trip is where and

15. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Shapiro Time Delay As with the deflection of light, we can find the weak field limit of this integral which would apply in the Solar System The first term in this expression is simply the expected Newtonian time delay, and the other terms are a relativistic correction (but what is wrong with the above?). For photons grazing the Solar surface we get

16. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Solar System Tests Chapter 10 discusses Solar System tests of general relativity, including measurements of the Parameterized-Post-Newtonian (PPN) parameters; these add higher terms to the metric and ‘extend’ relativity. For Einstein’s theory of relativity, these parameters must be exactly unity. While interesting, the contents of this chapter will not be examinable. However, you should read through the material. We will summarize the solar system tests.

17. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Perihelion Shift of Mercury Mercury is the closest planet to the Sun, with a semi-major axis of 58 £ 10 6 km and eccentricity of 0.21. The orbit of Mercury has been known to precess for quite a while. The vast majority of the precession is due to Newtonian effects. However, a residual precession of 42.98 § 0.04 “/century could not be explained. The prediction from Einstein’s analysis of the orbit in the weak- field limit predicts;

18. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Gravitational Lensing Einstein’s first prediction was that light would be deflected as it passed by massive objects. He calculated that a light ray grazing the Sun would be deflected by Made in 1916, this prediction could not be tested until the end of WWI. Eddington organized two expeditions to observed an eclipse in 1919. With the Moon blocking out the Sun, the positions of stars could be measured, agreeing (roughly) with Einstein’s prediction. Now measured to an accuracy of ~1%.

19. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Gravitational Redshift The final test proposed by Einstein in 1916 was the gravitational redshift. This was finally measured by the Pound-Rebka experiment in 1959 by firing gamma rays up and down a 22m tower at Harvard. Measuring a frequency change of 1 part in 10 15, their measurement agreed with t Einstein prediction with an uncertainty of 10%. Five years later, the accuracy was improved to a 1% agreement and now measurements can accurately agree to less than a percent accuracy.

20. http://www.physics.usyd.edu.au/~gfl/L ecture Lecture Notes 5 Shapiro Time Delay The Shapiro delay has also been measured using space probes, including Mariner in 1970 and Viking in 1976. The expected delay is of order 100s of microseconds over a total journey time of ~hrs, but atomic clocks are accurate to 1 part in 10 12. A recent measurement using the Cassini space probe found the agreement to be (Bertotti et al 2003)