Announcements

1. Announcements • Exam 3 is Thursday April 11. Tentatively will cover Chapters 7, 8 & 9 Sample questions will be posted soon • New links for General Relativity have been posted on the website. Check them out!

2. General Relativity deals with geometry

3. Basic geometries Flat Spherical Hyperbolic

4. The Metric is the formula for the distance between two points Spherical geometry Euclidian geometry Hyperbolic geometry

5. General Relativity uses Riemannian Geometry Riemannian geometries are locally flat. On a small enough scale they are Euclidian. On a larger scale, though, there is curvature

6. The metric for general relativity is the space-time interval (almost) The space-time interval for Special Relativity is for flat space so a = g = 1 and b = 0. The metric for curved space includes the spatial separation between the two events (gDx) plus the temporal separation (ac2Dt2) plus a cross term. The mathematics of General Relativity is all about finding a, b and g.

7. The Equations of General Relativity can look simple The left side is the geometry of space-time. The right side is the mass-energy distribution

8. The symbols Rmn and Tmn are short hand ways of writing differential equations

9. When the whole thing is seen, it isn’t so simple anymore

10. There’s more

11. and more

12. But what does it all mean? Mass-energy tell space-time what it’s geometry must be. Any object that moves into that “region” of space-time then follows the trajectory given by the shortest space-time interval between two space-time points.

13. Observational evidence for General Relativity: Light Bending Einstein’s Cross Watch Gravity Lensing video

14. More often, arcs result from gravitational lensing

15. General relativity says the binary pulsar should radiate gravity waves Watch Binary Pulsar, Binary Pulsar Merger and Neutron Star Merger Swift videos

16. Observed Spin-down of the binary pulsar system

17. LIGO looks for gravity waves Watch YouTube video at LIGO Gravity Wave Observatory video

18. LIGO is a big Interferometer

19. Two interferometers were built to be sure they detect gravity waves and not passing trucks

20. Black Holes

21. The Schwarzschild solution to the equations of general relativity • Spherical mass • Spherical symmetry gives metric radial (r) dependence only…no dependence on angles around sphere • Stationary with no rotation…static: metric should not depend on time • Only valid in vacuum outside the mass Karl Schwarzschild: 1873 – 1916. Worked out the solution in 1915 while serving as an artillery officer with the German army on the Russian front during WW I

22. The equation Schwarzschild solved Looks easy enough until you start working the D.E.’s

23. Schwarzschild’s Metric The shortest space-time interval between two events outside a spherical mass distribution

24. The Schwarzschild Radius Using the metric becomes

25. What does it mean? • Look at how RS/r affects the terms for various values of r greater than the radius of the object (i.e. RS/r<1) • Recall that Ds is the proper space-time interval between two events in space-time RS for the Sun is 2.95 km, for Earth it is 8.86 mm

26. “Far” from the mass r>>RS RS/r is very small so becomes This is the metric for ordinary spherical geometry “Flat” space-time

27. Solution is not “flat” geometry if r~RS Dt term gets small while Drterm gets large…space and time start to get warped by the mass inside the sphere

28. Gravitational Time Dilation For clock near object, RS/r is close to 1 so Ds is large. The outside observer (far from the black hole) sees the clock for someone near the event horizon (r ≈RS) to slow down. At the event horizon it stops completely.

29. Gravitational Length Contraction Robject≈ RS…”Compact” object Measure length by determining positions of ends simultaneously so Ds is zero For r close to RS the Dt term is small and the Drterm is large. Since 1/(1-(RS/r)) is small, Drmust be very large.

30. What happens when R = RS (at the Schwarzschild radius)? To an observer far from the black hole time for a person falling toward the black hole stops (Dt goes to zero) and the distance from the outside observer goes to infinity. In other words, the outside observer never sees the person falling toward the black hole reach the event horizon.

31. What happens inside the Schwarzschild Radius? r < RS Space becomes time-like (positive coefficient) and time becomes space-like (negative coefficient) So, you can move forward or backwards in time? The only direction in space (other that around) is toward the center. You can’t move away from the center, only towards it or around it.

32. Inside the Schwarzschild radius, the future is towards the center Inside event horizon Near event horizon At event horizon It is no longer possible to remain stationary. The very fabric of space-time falls towards the event horizon.

33. At the center is the singularity If you plug r = 0 into the Schwarzschild metric it complete breaks down. This tells us something is very wrong.

34. The distance at which the orbital speed is the speed of light is the photon sphere The radius of the photon sphere is 3/2 the Schwarzschild radius If an object has a mass, its minimum stable circular orbit is 3RS

35. Black holes have no hair • The event horizon of a black hole is perfectly spherical. • Only three quantities completely describe any black hole Mass Angular Momentum Electric Charge