1. Relativistic Laws Electromagnetism (Maxwell) is Lorentz-invariant Nuclear interactions have Lorentz-invariant form Quantum relativistic version (Quantum Field Theory 1940/1972. Gravity???? 1

2. Relativistic Gravity is no good. Whose ? When? Maxwell gives a relativistic version of in terms of electromagnetic fields We need a field theory for gravity but Maxwell’s idea will not work – mass is not conserved. Energy-Momentum is conserved but not invariant What is a gravitational force anyway?? 2

3. The Principle of Equivalence Observer in free fall – only gravity acts - there is no gravitational force. Stars and galaxies are in free-fall! Conversely, accelerating frames exhibit gravitational force in the absence of gravity Gravitation at one event can be completely cancelled by choosing a suitably accelerating frame – letting go Gravity is acceleration? Tidal Forces - the change in in space and time – are truly gravitational 3

4. Gravity is Acceleration 4

5. Gravitational Redshift 5

6. Equivalence Principle: lower clock runs slow Pound Rebka 1959 measure the effect over height of tower in Jefferson Lab: complete agreement GPS system needs to account for redshift 6

7. Gravity Small regions of spacetime, if observed by inertial (freely falling) observers, appear free of gravity Inertial determines acceleration. At any event are inertial observers with all velocities and their experiences are related by Lorentz transformations Tidal effects mean that inertial here and inertial there do not share same acceleration Given a velocity there is a unique inertial worldline starting with that velocity 7

8. Geometry Small regions of a curved space appear like flat space and can be described with usual coordinates Through any point we can draw straight lines going off in all directions and they are related by rotations Curvature means straight lines here and there are not related by rotation Given a point and an initial direction, there is a unique geodesic (straight line) starting at that point in that direction Shape of a space is encoded in distances between points as coordinate-invariant information - curvature 8

9. A Simple and Familiar Example Earth’s surface is a two-dimensional curved space To a good approximation, it is spherical. Every point is like any other Start at pole (might as well) in any direction and head due South along meridian Initially paths behave like straight lines on plane Farther out notice they are too near each other – Earth is curved – and eventually all meet at opposite pole Great circles are geodesics of a sphere Positive curvature means geodesics diverge less than in flat space 9

10. The Analogy Inertial worldlines Lorentz transformations Interval Gravity Geodesics Rotations Distance Curvature 10

11. Visualization 11

12. General Relativity Mass (Energy) is a source of spacetime curvature Inertial (free-fall) motion is along geodesics At small curvature and slow speeds reproduce Newton Complication: Gravitational energy is a source. Equations are nonlinear 12

13. First Results Bound (negative energy) Newtonian orbits are closed. After a period motion repeats Perturbations modify this. Mercury orbit precesses 1.5 ° /century Planetary contributions explain all but 43” Einstein uses effective potential to find relativistic deviation 43” GR essential in close binaries! 13

14. Gravity Probe B Precision Measured deviation from Newtonian predictions due to curved geometry and Earth rotation 14

15. Light Deflection Does Gravity act on light? Equivalence principle or mass-energy equivalence: yes Full GR calculation: Eddington 1919 eclipse measures deflection by Sun 15

16. 16 Gravitational Lensing Massive objects in line of sight act as lenses Use this to see dimmest objects Use lensing to learn about the lens: Exoplanet detection Dark Matter detection

17. Binary Pulsar PSR1913+16 In Aquila at Millisecond pulsar with period Periodic pulse delay with period Binary of two neutron stars of mass and System is very eccentric (perturbed by SN?) with periastron Perfect lab to study GR Pulse delay exhibits Doppler effect as well as gravitational redshift Precession of perihelion measured – agrees with GR 17

18. Gravitational Waves As neutron stars orbit GR predicts they will lose energy to gravitational waves Rate of period decrease consistent with GR In merge 18

19. Can We See Them? Gravitational wave detectors like LIGO expect to find evidence of violent neutron star mergers by detecting change in length of 3km laser by fm 19

20. Gravitational Redshift Near Earth we found Away from Earth expect something like In fact, observing from a great distance the correct expression is 20

21. Horizons – Black Holes What happens if you can get near ? Can you? Neutron stars are close As you approach horizon redshift Collapse past neutron degeneracy continues past horizon creating black hole Looking from afar you never get there Slow and dim as you near horizon – as seen from afar No light comes out! Distant light blueshifted as seen near horizon 21

22. Coming Closer If you can get to within a few relativistic effects become marked Stable orbits exist for so nothing orbits nearer At light in unstable circular orbit 22 Tidal forces can become extreme at small However less so for large mass

23. Coordinate Issues At horizon time stands still – as seen from afar Use a better clock – free falling in Horizon is in fact lightlike Once inside even lightspeed will not get you out 23

24. Inside the Horizon What happens to the stellar core that collapsed? Once inside horizon, within finite proper time (measured by your clock) reach singularity At singularity tidal acceleration diverges so matter ripped apart Divergence signals breakdown of equations. We don’t know. 24

25. Is This Real? Can We See Them? Technically, no. No light.. But in our time core stuck at forever We see the effects of dense massive objects With mass transfer have accretion disk to heated to - X-rays Cygnus X-1 is an X-ray source in close binary with type O supergiant Doppler produces mass for unseen X- ray source companion Flickers in ms: 25

26. Images 26

27. Black Hole in M31 27

28. Big Holes Stellar motions near center of Milky Way show compact object of mass Energetic jets and X-rays from SgR A* Most galaxies seem to have massive black holes in their centers Intermediate-mass black holes are new and under study 28

29. Black Hole Facts No Hair: Collapse loses all properties of star. Black hole characterized completely by mass, angular momentum, and electric charge Singularity is real (Hawking, Penrose). General Relativity is incomplete Cosmic censorship conjecture: singularities hidden inside horizons. Physics outside well- defined 29

30. Wormholes Look again at description of region near horizon. Describes two separate spacetimes touching for an instant Nothing can get through Try to modify solution to get big wormholes – not yet What’s on the other side? 30

31. Quantum Black Holes Hawking: Quantum effects near horizon leads to radiation with energy loss Hawking radiation is blackbody at Negative specific heat: hotter as loses energy Evaporate in for Microscopic black holes go faster if created? 31