1. Principle of Equivalence: Einstein 1907 Box stationary in gravity field Box falling freely Box accelerates in empty space Box moves through space at constant velocity

2. Equivalence Principle Special relativity: all uniformly moving frames are equivalent, i.e., no acceleration Equivalence principle: Gravitational field = acceleration freely falling frames in GR = uniformly moving frames in SR.

3. Tides Problem: Gravity decreases with distance => stretch… r1r1 r2r2 moon

4. Tides = gravity changes from place to place freely falling not freely falling ? ? ? ? Tides

5. Light rays and Gravity… Remember: gravity bends light… accelerating observer = gravity

6. Light Rays and Gravity II In SR: light rays travel on straight lines => in freely falling fame, light travels on straight lines BUT: to stationary observer light travels on curved paths => Maybe gravity has something to do with… curvature of space ?

7. Curved Spacetime Remember: Gravity warps time slow fast BUT: in spacetime, time and space are not separable => Both space and time are curved (warped) This is a bit hard to vizualize (spacetime already 4D…)

8. GR: Einstein, 1915 Einstein: mass/energy squeeze/stretch spacetime away from being “flat” Moving objects follow curvature (e.g., satellites, photons) The equivalence principle guarantees: spacetime is “locally” flat The more mass/energy there is in a given volume, the more spacetime is distorted in and around that volume.

9. GR: Einstein, 1915 Einstein’s “field equations” correct “action at a distance” problem: Gravity information propagates at the speed of light => gravitational waves r?

10. Imagine being an ant… living in 2D You would understand: left, right, forward, backward, but NOT up/down… How do you know your world is curved? Curvature in 2D…

11. In a curved space, Euclidean geometry does not apply: - circumference  2  R - triangles  180° - parallel lines don’t stay parallel <2  R R R 2R2R  =180  Curvature in 2D…

14. Geodesics To do geometry, we need a way to measure distances => use ant (let’s call the ant “metric”), count steps it has to take on its way from P1 to P2 (in spacetime, the ant-walk is a bit funny looking, but never mind that) Geodesic: shortest line between P1 and P2 (the fewest possible ant steps) P1P2 ant

15. To the ant, the geodesic is a straight line, i.e., the ant never has to turn In SR and in freely falling frames, objects move in straight lines (uniform motion) In GR, freely falling objects (freely falling: under the influence of gravity only, no rocket engines and such; objects: apples, photons, etc.) move on geodesics in spacetime. Geodesics

16. Experimental Evidence for GR If mass is small / at large distances, curvature is weak => Newton’s laws are good approximation But: Detailed observations confirm GR 1) Orbital deviations for Mercury (perihelion precession) Newton:Einstein:

17. 2)Deflection of light Experimental Evidence for GR

19. What happens as the star shrinks / its mass increases? How much can spacetime be distorted by a very massive object? Remember: in a Newtonian black hole, the escape speed simply exceeds the speed of light => Can gravity warp spacetime to the point where even light cannot escape it’s grip? That, then, would be a black hole. Black Holes

21. Time flows more slowly near a massive object, space is “stretched” out (circumference < 2  R) Critical: the ratio of circumference/mass of the object. If this ratio is small, GR effects are large (i.e., more mass within same region or same mass within smaller region) Black Holes ??? 1) massive2) small

22. GR predicts: If mass is contained in a circumference smaller than a certain size space time within and around that mass concentration qualitatively changes. A far away observer would locate this critical surface at a radius Gravitational time dilation becomes infinite as one approaches the critical surface. gravitational constant speed of light critical circumference mass Schwarzschild radius The Schwarzschild Radius

23. To a stationary oberserver far away, time flow at the critical surface (at R S ) is slowed down infinitely. Light emitted close to the critical surface is severely red-shifted (the frequency is lower) and at the critical surface, the redshift is infinite. From inside this region no information can escape red-shifted red-shifted into oblivion Black Holes

24. Inside the critical surface, spacetime is so warped that objects cannot move outward at all, not even light. =>Events inside the critical surface can never affect the region outside the critical surface, since no information about them can escape gravity. =>We call this surface the event horizon because it shields the outside completely from any events on the inside. Black Holes

25. Critical distinction to the Newtonian black hole: Nothing ever leaves the horizon of a GR black hole. Lots of questions… What happens to matter falling in? What happens at the center? Can we observe black holes anyway? And much, much more… Newton Einstein Black Holes