Outline. General Relativity. Black Holes as a consequence of GR. Gravitational redshift/blueshift and time dilation Curvature Gravitational Lensing

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1 Outline General Relativity Gravitational redshift/blueshift and time dilation Curvature Gravitational Lensing Black Holes as a consequence of GR

2 Waste Disposal It is decided that Earth will get rid of it s nuclear waste by shooting it into space. To make sure it doesn t come back, it has to either be shot into the Sun or out of the solar system. Which was is cheaper? Radius of Sun = km Orbital speed of Earth = 30 km/s

3 Escape Velocity Kinetic Energy: Escape Velocity: Orital Velocity:

4 General Relativity

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

6 Gravitational Doppler Shift and Time Dilation Total energy is always conserved Light gains energy (blueshifted) when falling towards a mass, loses energy (redshifted) when going away Time runs slower close to a mass compared to far away

7 Light rays and Gravity Remember: gravity bends light accelerating observer = gravity

8 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?

9 Tides Problem: r2 r1 moon Gravity decreases with distance => stretch

10 Tides Tides = gravity changes from place to place? not freely falling freely falling??? not freely falling

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

12 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.

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

14 Curvature in 2D 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?

15 Curvature in 2D In a curved space, Euclidean geometry does not apply: - circumference 2π R - triangles parallel lines don t stay parallel Σϕ=180 2πR R R <2πR

16 Curvature in 2D

17 Curvature in 2D

18 Plane Travel If I was going to fly from Madison, WI (43N 89W) to Lhasa, Tibet (29N 91E), should I fly southwest or southeast?

19 Plane Travel

20 Plane Travel

21 Curvature in 2D

22 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) P1 ant P2

23 Geodesics 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.

24 Geodesics Gravitational Lensing 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.

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30 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:

31 Black Holes

32 Black Holes 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.

33 Black Holes A Black Hole is a collapsed region of space Gravity curves space so much that close enough in light is bent so much it always falls in If you get close enough to a blakc hole, you can never get back out