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Outline

– Gravitational / and dilation – – Gravitational Lensing • Holes as a consequence of GR Waste Disposal

• It is decided that Earth will get rid of it’s nuclear waste by shooting it into . To make sure it doesn’t come back, it has to either be shot into the or out of the . Which was is cheaper?

• Radius of Sun = 7 × 105 km • Orbital speed of Earth = 30 km/s

• Kinetic :

• Escape Velocity:

• Orital Velocity: General Relativity Principle of Equivalence: Einstein 1907 g Box stationary in Box = accelerates in empty space g

Box falling freely =

g Box moves through space at constant velocity Gravitational Doppler Shift and • Total energy is always conserved • gains energy (blueshifted) when falling towards a , loses energy (redshifted) when going away

• Time runs slower close to a mass compared to far away Light rays and Gravity…

• Remember: gravity bends light…

accelerating observer = gravity 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 ? Tides

• Problem:

r2

moon r1

• Gravity decreases with distance => stretch…

Tides

• Tides = gravity changes from place to place

not freely falling ? freely falling ?? ?

not freely falling Curved

• Remember: Gravity warps time

BUT: in spacetime, time and space are not separable fast

=> Both space and time are curved (warped)

This is a bit hard to vizualize slow (spacetime already 4D…) GR: Einstein, 1915

• Einstein: mass/energy squeeze/stretch spacetime away from being “flat”

• Moving objects follow curvature (e.g., satellites, )

• The 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. GR: Einstein, 1915

• Einstein’s “field equations” correct “” problem:

Gravity information propagates at the

=> gravitational

r? 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 is curved? Curvature in 2D…

• In a , Euclidean does not apply:

- circumference ≠ 2π R - triangles ≠ 180° - parallel lines don’t stay parallel

2πR R R <2πR Σϕ=180° Curvature in 2D… Curvature in 2D… Plane Travel

• If I was going to fly from Madison, WI (43N 89W) to Lhasa, Tibet (29N 91E), should I fly southwest or southeast? Plane Travel Plane Travel Curvature in 2D…

• To do geometry, we need a way to measure distances => use ant (let’s call the ant “”), 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) • : shortest line between P1 and P2 (the fewest possible ant steps)

ant P1 P2 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 )

• 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 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.

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 (perihelion )

Newton: Einstein: Black Holes Black Holes

• What happens as the shrinks / its mass increases? How much can spacetime be distorted by a very massive object?

• Remember: in a Newtonian , 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

• A Black Hole is a collapsed region of space • Gravity 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