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General Relativity – Gravitational Redshift/Blueshift and Time Dilation – Curvature – Gravitational Lensing • Black Holes As a Consequence of GR Waste Disposal

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General Relativity – Gravitational Redshift/Blueshift and Time Dilation – Curvature – Gravitational Lensing • Black Holes As a Consequence of GR Waste Disposal Outline • General Relativity – Gravitational redshift/blueshift and time dilation – Curvature – Gravitational Lensing • Black 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 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 = 7 × 105 km • Orbital speed of Earth = 30 km/s Escape Velocity • Kinetic Energy: • Escape Velocity: • Orital Velocity: General Relativity Principle of Equivalence: Einstein 1907 g Box stationary in Box gravity field = accelerates in empty space g Box falling freely = g Box moves through space at constant velocity 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 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 Spacetime • 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, 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. GR: Einstein, 1915 • Einstein’s “field equations” correct “action at a distance” problem: Gravity information propagates at the speed of light => gravitational waves 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 world is curved? Curvature in 2D… • In a curved space, Euclidean geometry 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… 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) 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 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 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 Mercury (perihelion precession) Newton: Einstein: Black Holes 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. 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 .
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