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Curvature in 2D…

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…. Curvature in 2D…. Geodesics. To do geometry, we need a way to measure distances

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Curvature in 2D…

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  1. 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?

  2. Curvature in 2D…

  3. Curvature in 2D…

  4. Geodesics • To do geometry, we need a way to measure distances • In flat space, the shortest distance between P1 and P2 is a straight line • In curved space, the ant follows a • Geodesic: shortest line between P1 and P2 (the fewest possible ant steps) To the ant, the geodesic is a straight line, i.e., the ant neverhas toturn ant P1 P2

  5. Curvature and Clocks (time) I If both ants walk at constant velocity and leave P1 at the same time, then the ants arrive at point P2 at different times – the top ant gets their “first”. If both ants record the time when they first reach P2 – their clocks are no longer synchronized

  6. E = mc2 • Photons have an energy related mass and are affected by gravity. • The geometric surface shape of the Universe is determined by the mass distribution • Photons are therefore constrained to always travel on the surface of the Universe • Space is flat where there is no mass and curved where there is mass.

  7. Einstein: 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:

  8. Precession movieLightBend

  9. Experimental Evidence for GR 2) Deflection of light

  10. Experimental Evidence for GR

  11. 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.

  12. Black Holes

  13. Black Holes • To a stationary oberserver far away, time flow at the critical surface (at RS) 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

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