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 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 never has to turn ant P1 P2

Curvature and Clocks (time) 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

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.

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:

Precession movie LightBend

Experimental Evidence for GR 2) Deflection of light

Experimental Evidence for GR

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

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