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…

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)

P1 P2

ant

To the ant, the geodesic is a straight line,

i.e., the ant never has to turn

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

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 movieLightBend

2) Deflection of light

Experimental Evidence for GR

Experimental Evidence for GR

• 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 informationcan escape red-shifted

red-shifted intooblivion

Black Holes