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 never has to turn
ant P1 P2

5. 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

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. 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:

8. Precession movie LightBend

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