1. A recap of the main points so far… Observers moving uniformly with respect to each other do not agree on time intervals or distance intervals do not agree about the simultaneity or otherwise of events A moving clock appears to run slow – time dilation A moving ruler appears short – length contraction A moving object has an increased relativistic mass Mass can be converted into energy (and vice-versa) E = m c 2

2. The Twins Paradox

3. Symmetry As far as the observer on the spaceship is concerned, a clock left on Earth is running slow and lengths are contracted in the direction of motion. As far as the observer on Earth is concerned, a clock on the spaceship is running slow and lengths are contracted in the direction of motion. The frames of reference are equivalent?

4. Hammer and feather on the moon

5. Underlying principles of General Relativity The Equivalence Principle No difference between a steady acceleration and a gravitational field

6. Gravity and Acceleration cannot be distinguished

7. h V = a h/c

8. h Gravitational field Equivalence principle – this situation should be the same

9. Consequences I. Gravitational Redshift Size of effect (doppler shift equivalent speed) Earth 0.2 m/s Sun 0.6 km/s White Dwarf 70 km/s Neutron Star 100,000 km/s Black-hole infinite

10. Gravitational redshift of Sirius B Equivalent velocity 80 km/s

11. Corrections to GPS Orbital radius 26,000 km gh/c 2 = 0.0000000026 Clocks “on the ground” would lose about 0.2 ms per day (actually a bit less). Clocks on the GPS satellites run intentionally slowly!

15. Lensing of distant galaxies by a foreground cluster

18. QSO 2237+0305 The Einstein cross

19. Abell 1689 Plus derived Mass Map

20. Galaxies at 3, 6 and 11 billion light years away Einstein Rings