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

The Twins Paradox

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?

Hammer and feather on the moon

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

Gravity and Acceleration cannot be distinguished

h V = a h/c

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

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

Gravitational redshift of Sirius B Equivalent velocity 80 km/s

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

Lensing of distant galaxies by a foreground cluster

QSO The Einstein cross

Abell 1689 Plus derived Mass Map

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