1. How do we transform between accelerated frames? Consider Newton’s first and second laws: m i is the measure of the inertia of an object – its resistance to a change in its state of motion. m i is the inertial mass of an object. Now consider Newton’s Law of Gravitation: m g is a measure of the response of an object to gravitation. M g is the gravitational mass of an object.

2. Equivalence Principle Consider a man in an elevator, in two situations: 1) Elevator is in free-fall. Although the Earth is exerting gravitational pull, the elevator is accelerating so that the internal system appears inertial! 2) Elevator is accelerating upward. The man cannot tell the difference between gravity and a mechanical acceleration in deep space! m i = m g

3. But now for a paradox! We shine a light into a window of the elevator, while the car is accelerating. An inertial observer outside sees the light move on a straight-line trajectory. But the accelerated passenger feels either gravity or mechanical acceleration (he can’t tell which). She observes that the photons of light have mass- equivalent energy, and are therefore accelerated on a curved path!/

4. Einstein’s solution: space curves near a mass. Gravitational attraction is “straight” motion in a curved space. G    8  G  c 2   G  is the curvature of coordinate  with respect to coordinate ( , running 1  4). T  is the stress-energy tensor that sources curvature from the density of mass and energy in a region of space-time. ds 2 = c 2 (1-2MG/c 2 r)dt 2 - dr 2 /(1-2MG/c 2 r) - r 2 (d  2 +sin 2  d  2 )

5. The red shift of light Set dr = d  = d  = 0, and then divide by c, in equation (1), we obtain d  = (1-2MG/c 2 r) 1/2 dt as the elapsed proper time d  on  a  clock fixed at location r. We see that dt is not the proper elapsed time. Now apply equation (2) to our two clocks at distances r = r 1 and r = r 2 from a gravitating mass M, and compare the two proper time intervals, for the same coordinate time difference dt. We find d  1  d    = (1-2MG/c 2 r 1 ) 1/2  (1-2MG/c 2 r 2 ) 1/2

6. Gravitation lensing – the first test of general relativity Image of the rich galaxy cluster Abell 2218, taken with the Hubble Space Telescope. This cluster shows evidence of multiple lensing images, as well as numerous strong and weak arcs. Using information about the arcs and multiple images allows astronomers to reconstruct the mass distribution, which then gives us knowledge about the distribution of dark matter and allows us to estimate the matter density  0 of the universe.

7. Einstein Cross – lensing of a quasar by a nearby galaxy

8. Black holes The curvature of space-time increases with the concentration of gravitating mass. There is a limit to this process: when enough mass accumulates within a central core, the space curves within itself inside a critical radius – the Schwarzschild radius R s = 2MG/c 2. Matter and energy within this radius can never escape!

9. Black holes have been found in the cores of many galaxies

10. Near the surface of the black hole, neighboring stars are orbiting at v  c

11. Light from close-orbiting stars is shifted to the red as they move towards us, blue as they move away STIS measures a velocity of 880,000 miles per hour (400 kilometers per second) within 26 light-years of the galaxy's center, where the black hole dwells. This motion allowed astronomers to calculate that the black hole contains at least 300 million solar masses. (Just as the mass of our Sun can be calculated from the orbital radii and speeds of the planets.)

12. Black holes – AAU 1997