1. Physics 55 Monday, December 5, 2005 1.Course evaluations. 2.General relativity with applications to black holes, dark matter, and cosmology. 3.Hubble’s Law. 4.The Big Bang.

2. Effect of Gravity on Clocks

3. Time Slows Down Near Event Horizon Time is not absolute: observer far from black hole sees time slow down near black hole, person near black hole sees time speed up far from black hole. Note: blue shift of light falling into black hole means that someone near event horizon struck by gamma rays!

4. Two Ways to Travel Forward it Time 1.Travel in a rocket at high speed then return home. 2.Hang out near the event horizon of a black hole. Both methods require immense amounts of energy and are possible in principle but not doable with present technology.

5. Application of Equivalence Principle: Gravity Bends Light

6. Experimental Confirmation of Bending of Light Amount of angular deflection for starlight grazing Sun ~ 2’’, tiny but definite effect. Note: one could estimate deflection of photon by assuming it has gravitational mass equivalent to E/c 2 = hf/mc 2 and calculate gravitational force acting on this mass but you get half the above answer, need general relativity!

7. Einstein Cross, Einstein Ring: Bending of Light by Galactic Masses

8. Gravitational Lensing Used to Detect Dark Matter

9. Dark Matter

10. “Microlensing” Reveals Properties of “Massive Compact Halo Objects” or MACHOS MACHO are likely red stars, brown dwarfs, Jupiter-like objects

11. Weakly Interacting Massive Particles: WIMPS Observations suggest that MACHOs make up less than half the observed dark halo around a galaxy, need another explanation for dark matter. Physicists and astronomers currently believe that dark matter is made of “exotic” particles unlike protons, neutrons, electrons, or neutrinos. Dark matter does not interact with anything via the electromagnetic interaction, which is why we can’t see it using light. Also suggests that dark matter can not condense into a compact mass since it is unable to get rid of its thermal energy by emitting light via thermal emission.

12. Bending of Light, Gravity Itself Requires A Radical Change in Viewpoint One could try to understand bending of light as a particle being deflected by a force but Einstein suggested a much more radical and powerful insight: that there are no forces, just bending of spacetime, in which case light and masses follow curved paths in response to this bending. What is meant by “spacetime”? How can points in spacetime “bend” or “warp”?

13. Spacetime

14. Euclid’s Axioms 1.For every point P and every point Q not equal to P there exists a unique line that passes through P and Q. 2.For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and segment CD is congruent to segment BE. 3.For every point O and every point A not equal to O there exists a circle with center O and radius OA. 4.All right angles are congruent to each other. 5.For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l The troublemaker

15. Non-Euclidean Geometries What is the maximum “sum of interior angles” for a spherical geometry? Can you have a triangle with interior sum of 180 o on a sphere? What kind of geometry is the surface of a cylinder?

16. The Einstein Equations for the Spacetime Metric g ij (t,x,y,z) Einstein gravitational field equations are 10 coupled nonlinear partial differential equations for the metric g ij as a function of space and time. The metric determines the spacetime distance between two spacetime points that are extremely close to one another. The metric varies with space and time because of the presence of mass-energy. A constant metric would occur for an empty universe.