1. SINGULARITY THEOREMS Singularity = place where physics breaks down –usually, where some predicted quantity becomes infinite, e.g., curvature of spacetime –Oppenheimer-Snyder show for perfectly spherical collapse: what about more realistic case? (Hawking 1974) Penrose diagram Topological (“global”) methods Penrose (1964), Hawking & Penrose (1970) Separate spacetime into regions, look at possible connections (by light-rays) between regions (esp. future/past infinity) Look at convergence/divergence of light rays

2. Trapped surface: surface within which all (outgoing + ingoing) light rays converge Singularity theorem: Every “trapped surface” contains a singularity Singularity need not be a point –e.g., ring-shaped for rotating hole

3. WHAT KIND OF SINGULARITY? Many kinds possible: –Oppenheimer-Snyder: simple point, infinite radial stretch (+ transverse squeeze) –Belinsky-Khalatnikov-Lifshitz (1970): dynamic, chaotic curvature: random, violent stretching Quantum effects become important on scales smaller than ~ 10 -33 cm (oscillations faster than ~10 -43 sec)

4. ARE SINGULARITIES ALWAYS FATAL? Almost certainly, but some interesting issues Ring singularity: can you “miss it”? –route through wormhole? BKL chaotic singularity –really an instability –amazing idea (Ori 1991) : chaotic tidal forces die down with time... if you leave the black hole alone –but any disturbance (you falling in, a photon falling in …) kicks the chaos back into high gear. Ring sing. chaotic moving target, hard to miss

5. COSMIC CENSORSHIP HYPOTHESIS “All singularities that form in the Universe are surrounded by a horizon” –no “naked singularities” (white holes) from astrophysical collapse –Just a hypothesis! May not be true (possible counterexample found in computer simulation in 1991) Makes an exception for the Big Bang –Penrose & Hawking prove Big Bang had to emerge from a singularity Distinguish past from future (recall Penrose diagram) –Black hole: can only exist in future of a worldline – White hole: can only exist in past (e.g., big bang OK)

6. WORMHOLE (aka EINSTEIN-ROSEN BRIDGE) Formal solution of GR equations –horizon exactly at throat –can connects regions far apart in spacetime Does it allow time travel? The catch: solution is not static from point of view of observer trying to cross : Impossible to pass through without getting trapped in singularity

7. WHAT DO WORMHOLES CONNECT? Can provide an alternate (shorter?) route between two regions of spacetime –no limit on separation of regions in “normal” space –idea to use as time machine Could connect two separate Universes Too-shortlived to be useful unless… –held open by “exotic matter” –“exotic matter” = negative energy = huge tension –exotic matter has similar properties to vacuum fluctuations in presence of strongly curved space (as in Hawking radiation)