SINGULARITY THEOREMS Singularity place where physics breaks down

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? • 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 Penrose diagram (Hawking 1974)

• 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

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)

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

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)

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

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)