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

• 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)