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)