1. Hawking radiation, infinite redshifts, and black hole analogues
Ted Jacobson
University of Maryland
Gravity and Black Holes, Cambridge, July 4, 2017

2. Wheeler to Bekenstein (1971): “If I drop a teacup into a black hole,
I conceal from all the world the
increase of entropy.”
Begins with the concept of black hole entropy
Black hole: inside hidden by causality.
Black hole

3. Black hole entropy Generalized second law: ? Bekenstein, 1972
Natural to assign it entropy: BH horizon hides information. What information? Will return to that question.
Why the area?
Horizon area never decreases (classically).
Area is extensive, local (as opposed to, say, mass).
Minimum area increase is independent of the mass, spin, charge of the black hole (uses the uncertainty principle, and GR) ?

4. Bekenstein’s entropy BH has a thermodynamic temperature:
Validity of the GSL requires that this be a REAL temperature!
This real temperature became evident when Hawking considered
a black hole immersed in the fluctuating vacuum of quantum fields.
Stephen was not seeking this --- it took him by surprise. He was
checking the prediction, of Zeldovich and others, that a rotating
black hole would spontaneously radiate.

5. Event horizon & infinite redshift
Frequency decay rate κ,
called “surface gravity”
(for a spherical black hole).
Because it is lightlike. The lightlike limit of special relativistic time dilation: no time passes. The only time that is also a length, and length that is also a time, is zero!
So light from an infalling source would be infinitely redshifted.
This redshift is not mysterious, but there are two other very unusual effects present. One is the singularity, which is mysterious, but hidden from view. The other is the loss of time translation symmetry.
----- Meeting Notes (5/9/16 23:38) -----
Picture by Roger Penrose
The Road to Reality (Knopf, 2005)

6. The “time” translation symmetry is space translation
inside the horizon…
…so the conserved quantity
is momentum inside, and can
thus be negative.
This implies that the quantum
field vacuum is unstable,
which leads to …
Picture by Roger Penrose
The Road to Reality (Knopf, 2005)

7. Hawking radiation: Quantum field theory implies
(1974)
Quantum field theory implies
black holes have a temperature
= 62 nK Msun/M
The singularity is hidden from view by the event horizon, but there is new effect that is not hidden!
The vacuum fluctuation pairs are slowly peeled apart.
It would take a solar mass black hole 1058 years to evaporate. A mini black hole with the mass of a mountain would take only 1010 years – the age of the universe

8. Hawking radiation: Tidal peeling of vacuum Fluctuations corresponds
(1974)
Tidal peeling of vacuum
Fluctuations corresponds
to pair creation, with
partners occupying
negative energy states.
The singularity is hidden from view by the event horizon, but there is new effect that is not hidden!
The vacuum fluctuation pairs are slowly peeled apart.
It would take a solar mass black hole 1058 years to evaporate. A mini black hole with the mass of a mountain would take only 1010 years – the age of the universe

9. Infinite redshifts The ancestors of the Hawking radiation
have exponentially growing frequency,
For a solar mass black hole, a blueshift
of e100,000 after only 2 seconds.
Is this an absurd extrapolation?
1) Must we believe in this to believe in
the Hawking effect?
2) Do outgoing modes really arise in this way?
There is a cosmological transPlanckian puzzle as well. They should both find solution in quantum gravity.
NO! It suffices to assume that quantum gravity
adiabatically delivers the outgoing free-fall vacuum
at length scales l with LP << l << RBH.
2) Who knows?! It’s hard to see how anything else could
be Lorentz invariant and local.

10. Sonic analog black holes
(v -> v + c if c ≠ constant)
Motivated in part by this puzzle Unruh introduced back in 1980 a sonic analogy for a black hole. How is the trans-Planckian puzzle resolved here?
Picture from Schutzhold, CQG 25 (2008)

11. Acoustic metric

12. Sonic analog black holes
This opens the opportunity to study, in theory and in the laboratory, in a setting where the fundamental physics is fully understood:
the origin of the outgoing modes, given an atomic, short distance cutoff?
the adiabatic delivery of the outgoing vacuum?
the wider validity of Hawking’s prediction?
Motivated in part by this puzzle Unruh introduced back in 1980 a sonic analogy for a black hole. How is the trans-Planckian puzzle resolved here?

13. Superfluid helium-4 dispersion relation
Origin of outgoing modes
TJ, PRD, ‘96

14. supersonic relativistic subsonic
(e.g. Bose-Einstein condensates)
relativistic
subsonic
(e.g. superfluid He-4, gravity waves on water)

15. Regulated ancestry of Hawking pairs
There is a breakdown of WKB at the horizon, and mode-mixing occurs, so this picture of the history of particle & partner lines shouldn’t be taken too seriously.
relativistic
superluminal
sub-luminal
dissipative
(Robertson- Parentani, 2015)

16. Numerous studies with linear field theory have shown that
the Hawking effect persists, as long as:
The incoming field is in the ground state.
The wavenumber ‘cutoff’ is higher than the surface gravity.*
For the current state of the art, see Parentani & collaborators, 2012, 2014
Many candidates for experimental black hole analogues have been studied
theoretically and, for some, experimentally:
Superfluid 4He flows
Superfluid 3He-A textures
Time-dependent waveguides
Bose-Einstein condensates
trapped ion rings
nonlinear optics in fibers & fused silica
gravity waves on water …

17. Bose-Einstein Condensate dispersion relation

18. Origin of the outgoing modes in a BEC analog BH
sub-sonic
supersonic
supersonic
sub-sonic
horizon
flow

19. black & white hole pair flow positive vs. negative energy subsonic
supersonic
flow
black & white hole pair

20. “Black hole laser” In the presence of an inner horizon,
(S. Corley & TJ, ’99)
In the presence of an inner horizon,
supersonic dispersion can produce a
superradiant instability (like the rotating “black hole bomb”).
subsonic
supersonic
subsonic

21. Steinhauer’s experiment with a quasi-one dimensional BEC
containing a BH/WH cavity (2014)
Figures from theoretical modeling paper:
(arXiv )

22. Time series of images of condensate density (Steinhauer, 2014)

23. Model the condensate using the mean field, “Gross-Pitaevskii” equation:
M = atomic mass, N = number of atoms, Ψ = exp. val. of many body field operator
, a = s-wave scattering length.

24. Stimulated emission of classical Hawking radiation by
Cerenkov radiation from the white hole horizon.
Steinhauer parameters enhanced parameters
x 10
x 10
trap size twice as large
potential step half as deep

25. Stimulated emission of classical Hawking radiation by
Cherenkov radiation from the white hole horizon.
The standing wave grows quadratically with the density of the flow. Not an exponential instability after all.
It is at rest in the WH rest frame, but not in the BH rest frame! So the frequency iszero in the WH frame and nonzero in the BH frame.

26. Spectral analysis confirms the Cerenkov mechanism,
and rough agreement with Hawking “thermal” prediction
Frequency spectrum in the BH frame.
Each Bogoliubov mode is composed of a pair of modes with opposite frequency and wavevector. BCR and p are NEGATIVE energy modes, HR is positive energy.
Thermal, Hawking prediction:
Simulation “measurement”
agrees with this to within 25%

27. Concluding comments:
The Hawking effect extends to acoustic analogs, which regulate the infinite redshift, and can be studied in the laboratory.
The origin of the outgoing modes in spacetime remains (to me) totally puzzling.
Next talk: an experiment to measure entanglement of spontaneous, quantum
Hawking radiation.
The era of experimental Hawking radiation has dawned!