Optical black holes?

Matt Visser, Washington University, St. Louis
visser@tui.wustl.edu

The last few years have seen a lot of interest in condensed matter
analogues for classical Einstein gravity. The most well-developed
of these analog models are Unruh's acoustic black holes (dumb black
holes), but attention has recently shifted to the optical realm. The
basic idea is that in a dielectric fluid the refractive index, the
fluid velocity, and the background Minkowski metric can be combined
algebraically to provide an ``effective metric'' that can be used to
describe the propagation of electromagnetic waves. The most detailed
and up-to-date implementation of this idea is presented in very recent
papers by Leonhardt and Piwnicki [1-4], which are based on a thorough
re-assessment of the very early work of Gordon [5].
To get the flavour of the way the effective metric is set up, start
with a dispersionless homogeneous stationary dielectric with
refractive index *n* and write the electromagnetic equations of motion
as

If you write this in terms of the Minkowski metric and dielectric 4-velocity

Now promote the refractive index and 4-velocity to be slowly-varying functions of space and time. (Slowly varying with respect to the wavelength and frequency of the EM wave.) The preceding formula

with the (inverse) metric being proportional to

A more detailed calculation confirms this suggestion, and also lets you fix the overall conformal factor (it's unity, at least in Gordon's implementation). If you are only interested in ray optics then fixing the conformal factor is not important. Once you have this effective metric in hand, applying it is straightforward (even if the physical situation is unusual).

There are a few tricks and traps:

- 1.
- The 4-velocity is normalized using the Minkowski metric
,
and in this particular subfield it seems to have become conventional
to define
,
so that the index on the 4-velocity
is lowered with the Minkowski metric. (But for everything else you
raise and lower indices using the effective metric.) The metric itself
is then

- 2.
- The analogy with Einstein gravity only extends to the kinematic aspects of general relativity, not the dynamic. There is no analog for the Einstein equations of general relativity and trying to impose the Einstein equations is utterly meaningless.
- 3.
- If however, instead of using the energy conditions plus the Einstein equations, you place constraints directly on the Ricci curvature tensor or Einstein curvature tensor then you can still prove versions of the focusing theorems.
- 4.
- As in general relativity, the Riemann tensor and its contractions are still useful for characterizing the relative motion of nearby geodesics.
- 5.
- The Fresnel drag coefficient can be read off directly from the
contravariant components of the metric. Specifically

For low velocities this implies that the medium drags the light as though the medium had an effective velocity

The effective velocity of the medium is just the ``shift vector'' in the metric. The fact that the Fresnel drag coefficient drops out automatically should not surprise you at all since we are extracting all this from a manifestly Lorentz invariant formalism, and so you must get the same result as from the more usual approach based on the relativistic addition of velocities

- 6.
- There are optical analogs of the notions of ``trapped surface'', ``apparent horizon'', ``event horizon'', and ``optical black hole'' that are in exact parallel to those developed for the acoustic black holes [6,7].
- 7.
- If you somehow arrange an ``optical event horizon'' of this type, then there is near-universal agreement among the quantum field theory community that you should see Hawking radiation from this ``optical event horizon'', this radiation being in the form of a near-thermal bath of photons with a Hawking temperature proportional to the acceleration of the fluid as it crosses the horizon [6,7] -- this is a very exciting possibility, because we would love to be able to do some experimental checks on Hawking radiation.
- 8.
- You will be able to probe aspects of
*semiclassical quantum gravity*with this technique, but it won't tell you anything about quantum gravity itself. Because an effective metric of this type is not constrained by the Einstein equations it allows you only to probe*kinematic*aspects of how quantum fields react to being placed on a curved background geometry, but does not let you probe any of the deeper*dynamical*questions of just how quantum matter feeds into the Einstein equations to generate real spacetime curvature. Even though it should be kept in mind that these ``effective metric'' techniques are limited in this sense, they are still a tremendous advance over the current state of affairs. - 9.
- I should mention that I believe the original implementation of
Leonhardt and Piwnicki fails to generate genuine black holes, but that
this can be straightforwardly corrected [8]. Despite this technical
issue, which I believe causes problems for the particular toy model
they discussed, it is clear that the basic idea is fine -- it is
possible to form ``optical black holes'' by accelerating a dielectric
fluid to superluminal velocities (superluminal in the sense
*c*/*n*). Any region of superluminal fluid flow will be an ergo-region, and any surface for which the inward component of the fluid flow is superluminal will be a trapped surface.

*References:*

[1] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. **84**, 822-825 (2000).

[2] U. Leonhardt and P. Piwnicki, Phys. Rev. A **60**, 4301-4312 (1999).

[3] U. Leonhardt, *Spacetime physics of quantum dielectrics*,
`physics/0001064`.

[4] U. Leonhardt, http://www.st-and.ac.uk/~www_pa/group/quantumoptics/media.html

[5] W. Gordon, Ann. Phys. (Leipzig) **72**, 421 (1923).

[6] W. Unruh, Phys. Rev. Lett. **46**, 1351 (1981);
Phys. Rev. D **51**, 2827 (1995).

[7] M. Visser, Class. Quantum Grav. **15**, 1767 (1998);
see also `gr-qc/9311028`.

[8] M. Visser,
`gr-qc/0002011`

[9] L. V. Lau, *et al*, Nature (London) **397**, 594 (1999).