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

\begin{displaymath}\left(-n^2 {d^2\over dt^2} + \nabla^2 \right) F^{ab} =0.

If you write this in terms of the Minkowski metric $\eta^{ab}$ and dielectric 4-velocity Va, then

-n^2 (V^c\;\nabla_c)^2 + [\eta^{cd} + V^c\; V^d ] \; \nabla_c \nabla_d
\right\} F_{ab} = 0.

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 suggests that it is possible to write

\begin{displaymath}{1\over\sqrt{-g}} \;
\partial_c \left( \sqrt{-g}\; g^{cd} \; \partial_d F_{ab} \right) = 0,

with the (inverse) metric being proportional to

\begin{displaymath}g^{ab} \propto \eta^{ab} - (n^2-1) \; V^a \; V^b.

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:

The 4-velocity is normalized using the Minkowski metric $\eta_{ab}
\; V^a \; V^b = -1$, and in this particular subfield it seems to have become conventional to define $V_a = \eta_{ab} V^b$, 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

\begin{displaymath}g_{ab} =
\eta_{ab} - (n^{-2}-1)\; V_a \; V_b =
[\eta_{ab} + V_a \; V_b ] - n^{-2} \; V_a \; V_b.

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.
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.
As in general relativity, the Riemann tensor and its contractions are still useful for characterizing the relative motion of nearby geodesics.
The Fresnel drag coefficient can be read off directly from the contravariant components of the metric. Specifically

\begin{displaymath}g_{0i} = (n^{-2}-1)\;\gamma^2 \; \vec v.

For low velocities $\gamma\approx1$ this implies that the medium drags the light as though the medium had an effective velocity

\begin{displaymath}\vec v_{\mathrm{eff}} = (n^{-2}-1) \; \vec v.

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

\begin{displaymath}c_{\mathrm{dragged}} = { v + (c/n) \over 1 + {v (c/n)\over c^2 } }
\approx {c\over n} + (n^{-2}-1) \; v + O(v^2).

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].
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.
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.
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.
Finally, let me emphasize the fundamental experimental reason this is now all so interesting: experimental physicists have now managed to get refractive indices up to $n\approx 30,000,000$ which corresponds to $c/n \approx 10$ meters/second [9] -- and it is this experimental fact that holds out the hope for doing laboratory experiments in the not too distant future.


[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).

Jorge Pullin