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 .
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
If you write this in terms of the Minkowski metric
dielectric 4-velocity Va, then
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
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:
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
which corresponds to
meters/second  -- and it is this experimental
fact that holds out the hope for doing laboratory experiments in the
not too distant future.
- The 4-velocity is normalized using the Minkowski metric
and in this particular subfield it seems to have become conventional
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
- 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
- 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
- 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
- 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 . 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.
 U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822-825 (2000).
 U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301-4312 (1999).
 U. Leonhardt, Spacetime physics of quantum dielectrics,
 U. Leonhardt,
 W. Gordon, Ann. Phys. (Leipzig) 72, 421 (1923).
 W. Unruh, Phys. Rev. Lett. 46, 1351 (1981);
Phys. Rev. D 51, 2827 (1995).
 M. Visser, Class. Quantum Grav. 15, 1767 (1998);
see also gr-qc/9311028.
 M. Visser,
 L. V. Lau, et al, Nature (London) 397, 594 (1999).