General Relativity Applied to Non-Gravitational Physics G W Gibbons DAMTP, Cambridge YIPQS Symposium, Yukawa Institute, Kyoto, 8 Feb 2012
General Relativity, its mathematical techniques and conceptual frame- work are by now part of the tool kit of (almost) all theoretical physi- cists and at least some pure mathematicians. They have become part of the natural language of physics. Indeed parts of the subject are passing into Mathematics departments. It is natural therefore to ask to what extent can they illuminate other (non-relativistic) areas of physics. It is also the case that the relentless onward progress of technology makes possible analogue experiments illustrating basic ideas in General Relativity. In this talk I will illustrate this ongoing proces of Unification
As a topical example of the relentless progress of technology last month ∗ saw the demonstration in the laboratory some 40 years after the original prediction ! † of a very basic mechanism in semi-classical General Relativity: amplification of vacuum fluctuations in a time- dependent environment. This is the basis of all we believe about inflationary perturtbations, Hawking evaporation, Black Hole information “Pardadox?” and much of AdS/CFT etc etc. ∗ Wilson et al. Observation of the Dynamical Casimir effect, Nature 479 (2011) 376-379 † G. T Moore, Quantum Theory of Eletromagnetic Fields in Variable Length One- Dimensional Cavity. J. Math Phys 11 (1970) 2379-2691, S.A. Fulling Radiationa from a moving mirror in two dimensional spacetime : Conformal Anomaly, Proc Roy Soc A 348 (1976) 393-414
The idea of finding analogue models for General Relativistic effect is not new, but the pace has hotted up recently. Some important early work was done on cosmic strings modelled by vortices in superfluid Helium 4 and by Volovik ∗ , who noted that the order parameter of some phases of superfluid Helium 3 is a triad e i such that e i · e j = δ ij . More recently, the empahisis has shifted to the optics of metamaterials and most recently to graphene . There are also interesting analogies in liquid crystals. ∗ G. Volovk, The Universe in a Helium Droplet Oxford University Press (2003)
Let’s start with a very simple and sadly topical ∗ example: Shallow Water Waves If η = η ( t, x, y ) is the height of the water above its level when no waves are present and h = h ( x, y ) the depth of the water, then shallow water waves satisfy the non-dispersive wave equation: † ( ghη x ) x + ( ghη y ) y = η tt , where g is the acceleration due to gravity. From now on we adopt units in which g = 1 The wave operator coincides with the covariant D’ Alembertian √− g∂ µ ( √− gg µν ∂ ν η ) = 0 , 1 with respect to the 2 + 1 dimensional spacetime metric ds 2 = − h 2 dt 2 + h ( dx 2 + dy 2 ) . ∗ Tsunami † Einstein Equivalence Principle
Applying Ray theory and Geometrical Optics one writes η = Ae − iω ( t − W ( x,y )) , where A ( x, y ) is slowly varying. To lowest order W satisfies the Hamilton-Jacobi equation ∂y ) 2 = 1 ( ∂W ∂x ) 2 + ( ∂W h , and the rays are solutions of dx dt = h∂W ∂x .
Given any static spacetime metric ds 2 = − V 2 dt 2 + g ij dx i dx j , the projection x i = x i ( t ) of light rays, that is characteristic curves of the covariant wave equation or the Maxwell or the Dirac equations, onto the spatial sections are geodesics of the Fermat or optical metric given by 0 = g ij ds 2 V 2 dx i dx j In the special case of shallow water waves, the rays are easily seen to be geodesics of the metric o = dx 2 + dy 2 ds 2 . h
For a linearly shelving beach y > 0 . h ∝ y the rays are cycloids, and all ray’s strike the shore, i.e. y = 0, orthog- onally. For a quadratically shelving beach, h ∝ y 2 y > 0 , the rays are circles centred on the shore at y = 0 , and again every ray intersects the shore at right angles.
In fact the optical metric in this case is ds 2 = dx 2 + dy 2 y 2 which is Poincar´ e ’s metric of constant curvature on the upper half plane. If x is periodically identified, one obtains the the metric in- duced on a tractrix of revolution in E 3 ,sometimes called the Beltrami Trumpet ( i.e. H 2 /βZ . Note that the embedding can never reach the conformal boundary at y = 0.
The optical time for rays to reach the shore in the second example above is infinite. This reminds one of the behaviour of event hori- zons. In fact there is a rather precise correspondence. The Droste- Schwarzchild metric in isotropic coordinates (setting G = 1 = c ) is ds 2 = − (1 − m 2 r ) 2 2 r ) 2 dt 2 + (1 + m 2 r ) 4 ( dx 2 + dy 2 + dz 2 ) . (1 + m � x 2 + y 2 + z 2 . The isotropic radial coordinate r is related with r = to the Schwarzschild radial coordinate R by R = r (1 + m 2 r ) 2 . The Event Horizon is at R = 2 m , r = m 2 If we restrict the Schwarzschild metric to the equatorial plane z = 0 we obtain ds 2 = − (1 − m 2 r ) 2 2 r ) 2 dt 2 + (1 + m 2 r ) 4 ( dx 2 + dy 2 ) . (1 + m
The optical metric is 0 = (1 + m 2 r ) 6 2 r ) 2 ( dx 2 + dy 2 ) . ds 2 (1 − m and h = ( r − m 2 ) 2 r 4 2 ) 6 . ( r + m we get the analogue of a black hole : a circularly symmetric island whose edge is at r = m 2 and away from which the beach shelves initially in a quadratic fashion and ultimately levels out as r → ∞ . Since 1 dh 2 + 4 6 dr = r − > 0 r − m r + m h 2 2 the beach shelves monotonically.
To obtain a cosmic strings for which the optical metric is a flat cone with deficit angle δ = 2 πp p +1 one needs a submerged mountain with p h ∝ ( x 2 + y 2 ) p +1 , As p = ∞ , we get a parabola of revolution and the optical metric approaches that of an infinitely long cylinder. If p = 1 the mountain is conical, like a submerged volcano. In physical coordinates x, y the rays are bent, but one may introduce coordinates in which it is flat: 2 π ds 2 = d ˜ r 2 + ˜ φ 2 , r 2 d ˜ 0 ≤ ˜ φ ≤ p + 1 In these coordinates the rays are straight lines.
One could multiply these examples to cover such things as cosmic strings, moving water and vortices. To take into account the fact that the earth is round we replace E 2 by S 2 dx 2 + dy 2 → dθ 2 + sin 2 θdφ 2 (1) which gives Einstein’s Static Universe in 2 + 1 dimensions. To take into account that it is rotating, we replace the static, i.e. time-reversal invariant metric by a stationary metric dθ 2 + sin 2 θdφ 2 → dθ 2 + sin 2 θ ( dφ − Ω dt ) 2 (2) All of this can be illustrated using a (possibly rotating) ripple tank. Let’s pass from hydrodynamics to to Optics and Maxwell’s equations.
Maxwell’s source-free equations in a medium are − ∂ B curl E = ∂t , div B = 0 , + ∂ D curl H = div D = 0 , ∂t , or if ∗ − E i dt ∧ dx i + 1 2 ǫ ijk B i dx j ∧ dx k F = H i dt ∧ dx i + 1 2 ǫ ijk D i dx j ∧ dx k = G dF = 0 = dG ∗ ǫ ijk = ± , 0
In what follows it will be important to realise that these equations hold in any coordinate system and they do not require the introduction of a spacetime metric. However to “close the system”, one must relate F to G by means of a “constitutive equation”.
If the medium is assumed to be static, and linear then D i = ǫ ij E j B i = µ ij H j where ǫ is the dielectric permittivity tensor and µ ij the magnetic per- meability tensor. If they are assumed symmetric ǫ ij = ǫ ji µ ij = µ ji then E = 1 � � E i D i + H i B i may be regarded as the energy density and 2 S = E × H the energy current or Poynting Vector since Maxwell’s equations imply div S + ∂ E ∂t = 0 .
“In olden days a glimpse of stocking was thought of as something shocking” and certainly µ ij and ǫ ij were assume positive definite “but now” , with the advent of Nanotechnology and the construc- tion of metamaterials “ anything goes” . As long ago as 1964, V.G. Vestilago ∗ pointed out that isotopic substances with with µ ij = ǫ ij = ǫδ ij and or which µδ ij , µ < 0 , ǫ < 0 give rise to left-handed light moving in a medium with a negative refractive index ∗ Sov. Phys. Usp. 10 (1968) 509-514
In 2001 R.A. Shelby, D.R. Smith and S. Schutz ∗ produced this effect for microwave frequencies. In 2002 D.R. Smith, D. Schurig and J.B. Pendry † appeared to have produced this effect in the laboratory. ∗ Science 292 (2001) 77-79 † App Phys Lett 81 (2002) 2713-2715
Assuming a spacetime dependence proportional to an arbitrary func- tion of k · x − ωt , with ω > 0 one finds k × E = ω B , k × H = − ω D . k × E = ωµ H , k × H = − ωǫ E It is always the case that ( E , H , S ) form a right handed orthogonal triad but if both µ and ǫ are negative then ( E , H , k ) give form a left-handed orthogonal triad and so S and k are anti-parallel rather than parallel as is usually the case. Since the wave vector k must be continuous across a junction between a conventional medium and and an exotic medium with µ < 0 , ǫ < 0 , this give rise to backward bending light.
The speed of propagation v = 1 n , where n is the refractive index is given by v 2 = ω 2 k 2 = 1 µǫ with is natural to take the negative square root to get the refractive index n = − 1 √ µǫ .
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