Higher rank tensors in physics, the Kummer tensor density

Transcription

1 Higher rank tensors in physics, the Kummer tensor density Friedrich W. Hehl University of Cologne and University of Missouri, Columbia, GIF Workshop Jerusalem Monday, 20 October 2014, 10:00-11:00 h Many thanks to Yakov Itin, to Yaakov Friedman, and to Tzvi Carr for the organization of the workshop. See P. Baekler, A. Favaro, Y. Itin, FWH, The Kummer tensor density in electrodynamics and gravity, Annals of Physics (NY) 349, (2014) file KummerJerusalem02.tex

2 Higher rank tensors in physics, the Kummer tensor density Contents 1. Higher rank tensors in physics 1.1 Classical continuum/field theory, elasticity 1.2 Classical electrodynamics 1.3 General relativity (GR) 1.4 Interrelationships 2. The Kummer tensor density 2.1 Fresnel surface 2.2 Magnetoelectricity 2.3 Wave propagation, Tamm-Rubilar tensor density 2.4 Kummer surface and optics 2.5 Premetric Kummer tensor density newly defined, irreducible decomposition 2.6 Kummer tensor in GR 2.7 Kummer-Weyl tensor of the Kerr solution

3 1.1 Classical continuum/field theory, elasticity 3 1. Stress tensor σ ab, force piercing through a 2d area element (see Cauchy tetrahedron) 2. Strain tensor ε ab, proportional to the difference between metric of the deformed and undeformed state 3. Elasticity (Hooke) tensor c abcd, local and linear response 4. Incompatibility (Saint-Venant) tensor W abcd, also inc, its vanishing guarantees that the strain can be derived from a displacement field s a according to ε ab = (a s b) ; W is the Riemann tensor belonging to the strain ε ab as metric (strain space) 5. Stress functions (Maxwell-Morera) χ ab, σ = incχ; in 3d stress space, Riemann = Ricci = stress, stress functions determine the metric 6. Stress/hyperstress 3-forms (pierce through a hypersurface of dimension (n 1) in n dimensions): Σ α = 1 (n 1)! Σi 1...i n 1 αdx i 1 dx i n 1, α 1 β = (n 1)! Σ α i 1...i n 1 β dx i 1 dx i n 1. Act as sources in metric-affine gravity (variation with respect to coframe and linear connection, repsectively

4 1.2 Classical electrodynamics 1. Excitation tensor density H ij 2. Field strength tensor F ij 3. Electromagnetic response tensor densityχ ijkl 4. Optical activity tensor g abc, see D a = ε ab E b +g abc ce b with g abc = g bac ; piezoelectric tensor d abc, Modulus for the quantum Hall effect (QHE) with the constitutive law (in 3d) J ab = σ ab cd F cd, with the premetric 4th rank tensor density σ ab cd = σ ba cd = σ ab dc, 9 = 8+1, absolute dimension [σ] = q 2 /h = 1/resistance 6. Energy-momentum tensor T i j, see the stress tensor of elasticity (Maxwell: Another theory of electricity, which I prefer, denies action at a distance and attributes electric action to tensions and pressures in an all-pervading medium, these stresses being the same in kind with those familiar to engineers, and the medium deing identical with that in which light is supposed to be propagated. ) 7. Tamm-Rubilar and Kummer tensor densitiesg ijkl and K ijkl, respectively 8. Chevreton tensor H ijkl, the Bel-Robinson tensor of the electromagnetic energy-momentum

5 1.3 General relativity 1. Metric tensor g ij 2. Riemann-Christoffel tensor R ijk l ({ }), also curvature tensor R ijk l (Γ) of a space with linear conn. Γ ij k, special case: Riemann-Cartan space 3. Weyl tensor W ijkl, an irreducible piece of Riemann 4. Cotton-Bach tensor C ijk, conformal tensor for 3d, is also valid in n dimensions, see García et al.(2004) 5. Ricci tensor Ric ij 6. Bel/Bel-Robinson superenergy tensor B ijkl, distinguish Bel tensor, also for matter, Bel-Robinson only for vacuum 7. Lanczos tensor L ijk, tensor potential for curvature 8. Simon tensor S ab, Simon-Mars tensor M ijk ( non-kerrness ), expressed in curvature and Killing vectors 9. Killing-Yano tensors: a symmetric tensor a ij = a (ij), satisfying a ij;k +A jk;i+aki;j = 0, is a Killing tensor of order 2, an antisymmetric tensor f ij = f [ij], satisfying f ij;k +f ik;j = 0, is a Killing-Yano tensor of order two, corresponding generalizations to higher rank tensors; related to separability of Dirac equation in the Kerr metric 10. Kummer tensor density K ijkl, defined by Ruse and Zund 11. Wheeler-DeWitt supermetric tensor in the Hamiltonian formul. of GR: G abcd = 1 (h ach bd +h ad h bc h ab h cd ) 5

6 6 1.4 Interrelationships Stress and strain spaces of classical elasticity carry a Riemannian structure and are analogous to structures in GR (see the stress tensor in GR) The elasticity tensor c abcd is an anlog of the electromagnetic response tensor χ ijkl (see work of Itin and fwh) The analogy between the principal part (1) χ ijkl of the electromagnetic response tensor and a space of constant curvature in GR was noted by Post, Formal Structure of Electromagnetics (1962), for example The Tamm-Rubilar tensor of electrodynamics is related to the Kummer tensor of GR Interrelationship between the principal null vectors of Kummer and curvature tensor, see Baekler and Favaro (2013/14) These are just examples of analogies between higher rank tesnors of different disciplines; often, they are not only formal (group theoretically), but are due to a common physical root

7 2.1 Fresnel surface At each point inside a crystal, we have a ray vector and a wave covector (one-form): ray surface and its dual, the wave surface, Ray surface was first constructed by Fresnel (1822) and is conventionally called Fresnel surface, also used for the wave surface, If the permittivity tensor has its vacuum values, the Fresnel surface is described by 3 principal values. In the case when all of them are equal, the Fresnel surface is an ordinary 2-dimensional sphere. For two unequal parameters, the surface is the union of two shells, a sphere and an ellipsoid. These two shells touch at two points. In the figure, we display such a surface for a crystal with three different principal values of the permittivity: (ε ab ) = ( ε ε ε 3 ) (, and (µ 1 ab ) = µ ) It is a union of two shells that meet at 4 singular points.the wave surfaces are the characteristics of the corresponding partial differential equations describing the wave propagation.

8 Fresnel wave surface as the specific quartic surface (α 2 x 2 +β 2 y 2 +γ 2 z 2 )(x 2 +y 2 +z 2 ) [ α 2 (β 2 +γ 2 )x 2 +β 2 (γ 2 +α 2 )y 2 +γ 2 (α 2 +β 2 )z 2] +α 2 β 2 γ 2 = 0, with the 3 parametersα := c/ ε 1, β := c/ ε 2, γ := c/ ε 3 and c = vacuum speed of light [Schaefer (1932), image by Jaumann]. The upper half depicts the exterior shell with the funnel shaped singularities, the lower half the inner shell. The two optical axes are denoted by I and II. The two shells cross each other at four points forming cusps. The wave vectors are denoted by N and N. generalize the Fresnel surface to more general ε s and µ 1 s. 8

9 2.2 Magnetoelectricity In the 1960s, substances were found that, if exposed to a magnetic field B a, were electrically polarizedd a = α a bb b and, reciprocally, if exposed to an electric field E a, were magnetized,h a = β a b E b. These are small effects of the order 10 3 ε 0/µ 0, or smaller. Constitutive law in a 4-dimensional covariant and premetric way: Collect the fields D a and H a in the 4d electromagnetic excitation tensor density H ij (= H ji ) and the fields E a and B a in the electromagnetic fields strength tensor F ij (= F ji), with the coordinate indices i,j,k,... = 0,1,2,3. Maxwell s equations read jh ij = J j, [i F jk] = 0. Crystal responds locally and linearly. With the electromagnetic response tensor densityχ ijkl (x), we have: H ij = 1 2 χijkl F kl, where χ ijkl = χ jikl, χ ijkl = χ ijlk. Irreducible decomposition under GL(4, R): χ ijkl = (1) χ ijkl + (2) χ ijkl + (3) χ ijkl, 36 = Axion part (3) χ ijkl := χ [ijkl] = αǫ ijkl ; skewon part (2) χ ijkl := 1 2 (χijkl χ klij ); principal part (1) χ ijkl fulfills (1) χ ijkl = (1) χ klij and (1) χ [ijkl] = 0. 9

10 2.3 Wave propagation, Tamm-Rubilar (TR) tensor densityg ijkl [χ] 4d wave covector q i = (ω, q), with the frequencyω and the 3d wave vector q. Generalized Fresnel equation (2003), which is quartic in the wave covector: G ijkl [χ]q iq jq k q l = 0. G is totally symmetric, G ijkl [χ] = G (ijkl) [χ], and has 35 ind. comps.: G ijkl [χ] := 1 4! ǫmnpq ǫrstuχmnr(i χ j ps k χ l)qtu, cubic in χ. G[χ] depends only on (1) χ and (2) χ; (3) χ drops out. Itin (2009) developed a new and generally covariant method for deriving the generalized Fresnel equation ( dispersion relation ). Looks different from the expression above. Obukhov (2009) showed equivalence. Probably the most symmetric form... (Itin) is G ijkl [χ] = 1 3! χa(ij b χ acbdχ c kl)d. Left-dual: χ ij kl := 1 2 ǫijmnχmnkl, right-dualχ ij kl := 1 2 χijmn ǫ klmn. Was derived directly by Lindell (2005) by using a dyadic calculus, see Favaro (2012). The Kummer tensor densityk ijkl [χ] is what we find from the last expression for G by dropping the factor 1/3! and the symmetrization parentheses (......). 10

11 2.4 Kummer surface and optics In 1864, Kummer discovered a family of quartic surfaces that would later play a major role in the development of algebraic geometry. Kummer surfaces belong to the 3d real projective space RP 3, but can be derived from hypersurfaces in the 4d space R 4 {0}, by taking a section. This is usually regarded as a change of coordinates, from the homogeneous(t, x) to the inhomogeneous(1, x/t). For a dispersionless linear medium with zero skewon part, the Fresnel surface coincides with the general Kummer surface. Kummer s original motivation came from optics. He examined ray bundles of the second order, that is, ray bundles that have at one point two independent rays, like in a birefringent crystal. Given that rays in 4d space are projected in a 3d space, the tool for this analysis was projective geometry. We now consider the equality, for media with zero skewon piece, of Fresnel and Kummer surfaces. For electromagnetic waves, E a and B a are orthogonal. Similarly for H a and D a. Moreover, the electric and magnetic parts of the energy density are always equal. A premetric formulation is possible, ǫ ijkl F ijf kl = 0 E ab a = 0, ǫ ijkl H ij H kl = 0 H ad a = 0, H kl F kl = 0 E ad a = H ab a. These eqs. define geom. optics itself, see Hehl and Obukhov (2003).

12 A modern image of a Kummer surface, due to Rocchini and to be compared with the figure of Schaefer/Jaumann:

13 13 If I,J,M,N = 1,...,6 denote pairs of antisymmetric indices, we find ǫ IJ F IF J = 0, ǫ MNχ MI χ NJ F IF J = 0, χ IJ F IF J = 0. If the medium has a vanishing skewon part, these eqs. have a meaning in projective geometry. Eventually, we recognize that the Kummer surface emerged during studies of the light propagation in local and linear media. Thus, light stood at the cradle of the Kummer surface. The Fresnel surface of an electromagnetic response tensor densityχ ijkl, whose skewon part vanishes, χ ijkl χ klij = 0, is a Kummer surface. Moreover, every Kummer surface appears as the Fresnel surface of a generic electromagnetic response tensor density, whose skewon part vanishes.

14 2.5 Premetric Kummer tensor densityk ijkl [T ] newly defined, irreducible decomposition Doubly antisymmetric 4th rank tensor density T ijkl with T (ij)kl = 0 and T ij(kl) = 0. Irreducible decomposition of T ijkl under GL(4,R): T ijkl = (1) T ijkl + (2) T ijkl + (3) T ijkl. 36 = Motivated by the Tamm-Rubilar tensor density G ijkl [χ], we want to define a tensor density, which is cubic in the fourth rank tensor density T. With the diamond dual we define premetrically T ij kl = 1 2 T ijcd ǫ cdkl, K ijkl [T ] := T aibj T acbdt ckdl, with K ijkl [T ] = K klij [T ]. The irreducible decomposition turns out to be, K ijkl = (1) K ijkl + (2) K ijkl + (3) K ijkl + (4) K ijkl + (5) K ijkl + (6) K ijkl, 136 = 35 }{{} TR }{{} up to linear combs

15 2.6 Kummer tensor in GR A Kummer tensor belonging to the Riemann curvature tensor R ijkl of general relativity was defined by Zund (1969) explicitly and earlier by Ruse (1944) implicitly. If a Riemannian metric with Lorentz signature is given, we can define the scalar density of weight +1 as g, with g := detg rs. Then, T ijkl = ( g) 1 T ijkl, K ijkl = ( g) 1 K ijkl, η ijkl := gǫ ijkl, and the dual := g is built with this unit tensor. In GR, T ijkl is identified with the Riemann curvature tensor R ijkl. Algebraic symmetries of Riemann: R (ij)kl = 0, Rij(kl) = 0; Rijkl = R klij ; R[ijkl] = 0. Thus, R has 20 independent components. Then, the Kummer of Riemann reads (Zund): K ijkl [ R] = R aibj R acbd Rckdl. Now we have, besides the conventional pairwise symmetry K ijkl [ R] = K klij [ R] also K ijkl [ R] = K jilk [ R]. In vacuum gravity, only K ijkl [ C] will be relevant, with C ijkl as the Weyl tensor with its 10 independent components. We find for K ijkl [ C]: K ijkl = K klij, K ijkl = K jilk, K [ij](kl) = 0, K (kl)[ij] = 0. 15

16 2.7 Kummer-Weyl tensor in the Kerr solution 16 Some unpublished results: Coframe of Kerr in Boyer Lindquist coordinates(t, r, θ, φ), reads ϑ 0 = Σ (dt+asin2 θdφ), Σ ϑ 1 = dr, ϑ2 = Σdθ, ϑ 3 = sinθ [ adt+(r 2 +a 2 )dφ ]. Σ The structure functions are defined according to Σ = Σ(r,θ) := r 2 +a 2 cos 2 θ = (r) := r 2 2Mr +a 2. The coframe is orthonormal. Thus, the metric reads g = ϑ 0 ϑ 0 +ϑ 1 ϑ 1 +ϑ 2 ϑ 2 +ϑ 3 ϑ 3 or, in terms of local coordinates, ( ) g = dt 2 dr 2 +Σ +dθ2 + 2Mr ( dt asin 2 θdφ ) 2 +(r 2 +a 2 )sin 2 θdφ 2. Σ

17 For the Kummer scalar K[ C] := (1) K µ λ µλ [ C], we find K[ C] = 24M3 r (r 8 36r 6 a 2 cos 2 θ +126r 4 a 4 cos 4 θ Σ 9 84r 2 a 6 cos 6 θ +9a 8 cos 8 θ) = 24M3 r (r 2 3a 2 cos 2 θ) Σ 9 (r 6 33r 4 a 2 cos 2 θ +27r 2 a 4 cos 4 θ 3a 6 cos 6 θ), the Kummer axial scalar (or pseudo-scalar), turs out to be A[ C] := η αβγδ (6) K αβγδ [ C], A[ C] = 24M3 acosθ (9r 8 84r 6 a 2 cos 2 θ +126r 4 a 4 cos 4 θ Σ 9 36r 2 a 6 cos 6 θ +a 8 cos 8 θ) = 24M3 acosθ (3r 2 a 2 cos 2 θ) Σ 9 (3r 6 27r 4 a 2 cos 2 θ +33r 2 a 4 cos 4 θ a 6 cos 6 θ). The totally symmetric Kummer (35 independent components) S αβγδ [ C] := (1) K αβγδ [ C] = K (αβγδ) [ C] is, in a sense, complemetary to the Kummer axial scalar. It is related to the pricipal null vectors at this point. It has 35 independent components and can be represented as a symmetric matrix AB BA

18 K(r, θ) A(r, θ) Not unexpectedly, the axial scalar A[ C] is proportional to the Kerr parameter a, that is, to the specific angular momentum. The relation between the principal null vectors of the Kummer tensor (which is cubic in Riemann) and of the Riemann curvarure tensor will be in the center of our future interest. 18