(Super)-Energy for Arbitrary Fields and its Interchange: Conserved Quantities∗

Inspired by classical work of Bel and Robinson, a natural purely algebraic construction of super-energy (s-e) tensors for arbitrary fields is presented, having good mathematical and physical properties. Remarkably, there appear quantities with mathematical characteristics of energy densities satisfying the dominant property, which provides s-e estimates useful for global results and helpful in other matters. For physical fields, higher order (super)n-energy tensors involving the field and its derivatives arise. In Special Relativity, they provide infinitely many conserved quantities. The interchange of s-e between different fields is shown. The discontinuity propagation law in Einstein-Maxwell fields is related to s-e tensors, providing quantities conserved along null hypersurfaces. Finally, conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector. ∗This essay received an “honorable mention” in the 1999 Essay Competition of the Gravity Research Foundation. The importance of the Bel-Robinson and other super-energy (s-e) tensors [1, 2] is usually recognized, even though their physical interpretation remains somewhat obscure. Their mathematical usefulness is clear, a manifestation of which is their relevance in the proof of the global stability of Minkowski spacetime [3], or in the study of well-posed symmetric hyperbolic systems of differential equations including gravity (see, e.g. [4]). In this paper, I show how to generalize these important mathematical properties to arbitrary fields and, more importantly, try to shed some light into the physical meaning of s-e by finding non-trivial conservation laws which involve two different fields. First, let us present the procedure to construct the s-e of any given tensor [5]. Consider any tensor tμ1...μm as an r-fold (n1, . . . , nr)-form (n1 + . . . + nr = m) by separating the m indices into r blocks, each containing nA (A = 1, . . . , r) completely antisymmetric indices. This can always be done. Several examples are: Fμν = F[μν] is a simple (2)-form, while ∇ρFμν is a double (1,2)-form; the Riemann tensor Rαβλμ is a double symmetrical (2,2)-form (pairs can be interchanged); the Ricci tensor Rμν is a double symmetrical (1,1)-form, etcetera. Schematically, tμ1...μm will be denoted by t[n1],...,[nr] where [nA] indicates the A-th block with nA antisymmetrical indices. Duals can be defined by contracting each of the blocks with the volume element 4-form, obtaining the tensors (obvious notation, signature –,+,+,+): t ∗ [4−n1],...,[nr] , . . . , t [n1],..., ∗ [4−nr] , t ∗ [4−n1], ∗ [4−n2],...,[nr] , . . . , t ∗ [4−n1],..., ∗ [4−nr] . There are 1 + ( r 1 ) + . . .+ ( r r ) = 2r tensors in this set (including t[n1],...,[nr]). Let us define the “semi-square” (t[n1],...,[nr]×t[n1],...,[nr]) by contracting all indices but one of each block in the product of t with itself (after reordering indices if necessary) (t× t)λ1μ1...λrμr ≡ ( r ∏ A=1 1 (nA − 1)! ) tλ1ρ2...ρn1 ,...,λrσ2...σnr t ρ2...ρn1 , σ2...σnr μ1 ...,μr . The s-e tensor of t is 2r-covariant and defined as half of the sum of the 2r semisquares constructed with t[n1],...,[nr] and all its duals. Explicitly: 2Tλ1μ1...λrμr {t} ≡ ( t[n1],...,[nr] × t[n1],...,[nr] ) λ1μ1...λrμr + . . . . . . + ( t ∗ [4−n1],..., ∗ [4−nr] × t ∗ [4−n1],..., ∗ [4−nr] )