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For an excellent critical survey, including an exhaustive list of references, seeL. G. Suttorp andS. R. de Groot:Physica,39, 84 (1968).
L. Rosenfeld:Mem. Acad. Roy. Belg.,18, No. 6 (1940);F. J. Belinfante:Physica,7, 449 (1940).
The covariant derivative in the first term of equation (1) is to be taken by holding the tensorsp α,S λμ fixed by parallel propagation. ThusN ¦β =∂N/∂x β−(∂N/∂p μ)Γ μαβ p α−2(∂N/∂S λμ)Γ [μαβ S λ]α. Note also that the equations of motion cannot in general be cast in Hamiltonian form, and the extended phase volume (−g)1/2 d4 xdΩ is not conserved. Conventions: space-time signature (+ + + −), Riemann and Ricci tensors defined by 2A λ¦[μν]=A α R αλμν,R λμ=R αλμα, square brackets denote antisymmetrization.
W. G. Dixon:Nuovo Cimento,34, 317 (1964);Proc. Roy. Soc., A314, 499 (1970); A319, 509 (1970).
J. Madore:Ann. Inst. Henri Poincaré,11, 221 (1969);L. G. Suttorp andS. R. de Groot:Nuovo Cimento,65 A, 245 (1970);H. P. Künzle:Comm. Math. Phys.,27, 23 (1972).
Equations (5), (6) have to be supplemented by an equation determiningv β, which is in effect adefining equation for the world-line of the centre of mass. This is now generally taken to beS αβ p α=0, which, substituted into (5) and (6), yields an algebraic equation forv α.
The precise physical meaning ofT αβ(mat) is clear from the definition (7): in a local Lorentz frame, the componentT αβ(mat) is the flow of the α-component of material 4-momentum across a plane normal to the β-axis.
Ifm αβ=χS αβ, then the electric-dipole momentq β vanishes andH αβ is skew.
This electromagnetic-energy tensor is neither that ofAbraham orMinkowski, but is very close to the tensor introduced bySuttorp andde Groot (1). As these authors have stressed, the question: «What is the correct form of the energy tensor?» can be meaningfully applied only to thetotal tensorT αβ, since it is largely a matter of convention how one allocates the interaction energyF μα (H βμ −F βμ ) between matter and field. Cf. alsoChing-Wo Ng: B.Sc. (Hons.) thesis (University of Alberta) (1973).
For a special-relativistic derivation of (11b) see ref. (1). For the general-relativistic analysis seeW. Israel (paper in preparation).
H. Weyl:Phys. Rev.,77, 699 (1950);D. W. Sciama:Proc. Camb. Phil. Soc.,54, 72 (1958);T. W. B. Kibble:Journ. Math. Phys.,2, 212 (1961);A. Trautmann:Nature,242, 7 (1973).
C. G. van Weert:Proc. Kon. Ned. Akad. Wet. Amsterdam,73, 381 (1970).
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Work partially supported by a grant from the National Research Council of Canada.
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Israel, W. Electrodynamics, gravitation and spin. Lett. Nuovo Cimento 7, 860–864 (1973). https://doi.org/10.1007/BF02725404
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DOI: https://doi.org/10.1007/BF02725404