A Faradayan Principle for Selecting Classical Field Theories

Abstract

Faraday’s field concept presupposes that field stresses should share the axial symmetry of the lines of force. In the present article, the field dynamics is similarly required to depend only on field properties that can be tested through the motion of test‐particles. Precise expressions of this ‘Faradayan’ principle in field‐theoretical language are shown to severely restrict the form of classical field theories. In particular, static forces must obey the inverse square law in a linear approximation. Within a Minkowskian and Lagrangian framework, the Faradayan principle automatically leads to Maxwell’s theory of electromagnetism and to Einstein’s theory of gravitation, without appeal to the equivalence principle. A comparison is drawn between this, Feynman’s, and Einstein’s way to arrive at general relativity.

Acknowledgements

I am grateful to James McAllister and to two anonymous reviewers for their useful comments on an earlier version of this paper. I am of course responsible for any remaining flaws.

Notes

[1] On Faraday’s views, cf Gooding (1978); Darrigol (2000, chs. 1 and 3). The stress tensor of a single‐constant elastic body is σ_ij = K(∂ _i u_j + ∂ _j u_i + δ_ij ∂ _k u_k ), where u(r) denotes the elastic deformation.

[2] The test particles must have the symmetry of a geometrical point (isotropy) in order to exclude dipoles and higher multipoles.

[3] An interesting approach, proposed by Robert Wald (1986), consists in deriving Einstein’s free‐field equations as the only non‐linear extension of the Pauli‐Fierz equations of a spin‐two field that meets a requirement of perturbative consistency. This approach shares with the present one the advantage of not relying on the equivalence principle. However, its formal implementation is difficult; cf Straumann (2000, 17–19).

[4] The mere axial symmetry of the stress tensor would allow for Yukawa forces. The latter are excluded by further requiring that the stress tensor should be built from X.

[5] The divergent self‐interaction terms in the equations of motions disappear in the limit of a continuous dust.

[6] Evidently, this form of the matter‐dependent part of the action is not a realistic representation of macroscopic matter in general. It is nonetheless sufficient for our purpose, which is the analysis of the consequences of ideal measurability through test particles that interact only with the investigated field.

[7] Another possible form of the mass term is It will be used in the scalar case, in which it is slightly more convenient. The total action in the one‐particle case of course leads to a divergent self‐interaction. In order to avoid this divergence, one must think of a continuous dust in which each particle of the dust is only subjected to the (field‐mediated) action of the other particles.

[8] The limits of integration τ ₁ and τ ₂ must be infinitely large in order to include the whole trajectory of the particle.

[9] Although more complex forms of this action could be imagined, they would not be relativistic generalization of the interaction term in Equation 14 for the quasi‐static action.

[10] Adding a constant Λ to the curvature R in Equation 35 would lead to the cosmological term g_µν Λ on the left‐hand side of Einstein’s equation. With this modification, Newton’s theory no longer is the weak‐field, low‐velocity limit of Einstein’s theory.

[11] Feynman (1995, 66–69) gives a more rigorous argument of this kind. The reasoning is somewhat analogous to the way in which H. A. Lorentz derived the contraction of length from the Lorentz invariance of the Maxwell‐Lorentz equations before relativity theory. Norbert Straumann (2000, 24–25) astutely relies on the computable behaviour of a hydrogen atom.

[12] On Nordström’s theory and its role in the genesis of general relativity, cf Pais (1982, 232–237).

[13] Cf Landau and Lifchitz (1951, para. 94). This procedure is rooted in earlier efforts by Klein, Hilbert, and Einstein to derive the equations of gravitation from a variational principle.

[14] Suraj N. Gupta (1954) inaugurated this procedure. For the derivation of D ⁽²⁾, cf Feynman (1995, 43–4). The factor which equals one for the Minkowskian metric η, is only here for the sake of a forthcoming argument. A more usual form of the Pauli‐Fierz Lagrangian .

[15] Through a clever modification of the original action, Stanley Deser (1970) managed to obtain Einstein’s equations in one step of this kind only; cf Enrique Alvarez (1989, 562–564).

[16] The Condition i corresponds to the Bianchi identity of general relativity.

[17] For a proof of this point, cf Landau and Lifchitz (1951, 290–291).

[18] A more complex reasoning of this sort (without the benefit of the Weyl formula) is found in Feynman (1995, 82–87).

[19] On Einstein’s path toward general relativity, cf Pais (1982); Norton 1989a,1989b; Stachel 1989a, 1989b; Renn and Sauer 2003.

Additional information

Notes on contributors

Olivier Darrigol

Olivier Darrigol is at CNRS: Rehseis.