Maxwells Equations, Universality, and Relativity D. H. Sattinger - PDF document

Maxwell’s Equations, Universality, and Relativity D. H. Sattinger Department of Mathematics University of Arizona Notes of a plenary lecture at the 34 th Western States Mathematical Physics Conference, California Institute of Technology, February 15-16, 2016 Organizers Rupert Frank and Gang Zhou Courtesy: Master and Fellows, Trinity College, Cambridge dsattinger@math.arizona.edu http://math.arizona.edu/ ∼ dsattinger/ 1

Abstract The success of Einstein’s theory of General Relativity proves con- clusively that gravitation is a manifestation of the curvature of space- time. But in this talk we discuss a recent mathematical derivation of Maxwell’s equations showing that they are universal – depending on a parameter µ , they hold for all force fields, attractive or repulsive, generated by a material source such as charge or mass. This puts gravitation and electromagnetism on an equal footing on flat, Minkowski space-time. Maxwell’s equations for gravitation have long been held to be a formalism, irrelevant to gravitation. But we show that they are fundamental to the determination of the energy- momentum tensor in Einstein’s equations, and that General Relativity can be reformulated in terms of Maxwellian fields, rather than specific force fields. The proof of universality requires the conservation of material and Einstein’s two postulates of special relativity, and makes substantive use of Hodge theory. 1 Einstein’s Premise Einstein’s theory of general relativity introduced a new paradigm into physics. The dynamics of a charged particle in an electromagnetic field is governed by Newton’s second law of motion, the force given by the Lorentz force. But Einstein’s theory is geometric, and the dynamics of a particle are given by the geodesics of the metric tensor on a semi-Riemannian manifold, generated by the presence of mass. Nevertheless, in a recent paper [14], I proved that the electro- magnetic field equations are universal, in the sense that they con- stitute a general mathematical theory, apply to all fields generated by a material source, and are valid for both attractive and repulsive fields. Maxwell’s equations therefore put gravitation and electromag- netism on an equal footing on flat space-time; and the relationship of Maxwell’s linear field theory to Einstein’s nonlinear geometric theory must be explained. We shall show in this section that the two theories fit naturally together, but that their unification leads to a fundamen- tally new paradigm in the theory of relativity: namely the equations of both special and general relativity are universal, and are formulated in terms of Maxwellian fields, rather than specific force fields. 2

Prior attempts to build a relativistic theory of gravitation were based on an application of Maxwell’s equations. Maxwell himself mentioned the idea briefly in his original tract on electrodynamics, but abandoned it because of the implications of the negative energy of an attractive field. Nevertheless, efforts to do so were proposed by Heaviside (1893 [5]), Lorentz (1900 [9]), and Poincar´ e (1905, [12]). There was a good deal of debate concerning a relativistic treatment of gravitation based on Lorentz invariance in the years leading up to Einstein’s publication of his work in 1915. Those efforts were abandoned with the success of Einstein’s theory; and the notion that Maxwell’s equations might be relevant to gravita- tion has been strenuously rejected. (See Pais [11], Chapter 13, as well as the discussion in [13]). But we shall see that Maxwell’s equations for gravitation are essential in determining the energy-momentum tensor in the Einstein Field equations of general relativity. Special relativity is built on the equivalence of inertial frames [2]; while Einstein’s general theory of relativity is built on the Principle of Equivalence, which he introduced in 1907 [3]. In that paper he pro- posed that the laws of physics should be valid in all frames of reference, including non-inertial (accelerated) frames. He posited two identical clocks, one accelerating at a constant rate along the negative real axis, and one on the right, stationary, but in a constant gravitational field. If the laws of physics are the same for both observers, then the two clocks ought to keep the same time. He showed that the rate of the clock on the right is determined by the potential of the gravitational field. This ultimately becomes the coefficient g 44 in his metric tensor in his general theory of relativity, and is the basis for gravitational red-shift. Einstein showed that the assumption of general invariance un- der general coordinate transformations, together with the Principle of Equivalence, leads by purely mathematical considerations to a ge- ometric field theory, in which Newton’s law of motion is replaced by the geodesics on a semi-Riemannian manifold. But Einstein’s theory is incomplete, for the source term for the field equations is not determined by the Principle of Equivalence. Einstein introduced the notion of an energy-momentum tensor as the source, in analogy with the material source in Maxwell’s equations, and dis- cusses two physical examples – the energy-stress tensor of a frictionless adiabatic gas, and the Maxwell stress tensor for the electromagnetic field. Since Maxwell’s equations were seen as specific to electromag- 3

netism, while Einstein identified the geometry of space-time with the gravitational field, he did not use the Maxwell stress tensor in his equations. Einstein uses the word gravitation in his 1907 paper, but his argu- ment is entirely mathematical and easily extends to any conservative force field. Why then should the Principle of Equivalence apply to the gravitational but not the electromagnetic field? And why should the proposition that the laws of physics be the same in any coordi- nate system apply only to gravity and not to electromagnetism? He says nothing about this, and builds the theory of general relativity on the premise that “gravitation occupies an exceptional position with regard to other forces, particularly the electromagnetic forces . . . ” That premise is contradicted by the proof of universality, which we shall discuss in § 2. The natural mathematical language for general relativity is tensor analysis; and Einstein gives a succinct exposition of this machinery in [4]. Tensor analysis is the natural language of differential geome- try, and Hilbert’s approach to general relativity is grounded in that discipline. The notion of intrinsic geometry is fundamental to the sub- ject. Natural objects are ones which do not depend on the choice of coordinates. Gauss’ Theorem Egregium , for example, states that the Gaussian curvature on a 2-dimensional surface embedded in R 3 de- pends only on the metric tensor, not on the embedding of the surface, and not on the coordinate system. Hilbert [7] obtains the field equations of general relativity as an action principle, δ H = 0, H = K + L , where K √ g d Ω , L √ g d Ω . �� �� K = L = Here K is the Riemannian curvature invariant, better known as the Ricci scalar curvature R ; L is the Lagrangian of a physical system with coordinates q i and their derivatives q i,j , invariant under the group of diffeomorphisms of space-time; and √ g d Ω, is the invariant volume element, where g = det || g ij || and d Ω = dx 1 dx 2 dx 3 dx 4 . Hilbert calculates all the invariants and finds that the simplest case is that in which L is the Lagrangian for Maxwell’s equations, where q i = A i , the Maxwell 4-potential. We write L = F ij F ij (+ µA i J i ) , F ij = g ik g jl F kl . F ij = A j,i − A i,j , 4