Side-stepping ‘futility’: Tim Palmer on energy-momentum in general relativity

Harvey Brown

University of Oxford, UK

I came to know Tim over a decade ago when he started to make an appearance at the Thursday seminars in philosophy of physics in Oxford and discuss with me and others his highly original and unorthodox views on the foundations of quantum mechanics. Tim’s ideas in this field have developed a lot over the years, as a number of sophisticated and provocative papers have shown. But his scientific interests outside of climate physics are not confined to quantum mechanics, as his fascinating book The Primacy of Doubt testifies. One topic the book does not cover, however, is Tim’s work on Einstein’s theory of gravity, or the general theory of relativity. Even here, Tim was to demonstrate his originality and technical skills, and the story takes us back to his DPhil studies in Oxford.

Tim studied for his doctorate in the 1970s under the supervision of Dennis Sciama, famous both for his wide-ranging expertise in theoretical physics, particularly cosmology, and for having supervised such luminaries as Stephen Hawking, Martin Rees and David Deutsch. After much effort, Tim decided that Sciama’s original suggestion to work on a problem in the physics of black holes was technically too knotty. Disappointed and somewhat anxious about where his thesis was going, he discovered the 1970 work of W. G. Dixon at Cambridge, who studied the equations of motion of extended bodies in general relativity by exploiting the formalism of bi-tensors originally introduced by Synge (1960) and DeWitt and Brehme (1960). It occurred to Tim that this formalism held the key to solving another long-standing conceptual problem in Einstein’s theory of gravity.

In the course of developing his theory in the period from 1905 – the year he published his special theory of relativity – to 1915, Einstein maintained the law of conservation of energy-momentum as a primary desideratum. But he realised that the energy-momentum tensor of matter fields in his final theory did not satisfy the continuity equations that its counterpart for electromagnetism did in the special theory: its covariant divergence vanished rather than its ordinary divergence. This meant it could not be integrated over space to provide an integral conservation law. This was not altogether surprising: matter interacts with the gravitation field in Einstein’s theory. But Einstein’s 1915 field equations did not yield an energy-momentum tensor for pure gravity built out of the metric field and its derivatives. Einstein realised that a conservation law could however be constructed – a vanishing (ordinary) divergence – involving the sum of the matter energy-momentum tensor and a ‘non-covariant’ object representing energy-momentum of gravity. This object is called the Einstein pseudo-tensor.

It did not take long for important figures in physics to criticise Einstein’s reasoning. Such luminaries as Hermann Weyl, Wolfgang Pauli and Erwin Schrödinger were alarmed by the fact that at any point in space-time, a coordinate system, or frame of reference, can be found relative to which Einstein’s pseudo-tensor vanishes, meaning that no objective status for the value of gravitational energy-momentum at that point can be given. In their celebrated 1973 textbook on general relativity, Charles Misner, Kip Thorne and John Wheeler described the ongoing searches for a local definition of gravitational energy-momentum as futile.

The ‘non-covariant’ nature of the Einstein pseudo-tensor is not its only awkward feature. Two others follow from the work of the Göttingen mathematicians David Hilbert, Felix Klein and especially Emma Noether who were following closely the development of Einstein’s thinking. The first feature is that when, for example, matter is absent, the pseudo-tensor is the sum of a term which vanishes ‘on-shell’, i.e. when the Einstein field equations hold, and a term in the mathematical form of a curl. It turned out that this latter term is far from unique. Indeed an infinity of such terms exist in principle, and a number of alternative proposals have appeared in the literature over the years, the most famous of which is probably the pseudo-tensor due to Lev Landau and Evgeny Lifschitz. Finally, when the (ordinary) divergence of the pseudo-tensor is considered, the only interesting term, given the Einstein field equations, is the divergence of said curl, and this vanishes identically! The Göttingen mathematicians asked: where then is the physics in this “improper” gravitational conservation law? The issue is subtle and still being debated today.

In a 1978 paper in Physical Review D, based on work he did in his DPhil thesis, Tim proposed a way around the “futility” described by Mister, Thorne and Wheeler. He was fully aware that in special cases, such as spacetimes with symmetries (giving rise to ‘Killing vectors defining directions in spacetime along which the metric is conserved) and asymptotically flat spacetimes, meaningful notions of gravitational energy-momentum can be constructed. But what Tim wanted was a conservation law that holds in generic spacetimes consistent with Einstein’s field equations, and to this end he exploited the formalism of bi-tensors.

Briefly, what Tim was able to demonstrate is that the Einstein’s metric – the fundamental dynamical object in general relativity – can be decomposed into a sum of two bi-tensors, i.e. tensors that are defined for pairs of spacetime points, not single points. When one of these points in fixed, in the case of one of these bi-tensors, its values at arbitrary points in the normal neighbourhood of the fixed point define a flat metric. As with Minkowski spacetime in special relativity, this flat metric has symmetries and hence Killing vectors – one might call these hidden symmetries in a generic spacetime. These in turn can be used to construct an expression for the sum of the matter energy-momentum tensor contracted with a Killing vector and a non-tensorial object representing gravitational energy-momentum whose definition is reminiscent of Einstein’s pseudo-tensor. The key difference is that Tim’s object, although dependent on the choice of an appropriate Killing vector, has a single index, and this allows for the vanishing covariant divergence of the mentioned sum to be integrated and produce an integral conservation law. A key property of this ingenious construction is that the conserved quantity is unique (using Synge’s ‘world-function’ to define the hidden flat metric). It also has the nice property that the radiative part of the total gravitational energy-momentum associated with a region of space vanishes when the spacetime has a Killing vector; in the case of a stationary gravitational field one does not expect radiation of gravitational waves.

Tim’s proposal not so much overcomes the futility posed by Mister, Thorne and Wheeler, as sidesteps it. This is because at any point in spacetime, although as in Einstein’s proposal there no objective value of gravitational energy-momentum, there is one for an appropriately defined neighbourhood of any point. Energy-momentum is ‘quasi-local’.

There is today a significant literature on quasi-local definitions of energy-momentum in general relativity, one of which is due to Sir Roger Penrose. But it seems Tim’s 1978 proposal was the first, and it represents a significant achievement for a doctoral student.

In his book "Primacy of Doubt", Tim mentions a chance meeting he had at the end of his doctoral studies with the meteorologist Raymond Hide which triggered an interest in climate physics. Tim goes on to describe the agony he went through for days trying to decide between taking up the offer of a post-doc with Stephen Hawking’s gravity theory group in Cambridge, and an offer to join the Met Office. There is no doubt that climate science’s great gain by Tim’s decision was Stephen Hawking’s and the relativity community’s loss. One might say it was a case of `gravity waves’ in atmospheric physics winning out over those in Einstein’s theory of general relativity.

Happy birthday, Tim.