Luke Butcher

Theoretical Physicist and Sometimes Playwright

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Gravitational Waves

There is no question that gravitational waves carry energy, but it has been notoriously difficult to describe where in spacetime this energy resides. This problem was so widely believed to be unsolvable that the prevailing wisdom (expressed most famously in the textbook Gravitation) held that trying to localise gravitational energy was "looking for the right answer to the wrong question". During my PhD I sought a right question to ask, and was lead to a remarkable new description of the energy, momentum and angular-momentum carried by weak gravitational fields: \begin{align}\kappa \bar{\tau}_{\mu\nu}&= \tfrac{1}{4} \partial_{\mu}h_{\alpha\beta} \partial_{\mu}\bar{h}^{\alpha\beta},\\ \kappa s^\alpha_{\phantom{\alpha}\mu\nu} &= 2 \bar{h}_{\beta[\nu}\partial^{[\alpha} \bar{h}_{\mu]}{}^{\beta]}.\end{align} In the above formulae, \(h_{\mu\nu}\equiv g_{\mu\nu}- \eta_{\mu\nu}\) is the gravitational field, and \(\tau_{\mu\nu}\) and \(s^\alpha{}_{\mu\nu}\) are tensors that quantify the energy, momentum, and spin of \(h_{\mu\nu}\) at every point in space.

These tensors fulfil all their expected roles: (i) they account for the energy, momentum and angular-momentum exchanged between gravity and matter, (ii) they can be derived by symmetry arguments, and (iii) they describe gravitational self-interaction, so that gravitational energy curves space just like the energy of matter. Moreover, \(\tau_{\mu\nu}\) and \(s^\alpha{}_{\mu\nu}\) display a variety of desirable properties that have not been seen in previous attempts to localise gravitational energy. In particular, \(\tau_{\mu\nu}\) is locally positive and causal (for transverse-traceless \(h_{\mu\nu}\)) meaning that the gravitational energy-density is never negative, and never flows faster than light. Crucially, these properties also lead to a program that naturally removes the gauge ambiguity from the description. Interested readers should take a look at the introductory paper; this was followed by a treatment of spin, and further theoretical developments. This work has also been used to produce some hypnotic animations of gravitational energy flowing away from compact sources.


Our present understanding of gravity allows spacetime to be curved into strange shapes, at least in principle. One particularly interesting prospect is for space to be warped into a tube-like shortcut called a wormhole, which could permit faster-than-light communications and time-travel. Unfortunately, wormholes require exotic matter with negative energy to survive — without it, the throat will collapse before anything can cross from one side to the other.

I have been examining whether the negative energy needed by the wormhole can in fact be generated by the wormhole itself. Can the curvature of a wormhole cause a quantum field to occupy a state with negative energy, and will this energy be enough to keep the wormhole open? wormhole My first paper on this subject focussed on the Casimir energy \(\rho^\mathrm{Casimir}\) produced by wormholes which are very long and thin (\(L\gg a\)). Although this negative energy was not enough to completely stabilise the wormhole, preliminary calculations suggested that it may keep the wormhole open just long enough for a small amount of light to squeeze through. This work received some coverage in the popular press. For a slightly more detailed explanation of this research, see my summary for a general audience. A more technical overview is available in the form of a poster.

It's not all good news for science-fiction fans, however. My latest paper proves that one particular idea for a stable wormhole (supported by a non-minimally coupled classical scalar field) cannot work.