Resistive Plasmas

All models are wrong, but some are useful.

We're interested in numerical simulations of neutron stars, for predicting observable signals (for example from gravitational waves, or in X-ray) from the inspiral and merger. This quote from George Box is the core of the steps we take. It's impossible to model fundamental particle interactions when your grid cells are tens to hundreds of metres across, so we approximate. The question we tackle in these three papers is how much our choice of approximation matters, and how we can make it better.

The standard approximation is ideal (magneto)hydrodynamics: the Living Reviews of Martí and Müller or Font show just how far this approximation can take us. Here all the particles move together, without viscosity or resistance, and the magnetic field is locked to the fluid. As we need some relative flow between the charged particles to generate the magnetic field in the first place, it's clear that some substantial simplification has gone on here: what's not clear is how much it matters.

There's two ways to build a more "realistic" model: add in the extra physics by hand, or build a complex but consistent model from first principles. As relativity has some really nice features, such as energy conservation and causality, that we don't want to lose, we took the second route.

We started from a variational approach to relativistic fluids. While they've been around a long time, variational approaches that include multiple fluids are newer, and building a model that includes resistance and dissipation directly from the variation is very recent. By including charge we can build a model that includes multiple interacting fluids that interact via scattering, with particle creation and resistance between their relative flows.

There are three papers describing the work so far. In the first we develop the formalism, adding charges into the action and derive the equations of motion. In principle this tells us how to compute, ab initio, the resistance between the flows and the particle creation rates. In reality this isn't possible: what we could imagine measuring isn't the information we need. Instead, we use these results to derive the essential constraints that must apply to consistent phenomenological models.

The second paper goes deeper into building phenomenological models. Using a "1+3" or fibration approach (as in cosmology, for example) it builds a sequence of ever more complex approximate models, with the aim of seeing how far we need to go to model a hot, magnetized neutron star. By the end we have models with heat flow and resistivity, including a consistent derivation of the thermo-electric effect, all in nonlinear GR.

The third paper takes the orthogonal approach, and builds a "3+1" or foliation model as used in nearly all numerical relativity simulations. By concentrating on slightly simpler models we derive consistent balance laws for hot resistive plasmas including resistivity, as needed for nonlinear simulations of neutron star mergers. However, there's still substantial work needed here, as we show that there are significant technical issues with including the entrainment effect, which is one of the most interesting things to study in numerical evolutions.

Work on these three papers has occupied a lot of time for Nils Andersson, Greg Comer, Kyriaki Dionysopoulou, and myself, over the past few years. It's a significant step forward in the frameworks needed for numerical relativity. However, there's a lot more work to do, particularly in actually getting numerical simulations from these equations. Kiki in particular has made great steps in this direction, and the first results should be public soon.