Einstein’s theory of general relativity describes our universe as a four-dimensional spacetime. Three spatial dimensions that we observe in our everyday life plus that of time. The fundamental overhaul in our understanding of nature, presented by Einstein, is that this spacetime, within which we live, curves and moulds in the presence of matter and energy. For the umpteenth time, imagine the classic rubber sheet analogy representing the fabric of spacetime, add a mass on top, representing a planet, star or black hole and the sheet will curve in its presence. Spacetime is reactive and spacetime is malleable.
Einstein’s understanding of this behaviour of space and time is cemented in his field equations, which represent the evolution of a spacetime given an initial set up of masses or energy. However the universe is complex and spacetime can be very messy – it can contain chaotic galaxies, multi-planetary systems, exploding stars and even colliding black holes. Einstein’s equations cannot be exactly solved for such detailed, interacting systems – so there exist but a finite collection of exact solutions to the field equations representing physical setups in nature that we can analytically work with. A collection of black hole’s is one such class of solutions – and the nature of these solutions will be our focus for today.
The Schwarzchild solution to the Einstein field equations represents a static (not rotating) black hole in the presence of nothing else. One black hole, alone in the spacetime – isolated and simple enough, giving an exact solution. The Kerr solutions to the field equations are the same setup except the black hole is spinning. A black hole can’t do much else really – in fact the only three ingredients that characterise a black hole is its mass, its rotation and whether it has any charge. This is summarised as the ‘No Hair Theorem‘ – all other information about a black hole disappears and for some reason according to John Wheeler, this is akin to it being bald… not so sure about his reasoning there but I digress, such lack of hair is a story for another time. Now a question at the forefront of mathematical physics is whether these black hole solutions are stable. Whether if we perturb the black hole a little bit, it will settle back to down a stable system which can be represented by the previous solution. Proving this would be proving the ‘Black Hole Stability Conjecture’. But what is meant here by perturb? Let me explain.
Imagine a still pond, undisturbed with flat water. Then imagine this pond is disturbed or perturbed by an external object, for example by throwing in a pebble. A perturbation is essentially an external influence on a system which causes a disturbance to its original state. Perturbations are created by the injection of energy in some form or another and as we know from Einstein, energy is akin to mass and mass causes curvature of spacetime. So if we throw a pebble into a pond, the water is disturbed and energy is propagated outwards in the form of ripples. Similar idea with a black hole, throw in some kind of object or add some form of energy and spacetime will ripple outward. Instead of water waves forming these ripples, in spacetime the ripples take the form of gravitational waves and these waves warp the surrounding spacetime just as the water waves warp the surrounding water. What mathematical physicists want to prove, is that regardless of the size of the perturbation, if we are dealing with a static or rotating black hole (provided it isn’t rotating too fast) then it will eventually settle back down into its original stable state.
Merger of two black holes – a highly perturbed system creating strong gravitational waves.
The key to this endeavour is to show that the gravitational waves produced in the perturbation all eventually fully decay – as they ripple outwards they become smaller and smaller until they finally have no effect on the surrounding spacetime. We need to measure the size of these waves to make sure they are getting smaller/decaying, so for this we need coordinates – as coordinates allow us to measure distances between points. These calculations can all be done but it’s not that easy… there is a catch. General relativity is diffeomorphism invariant. Diffeo-what I hear you say? Horrible phrase but really not that complicated at all – basically all it means is that it doesn’t matter what coordinate system you choose to use in your work. Think back to the GCSE maths days, different coordinates exist to better describe different physical systems. Euclidean x,y,z coordinates are often used when dealing with straight lines or 3D objects whilst spherical coordinates r, θ, ϕ are often used when dealing with curved lines or 3D objects. But the point is the world is does not have such coordinates naturally built in – they are mathematical constructs designed by humans to help us with whatever our calculation at hand is. However, in this case a particular choice of coordinates can obscure whether the gravitational waves are decaying. It wouldn’t be a wrong choice that you made when choosing the coordinates you did, but it would not have been the best possible choice given your “are black holes stable?” aims. Therefore, the problem is making the right coordinate choice in order to examine black hole stability, and making the best choice in mathematics is not obvious…
Mathematicians across the world are working on this proof in order to show black holes are stable solutions to the Einstein field equations and there is excitement in the community as some groups seem to be getting close. Proving the stability of black holes will give us a greater understanding of their behaviour and given some of the universe’s most specular phenomena occur around these astronomical beasts, the more we know about them with certainty, the better equipped we are to tackling some of nature’s biggest outstanding questions.