Solving the Einstein Equation

The Einstein equation encapsulates the nature of a spacetime. If we can solve this equation for a particular system we have at our disposal a full understanding of its initial state, as well as how it will evolve throughout all future moments. We understand the forms of the contained matter, where they are located and how they will move, interact and change over time. However, solving this equation is no easy feat and it can only be done exactly for simple systems, after which we require clever approximations.

The Einstein equation

Let me first clarify what I mean by a particular system. I make an extreme understatement when I say the universe is complicated. We can acquire some mental sanity by breaking it down and grouping its components into different systems. For example, you sitting in your living room reading this, could be treated as a system and a mathematical model could be designed to measure the evolution of this system. The model could measure effects such as the fluctuating temperature due to the incoming and outgoing heat flows, or the change in mass as more objects enter or leave the room. We can have much bigger systems, such as the solar system with models that track the dynamics of the planets around the sun. Or, we could have a system describing two black holes about to collide in a far corner of the universe. We state from the off-set what is included in our system and assume influence from no outside objects within our model, mathematically this is done through the inclusion of constraints and boundary conditions. 

The Einstein equation is most often used to describe the evolution of relativistic systems on the cosmological scale. Think black holes, exploding stars and the big bang. The deceptively simple looking equation is actually comprised of ten distinct highly non-linear partial differential equations which describe the behaviour of the system. To write each of the ten equations out in full, in their most general form without any grouping of the terms, would literally take hundreds of pages. Solving it, is therefore an incredibly daunting task.

Beginning to expand the Einstein tensor term into its constituents.

To be able to find exact solutions to this equation is extremely rare and only occurs for a handful of simple systems in which a large number of these terms fall away to zero. For example, if we are looking at a system with an isolated body of mass sitting in an otherwise empty space-time, the term T in the above equation (the Stress-Energy tensor) disappears, greatly simplifying the equations with now nothing on the right hand side. You may wonder how could this could represent any interesting physical situation in nature, but remember our thinking in systems. An isolated black hole, far away from any other matter can be well represented by this scheme due to the negligible influence from far away bodies and thus their exclusion from the system. Symmetry in a system also provides a big helping hand, if a system has spatial (or far less common temporal) symmetry this will also make the equations much more tractable due to a large cancellation of terms. Think back to the living room example if this isn’t immediately obvious, if the left half of my living room is identical in every way to the right half, my model needs only do half the work to represent the system. 

There then exist systems which are not exact solutions to the Einstein equation, but to which we can find approximate solutions due to certain simplifications. If the gravitational field is weak or the speed of the bodies in the system is substantially slower than the speed of the light, a number of the terms in the equations become very small and can essentially be ignored. Such approximations allow us, to a high a degree of accuracy, to model the dynamics of planets, certain binary neutron stars and particular emissions of gravitational waves. However, the exact and approximately solvable cases only represent a fraction of the systems we’d like to model in the universe and unfortunately, as well as obviously, the most interesting and physically realistic cases are the most complex. 

To examine gravitational waves from colliding black holes or supernovae, to model relativistic phenomena such as active galactic nuclei or to follow spacetime singularities, we work in the regime of strong gravity and require the ability to solve the Einstein equation in its full, almighty form. The Einstein equation must be solved everywhere in the system, tracking the matter at each point, how fast it is moving, the pressures and stresses and the resulting warping of the surrounding spacetime. The changes at one point in the system then affect the spacetime geometry at every other point. All the terms laid out in the hundreds of pages must be inputted into the equation to ensure accurate results. Thankfully for the human brains of all relativists, we now have the technology to pass this less than enviable job onto computers. This is the field of Numerical Relativity, the ability to provide approximate solutions to the Einstein equation for systems using (as the name suggests) numerical methods.

Simulation of merging black holes radiating gravitational waves. NASA/Ames Research Center/C. Henze

Researchers have developed methods to computationally evolve systems by inputting the data describing the system at an initial moment and then using finite differencing numerical methods to evolve the state forward step by step in time. However, numerical discretisation is a double edged sword. The smaller the taken steps the smaller the introduced error and the smaller the deviation from the true solution. Small steps however come at the cost of runtime. If the problem is discretised up into smaller chunks, there obviously exists a larger number of them for the computer to process, taking a longer time to spit out a complete simulation. A main job of the numerical relativist is to ensure the inputted data describing the system at an initial time was well-posed. This meaning, it will not lead to numerical instabilities when the computer evolves it and will ultimately present an accurate approximation for the system’s behaviour over time. 

This problem is known as the initial value problem and the nature of the theory of general relativity, to which Einstein’s equation belongs, provides a high entry barrier for acceptable initial data due to the theory being gauge invariant. What this means is that the model that describes the behaviour of your system must be independent of the coordinates you choose to do the modelling! Coordinates are after all just a mathematical choice to make certain aspects of your calculation easier, but more on this subtlety another time. This gauge invariance means the data must be subject to certain  mathematical constraints and boundary conditions which ensures criticial information about the physical situation trying to be modelling is appropriately included. Takeaway message, although we give the brute work to computers, we must still work extremely hard to formulate what we feed in before we can kick back and let the algorithm chug on. I hope to do a more detailed post on the formulation of Numerical Relativity soon.

For a beautiful visualisation of a numerical simulation of two merging black-holes, with asymmetric masses and the extra complications of orbital precession (GW190412) see the following video from the Albert Einstein Institute.

Such intricate behaviour, all ultimately encapsulated in the equation given at the top of this post.

Feature Photo Credit: NASA/VICTOR TANGERMANN

2 responses to “Solving the Einstein Equation

  1. Great stuff, that I don’t really understand. My maths is a bit old now. But just remember that any model is just an approximation of reality, Godel told us that. And this only skates the surface of things, not the interiors. Humbling thought, that.

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