SymplecticEinstein
A new perspective on Einstein’s field equations
The symplectic geometry of anti-selfdual Einstein metrics Project Objectives
moving through a 3-dimensional space, but rather a 6-dimensional world. This means 3 dimensions telling you the position of the ball, and 3 dimensions telling you the momentum of the ball. This 6-dimensional space has something called a symplectic structure, which weaves together position and momentum in an intricate way,” he outlines. The specific 6-manifold which is produced from the 4-manifold is called a symplectic Fano. “The Riemannian metric and the symplectic structure both come from a geometric object called a definite connection,” continues Dr Fine.
Riemannian geometry and Einstein’s field equations are both extremely important in the study of pure mathematics. We spoke to Dr Joel Fine, the Principal Investigator of the Symplectic Einstein project, about his work in reformulating Einstein’s equations in dimension 4, work which is opening up new avenues of investigation. The Einstein field equations remain a source of great interest in both physics and pure mathematics over a century after they were published. As the Principal Investigator of the Symplectic Einstein project, Dr Joel Fine is working to develop a new formulation of Einstein’s equations in dimension 4. “There are two distinct, ultimately related, problems in this work. One is, can we find a fourdimensional solution to Einstein’s equations in Riemannian geometry? Then, if we find such a solution, what can it tell us about the geometric space in which it lives?” he outlines. In his research, Dr Fine uses gauge theory to derive metrics, a form of distance function, with the aim of building a deeper understanding of abstract geometric spaces. “Einstein’s equations are extremely useful in helping us to understand geometry, which is my main motivation for studying them. I’m taking ideas from physics and using them to solve problems in mathematics.” he explains. Manifolds A number of these problems centre on the properties of geometric structures called manifolds, such as the surface of a ball, which is an example of a 2-dimensional manifold. If we take a finite 2-dimensional surface, with two different possible directions of movement and on which any loop can be shrunk without
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causing damage, then it can be proved to be the surface of a sphere. “The way to prove it is to use the information to construct a metric that is completely round. This metric is positively curved to the same extent in every single direction, and the only surface that has such a property is a 2-dimensional sphere,” says Dr Fine. The same statement can be made about a 3-dimensional manifold on a sphere, following Perelman’s proof of Thurston’s geometerization
in 4 dimensions. “The more information we can get the better. I want to find hypotheses under which you can find solutions to Einstein’s equations, and to find situations in which you can say something about those manifolds once you know that you’ve solved Einstein’s equations,” he outlines. There is currently no kind of roadmap or guide for this work. “We don’t have a 4-dimensional analogue of Thurston’s geometerization conjecture. We don’t have a candidate for
Einstein’s equations are extremely useful in helping us to understand geometry, which is my main motivation for studying them. I’m taking ideas from physics and using them to solve problems in mathematics. conjecture. “Thurston conjectured – and then Perelman proved – that you can break any 3-dimensional manifold into separate pieces, and each of these pieces houses one of these special geometries,” explains Dr Fine. The focus of Dr Fine’s attention in the project is now on 4-dimensional manifolds, where classifying the solutions to Einstein’s equations is a complex task. While it has been demonstrated that Einstein’s equations can be solved in 3 dimensions given the right hypothesis, Dr Fine says this is not the case
the different sorts of pieces that you might be able to break a 4-dimensional manifold into,” explains Dr Fine. A class of 4-manifolds with a certain type of geometry that produces a Riemannian metric on the 4-manifold has been identified. Additionally, these 4-manifolds also produce a type of symplectic 6-manifold, which is another important dimension of Dr Fine’s research. “We should perhaps think about physics as taking place not in a 3-dimensional world, for example with a ball
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Definite connection This definite connection can be changed until the metric that it provides solves Einstein’s equations. The point can be illustrated by taking the example of a deflated football. “Imagine that you try to inflate the ball into a round sphere, so that you can recognise it. You do this with your definite connection – and you can change your definite connection until the metric that it gives you solves Einstein’s field equations,” explains Dr Fine. At the same time, this deformation will turn the original symplectic structure in 6 dimensions into one that can be more easily recognised. “It’s on a list which was produced by algebraic geometers around 30 years ago, it’s among the first Fano manifolds to be identified,” continues Dr Fine. “This would prove that the symplectic Fanos coming from this construction are never more exotic than those that the algebraic geometers already knew about. It’s a step towards classifying symplectic Fano manifolds.” A definite connection provides a Riemannian metric, which opens up new avenues in both maths and physics research, for example in developing a quantum field theory description of gravity. However, Dr Fine’s primary interest is in solving mathematical problems, such as developing an existence theory for Poincaré-Einstein metrics.
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“Poincaré-Einstein metrics are a very specific type of solution to Einstein’s equations. These spaces are not finite in extent, they extend out to infinity,” he explains. The further these spaces extend out towards infinity, the more closely they resemble hyperbolic geometry, which is negatively curved. “What you would normally think of as straight lines run away from each other very quickly. There’s an enormous amount of space out in infinity in these manifolds,” says Dr Fine. “These types of metric are a very important family of solutions to Einstein’s equations.” These Poincaré-Einstein spaces are again associated with 6-dimensional symplectic manifolds, this time of Calabi-Yau type, rather than Fano. By studying the symplectic geometry of this 6-dimensional manifold, Dr Fine aims to build a deeper understanding of the behaviour of the Einstein metrics. “I apply modern techniques in symplectic geometry, that tell us things about the symplectic manifold. Then I translate them back into new and interesting facts about the 4-dimensional manifold,” he outlines. The nature of this research means it is very difficult to predict the likely outcomes; however, Dr Fine says that the project has helped pave the way for further investigation. “With this kind of project it’s not necessarily about closing the door on a subject, the funding bodies would rather you open the door to lots of other researchers,” he says. “Some of the original goals in the project are still quite distant, but I’ve also found other things that were unexpected and very interesting.” A good example is Dr Fine’s achievement, together with his colleague Bruno Premoselli, in finding the first negatively curved solution to Einstein’s equations which is not just locally symmetric. “This was completely unexpected, and it has definitely opened up a new area of research, that people can try and understand these negatively curved Einstein metrics,” says Dr Fine.
My goal is to better understand 4-dimensional solutions to Einstein’s equations. I hope to do this by exploiting a new formulation of these equations which makes it possible to exploit ideas from different seemingly unrelated areas of geometry.
Project Funding
The main source of funding is the ERC grant (646649 SymplecticEinstein). Parts have also been funded by the FNRS MIS grant F.4522.15.
Contact Details
Project Coordinator, Dr Joel Fine Department of Mathematics, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Brussels, Belgium. T: +0032 (0) 2 650 58 42. E: joel.fine@ulb.ac.be W: http://homepages.ulb.ac.be/~joelfine W: http://geometry.ulb.ac.be
Dr Joel Fine
Dr Joel Fine is a member of the Differential Geometry Group at Université Libre de Bruxelles. He gained his degree at the University of Oxford, before going on to further study at the Imperial College London, where he worked with Fields Medallist Sir Simon Donaldson. He specialises in geometric analysis and the study of Einstein’s equations.
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