research: DEFORMATIONS OF GENERAL RELATIVITY
Ever since Newton realized that the same force that pulled an apple to the ground also tethered the Moon in its orbit around the Earth, scientists have been fascinated with gravity. In 1916, Einstein revolutionised the way we look at space and time, by revealing that Gravity was Geometry. The decades since have brought us black holes and the Big Bang and myriad other wonders. In a few years, an upgraded gravitational wave observatory, Advanced LIGO, will open, and we will have a new window into the cosmos, peering far back in time. On theoretical fronts, some scientists pursue a quantum theory of gravity, whether in the form of String Theory or Loop Quantum Gravity or more exotic theories, while others, keeping in mind that Physics is ultimately about the real world and measurements, look for phenomenological theories of quantum gravity, which will help us identify when gravity starts to deviate from the predictions of our classical theory of General Relativity. It is an exciting time to be working in gravity, whether classical or quantum, and this is a world I would like to open up to my students. To give you an idea of where I myself work, in this vast frontier, I must first start with a little background.
General Relativity is famous for having a remarkably simple and short expression for its Lagrangian and action:
General Relativity is famous for having a remarkably simple and short expression for its Lagrangian and action:
This looks remarkably clean, but that is only because we have swept details under the rug. If we were to unpack the definition of R, we would find a mess of partial derivatives and Christoffel symbols. The reason we condense everything into a simple R is because it has a simple geometrical interpretation; R is the curvature of the space-time. Seen in this light, the equation above acquires a pleasing meaning-- we know from the action principle that Nature searches for the Least Action--or to be more precise, and extremum of the action S. Here that means that empty space-time tries to find an extremum of the curvature. But students who have taken a Classical Mechanics course know that besides the Lagrangian formulation, there is also the Hamiltonian formulation. Does the Hamiltonian of General Relativity have a similarly pleasing interpretation?
On the surface of it, it seems an unusual thing to look for the Hamiltonian formulation of General Relativity. After all, shortly after Einstein's formulation of Special Relativity, didn't Minkowski declare, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."? Doesn't the Hamiltonian formulation seem to privilege time, encouraging a divorce? Yet there is no denying that time and time ordering hold a special place in quantum mechanics and quantum field theory, and that questions about the Arrow of Time still abound. And perhaps more fundamentally, the Hamiltonian view is closer to how we see the world. We do not plan our paths and trajectories thinking - “I must extremize my action!”. We have initial conditions, and as time passes, we respond to the changing forces and factors in our environment. Maybe treating Time with a little extra attention will reward us with insights into these questions.
Unlike the Lagrangian formulation of General Relativity, where we deal with the entire space-time at once, in the Hamiltonian formulation, we start with an initial three dimensional slice of space, and see how it evolves in time. With this picture in mind, after several manipulations, we can write the Hamiltonian for General Relativity (GR) as
On the surface of it, it seems an unusual thing to look for the Hamiltonian formulation of General Relativity. After all, shortly after Einstein's formulation of Special Relativity, didn't Minkowski declare, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."? Doesn't the Hamiltonian formulation seem to privilege time, encouraging a divorce? Yet there is no denying that time and time ordering hold a special place in quantum mechanics and quantum field theory, and that questions about the Arrow of Time still abound. And perhaps more fundamentally, the Hamiltonian view is closer to how we see the world. We do not plan our paths and trajectories thinking - “I must extremize my action!”. We have initial conditions, and as time passes, we respond to the changing forces and factors in our environment. Maybe treating Time with a little extra attention will reward us with insights into these questions.
Unlike the Lagrangian formulation of General Relativity, where we deal with the entire space-time at once, in the Hamiltonian formulation, we start with an initial three dimensional slice of space, and see how it evolves in time. With this picture in mind, after several manipulations, we can write the Hamiltonian for General Relativity (GR) as
H and Di are known, somewhat confusingly, as the super-Hamiltonian and the super-momentum (or Diffeomorphism constraint), respectively. Their mathematical expressions are not particularly illuminating, but their physical actions are quite simply interpreted. Just like how in quantum mechanics, ordinary momentum generates spatial translations, or Energy generates time translations, Di generates spatial diffeomorphisms; it moves around points in the spatial slice, without moving off the slice itself. Similarly, H generates time evolution, pushing points on the spatial slice perpendicular to the slice, moving them forward in time. You can picture a horizontal sheet of paper, fluttering and spinning as it falls through the air. The horizontal motion of the paper is like the spatial diffeomorphisms; the fluttering and the falling, perpendicular to the paper, are like the time evolution. This is a nice geometrical picture of the Hamiltonian, but we get more; it turns out that H and D satisfy certain relations under Poisson brackets. Crucially, they are closed under the Poisson bracket; the bracket of two Ds is another D, the bracket of an H and a D is another H, and the bracket of two Hs is a D. You don't get anything new when you take the Poisson bracket of any two linear combinations of H and D.
Now we are at the core of the story. In the 1970s, Kuchar showed a remarkable result. Suppose all you knew was that the Hamiltonian of General Relativity was composed of two (unknown) objects H and D, and that they closed in that particular way under the Poisson bracket. With just that knowledge, under some very simple assumptions, you could reconstruct the full Hamiltonian of General Relativity. This was truly astonishing. It shone a new light on the idea that Gravity was Geometry; it said that if all you knew was the way a spatial slice moved through space-time, you were set; you could reconstruct General Relativity.
When I first learned of this result, I was electrified. It has long been suspected that one of the central assumptions of GR--that space-time can be described by a manifold--was wrong. In a manifold, you can always zoom in and distinguish features at smaller and smaller length scales. But what we know of Quantum Mechanics, Special Relativity and General Relativity suggests that this is not true. To resolve features on a small length scale, you need a particle with a small wavelength and high frequency--and high energy. However, if you cram too much energy into a small region, it forms a black hole--and now you are back where you started, since cannot find out what happens inside a black hole. Thus there is a a minimum length scale in space-time below which you cannot resolve any details. This means that space-time is not an ordinary manifold.
If space-time, is not a manifold, the way you push around a slice of space in space-time should be different--and this is reflected in the behaviour of H and D. This is part of what I research. I have modified the way H and D close under Poisson brackets, and then using Kuchar's methods, I have reconstructed the explicit forms of H and D. Why is this important? Using these methods, we can investigate what physical effects result from changing the fundamental manifold structure of space-time. For instance, working with Prof. Martin Bojowald, initial investigations have turned up fascinating implications for the early universe. Our results suggest two important changes to the Big Bang picture. One is that inhomogeneities in the early universe are not important to the evolution of the universe, and that the homogeneous universe avoids a Big Bang singularity. The second result is even stranger--it implies that ‘before’ the universe began, there was region of four dimensional space, with no time; a 4+0 space-time. But somewhere in that four dimensional space, there was a change, and Time was born, and we went from a 4+0 space to a 3+1 space-time. Thus Time, and the Universe, had a ‘beginning-less beginning’.
These results are still under investigation, and there is still much work required to tease out which assumptions are necessary and which can be relaxed; and also to see how such novel physics can manifest itself in experiments today. They are marvelous ideas, but how can they be made accessible to undergraduates, in such a way that they can meaningfully contribute to the research? If the structure of space-time is not a manifold, this will affect everything that lives on that space-time: fermions like electrons and neutrinos, vector boson fields like the electromagnetic field and scalar fields like the recently discovered Higgs boson. Investigating these changes is an open issue, and accessible to students who have studied--or are willing to study--Electromagnetism and Hamiltonian methods in Classical Mechanics; slightly more advanced projects are available to those who wish to learn a bit of the Dirac Theory of the electron. There are also toy models of space-time in which calculations are much simpler; these are useful in building intuition about more complicated structures of space-time. Students who participate in these projects will gain useful exposure to Hamiltonian methods in Mechanics and Electromagnetism, the Dirac theory of simple fermions, and various mathematical techniques. They will learn to read scientific papers and texts and extract information; to collaborate and cooperate in order to do calculations, understand their physical meaning, and crucially--ask questions. Most importantly, they will take their first steps from the arena of Physics courses--where no matter how challenging the course, you are reasonably sure an answer exists--to the arena of Physics research, where framing the question, the choice of methods to achieve the answer, even the very existence of the answer--all these are up in the air. These are general skills which will be useful no matter what their future path, whether it is graduate studies or life outside academia. I would be happy to mentor them on their journey, and hope to gain some of them as fellow travelers on my own path to understand a little better the nature of the space and time in which we exist and move.
Now we are at the core of the story. In the 1970s, Kuchar showed a remarkable result. Suppose all you knew was that the Hamiltonian of General Relativity was composed of two (unknown) objects H and D, and that they closed in that particular way under the Poisson bracket. With just that knowledge, under some very simple assumptions, you could reconstruct the full Hamiltonian of General Relativity. This was truly astonishing. It shone a new light on the idea that Gravity was Geometry; it said that if all you knew was the way a spatial slice moved through space-time, you were set; you could reconstruct General Relativity.
When I first learned of this result, I was electrified. It has long been suspected that one of the central assumptions of GR--that space-time can be described by a manifold--was wrong. In a manifold, you can always zoom in and distinguish features at smaller and smaller length scales. But what we know of Quantum Mechanics, Special Relativity and General Relativity suggests that this is not true. To resolve features on a small length scale, you need a particle with a small wavelength and high frequency--and high energy. However, if you cram too much energy into a small region, it forms a black hole--and now you are back where you started, since cannot find out what happens inside a black hole. Thus there is a a minimum length scale in space-time below which you cannot resolve any details. This means that space-time is not an ordinary manifold.
If space-time, is not a manifold, the way you push around a slice of space in space-time should be different--and this is reflected in the behaviour of H and D. This is part of what I research. I have modified the way H and D close under Poisson brackets, and then using Kuchar's methods, I have reconstructed the explicit forms of H and D. Why is this important? Using these methods, we can investigate what physical effects result from changing the fundamental manifold structure of space-time. For instance, working with Prof. Martin Bojowald, initial investigations have turned up fascinating implications for the early universe. Our results suggest two important changes to the Big Bang picture. One is that inhomogeneities in the early universe are not important to the evolution of the universe, and that the homogeneous universe avoids a Big Bang singularity. The second result is even stranger--it implies that ‘before’ the universe began, there was region of four dimensional space, with no time; a 4+0 space-time. But somewhere in that four dimensional space, there was a change, and Time was born, and we went from a 4+0 space to a 3+1 space-time. Thus Time, and the Universe, had a ‘beginning-less beginning’.
These results are still under investigation, and there is still much work required to tease out which assumptions are necessary and which can be relaxed; and also to see how such novel physics can manifest itself in experiments today. They are marvelous ideas, but how can they be made accessible to undergraduates, in such a way that they can meaningfully contribute to the research? If the structure of space-time is not a manifold, this will affect everything that lives on that space-time: fermions like electrons and neutrinos, vector boson fields like the electromagnetic field and scalar fields like the recently discovered Higgs boson. Investigating these changes is an open issue, and accessible to students who have studied--or are willing to study--Electromagnetism and Hamiltonian methods in Classical Mechanics; slightly more advanced projects are available to those who wish to learn a bit of the Dirac Theory of the electron. There are also toy models of space-time in which calculations are much simpler; these are useful in building intuition about more complicated structures of space-time. Students who participate in these projects will gain useful exposure to Hamiltonian methods in Mechanics and Electromagnetism, the Dirac theory of simple fermions, and various mathematical techniques. They will learn to read scientific papers and texts and extract information; to collaborate and cooperate in order to do calculations, understand their physical meaning, and crucially--ask questions. Most importantly, they will take their first steps from the arena of Physics courses--where no matter how challenging the course, you are reasonably sure an answer exists--to the arena of Physics research, where framing the question, the choice of methods to achieve the answer, even the very existence of the answer--all these are up in the air. These are general skills which will be useful no matter what their future path, whether it is graduate studies or life outside academia. I would be happy to mentor them on their journey, and hope to gain some of them as fellow travelers on my own path to understand a little better the nature of the space and time in which we exist and move.
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