Tuesday, Dec. 8th
Norbert Bodendorfer, Univ. Warsaw
Title: Quantum symmetry reductions based on classical gauge fixings
PDF of the talk (1.4MB)
Audio [.wav 35MB]
YouTube.
Tuesday, Nov. 10th
Jedrzej Swiezewski, Univ. Warsaw
Title: Developments on the radial gauge
PDF of the talk (4MB)
Audio [.mp3 40MB]
by Steffen Gielen, Imperial College
A few months ago, physicists around the world celebrated the centenary of the field equations of
general relativity, presented by Einstein to the Prussian Academy of Sciences in November 1915.
Arriving at the correct equations was the culmination of an incredible intellectual effort by Einstein,
driven largely by mathematical requirements that the new theory of gravitation (superseding
Newton's theory of gravitation, which proved ultimately incomplete) should satisfy. In particular,
Einstein realized that its field equations should be generally covariant – they should take the same
general form in any coordinate system that one chooses to use for the calculation, say whether one
uses Cartesian, cylindrical, or spherical coordinates. This property sets the equations of general
relativity apart from Newton's laws of motion, where changing coordinate system can lead to the
appearance of additional “forces” such as centripetal or Coriolis forces.
Many conferences were held honoring the anniversary of Einstein's achievement. What was discussed at those conferences was partially the historical context, the beauty of the form of the equations, or the precise mathematical and conceptual significance of general covariance. However, the most important legacy of general relativity and the main inspiration for modern research have been the new physical phenomena that appear in general relativity but not in Newtonian gravity: black holes are regions of spacetime where gravity becomes so strong that not even light can escape; the strong gravitational field outside a black hole leads to a time dilation so strong that an hour nearby a black hole can correspond to years on Earth, as used recently in the film Interstellar; and we now believe that the universe as a whole is expanding, and has been since the Big Bang which is thought of as the beginning of space and time.
In order to understand these dramatic consequences of the Einstein equations, physicists had to find solutions to these equations. This is rather challenging in general: the Einstein equations are complicated differential equations for ten functions, depending on one time and three space dimensions, that encode the gravitational field of spacetime. Furthermore, the conceptually appealing property of general covariance means that apparently different solutions of the equations can be simply the same physical configuration looked at in different coordinates. Indeed, both issues – finding solutions to the equations at all and understanding their meaning – were challenges in the early days of the theory, when physicists tried to make sense of Einstein's equations.
Despite this formidable challenge, the Prussian lieutenant of the artillery Karl Schwarzschild, while serving on the Eastern front in World War I, was able to derive an exact solution of Einstein's equations in vacuum within weeks of their publication, much to the surprise of Einstein himself. This solution, now called the Schwarzschild solution, describes a black hole, and is one of the most important solutions of general relativity. What Schwarzschild did in order to solve the equations was to assume a symmetry of the solution: he assumed that the configuration of the gravitational field should be spherically symmetric. In spherical coordinates, where each point in space is specified by one radial and two angular coordinates, it should be independent of any change in the angular directions. This means that one describes space as a collection of regular, concentric spheres. What Schwarzschild found was that the spheres did not have to be glued together to simply give normal flat space, but one could form a curved geometry out of them, with curvature increasing as one heads towards the centre (eventually forming a black hole), while still solving Einstein's equations. To be able to do the calculation, Schwarzschild had to choose a particularly suitable coordinate system, hence exploiting the property of general covariance in his favor.
This strategy of finding solutions is typical for practitioners of general relativity: cosmological solutions could similarly be found by assuming that the universe looks exactly the same at each point and in each direction in space (in mathematical terms, it is homogeneous and isotropic), and only changes in time. This reduces the problem of solving Einstein equations to a much simpler task, and explicit solutions could be written down, again in a suitable coordinate system. These simplest solutions already exhibit the main features of our universe (overall expansion and an initial Big Bang singularity) and are fairly realistic – indeed our Universe seems to display only small variations between different large-scale regions, and at the very largest scales is, within an approximation, well described by a geometry that simply looks the same everywhere in space.
Many conferences were held honoring the anniversary of Einstein's achievement. What was discussed at those conferences was partially the historical context, the beauty of the form of the equations, or the precise mathematical and conceptual significance of general covariance. However, the most important legacy of general relativity and the main inspiration for modern research have been the new physical phenomena that appear in general relativity but not in Newtonian gravity: black holes are regions of spacetime where gravity becomes so strong that not even light can escape; the strong gravitational field outside a black hole leads to a time dilation so strong that an hour nearby a black hole can correspond to years on Earth, as used recently in the film Interstellar; and we now believe that the universe as a whole is expanding, and has been since the Big Bang which is thought of as the beginning of space and time.
In order to understand these dramatic consequences of the Einstein equations, physicists had to find solutions to these equations. This is rather challenging in general: the Einstein equations are complicated differential equations for ten functions, depending on one time and three space dimensions, that encode the gravitational field of spacetime. Furthermore, the conceptually appealing property of general covariance means that apparently different solutions of the equations can be simply the same physical configuration looked at in different coordinates. Indeed, both issues – finding solutions to the equations at all and understanding their meaning – were challenges in the early days of the theory, when physicists tried to make sense of Einstein's equations.
Despite this formidable challenge, the Prussian lieutenant of the artillery Karl Schwarzschild, while serving on the Eastern front in World War I, was able to derive an exact solution of Einstein's equations in vacuum within weeks of their publication, much to the surprise of Einstein himself. This solution, now called the Schwarzschild solution, describes a black hole, and is one of the most important solutions of general relativity. What Schwarzschild did in order to solve the equations was to assume a symmetry of the solution: he assumed that the configuration of the gravitational field should be spherically symmetric. In spherical coordinates, where each point in space is specified by one radial and two angular coordinates, it should be independent of any change in the angular directions. This means that one describes space as a collection of regular, concentric spheres. What Schwarzschild found was that the spheres did not have to be glued together to simply give normal flat space, but one could form a curved geometry out of them, with curvature increasing as one heads towards the centre (eventually forming a black hole), while still solving Einstein's equations. To be able to do the calculation, Schwarzschild had to choose a particularly suitable coordinate system, hence exploiting the property of general covariance in his favor.
This strategy of finding solutions is typical for practitioners of general relativity: cosmological solutions could similarly be found by assuming that the universe looks exactly the same at each point and in each direction in space (in mathematical terms, it is homogeneous and isotropic), and only changes in time. This reduces the problem of solving Einstein equations to a much simpler task, and explicit solutions could be written down, again in a suitable coordinate system. These simplest solutions already exhibit the main features of our universe (overall expansion and an initial Big Bang singularity) and are fairly realistic – indeed our Universe seems to display only small variations between different large-scale regions, and at the very largest scales is, within an approximation, well described by a geometry that simply looks the same everywhere in space.
Loop quantum gravity is an approach at a quantization of general relativity, aiming to extend general relativity by making it compatible with quantum mechanics. What distinguishes it from other approaches is that the main property of general relativity, general covariance, is taken as a central guiding principle towards the construction of a quantum theory. In some respects, the status of loop quantum gravity can be compared to the early days of general relativity: while it is now known that a quantum theory compatible with general covariance can be constructed, and its mathematical structure is well understood, one now needs to understand the new physical phenomena implied by the quantization, beyond general relativity. Just as in the time after November 1915, today's physicists should find explicit solutions to the equations of loop quantum gravity that can be used to study the physical implications of the (relatively) new framework.
One of the main successes of loop quantum gravity has been its application to cosmology. Homogeneous solutions of the Einstein equations that approximately describe our universe have been shown to receive modifications once loop quantum gravity techniques are used, leading to a resolution of the Big Bang singularity by a Big Bounce, and potentially observable quantum effects. However, the resulting models of the universe are not solutions of the full theory of loop quantum gravity: rather, they arise from quantization of a reduced set of solutions of classical general relativity with loop quantum gravity techniques. There is no reason, in general, to expect that these are exact solutions of loop quantum gravity. Quantum mechanics is funny: quantization can lead to many inequivalent theories, depending on how one decides to do it. By assuming that the universe is homogeneous from the outset, one obtains a quantum theory of only a finite, rather than an infinite number of “degrees of freedom”. It is well known that quantum theories can behave differently depending on whether they have a finite or infinite number of degrees of freedom.
In their ILQGS seminars, Jedrzej and Norbert presented work towards resolving this tension. Namely, they presented an approach in which, similar to how Schwarzschild and contemporaries proceeded 100 years ago, one identifies a suitable coordinate system in which the spacetime metric, representing the gravitational field, is represented. In a quantum theory where general covariance is implemented fundamentally, this means one has to perform a “gauge-fixing”; the freedom of changing the coordinate system must be “fixed” consistently in the quantum theory. Gauge-fixings mean that one works with fewer variables, and has to worry less about different but physically equivalent solutions that are only related by changes in the coordinate system. Achieving them is often quite hard technically. Together with collaborators in Warsaw, Jedrzej and Norbert have made progress on this issue in recent years.
The second step, after a convenient coordinate system (think of spherical coordinates for treating the Schwarzschild black hole) has been chosen, is to do a “symmetry reduction” in the full quantum theory: rather than on the most general quantum universes, one now focusses on those that have a certain symmetry property. Norbert showed a detailed strategy for how to do this. One identifies an equation satisfied by all classical solutions with the desired symmetry, such as isotropy (i.e. looking the same in all directions). The quantum version of this equation is then imposed in loop quantum gravity, leading to a full quantum definition of symmetries like “isotropy” or “spherical symmetry” in loop quantum gravity. The obvious applications of the mechanism, which are being explored at the moment, are identifying cosmological and black hole solutions in loop quantum gravity, studying their dynamics, and verifying whether the resulting effects are in accord with what has been found in the simpler finite-dimensional quantum models described above. In particular, one would like to know whether singularities inside black holes and at the Big Bang, where Einstein's theory simply breaks down, can be resolved by quantum mechanics, as is hoped.
Jedrzej also showed how the methods developed in different “gauge-fixings” for classical general
relativity could be used to resolve a disputed issue in the context of the AdS/CFT correspondence in
string theory, where one faces a similar problem of fixing the huge freedom under changes in the
coordinate system in order to identify the invariant physical properties of spacetime. In particular, a
certain choice of gauge-fixing has been discussed in AdS/CFT, which leads to unfamiliar
consequences such as non-locality in the gauge-fixed version of the theory. The tools developed by
Jedrzej and collaborators could be used to clarify precisely how this non-locality occurs. They
hence provide a somewhat unusual example of the application of methods developed for loop
quantum gravity in a string theory-motivated context, clearly a positive example that can inspire
more work on closer connections between methods used in these different communities.
