Friday Seminar: Quantum energy Inequalities and Local Covariance
Energy inequalities, also called energy conditions, play an important role in General Relativity. The cornerstone of GR are the Einstein equations, which describe how the spacetime geometry is determined by the presence of matter within it. The mathematical object that describes "matter" is the energy-momentum tensor Tab (also called stress-energy tensor) , and its main components are the density of energy and its pressure. Energy conditions are restrictions set on it to make it represent "physically realistic" matter: for example, the condition that energy density must be positive, or that energy density must be greater than pressure. They are needed when we want to prove the validity of a theorem for any kind of spacetime with physically realistic matter of any kind in it: for example, that too much matter too close together will result in a black hole, or that spacetimes with "closed timelike curves" (aka time machines) cannot be produced. One can write down solutions to Einstein's equations in which black holes are avoided, or in which there are closed timelike curves, but they always require "exotic" matter that violates the energy conditions.
The catch is that energy conditions are satisfied for matter as described in classical physics. Taking into account quantum theory, energy can fluctuate wildly and the object that appears in the Einstein equations is actually the mean value or expectation value of the stress-energy tensor, [Tab] (I am using brackets instead of the usual < > for the "average", as the latter is not displayed properly). It is an easy matter to show (an outline of the proof was made in the seminar) that for a quantum field this expectation value can and often will not satisfy the energy conditions. For example, in the Casimir effect, the [Tab] of the quantum electromagnetic field in its vacuum state between two conducting but uncharged plates violates the weak energy condition (WEC), one of the most widely used energy conditions, that says in intuitive terms that any observer moving along any possible timelike (i.e. not attaining the speed of light) path will never measure energy to be negative. This is not pure speculation, as theory also predicts that this vacuum energy should give rise to a tiny force between the plates, a force which has been measured.
So the questions that comes is: can the classical energy inequalities be replaced by quantum energy inequalities, conditions on [Tab] which are weaker than the classical ones but sufficent to prove all the theorems we would like to prove excluding "exotic" solutions to the Einstein equations? Chris Fewster has focused his research on this question, following pioneering work by Ford and Roman, and gave us yesterday an overview of his recent results.
Quantum energy inequalities replace a purely local condition on [Tab] at each point for an averaged condition on integrals of [Tab] over spacetime regions. They state that the integral of contracted with a sampling tensor function fab must be greater than a certain quantity; in other words, they permit negative energies but put a bound on "how negative" they can get to be. It is hoped that this sort of condition will be enough to prove all the desired results, but nobody knows for sure. ("Desired" depends on the point of view, of course; the possibility of creating wormholes, warp drives and time machines sounds fun to me! But it would create lots of complications for physics, especially the latter ones for reasons well-known to all science fiction fans.) Using powerful techniques from algebraic quantum field theory based on a principle of local covariance developed in this paper (which as I understand them roughly state that when a spacetime can be locally embedded in another one then expectation values of quantum observables can be calculated in either of both spaces) Chris and his collaborators have proved a variety of interesting results for quantum inequalities in certain kinds of curved spacetimes.
In particular, they have proved that a certain bound for the amount of negative energy measured by an observer can be established in any spacetime which has a Minkowskian (flat) region within it. This applies for example to the Casimir effect, in which the metric between the two plates is flat. Using similar techiques a collaborator has proved here similar results for spacetimes with a part isometric to a part of Scwarzschild, which applies for the exterior of stars and other compact spherically symmetric objects.
This is where the connection with my previous work appears. In my research project for my first degree I calculated the vacuum energy [Tab] of quantum fields in weak gravitational fields were the Newtonian approximation of GR holds, considering in particular the case of spherical stars. The results, which were summarized in this publication (my first and so far only paper; another post will tell you how matters are progressing with my second one) included calculating exactly the amount of negative energy measured by an observer around such a star. (I hasten to say that this was of purely theoretical interest: the energy is much too small to be really detectable by realistic experiments). So when we went for dinner to the pub with Chris and all our research group after the talk, I had an opportunity of mentioning these results to him and passing the reference, which he wasn't aware of. I hope something interesting can come out of this...
UPDATE: Post reread on Sunday morning, many typos corrected. They were caused by tiredness at the moment of original writing (which finished about 3 am)