16 research outputs found
Energy conditions in general relativity and quantum field theory
This review summarizes the current status of the energy conditions in general
relativity and quantum field theory. We provide a historical review and a
summary of technical results and applications, complemented with a few new
derivations and discussions. We pay special attention to the role of the
equations of motion and to the relation between classical and quantum theories.
Pointwise energy conditions were first introduced as physically reasonable
restrictions on matter in the context of general relativity. They aim to
express e.g. the positivity of mass or the attractiveness of gravity. Perhaps
more importantly, they have been used as assumptions in mathematical relativity
to prove singularity theorems and the non-existence of wormholes and similar
exotic phenomena. However, the delicate balance between conceptual simplicity,
general validity and strong results has faced serious challenges, because all
pointwise energy conditions are systematically violated by quantum fields and
also by some rather simple classical fields. In response to these challenges,
weaker statements were introduced, such as quantum energy inequalities and
averaged energy conditions. These have a larger range of validity and may still
suffice to prove at least some of the earlier results. One of these conditions,
the achronal averaged null energy condition, has recently received increased
attention. It is expected to be a universal property of the dynamics of all
gravitating physical matter, even in the context of semiclassical or quantum
gravity.Comment: 61 pages, 6 figures, Topical review accepted by Classical and Quantum
Gravit
The Return of the Singularities: Applications of the Smeared Null Energy Condition
The classic singularity theorems of General Relativity rely on energy
conditions that can be violated in semiclassical gravity. Here, we provide
motivation for an energy condition obeyed by semiclassical gravity: the smeared
null energy condition (SNEC), a proposed bound on the weighted average of the
null energy along a finite portion of a null geodesic. We then prove a
semiclassical singularity theorem using SNEC as an assumption. This theorem
extends the Penrose theorem to semiclassical gravity. We also apply our bound
to evaporating black holes and the traversable wormhole of
Maldacena-Milekhin-Popov, and comment on the relationship of our results to
other proposed semiclassical singularity theorems.Comment: 28 pages, 7 figure
Multi-step Fermi normal coordinates
We generalize the concept of Fermi normal coordinates adapted to a geodesic
to the case where the tangent space to the manifold at the base point is
decomposed into a direct product of an arbitrary number of subspaces, so that
we follow several geodesics in turn to find the point with given coordinates.
We compute the connection and the metric as integrals of the Riemann tensor. In
the case of one subspace (Riemann normal coordinates) or two subspaces, we
recover some results previously found by Nesterov, using somewhat different
techniques.Comment: 9 pages, 4 figure
Quantum inequality for a scalar field with a background potential
Quantum inequalities are bounds on negative time-averages of the energy density of a quantum field. They can be used to rule out exotic spacetimes in general relativity. We study quantum inequalities for a scalar field with a background potential (i.e., a mass that varies with spacetime position) in Minkowski space. We treat the potential as a perturbation and explicitly calculate the first-order correction to a quantum inequality with an arbitrary sampling function, using general results of Fewster and Smith. For an arbitrary potential, we give bounds on the correction in terms of the maximum values of the potential and its first three derivatives. The techniques we develop here will also be applicable to quantum inequalities in general spacetimes with small curvature, which are necessary to rule out exotic phenomena
A singularity theorem for Einstein-Klein-Gordon theory
Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking's hypotheses, an important example being the massive Klein-Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein-Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein-Klein-Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete
Quantum inequality in spacetimes with small curvature
Quantum inequalities bound the extent to which weighted time averages of the renormalized energy density of a quantum field can be negative. They have mostly been proved in flat spacetime, but we need curved-spacetime inequalities to disprove the existence of exotic phenomena, such as closed timelike curves. In this work we derive such an inequality for a minimally-coupled scalar field on a geodesic in a spacetime with small curvature, working to first order in the Ricci tensor and its derivatives. Since only the Ricci tensor enters, there are no first-order corrections to the flat-space quantum inequalities on paths which do not encounter any matter or energy
Averaged null energy condition in a classical curved background
The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative. Exotic spacetimes, such as those allow wormholes or the construction of time machines are possible in general relativity only if ANEC is violated along achronal geodesics. Starting from a conjecture that flat-space quantum inequalities apply with small corrections in spacetimes with small curvature, we prove that ANEC is obeyed by a minimally-coupled, free quantum scalar field on any achronal null geodesic surrounded by a tubular neighborhood whose curvature is produced by a classical source