1994/11/21 by Ulvi Yurtsever
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cauchy stress tensor #Classical mechanics #Cosmology and Gravitation Theories #Energy condition #General relativity #Geodesic #Geodesics in general relativity #Mathematical analysis #Mathematical physics #Mathematics #Minkowski space #Negative energy #Null (SQL) #Physics #Pointwise #Pure mathematics #Quantum #Quantum Electrodynamics and Casimir Effect #Quantum field theory #Quantum field theory in curved spacetime #Quantum gravity #Quantum mechanics #Scalar field #Spacetime #Tensor (intrinsic definition) #gr-qc
paper · pdf · doi:10.1103/physrevd.51.5797
published as Phys.Rev. D51 (1995) 5797-5805 · 20 pages
arxiv created 1994/11/21 · openalex publication_date 1995/05/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For a large class of quantum states, all local (pointwise) energy conditions widely used in relativity are violated by the renormalized stress-energy tensor of a quantum field. In contrast, certain nonlocal positivity constraints on the quantum stress-energy tensor might hold quite generally, and this possibility has received considerable attention in recent years. In particular, it is now known that the averaged null energy condition, the condition that the null-null component of the stress-energy tensor integrated along a complete null geodesic is non-negative for all states, holds quite generally in a wide class of spacetimes for a minimally coupled scalar field. Apart from the specific class of spacetimes considered (mainly two-dimensional spacetimes and four-dimensional Minkowski space), the most significant restriction on this result is that the null geodesic over which the average is taken must be achronal. Recently, Ford and Roman have explored this restriction in two-dimensional flat spacetime, and discovered that in a flat cylindrical space, although the stress energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (nonachronal) null geodesics, when the ``Casimir-vacuum'' contribution is subtracted from the stress-energy the resulting tensor does satisfy the ANEC inequality. Ford and Roman name this class of constraints on the quantum stress-energy tensor ``difference inequalities.'' Here I give a proof of the difference inequality for a minimally coupled massless scalar field in an arbitrary (globally hyperbolic) two-dimensional spacetime, using the same techniques as those we relied on to prove the ANEC in an earlier paper with Wald. I begin with an overview of averaged energy conditions in quantum field theory.