Topological censorship

Abstract

All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from J ( − to J ( + is homotopic to a topologically trivial curve from J ( − to J ( + . (If the Poincaré conjecture is false, the theorem does not prevent one from probing fake 3-spheres).

Universal bounds on operator dimensions from the average null energy condition

“…The first two cases where interesting results can occur are for operators with spin (6, 1) and (7,1). In both these cases we computed all matrix elements of E analytically.…”

“…Therefore although we have two operators h and h † , only one Fourier transform is required. 7 Here we are using the fact that in our spinor conventions the index names indicate the transformation properties under this SO (2). See appendix A.…”

“…This is the averaged version of the null energy condition, which appears as a crucial assumption in many classical theorems of general relativity [7].…”

“…(The full analytic form of this matrix for the case of operators in (6, 1) and (7,1) representations is given in Mathematica files included with the submission of this paper. See appendix D for more details.)…”

“…7 The above argument tells us that we only need to consider pairs of polarizations corresponding to conjugate components:…”