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Quantum field theory
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analysis ANEC approximation arbitrary asymptotically flat region averaged null energy black hole calculation causal curve chronology horizon chronology protection chronology-violating region classical closed timelike curves conjecture consider constraints construction coordinate cosmic strings cosmological constant defined definition denotes discussion dominant energy conditions Einstein equations Einstein field equations Einstein tensor energy density Euclidean event horizon Exercise exotic matter finite fluctuations formalism function geometry globally gravitational field Hawking and Ellis hypersurfaces implies infinity integral linearized Lorentzian metric Lorentzian wormholes machine manifold mass Minkowski space monograph Note notion null curve null energy condition null geodesics paradoxes parameter Planck length Planck scale quantization quantum field theory quantum gravity quantum mechanics radial radius Reissner-Nordstrom relativity renormalized result Riemann tensor scalar Schwarzschild second fundamental form solution spacelike spacetime containing spatial spherical symmetry stress-energy tensor surface technical theorem tidal time-shift topology topology change traversable wormhole universe vacuum violated Vleck determinant wormhole mouths wormhole throat