Quantum Inequalities in Curved Two Dimensional Spacetimes

TL;DR

This paper demonstrates that quantum inequalities for negative energy densities hold in two-dimensional curved spacetimes with conformal invariance, but these bounds diverge near horizons or singularities, implying unconstrained negative energies in these regions.

Contribution

It establishes quantum inequalities for conformally invariant fields in 2D curved spacetimes and analyzes their behavior near horizons and singularities.

Findings

Quantum inequalities hold for conformally invariant fields in 2D curved spacetimes.

Bounds on negative energies diverge near horizons and singularities.

Negative energies become unconstrained close to horizons or initial singularities.

Abstract

In quantum field theory there exist states for which the energy density is negative. It is important that these negative energy densities satisfy constraints, such as quantum inequalities, to minimize possible violations of causality, the second law of thermodynamics, and cosmic censorship. In this paper I show that conformally invariant scalar and Dirac fields satisfy quantum inequalities in two dimensional spacetimes with a conformal factor that depends on $x$ only or on $t$ only. These inequalities are then applied to two dimensional black hole and cosmological spacetimes. It is shown that the bound on the negative energies diverges to minus infinity as the event horizon or initial singularity is approached. Thus, neglecting back reaction, negative energies become unconstrained near the horizon or initial singularity. The results of this paper also support the hypothesis that the quantum interest conjecture applies only to deviations from the vacuum polarization energy, not to the total energy.