Quantum vacuum energy in general relativity

The vacuum energy of a quantized field in a non-Minkowskian spacetime is discussed. The approach taken emphasizes the analogy between this vacuum energy and the energy of the vacuum state of the quantized electromagnetic field in the presence of a pair of parallel conducting plates, the Casimir energy. The energy of the vacuum state of a quantized scalar field in a one-dimensional box of length $L$ (the spacetime manifold $S^1\times R$) is shown to be $-\frac{\pi \hslash c}{6L}$. A massless, conformal scalar field in the Einstein universe ($S^3\times R$) also possesses a nonzero vacuum energy. The vacuum energy density in this case is $\rho =\hslash c{\left(480\mathrm{}{\pi }^2{a_0}^4\right)}^{-1\mathrm{}}$, where $a_0$ is the radius of the universe. The pressure $P$ is $\frac{1}{3}\rho$, so the energy-momentum tensor associated with these zero-point fluctuations is of the same form as that for classical radiation. It is shown that a closed Robertson-Walker universe has the same vacuum energy density and pressure as a static universe of instantaneously equal radius. The electromagnetic, neutrino, and minimally coupled scalar fields in the Einstein universe cannot be treated successfully by these techniques. Finally, the vacuum energy of a scalar field in the presence of a linearized plane gravitational wave is discussed. It is shown that for a certain choice of vacuum state, which is an eigenstate of the Hamiltonian so no pair production occurs, the vacuum energy and pressure vanish. This result holds for both the conformal and nonconformal energy-momentum tensors.