Negative energy seen by accelerated observers

The sampled negative energy density seen by inertial observers, in arbitrary quantum states is limited by quantum inequalities, which take the form of an inverse relation between the magnitude and duration of the negative energy. The quantum inequalities severely limit the utilization of negative energy to produce gross macroscopic effects, such as violations of the second law of thermodynamics. The restrictions on the sampled energy density along the worldlines of accelerated observers are much weaker than for inertial observers. Here we will illustrate this with several explicit examples. We consider the worldline of a particle undergoing sinusoidal motion in space in the presence of a single mode squeezed vacuum state of the electromagnetic field. We show that it is possible for the integrated energy density along such a worldline to become arbitrarily negative at a constant average rate. Thus the averaged weak energy condition is violated in these examples. This can be the case even when the particle moves at nonrelativistic speeds. We use the Raychaudhuri equation to show that there can be net defocusing of a congruence of these accelerated worldlines. This defocusing is an operational signature of the negative integrated energy density. These results in no way invalidate nor undermine either the validity or utility of the quantum inequalities for inertial observers. In particular, they do not change previous constraints on the production of macroscopic effects with negative energy, e.g., the maintenance of traversable wormholes.

Figure 1

The figure illustrates that maximum negative energy density is obtained when we set $\omega =\Omega$ and $\delta =\pi$. The dotted line represents the Lorentz factor in Eq. (21), $[1-A^2{sin}^2(\Omega t){]}^{-1/2}$, while the solid line represents the cosine term, $-\cos (2\Omega t-\delta )=\cos (2\Omega t)$ both graphed as functions of time. Here we have chosen $A=0.2$ and $\Omega =2$. (Note that the graph of the cosine term has been rescaled and that for the Lorentz factor has been displaced downward by 0.9).

Figure 2

The integrated energy density multiplied by the quantization volume, $IV$, seen by an accelerated observer who is moving perpendicularly to the direction of wave propagation, is plotted as a function of his proper time, $\tau$, for the parameters $r=0.01$, $A=0.9$, and in units where $\Omega =1$.

Figure 3

The integrated energy density multiplied by the quantization volume, $IV$, seen by an accelerated observer who is moving parallel to the direction of wave propagation, is plotted as a function of his proper time, $\tau$, for the parameters $r=0.01$, $A=0.9$, and $\omega =2$, in units where $\Omega =1$. Note that the accumulated negative energy density grows much faster than in the case of the perpendicularly moving observer, due to the linear Doppler shift term in the energy density. We also see extra structure in this curve as well, because the expression for the energy density is more complicated than in the perpendicular motion case.