Tidal and nonequilibrium Casimir effects in free fall

In this work, we consider a Casimir apparatus that is put into free fall (e.g., falling into a black hole). Working in $1+1\mathrm{D}$, we find that two main effects occur: First, the Casimir energy density experiences a tidal effect where negative energy is pushed toward the plates and the resulting force experienced by the plates is increased. Second, the process of falling is inherently nonequilibrium and we treat it as such, demonstrating that the Casimir energy density moves back and forth between the plates after being “dropped,” with the force modulating in synchrony. In this way, the Casimir energy behaves as a classical liquid might, putting (negative) pressure on the walls as it moves about in its container. In particular, we consider this in the context of a black hole and the multiple vacua that can be achieved outside of the apparatus.

Figure 1

Just as particles in a falling box experience a tidal force which forces them to the sides (right), the negative Casimir energy experiences a similar tidal force (left). The result, much like the box, is an increase in (negative) Casimir pressure on the two plates. Additionally, a falling Casimir apparatus is inherently dynamical, and excitations will be created (not pictured) both outside and inside the box.

Figure 2

The series of coordinate transformations made to define coordinates where the plates’ coordinates are at constant coordinate positions. The map $(p,\mathbb{1})$ represents the mapping $(u,v)\mapsto (p(u),v)$ (and similarly for the other mappings). We first “straighten” out plate $A$ with $(p,\mathbb{1})$, then plate $B$ with $(H,H)$. After that, there is an infinite-dimensional space of solutions that keep the plates at constant coordinate position but are nontrivial transforms $(Q,Q)$. In an initially static model a unique $Q$ can be determined by causality as explained in Fig. 4 and associated text.

Figure 3

This shows how the Eqs. (22) and (23) associate a coordinate at one point in space to another point by reflections off both mirrors. The operations should be read top to bottom. This describes the coordinate transformation $H$ that puts the plates on constant coordinates after we transform $(u,v)\mapsto (\widehat{u},\widehat{v})=(H\circ p(u),H(v))$.

Figure 4

In order to get physically sensible results, in the past of region C we assume our space-time is static, and therefore we can define a coordinate system in which the plates are initially at rest with a vacuum $|0⟩$ defined with respect to the vector field ${\partial }_t$. (Imagine dropping plates from above a black hole as we will consider in Sec. 6.) Then at (0, 0) [i.e., $t=0$] plate $A$ begins to move and at $(t_0-L,t_0+L)$ [i.e., $t=t_0$] plate $B$ begins to move. We can solve Eq. (22) using these initial conditions. On the other hand, if we solve Eq. (22) in the future of region C without specifying the initial conditions, we can use the function $Q$ in Eq. (25) to implement those initial conditions. The result is Eqs. (36) and (38).

Figure 5

Plots of the static and dynamic contributions to $⟨T_{\mu \nu }⟩$ [progression in time is shifted vertically by $+\tau /2$]. (left) The Casimir energy is spread out in space and once it begins falling, it starts to slosh back and forth between the plates. (right) The dynamic Casimir effect creates small energy bursts at the point where the plates are dropped, which then bounce back and forth between the plates.