Time-traveling billiard-ball clocks: A quantum model

General relativity predicts the existence of closed timelike curves (CTCs), along which an object could travel to its own past. A consequence of CTCs is the failure of determinism, even for classical systems: one initial condition can result in multiple evolutions. Here we introduce a quantum formulation of a classic example, where a billiard ball can travel along two possible trajectories: one unperturbed and one, along a CTC, where it collides with its past self. Our model includes a vacuum state, allowing the ball to be present or absent on each trajectory, and a clock, which provides an operational way to distinguish the trajectories. We apply the two foremost quantum theories of CTCs to our model: Deutsch's model (D-CTCs) and postselected teleportation (P-CTCs). We find that D-CTCs reproduce the classical solution multiplicity in the form of a mixed state, while P-CTCs predict an equal superposition of the two trajectories, supporting a conjecture by Friedman et al. [Phys. Rev. D 42, 1915 (1990)].

Figure 1

Two possible spatial trajectories of a classical billiard ball through CTC-wormhole spacetimes with the same initial data (i.e., position and velocity) in a two-dimensional slice of three-dimensional space. (a) A chronology-respecting history (evolution $A$) in which the billiard ball evolves freely between the CTC and wormhole; (b) a chronology-violating history (evolution $B$) in which the billiard ball is struck onto the CTC path by its future self, thereby causing it to time travel into its past and subsequently strike its past self onto the CTC. The initial and final conditions for both evolutions are, respectively, exactly the same, meaning that the histories are (classically) indistinguishable.

Figure 2

Based on the simplified billiard-ball paradox in Fig. 1, these diagrams in $(1+1)$-dimensional spacetime visualize the basic idea of the model under study in the reference frame of the clock. (a) The history where the clock travels through the CTC-wormhole region with no interaction; (b) the history of a clock which elastically interacts with itself. The times as measured by the clocks at specific points along the histories are given in brackets next to their analog clock symbols. These clocks begin at time $t=0$ and, due to their initial velocity, measure the proper times $t_A=t_{\mathrm{in}}+t_{\mathrm{out}}$ and $t_B=t_{\mathrm{in}}+\mathrm{\Delta }t+t_{\mathrm{out}}=t_A+\mathrm{\Delta }t$ for evolutions $A$ and $B$, respectively. This means that we denote the duration of the segment on the CTC in evolution $B$ to be $\mathrm{\Delta }t$, while the inbound and outbound times to and from the wormhole axis (dashed line) are $t_{\mathrm{in}}$ and $t_{\mathrm{out}}$, respectively. It is important to note that the final position bears no information about the path the ball has taken.

Figure 3

Quantum circuit models of the two trajectories under study. (a) The evolution $A$ in Fig. 2; (b) the evolution $B$ in Fig. 2.

Figure 4

The vacuum-based quantum circuit formulation of the billiard-ball paradox (see Fig. 1) under study. The connected crosses between the two channels represent our prescribed vacuum-excluding $\mathrm{SWAP}$ gate.

Figure 5

D-CTCs CV clock probabilities (unitless) for the vacuum model plotted against the dimensionless time ratio. The CTC multiplicity parameter was taken to be $g=\frac{1}{3}$, and the lines correspond to varying values of $\sqrt{\mathrm{\Omega }}$.

Figure 6

D-CTCs CR clock probabilities (unitless) for the vacuum model plotted against the dimensionless time ratio. The CTC multiplicity parameter was taken to be $g=\frac{1}{3}$, and the lines correspond to varying values of $\sqrt{\mathrm{\Omega }}$.

Figure 7

P-CTCs clock probabilities (unitless) for the vacuum model plotted against the dimensionless time ratio. The lines correspond to varying values of $\sqrt{\mathrm{\Omega }}$.

Figure 8

P-CTCs populations (unitless) for the vacuum model plotted against the dimensionless time ratio. The lines correspond to varying values of $\sqrt{\mathrm{\Omega }}$.