Quantum dynamics of the Einstein-Rosen wormhole throat

We consider the polymer quantization of the Einstein wormhole throat theory for an eternal Schwarzschild black hole. We numerically solve the difference equation describing the quantum evolution of an initially Gaussian, semiclassical wave packet. As expected from previous work on loop quantum cosmology, the wave packet remains semiclassical until it nears the classical singularity at which point it enters a quantum regime in which the fluctuations become large. The expectation value of the radius reaches a minimum as the wave packet is reflected from the origin and emerges to form a near-Gaussian but asymmetrical semiclassical state at late times. The value of the minimum depends in a nontrivial way on the initial mass/energy of the pulse, its width, and the polymerization scale. For wave packets that are sufficiently narrow near the bounce, the semiclassical bounce radius is obtained. Although the numerics become difficult to control in this limit, we argue that for pulses of finite width the bounce persists as the polymerization scale goes to zero, suggesting that in this model the loop quantum gravity effects mimicked by polymer quantization do not play a crucial role in the quantum bounce.

Figure 1

A schematic of the geometrodynamics of the time evolution of the throat of the Einstein-Rosen bridge as a function of proper time of a comoving observer is shown.

Figure 2

The time evolution of wave packet, $t=0$, is shown. In this figure $\mu =1$, $\delta =0.05$, ${\varphi }_0=500$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 3

The time evolution of wave packet, $t=20$, is shown. In this figure $\mu =1$, $\delta =0.05$, ${\varphi }_0=500$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 4

The time evolution of wave packet, $t=35$, is shown. In this figure $\mu =1$, $\delta =0.05$, ${\varphi }_0=500$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 5

The time evolution of wave packet, $t=60$, is shown. In this figure $\mu =1$, $\delta =0.05$, ${\varphi }_0=500$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 6

The evolution of $⟨\varphi ⟩$ as a function of proper time $t$ is shown. The top curve (blue, solid) is for the full polymer theory whereas the bottom curve (red, dashed) is for the effective theory. The former generally produces a bounce radius that is slightly larger than the latter. In this figure $\mu =1$, $\delta =0.05$, ${\varphi }_0=500$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 7

The value of the standard deviation of $\varphi$ at the bounce, $\Delta {\varphi }_b$, as a function of the initial value of the standard deviation $\Delta {\varphi }_0$ is shown. In this figure $\mu =1$, ${\varphi }_0=1000$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 8

The value of $\Delta \varphi /⟨\varphi ⟩$ as a function of the time is shown. In this figure $\mu =1$, $\delta =0.05$, ${\varphi }_0=500$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 9

The value of the minimum of $⟨\varphi ⟩$ as a function of $\Delta {\varphi }_b$ is shown. For all the data points $\mu =1$, ${\varphi }_0=1000$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$. Although $\Delta {\varphi }_b$ cannot be made arbitrarily small in the full quantum theory, the figure strongly suggests that formally in the limit $\Delta {\varphi }_b\to 0$, ${\varphi }_{\min }$ approaches the value predicted by the effective theory, which for given values of parameters is $40l_{\mathrm{Pl}}^2$ [cf. Eq. (3.5)].

Figure 10

The value of the minimum of ${\varphi }_{\min }$ as a function of $\Delta {\varphi }_0$ is shown. For all the data points $\mu =1$, ${\varphi }_0=1000$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.

Figure 11

The value of ${\varphi }_{\min }$ as a function of $\mu$ is shown. All the data points correspond to the same the initial wave packet with a physical width $\Delta {\varphi }_0\approx 17.7$, ${\varphi }_0=80$, and ${\pi }_0=-1\times {10}^{-7}\approx 0$.