Matter-Driven Change of Spacetime Topology

Using Monte Carlo computer simulations, we study the impact of matter fields on the geometry of a typical quantum universe in the causal dynamical triangulations (cdt) model of lattice quantum gravity. The quantum universe has the size of a few Planck lengths and the spatial topology of a three-torus. The matter fields are multicomponent scalar fields taking values in a torus with circumference $\delta$ in each spatial direction, which acts as a new parameter in the cdt model. Changing $\delta$, we observe a phase transition caused by the scalar field. This discovery may have important consequences for quantum universes with nontrivial topology, since the phase transition can change the topology to a simply connected one.

Figure 1

Left: a torus (opposite sides identified) with a pinch. The region in red is the region where $\varphi$ changes from 0 to $\delta$. In the blue part it stays constant. $\varphi$ is constant in the horizontal direction. Right: a torus where $\varphi$ is constant in the horizontal direction and uniformly increases from 0 to $\delta$ from bottom to top.

Figure 2

The projection of four-volume, as defined by (17), on one spatial direction ($x$, $y$, or $z$) for a typical cdt configuration with small jump magnitude ($\delta =0.1$). The horizontal axis is $\overline{\varphi }/\delta$.

Figure 3

The projection of four-volume, as defined by (17), on the $xy$ plane for a typical cdt configuration with small jump magnitude ($\delta =0.1$). Different colors correspond to different times $t$ of the original (lattice) time foliation.

Figure 4

The projection of four-volume, as defined by (17), on one spatial direction ($x$, $y$, or $z$) for a typical cdt configuration with large jump magnitude ($\delta =1.0$). The horizontal axis is $\overline{\varphi }/\delta$.

Figure 5

The projection of four-volume, as defined by (17), on the $xy$ plane for a typical cdt configuration with large jump magnitude ($\delta =1.0$). Different colors correspond to different times $t$ of the original (lattice) time foliation.