Extended Rindler spacetime and a new multiverse structure

This is the first of a series of papers in which we use analyticity properties of quantum fields propagating on a spacetime to uncover a new multiverse geometry when the classical geometry has horizons and/or singularities. The nature and origin of the “multiverse” idea presented in this paper, that is shared by the fields in the standard model coupled to gravity, are different from other notions of a multiverse. Via analyticity we are able to establish definite relations among the universes. In this paper we illustrate these properties for the extended Rindler space, while black hole spacetime and the cosmological geometry of mini-superspace (see Appendix B) will appear in later papers. In classical general relativity, extended Rindler space is equivalent to flat Minkowski space; it consists of the union of the four wedges in $(u,v)$ light-cone coordinates as in Fig. 1. In quantum mechanics, the wavefunction is an analytic function of $(u,v)$ that is sensitive to branch points at the horizons $u=0$ or $v=0$, with branch cuts attached to them. The wave function is uniquely defined by analyticity on an infinite number of sheets in the cut analytic $(u,v)$ spacetime. This structure is naturally interpreted as an infinite stack of identical Minkowski geometries, or “universes”, connected to each other by analyticity across branch cuts, such that each sheet represents a different Minkowski universe when $(u,v)$ are analytically continued to the real axis on any sheet. We show in this paper that, in the absence of interactions, information does not flow from one Rindler sheet to another. By contrast, for an eternal black hole spacetime, which may be viewed as a modification of Rindler that includes gravitational interactions, analyticity shows how information is “lost” due to a flow to other universes, enabled by an additional branch point and cut due to the black hole singularity.

Figure 1

Four regions of the map ($u$, $v$) to ($t$, $y$).

Figure 2

Increasing $t(\tau )$ in I or III.

Figure 3

Decreasing $t(\tau )$ in I or III.

Figure 4

$V(y)$ and kinetic energy. The particle cannot move to the region $y>y_{*}$.

Figure 5

A path on $0^{th}$ sheet in analytic $u$ -plane.

Figure 6

A path on $0^{th}$ sheet in analytic $v$ -plane.

Figure 7

Figure 7—A path on the real $(u,v)$ plane in the (0,0) universe.

Figure 8

Incoming/reflected waves at $y\to -\infty$ (regions II&IV); vanishing wavefunction at $y\to +\infty$ in I & III.

Figure 9

Magnitudes of incoming and outgoing fluxes at the horizons of each region. $\text{Blue arrows}=+\mathrm{sign}$, and $\text{red arrows}=-\mathrm{sign}$. For example, for region I, at the future horizon $(v=0)$, the outgoing particle current is proportional to $|a_{1-}^2(\omega )|$ (red) and the incoming antiparticle current is proportional to ${|b_{1-}(\omega )|}^2$ (blue). Similarly at the past horizon of region I $(u=0)$, the incoming particle current is proportional to ${|a_{1+}(\omega )|}^2$ (blue) and the outgoing antiparticle current is ${|b_{1+}^{†}(\omega )|}^2$ (red). Similar in and out currents are indicated for each region.

Figure 10

The $({\varphi }_{\gamma }(\tau ),h_{\gamma }(\tau ))$ field space.

Figure 11

Cosmological crunch and bang with antigravity in between.