Drawing conformal diagrams for a fractal landscape

Generic models of cosmological inflation and the recently proposed scenarios of a recycling universe and the string theory landscape predict spacetimes whose global geometry is a stochastic, self-similar fractal. To visualize the complicated causal structure of such a universe, one usually draws a conformal (Carter-Penrose) diagram. I develop a new method for drawing conformal diagrams, applicable to arbitrary 1+1-dimensional spacetimes. This method is based on a qualitative analysis of intersecting lightrays and thus avoids the need for explicit transformations of the spacetime metric. To demonstrate the power and simplicity of this method, I present derivations of diagrams for spacetimes of varying complication. I then apply the lightray method to three different models of an eternally inflating universe (scalar-field inflation, recycling universe, and string theory landscape) involving the nucleation of nested asymptotically flat, de Sitter and/or anti-de Sitter bubbles. I show that the resulting diagrams contain a characteristic fractal arrangement of lines.

Figure 1

A conformal diagram of the 1+1-dimensional Minkowski spacetime. Dashed lines show lightrays emitted from points $A,B$. The curved line is the trajectory of an inertial observer moving with a constant velocity, $x^1=0.3x^0$.

Figure 2

The local correspondence between straight lines drawn at 45° angles in the conformal diagram (left) and lightrays in the physical spacetime (right).

Figure 3

Construction of the conformal diagram for the Minkowski spacetime. The point $E$ cannot belong to the diagram because there are no lightrays intersecting at $E$.

Figure 4

A Cauchy line must have a slope between $-45°$ and 45° (left). A timelike boundary must be a curve with a slope between 45° and 135° (right).

Figure 5

Construction of a conformal diagram for a part of de Sitter spacetime delimited by thick lines (left). The upper boundary of the diagram (right) is a horizontal line.

Figure 6

The local direction of the infinite boundary (thick line) is found by determining the rightmost rays $r_X$, $r_{X^{\prime }}$ for nearby rays $X,X^{\prime }$. The direction is at 45° angle (a) when $r_X=r_{X^{\prime }}$ and horizontal (b) when $r_X\ne r_{X^{\prime }}$.

Figure 7

Construction of the conformal diagram for the spacetime of a collapsing star. The thick dotted line $CF$ is a Schwarzschild singularity.

Figure 8

A conformal diagram for the future part of the de Sitter spacetime with flat spatial sections. The curved line represents the spatially infinite Cauchy surface $t=t_0$. The local behavior of lightrays is the same as in Fig. 5.

Figure 9

Left: The domain of Minkowski spacetime between two causally separated observers $A,B$ moving with a constant proper acceleration in opposite directions. The dashed lines show that the two observers cannot see each other. Right: A closed Minkowski universe. The thick lines represent the identified boundaries $x=x_{1,2}$. A lightray (dashed line) crosses the line $x=x_{1,2}$ infinitely many times.

Figure 10

A spacetime with two collapsing stars. The point $T$ is a future timelike infinity reached by an observer (thick curve) remaining between the two black holes. The lines $AT$ and $TB$ represent BH singularities and the dashed lines are the BH horizons.

Figure 11

The future part of a de Sitter spacetime with a collapsing star. The line $AB$ represents the BH singularity; the dashed line is the BH horizon.

Figure 12

Diagrams for a maximally extended Schwarzschild-de Sitter spacetime. The conventional, unbounded diagram (top) and an equivalent diagram having a finite extent (bottom) are related by a conformal transformation. The thick dotted lines represent Schwarzschild singularities. Both diagrams contain infinitely many Schwarzschild and de Sitter regions.

Figure 13

A thermalized bubble in a de Sitter spacetime with flat spatial sections. The dashed lines are equal-temperature surfaces in the thermalized region (shaded).

Figure 14

Null geodesics (dashed lines) in a de Sitter spacetime with a bubble (shaded). The last lightrays not entering the bubble are labeled $A$ and $B$. Question marks indicate that the behavior of lightrays in the bubble interior is yet to be investigated.

Figure 15

A conformal diagram for the future part of de Sitter spacetime with a dust-dominated bubble. The bubble walls are the vertical curves ending at $C$ and $D$; the (spacelike) dotted line symbolizes the nucleation event.

Figure 16

The bubble interior (shaded) viewed in the flat coordinates $(\tau ,\zeta )$. The dotted lines show the continuation of the bubble wall to times before the bubble nucleation ($\tau =0$). The dashed lines are lightrays entering the bubble from opposite sides; such lightrays always intersect.

Figure 17

The bubble interior as in Fig. 16 but with a dark energy-dominated subdomain (darker shade). The rays labeled $A,B$ asymptotically coincide with the bubble walls.

Figure 18

A conformal diagram for the future part of de Sitter spacetime with a dark energy-dominated bubble interior. The bubble walls are shown as in Fig. 15. The points $D,E$ represent the asymptotic position of the bubble walls.

Figure 19

A conformal diagram for the future part of de Sitter spacetime with two dust-dominated bubbles.

Figure 20

Construction of a random Cantor set. Arrows show the bubble nucleation sites randomly chosen at each timestep. The (comoving) bubble size decreases geometrically with time and the number of nucleation sites increases. After infinitely many steps, there remains a fractal set of points (sketched below).

Figure 21

A conformal diagram for the future part of an eternally inflating spacetime with a fractal arrangement of dust-dominated bubbles. (A computer simulation was used to position the bubbles.)

Figure 22

A flat spacetime with a nucleated de Sitter bubble. The bubble appears as a black hole from outside (cf. Figure 7) but contains a de Sitter patch in its interior (shaded). The line $AB$ is the bubble wall, $EA$ is the nucleation surface, $DF$ is the BH singularity, $BC$ is the de Sitter asymptotic future, and the dashed line is the BH horizon. The line $F^{\prime }{F}^{\prime \prime }$ is a trajectory accelerating away from the BH.

Figure 23

A future part of the dS spacetime with a dS bubble having a higher expansion rate (cf. Figure 22). The initial radius of the bubble is assumed to be smaller than the background dS horizon.

Figure 24

A future part of the dS spacetime with a nucleated dust-dominated bubble and a nested dS bubble (cf. Figure 23). The black hole horizons, bubble walls, and nucleation surfaces are shown as in Figs. 15, 22.

Figure 25

A sketch of a diagram for the future part of a recycling dS spacetime, featuring a random arrangement of nested bubbles of all types. Schwarzschild singularities are not shown.