Is Spacetime Countable?

Abstract

Is there a number for every bit of spacetime, or is spacetime smooth like the real line? The ultimate fate of a quantum theory of gravity might depend on it. The troublesome infinities of quantum gravity can be cured by assuming that spacetime comes in countable, discrete pieces which one could simulate on a computer. But, perhaps there is another way? In this essay, we propose a picture where scale is meaningless so that there can be no minimum length and, hence, no fundamental discreteness. In this picture, Einstein’s Special Relativity, suitably modified to accommodate an expanding Universe, can be reinterpreted as a theory where only the instantaneous shapes of configurations count.

Not everything that counts can be counted and not everything that can be counted counts.

–Albert Einstein

Appendices

Möbius Transformations and the Lorentz Transformations

The Möbius transformations are defined as:

$$\begin{aligned} \zeta \rightarrow \frac{a\zeta + b}{c\zeta + d}, \end{aligned}$$

(14.4)

where $a, b, c, d$ are complex numbers obeying $ac - bd \ne 0$. This group is well-known to be isomorphic to the projective special linear group PSL$(2,\mathbb {C})$, which, in turn, is isomorphic to the orthochronous Lorentz group SO$^+(3, 1)$. It is this property that we exploit in Appendix B. For more info on the Möbius transformations and for visualizations which inspired our diagrams on stereographic projection, see [10].

De Sitter Inertial Observers to Scale Invariant Particles

For a much more detailed account of the material presented here, see the technical paper [6].

We are inspired by the Shape Dynamics formulation of gravity, as presented in [1], where equivalence with GR is manifest in Constant Mean Curvature (CMC) slicings of solutions to the Einstein equations. For dS$^{d, 1}$ spacetime, the CMC slices are constant $t$ hypersurfaces in the ambient $\mathbb {R}^{d+1, 1}$ and have $\mathcal S^{d}$ topology. To see this, we can use a convenient choice of coordinates for the embedding:

$$\begin{aligned} t&= \ell \sinh \varphi&x^I&= \ell \cosh \varphi \, \tilde{x}^I, \end{aligned}$$

(14.5)

where $I = 1,\ldots ,(d+1)$ and $\tilde{x}^I \tilde{x}^J \delta _{IJ} = \tilde{x}^2 = 1$. Using these coordinates, the induced metric is

$$\begin{aligned} ds^2 = -\ell ^2 d\varphi ^2 + \ell ^2 \cosh ^2\varphi \, d\Omega ^2, \end{aligned}$$

(14.6)

where $d\Omega ^2$ is the line element on the unit $d$-sphere. Since the spatial metric is conformal to the metric on the unit sphere (which is homogeneous), it is clear that this slicing must be CMC.

We now consider a useful set of coordinates:

$$\begin{aligned} x^\pm&= x^0 \pm x^{d+1}&X^i&= \frac{x^i}{x^0 - x^{d+1}}\,, \end{aligned}$$

(14.7)

where $i = 1,\ldots ,d$. The $x^\pm$ are just light-cone coordinates in the ambient space. We can single out one of these, namely $x^-$, as a convenient time variable and write the other $x^+ = \frac{x^2 - \ell ^2}{x^0 - x^{d+1}} = \frac{1}{x^-} \left( \frac{X^2}{(x^-)^2} - \ell ^2\right)$ using the definition of de Sitter spacetime. The $X^i$’s are a convenient choice of spatial coordinates because, as can be shown with a straightforward calculation, in the limit as $t \rightarrow \pm \infty$ (i.e., the conformal boundary of spacetime), they are just giving the stereographic projection of coordinates on the constant-$t$ hypersurfaces onto a Euclidean plane:

$$\begin{aligned} X^i \rightarrow \frac{\tilde{x}^i}{1 - \tilde{x}^{d+1}}. \end{aligned}$$

(14.8)

The utility of these coordinates becomes obvious when one considers the action of the ambient Lorentz transformations $x^\mu \rightarrow \Lambda ^\mu _\nu x^\nu$ on the new coordinates. Indeed, near the conformal boundary, it can be shown that $x^- \rightarrow x^-$ and that the $X^i$ transform under the full conformal group.

This last property allows us to define a scale-invariant theory holographically using the action principle for massive particles following bulk geodesics. To see how this can be done, consider the action for a single particle of mass $m$ following a geodesic in dS

$$\begin{aligned} S(X^i_{in}, X^i_{out}) = \lim _{t_0 \rightarrow \infty } \int \limits _{-t_0}^{t_0} d t \left[ m \sqrt{ - \eta _{\mu \nu } \dot{x}^\mu \dot{x}^\nu } + \lambda \left( \eta _{\mu \nu } x^\mu x^\nu - \ell ^2 \right) \right], \end{aligned}$$

(14.9)

where $X^i_{in}$ and $X^i_{out}$ are the asymptotic values of the coordinates $X^i$ on the past and future conformal boundary. The Lagrange multiplier $\lambda$ enforces the constraint keeping the particle on the dS hyperboloid. If we evaluate this along the classical solution while carefully taking the limit, $S$ becomes of a function of the asymptotic values of $X^i$. Moreover, as was just indicated, it is also conformally invariant. This means that it can be interpreted as the Hamilton–Jacobi function of some holographically defined conformally invariant theory.

In [6], $S$ is explicitly computed in this limit. The result is

$$\begin{aligned} S = \frac{m\ell }{2} \left[ \ln \left( \frac{(X_{in} - X_{out})^2}{\epsilon ^2} - 2 \right) + \mathcal O(\epsilon ^4) \right], \end{aligned}$$

(14.10)

where $\epsilon = \ell /t \rightarrow 0$ as $t \rightarrow \infty$. This behaves exactly like the Hamilton–Jacobi functional of a reparametrization invariant theory with potential equal to $V = \frac{1}{X^2}$, which is well-known to be scale invariant. We see that a free massive particle in dS spacetime can be equivalently described by a scale-invariant particle in a reparametrization invariant theory. Furthermore, the bulk dS isometries map explicitly to conformal transformations in the dual theory, as advertised.