Spacetime with zero point length is two-dimensional at the Planck scale

Abstract

It is generally believed that any quantum theory of gravity should have a generic feature—a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length $\ell _0$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell ,\ell _0)$ in a D-dimensional spacetime of a region of size $\ell$ scales as $V_D(\ell , \ell _0) \propto \ell _0^{D-2} \ell ^2$ when $\ell \sim \ell _0$, while it reduces to the standard result $V_D(\ell ,\ell _0) \propto \ell ^D$ at large scales ($\ell \gg \ell _0$). The appropriately defined effective dimension, $D_\mathrm{eff}$, decreases continuously from $D_\mathrm{eff}=D$ (at $\ell \gg \ell _0$) to $D_\mathrm{eff}=2$ (at $\ell \sim \ell _0$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

Appendix

In this appendix we will briefly illustrate how the prescription of qmetric works by describing the key steps in computing R and K for the qmetric; the general derivation and expressions can be found in [10]. Here, we will sketch an easier method based on using the synchronous frame for the background metric (the details will be given in [12]). The qmetric line element in the synchronous frame turns out to be,

$$\begin{aligned} \mathrm {d}s^{2}=\frac{1}{A(\sigma )}\left[ \mathrm {d}\sigma ^{2}+A^{2}(\sigma )\left( \frac{\Delta _{\phantom {F}}}{\Delta _{\mathcal F}}\right) ^{2/D_{1}}h_{\alpha \beta } \mathrm {d}x^{\alpha } \mathrm {d}x^{\beta }\right] ;\quad A=\frac{\sigma ^{2}+\ell _0^{2}}{\sigma ^{2}} \end{aligned}$$

(13)

where $D_{1}=D-1$. Then the Ricci biscalar $R[q_{ab}]$ can be computed using the usual formula which relates the metric to Ricci scalar with all derivatives acting on x. We then get:

$$\begin{aligned} R_{q}=&\frac{1}{A}R_{\Sigma } \left( \frac{\Delta _{\phantom {F}}}{\Delta _{\mathcal F}}\right) ^{-2/D_1}s - \frac{D_1D_2}{\mathcal F(\sigma ^{2})}+4(D+1)\dfrac{d\ln \Delta _{\mathcal F}}{\mathrm {d}\mathcal F}\nonumber \\&-\frac{\mathcal F}{\sigma ^{2}}\left( K_{ab}K^{ab}-\frac{1}{D_{1}}K^{2}\right) +4 \mathcal F \left[ -\frac{D}{D_{1}}\left( \frac{d\ln \Delta _{\mathcal F}}{\mathrm {d}\mathcal F}\right) ^{2}+2\dfrac{\mathrm {d}^{2}\ln \Delta _{\mathcal F}}{\mathrm {d}\mathcal F^{2}}\right] \end{aligned}$$

(14)

where again $D_{2}$ stands for $D-2$ and $R_{\Sigma }$ stands for the curvature scalar on $\sigma =\text {constant}$ surface. In the final expression we interpret $\sigma (x,x')$ as the geodesic distance in the background metric. Then using expressions for derivatives of the van Vleck determinant, we first take the coincidence limit $\sigma ^{2}\rightarrow 0$ keeping $\ell _0$ finite to obtain (with $\mathcal {S}\equiv R_{ab}n^{a}n^{b}$):

$$\begin{aligned} \lim _{\sigma \rightarrow 0}R_{q}=D\mathcal {S}+\frac{2}{3}(D+1)\left( n^{a}\nabla _{a}\mathcal {S}\right) \ell _0+\mathcal {O}(\ell _0^{2}) \end{aligned}$$

(15)

If we now take $\ell _0\rightarrow 0$ we get,

$$\begin{aligned} \lim _{\ell _0\rightarrow 0}\lim _{\sigma \rightarrow 0}R_{q}=D\mathcal {S} \end{aligned}$$

(16)

where all the objects on the right hand side is being determined by the bare metric $g_{ab}$ (see also [10]). Note that if we first take the limit $\ell _0\rightarrow 0$ in Eq. (14), we would obtain the Ricci scalar for the bare metric and the second limit of $\sigma ^2\rightarrow 0$ becomes vacuous. Hence the two limits (a) $\sigma \rightarrow 0$ and (b) $\ell _0\rightarrow 0$ do not commute, as noted in Eq. (4).

Similar result holds for trace of extrinsic curvature K of the equi-geodesic surface evaluated for the qmetric. There we readily obtain leading terms in orders of $\ell _0^{2}$ to be,

$$\begin{aligned} \lim _{\sigma \rightarrow 0} K_{q}=\frac{D_{1}}{\ell _0}-\frac{\ell _0}{3}\mathcal {S} \end{aligned}$$

(17)

In which $\mathcal {S}=R_{ab}n^{a}n^{b}$, evaluated for the bare metric, appears again. This quantity $\mathcal {S}$ is the entropy density of the null surfaces used in the emergent gravity paradigm [24–26]. The first term in Eq. (17) is a zero point entropy density of the spacetime and is closely related to the possible solution to the cosmological constant problem [11, 27].