Planck Length and Metric Geometry

1. The Planck Length Problem

The Planck length is the following combination of fundamental constants, having the dimension of length:

$${\ell }_{Pl}=\sqrt{{\displaystyle \frac{\hslash G}{c^3}}}.$$

Numerical value of the Planck length is ${\ell }_{Pl}\approx 1.61·{10}^{-33}cm$.

The physical meaning of the Planck length is as follows. This is a scale on which it is fundamentally impossible to consider the theory of gravity without taking into account the quantum effects [1], since it is on the Planck scale that the values with the dimension of length inherent for gravity theory (the Schwarzschild radius of a spherically symmetric black hole) coincide with those for quantum theory (the Compton wavelength). Really, the Compton wavelength is given by the expression

$${\lambda }_C={\displaystyle \frac{\hslash }{mc}},$$

and the Schwarzschild radius is

$$r_g={\displaystyle \frac{2Gm}{c^2}}.$$

It is easy to see that the equality takes place:

$${\ell }_{Pl}^2={\displaystyle \frac{{\lambda }_C{r}_g}{2}}.$$

The appearance of a black hole on Planck scales does not allow us to obtain information about the structure of space on scales smaller than the Planck length.

In [2], it was conjectured that this kind of effect is associated with a fundamental change in the geometry of space on the Planck scale. Namely, the existence of unmeasurable regions of space is the result of a violation of Archimedes’ axiom (the axiom of measurability) in Euclidean geometry. A conjecture about the non-Archimedean nature of space on Planck scales was formulated. This gave rise to the development of a new field in mathematical physics [3,4].

However, the question of the mechanism of changing the metric from Archimedean to non-Archimedean remains open. In this paper, an attempt is made to construct a model of metric change using the apparatus of metric geometry. Namely, a geodesic in the Gromov-Hausdorff space connecting ultrametric and ordinary metric spaces will be explicitly constructed. As a model example of an ultrametric space, we will consider the set ${\mathbb{Z}}_p$ of p-adic integers with a metric generated by the standard p-adic norm; as a model example of an ordinary metric space, we will choose the unit segment $[0,1]\subset \mathbb{R}$ with a standard metric generated by the absolute value.

2. Metric Geometry. Basic Notions

A metric space is a pair $X=(X,d_X)$, where X is a set, $d_X$ is a metric on X, that is, a mapping $d_X:X\times X\to [0,\infty )$ satisfying the conditions:

• $d_X(x,x^{\prime })=0\iff x=x^{\prime }$;
• $d_X(x,x^{\prime })=d_X(x^{\prime },x)$;
• $d_X(x,x^{\prime \prime })\le d_X(x,x^{\prime })+d_X(x^{\prime },x^{\prime \prime })$.

If $d_X$ satisfies the condition $d_X(x,x")\le max\{d_X(x,x^{\prime }),d_X(x^{\prime },x")\}$ then this is ultrametric, the space $(X,d_X)$ is ultrametric (or non-Archimedean).

Important examples for the future are the following.

• $\mathbb{I}=[0,1],d_{\mathbb{I}}(x,x^{\prime })=|x-x^{\prime }|$ – Archimedean space;
• ${\mathbb{Z}}_p,d_{{\mathbb{Z}}_p}(x,x^{\prime })={|x-x^{\prime }|}_p$ – non-Archimedean space;
• ${\Delta }_m=\{x_1,\dots x_m\},d_{{\Delta }_m}(x_i,x_j)=1,i\ne j,i,j=1,2,\dots ,m$ – simplex.

We define two operations on metric spaces: direct product and dilation.

Direct product $(X\times Y,d_{X\times Y})$ of the metric spaces X and Y is the Cartesian product of $X\times Y$ with the metric given by the expression

$$d_{X\times Y}\left((x,y),(x^{\prime },y^{\prime })\right)=max\left\{d_X(x,x^{\prime }),d_Y(y,y^{\prime })\right\}.$$

Let $\lambda \in {\mathbb{R}}_{+}$ be a positive real number. The space $\lambda X$ obtained from the space $(X,d_X)$ by dilation the metric has the form:

$$\lambda X=(X,\lambda d_X).$$

Let $(X,d)$ be a metric space and $H=H\left(X\right)$ be a set of compact subsets of X. We define the metric (Hausdorff metric) $d_H$ on H.

Let $A,B\in H\left(X\right)$,

$$d_H(A,B)=inf\{ϵ>0:B\subset U_{ϵ}\left(A\right)\mathrm{and}A\subset U_{ϵ}\left(B\right)\},$$

where $U_{ϵ}\left(A\right)=\{x\in X:d(x,A)\le ϵ\}$.

$(H\left(X\right),d_H)$ is a metric space, and it is true that $H\left(X\right)$ is compact if and only if X is compact.

By means of $GH$, we denote the set of isometric classes of compact metric spaces. We introduce the metric on the set $GH$ as follows [5,6].

The realization of the pair $X,Y$ of compact metric spaces is called the triple $(Z,X^{\prime },Y^{\prime })$, where Z is a metric space, $X^{\prime }\subset Z,Y^{\prime }\subset Z$, $X,Y$ are isometric to $X^{\prime },Y^{\prime }$, respectively, and $d_Z{{|}_{X^{\prime }}=d_X,d_Z|}_{Y^{\prime }}=d_Y$.

$$d_{GH}(X,Y)=\underset{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{X},\mathrm{Y}}{inf}{d}_H(X^{\prime },Y^{\prime }).$$

$(GH,d_{GH})$ is a complete separable metric space.

3. Calculation of Distances

The following Theorems are valid.

Theorem 1.

$$d_{GH}(\mathbb{I},{\mathbb{Z}}_p)={\displaystyle \frac{1}{2}}.$$

Theorem 2.

Let X be a connected compact metric space, $\mathrm{diam}X=1$. Then we have:

$$d_{GH}(X,{\mathbb{Z}}_p)={\displaystyle \frac{1}{2}}.$$

Theorem 3.

Let k be a positive integer such that the inequalities $p^k<q<p^{k+1}$ are satisfied. Then equality is valid:

$$2d_{GH}({\mathbb{Z}}_p,{\mathbb{Z}}_q)=1-{\displaystyle \frac{1}{p^k}}.$$

A subset of $R(X,Y)\subset X\times Y$ of the direct product of the sets X and Y is called a correspondence if the projections of this subset onto the components of the product are surjective: ${\mathrm{pr}}_{X}R(X,Y)=X,{\mathrm{pr}}_{Y}R(X,Y)=Y$.

The distortion $\mathrm{dist}R(X,Y)$ of a corresponence $R(X,Y)$ is the following number:

$$\mathrm{dist}R(X,Y)=\underset{(x,y),(x^{\prime },y^{\prime })\in R(X,Y)}{sup}|d_X(x,x^{\prime })-d_Y(y,y^{\prime })|$$

.

The following statement [6] is true:

$$d_{GH}(X,Y)={\displaystyle \frac{1}{2}}\underset{correspondencesR(X,Y)}{inf}\mathrm{dist}R(X,Y).$$

This statement provides a convenient way to calculate distances in the Gromov-Hausdorff space. Here are some simple examples.

Example 1.

Let $R(X,Y)=X\times Y$, then

$$\mathrm{dist}R(X,Y)=max\{\mathrm{diam}X,\mathrm{diam}Y\}.$$

Therefore,

$$2d_{GH}(X,Y)\le max\{\mathrm{diam}X,\mathrm{diam}Y\}.$$

Example 2.

$2d_{GH}(X,{\Delta }_1)=\mathrm{diam}X.$ Using the triangle inequality

$$d_{GH}(X,{\Delta }_1)\le d_{GH}(X,Y)+d_{GH}(Y,{\Delta }_1),$$

we get:

$$2d_{GH}(X,Y)\ge |\mathrm{diam}X-\mathrm{diam}Y|.$$

Example 3.

Let $f:X\to Y$ be surjective. Then the graph $\left\{\right(x,f\left(x\right)),x\in X\}$ is a correspondence.

There are two important points:

• there is (not unique) optimal correspondence

  $$R_{opt}(X,Y):2d_{GH}=\mathrm{dist}{R}_{opt}(X,Y);$$
• to calculate distances in the Gromov-Hausdorff space, it is enough to consider only closed correspondences.

Proof of Theorem 1. Let $R({\mathbb{Z}}_p,\mathbb{I})$ be an arbitrary closed correspondence. Let’s consider ${\mathbb{Z}}_p$ as a disjoint union of p balls of radius $1/p$, ${\mathbb{Z}}_p={\bigsqcup }_{i=1,\dots ,p}{B}_{1/p}^i$. The family of subsets $\mathbb{I}$ of the form $\{{\mathrm{pr}}_{\mathbb{I}}R(B_{1/p}^i,\mathbb{I}),i=1,\dots ,p\}$ forms a covering of the segment $\mathbb{I}$ by closed subsets. Since $\mathbb{I}$ is connected, at least two sets of our coverage have a common point. The projections on ${\mathbb{Z}}_p$ of the preimages of this common point lie in different balls $B_i$ and $B_j$. Therefore, the distance in ${\mathbb{Z}}_p$ between the projections of the preimages is equal to one. Thus, $\mathrm{dist}R({\mathbb{Z}}_p,\mathbb{I})\ge 1$. Since this is true for any correspondence, choosing the optimal one yields $2d_{GH}({\mathbb{Z}}_p,\mathbb{I})\ge 1$. On the other hand, $2d_{GH}({\mathbb{Z}}_p,\mathbb{I})\le max\{\mathrm{diam}{\mathbb{Z}}_p,\mathrm{diam}\mathbb{I}\}=1$.

Since we used only the connectivity of the space $\mathbb{I}$, the same proof works in the case of Theorem 2.

Proof of Theorem 3.

Let N be a positive integer. Let’s consider ${\mathbb{Z}}_p$ as a disjoint union of $p^N$ balls of radius $ϵ=p^{-N}$. We will choose one point in each ball of the constructed partition. The set $X_N^{\left(p\right)}$ obtained in this way, consisting of $p^N$ points, is provided with the metric $d_N$, induced by the metric on ${\mathbb{Z}}_p$. As a result, we get the metric space $(X_N^{\left(p\right)},d_N)$. It is useful to note that the space $X_1^{\left(p\right)}$ is nothing but a simplex of ${\Delta }_p$. Note that the Hausdorff distance between ${\mathbb{Z}}_p$ and $X_N^{\left(p\right)}$ is equal to $ϵ$. This immediately implies the validity of the evaluation of $d_{GH}({\mathbb{Z}}_p,X_N^{\left(p\right)})\le ϵ$ (it suffices to consider the realization of the pair $({\mathbb{Z}}_p,X_N^{\left(p\right)})$ of the form $Z={\mathbb{Z}}_p=Y^{\prime },X^{\prime }=X_N^{\left(p\right)}$).

The following simple Lemma follows from the triangle inequality.

Lemma 1.

For any metric compact X, the inequality holds:

$$\left|d_{GH}(X,{\mathbb{Z}}_p)-d_{GH}(X,X_N^{\left(p\right)})\right|\le p^{-N}.$$

Let $MST\left(X_N^{\left(p\right)}\right)$ be the minimum spanning tree of a finite metric space $X_N^{\left(p\right)}$. By means of $\sigma \left(X_N^{\left(p\right)}\right)$, we denote the mst-spectrum of the space $X_N^{\left(p\right)}$, that is, the sequence of edge lengths of the minimum spanning tree in decreasing order. The following Lemma is valid.

Lemma 2.

$$\sigma \left(X_N^{\left(p\right)}\right)=\left\{\underset{p-1}{\underbrace{1,\dots ,1}}\underset{p(p-1)}{\underbrace{{\displaystyle \frac{1}{p}},\dots ,{\displaystyle \frac{1}{p}}}},\underset{p^2(p-1)}{\underbrace{{\displaystyle \frac{1}{p^2}},\dots ,{\displaystyle \frac{1}{p^2}}}}\dots ,\underset{p^{N-1}(p-1)}{\underbrace{{\displaystyle \frac{1}{p^{N-1}}},\dots ,{\displaystyle \frac{1}{p^{N-1}}}}}\right\}.$$

Let’s decompose ${\mathbb{Z}}_p$ into a disjoint union of p balls of radius $1/p$: ${\mathbb{Z}}_p={\bigsqcup }_i^p{B}_{1/p}^i$. In each of the partition balls, we will choose one element from the set $X_N^{\left(p\right)}$. The pairwise distances between the various elements of this set are equal to one, that is, it is a simplex ${\Delta }_p$. It follows directly from this that $MST\left(X_N^{\left(p\right)}\right)$ has exactly $p-1$ edge of length 1. Now each of the balls $B_{1/p}^i,i=1,2,\dots ,p$ of our partition let’s decompose into disjoint union of p balls of radius $1/p^2$ (in total, we get $p^2$ balls of radius $1/p^2$) and let’s do a similar reasoning for each of these balls. Continuing these arguments N times, we obtain the statement of the Lemma.

To further prove of Theorem 3, we first prove the estimate from below:

$$1-{\displaystyle \frac{1}{p^k}}\le 2d_{GH}({\mathbb{Z}}_p,{\mathbb{Z}}_q).$$

From the triangle inequality for the spaces ${\mathbb{Z}}_p,{\mathbb{Z}}_q,{\Delta }_{p^k}$ we obtain:

$$d_{GH}\left({\mathbb{Z}}_p{\mathbb{Z}}_q\right)\ge d_{GH}({\Delta }_{p^k},{\mathbb{Z}}_q)-d_{GH}({\Delta }_{p^k},{\mathbb{Z}}_p).$$

Next, we will use the results of [7] (Theorem 3.3). The above theorem states, in particular, the following:

$$2d_{GH}({\Delta }_m,X)=max\{{\sigma }_1-1,{\sigma }_m,1-{\sigma }_{m-1}\},$$

where X is a finite ultrametric space consisting of n points, and $1<m<n$.

Let’s choose a positive integer N, $q<p^N$. Then the equality

$$2d_{GH}({\Delta }_{p^k},X_N^{\left(q\right)})=1$$

is valid, because ${\sigma }_1={\sigma }_{p-1}={\sigma }_{p^k}=1$. In addition, the following equality is true

$$2d_{GH}({\Delta }_{p^k},X_N^{\left(p\right)})=1/p^k,$$

because in this case ${\sigma }_1={\sigma }_{p-1}=1$ and ${\sigma }_{p^k}=1/p^k$. Taking into account the last equalities for sufficiently large N, we obtain the required estimate from below.

To obtain an estimate from above, we construct the correspondence $R({\mathbb{Z}}_p,{\mathbb{Z}}_q)$ explicitly and calculate its distortion.

Let’s represent the number q as the sum of positive integers of the following form:

$$q=q_1+q_2+\dots +q_{p^k},1\le q_i\le p,i=1,2,\dots ,p^k.$$

Note that in this representation, at least one of the terms is not equal to 1 (since $p^k<q$).

Let’s decompose ${\mathbb{Z}}_p$ into a disjoint union of $p^k$ balls of radius $p^{-k}$:

$${\mathbb{Z}}_p={\bigsqcup }_{i=1}^{p^k}{B}_{p^{-k}}^i.$$

Let’s represent ${\mathbb{Z}}_q$ as a disjoint union of balls of radius $q^{-1}$ in accordance with the above decomposition of the number q:

$${\mathbb{Z}}_q={\bigsqcup }_{i_1=1}^{q_1}{B}_{q^{-1}}^{i_1}{\bigsqcup }_{i_2=1}^{q_2}{B}_{q^{-1}}^{i_2}\dots {\bigsqcup }_{i_1=1}^{q_{p^k}}{B}_{q^{-1}}^{i_{p^k}}.$$

Since any compact totally disconnected spaces are homeomorphic, there exists a homeomorphism $\varphi :{\mathbb{Z}}_q\to {\mathbb{Z}}_p$ such that for all $j=1,2,\dots p^k$ the conditions

$$\varphi \left({\bigsqcup }_{i_j=1}^{q_j}{B}_{q^{-1}}^{i_j}\right)=B_{p^{-k}}^j$$

are fulfilled.

As the desired correspondence, $R({\mathbb{Z}}_q,{\mathbb{Z}}_p)$ let’s take the graph of the map $\varphi$.

We’ll show that the distortion of this correspondence is $1-{\displaystyle \frac{1}{p^k}}$.

Let $x,x^{\prime }\in {\mathbb{Z}}_q:{|x-x^{\prime }|}_q\le {\displaystyle \frac{1}{q}}$, then the inequality $|\varphi \left(x\right)-\varphi \left(x^{\prime }\right){|}_p\le {\displaystyle \frac{1}{p^k}}$ is fulfilled by the definition of the map $\varphi$. Indeed, the inequality $|x-x^{\prime }{|}_q\le {\displaystyle \frac{1}{q}}$ means that x and $x^{\prime }$ lie inside a ball of radius $1/q$, and the image of each such ball lies inside a ball of radius $1/p^k$ in ${\mathbb{Z}}_p$. Therefore, for all such x and $x^{\prime }$, the inequality $\left||x-x^{\prime }{|}_q-{|\varphi \left(x\right)-\varphi \left(x^{\prime }\right)|}_p\right|\le p^{-k}$ holds.

Now let x and $x^{\prime }$ lie in different balls of radius $1/q$ in ${\mathbb{Z}}_q$ (in this case, $|x-x^{\prime }{|}_q=1$). There are two possible cases here. The first is when x and $x^{\prime }$ lie in different groups of balls, and the second is when they lie in the same group of balls.

In the first case, we have $|\varphi \left(x\right)-\varphi \left(x^{\prime }\right){|}_p\ge p^{-k+1}$, since $\varphi \left(x\right)$ and $\varphi \left(x^{\prime }\right)$ lie in different balls of radius $p^{-k}$ in ${\mathbb{Z}}_p$. Therefore, the inequality

$$\left||x-x^{\prime }{|}_q-|\varphi \left(x\right)-\varphi \left(x^{\prime }\right){\left|\right|}_p\right|\le 1-p^{-k+1}$$

holds.

In the second case, $|\varphi \left(x\right)-\varphi \left(x^{\prime }\right){|}_p\le p^{-k}$, since $\varphi \left(x\right)$ and $\varphi \left(x^{\prime }\right)$ lie in the same ball of radius $p^{-k}$ in ${\mathbb{Z}}_p$.

Now we will impose an additional condition on the map $\varphi$. As noted earlier, in our partition of a set consisting of q balls of radius $1/q$ in ${\mathbb{Z}}_q$ into $p^k$ groups of balls, there are groups (at least one) consisting of $q_k$ balls such that the inequalities $2\le q_k\le p$ are satisfied. The image of each such group under the map $\varphi$, is a ball of radius $p^{-k}$ in ${\mathbb{Z}}_p$ (each group has its own). Let’s decompose this ball into a disjoint union of p balls of radius $p^{-k-1}$, and divide this set into $q_k$ groups (recall that $q_k\le p$). We will construct the map $\varphi$ in such a way that each of the $q_k$ balls is mapped into its own group. In this case, if x and $x^{\prime }$ lie in different balls from the group of $q_k$ balls, then their images lie in different balls of radius $p^{-k-1}$ inside a ball of radius $p^{-k}$ and, thus, $|\varphi \left(x\right)-\varphi \left(x^{\prime }\right){|}_p=p^{-k}$.

Thus, we have obtained the following properties of the map $\varphi$:

$$\left||x-x^{\prime }{|}_q-{|\varphi \left(x\right)-\varphi \left(x^{\prime }\right)|}_p\right|\le {\displaystyle \frac{1}{p^k}},\mathrm{if}|x-x^{\prime }{|}_q\le {\displaystyle \frac{1}{q}},$$

and in the case of $|x-x^{\prime }{|}_q=1$:

$$\left||x-x^{\prime }{|}_q-{|\varphi \left(x\right)-\varphi \left(x^{\prime }\right)|}_p\right|\le 1-{\displaystyle \frac{1}{p^{k-1}}}$$

or

$$\left||x-x^{\prime }{|}_q-{|\varphi \left(x\right)-\varphi \left(x^{\prime }\right)|}_p\right|=1-{\displaystyle \frac{1}{p^k}}.$$

It follows directly from the last formulas that the graph of the constructed map $\varphi$ has a distortion equal to $1-{\displaystyle \frac{1}{p^k}}$. Therefore, the inequality is valid $2d_{GH}\left({\mathbb{Z}}_p,{\mathbb{Z}}_q\right)\le 1-{\displaystyle \frac{1}{p^k}}$. The theorem has been proved.

4. Geodesics

Note that the correspondence constructed during the proof of theorem 3 (the graph of the map $\varphi :{\mathbb{Z}}_q\to {\mathbb{Z}}_p$) is optimal.

It is not difficult to construct an optimal correspondence $R({\mathbb{Z}}_p,\mathbb{I})$ between ${\mathbb{Z}}_p$ and the unit interval $\mathbb{I}$.

As such a correspondence, consider the graph of the Monna map.

Let ${\mathbb{Z}}_p\ni x=x_0+x_{1}p+\dots x_k{p}^k+\dots$. The Monna map $\mu :{\mathbb{Z}}_p\to \mathbb{I}$ is given by the expression

$$\mu \left(x\right)={\displaystyle \frac{1}{p}}(x_0+x_1{p}^{-1}+\dots +x_k{p}^{-k}\dots ).$$

Calculate the distortion of the Monna map’s graph. Let $x,x^{\prime }\in {\mathbb{Z}}_p:{|x-x^{\prime }|}_p=p^{-n}$. This means that $x_0=x_0^{\prime },x_1=x_1^{\prime },\dots ,x_{n-1}=x_{n-1}^{\prime },x_n\ne x_n^{\prime }$. Then the inequality is valid

$$|\mu \left(x\right)-\mu \left(x^{\prime }\right)|\le p^{-n}.$$

Therefore, for all $x,x^{\prime }:|x-x^{\prime }|<1$, the estimate

$$\left||x-x^{\prime }{|}_p-|\mu \left(x\right)-\mu \left(x^{\prime }\right)|\right|\le 1/p$$

is valid.

Now let $|x-x^{\prime }{|}_p=1$, that is, $x_0\ne x_0^{\prime }$. Let $x_0>x_0^{\prime }$ be for certainty, then ${(x-x^{\prime })}_0=x_0-x_0^{\prime }$ and the inequalities are valid

$${\displaystyle \frac{x_0-x_0^{\prime }}{p}}\le |\mu \left(x\right)-\mu \left(x^{\prime }\right)|\le {\displaystyle \frac{x_0-x_0^{\prime }+1}{p}}.$$

It immediately follows that the distortion of the Monna map’s graph is $1-{\displaystyle \frac{1}{p}}$. Taking into account theorem 1, it can be concluded that the Monna map’s graph defines the optimal correspondence between ${\mathbb{Z}}_p$ and $\mathbb{I}$.

Our task is to construct a geodesic connecting ${\mathbb{Z}}_p$ and $\mathbb{I}$ in the Gromov-Hausdorff space. To do this, we will use the following result from the paper [8]:

Proposition 1.

Let $(X,d_X),(Y,d_Y)$ be compact metric spaces, then for any optimal correspondence $R_{opt}(X,Y)$ there is a family of compact metric spaces $R_t$ such that $R_0=X$, $R_1=Y$ and for $t\in (0,1)$$R_t=\left(R_{opt}(X,Y),d_t\right)$, where

$$d_t\left((x,y),(x^{\prime },y^{\prime })\right)=(1-t)d_X(x,x^{\prime })+t{d}_Y(y,y^{\prime })$$

defines the shortest curve in $GH$ connecting the spaces X and Y.

Thus, the following statement is true.

Theorem 4.

The family of spaces $({\mathbb{Z}}_p,d_0=|·{|}_p)$, $({\Gamma }_{\mu },d_t)$, $(\mathbb{I},d_1=|·\left|\right)$, where ${\Gamma }_{\mu }\subset {\mathbb{Z}}_p\times \mathbb{I}$ denotes the graph of the Monna map,

$$d_t\left((x,\mu (x\left)\right),(y,\mu (y\left)\right)\right)={(1-t)|x-y|}_p+t|\mu \left(x\right)-\mu \left(y\right)|,t\in (0,1),$$

defines the shortest curve connecting ${\mathbb{Z}}_p$ and $\mathbb{I}$ in the Gromov-Hausdorff space.

A geodesic connecting ${\mathbb{Z}}_p$ and ${\mathbb{Z}}_q$ is constructed in a similar way. To do this, instead of the graph of the Monna map, we need to take the graph of the map $\varphi$ from the proof of theorem 3.