Multidimensional singularity

A multidimensional singularity is a hypothetical object that arises as a result of the gravitational collapse of material objects in a multidimensional space-time.

One of the difficulties of the general relativity (GR) is the problem of singularities, which actually arose from the moment Fridmann obtained non-stationary cosmological solutions of the equations GR and became even more acute in connection with the problem of relativistic gravitational collapse.Singularity denotes a state of infinite density of matter, which indicates the inadequacy of GR.

From a physical and mathematical point of view, this problem is solved in a multidimensional space. Let us consider a 3-dimensional object (Fig. 1). The number of atoms of matter in the lattice nodes occupies a volume of $V$. Let the same number of atoms of matter be placed in a 2-dimensional space, i.e. on a plane. The atoms of matter will occupy an area of ​​$S$ with a side of a square of $a(2)$. It is clear that $a(2)>a(3)$, where $a(3)$ is the side of a 3-dimensional cube. The same number of atoms of a substance placed in a one-dimensional space will stretch in length $a(1)$ in the form of a row, and

$a(1)>a(2)>a(3)$

Here $a(n)=d*t$, where $d$ is the lattice step; $t$ is the number of lattice steps. It is intuitively clear that as the number of dimensions of space increases, for the same number of atoms of matter we will need an $n$-dimensional volume with an ever smaller side $a(n)$ of the corresponding $n$-dimensional "cube", that is,

$a(1)>a(2)>\cdots >a(k)>\cdots >a(n)$

It is easy to show that $a(n)$ and $a(k)$ are related by the following relation

$a(n)={a(k)}^{k/n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$

Indeed, (1) follows from the equality of quantity ("volume") atoms of matter in a given $n$-dimensional space

$V(1)=V(2)=\cdots =V(k)=\cdots =V(n)$

where $V(n)$ are the "volumes" of $n$-dimensional spaces that contain the same (equal) number of units of matter — atoms located at the nodes of $n$-dimensional cubic lattices with a step of $d$ in a given $n$-dimensional space. We can imagine that the distance $d$ between particles (atoms) becomes smaller and smaller. The chains of particles in the direction of each coordinate axis become what we call a continuum. And our rows of atoms turn into solid lines $V(1)$, planes $V(2)$, volumes $V(3)$ and so on up to $V(n)$.

And since

$V(1)=a(1)^{1};V(2)=a(2)^{2};\cdots ;V(k)=a(k)^{k};\cdots ;V(n)=a(n)^{n};$

then (1) follows from here.

For 3-dimensional space, from (1) we obtain the following relation

$a(n)=a(3)^{3/n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$

An interesting conclusion follows from relation (2). Suppose we need to fit the entire observable Universe together with its matter into an elementary $n$-dimensional "cube" with a side equal to 10 Planck length units $l_{pl}=10^{-33}$ cm. How many dimensions of space do we need for this?

The size of the observable Metagalaxy is $10^{28}$ cm, or, in Planck length units $10^{28}/10^{-33}=10^{61}l_{pl}$. From relation (2) we have

$10^{1}l_{pl}={(10^{61}l_{pl})}^{3/n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$

From (3) it is clear that even with 183 dimensions of space the entire observable Metagalaxy can be placed in a 183-dimensional "cube" with a side equal to $10\,l_{pl}$, that is, in fact, at a point (183-dimensional). Moreover, the density of matter in such a "cube" will remain equal to the density of matter located in the 3-dimensional space of the observable Metagalaxy.

Indeed, the density of matter in $n$-dimensional space is determined as follows

$\rho (n)={\frac {M}{{V}(n)}}$

where $M$ is the mass of the substance of the observed Metagalaxy, $V(n)$ is the volume of $n$-dimensional space, $\rho (n)$ is the density of matter in $n$-dimensional space. And since, by the condition, $V(3)=V(183)$, then $\rho (3)=\rho (183)$.

A clear example: rolling up a one-dimensional thread of length $r_{1}$ into a flat two-dimensional "rug" in the form of a spiral of diameter $r_{2}$ or into a three-dimensional ball of diameter $r_{3}$. It is clear that $r_{1}>r_{2}>r_{3}$, that is, the compactness of the placement of the thread increases with the increase in the dimension of space, but the density of the placement of the thread substance remains the same (the atoms of the thread substance will still be located at a distance of $d$ from each other in the direction of each $n$-th coordinate axis, (see Fig. 1).

If we consider not a multidimensional "cube" but, for example, a multidimensional "sphere", etc., then equation (1) on the right must be multiplied by the appropriate coefficient.

In the Newtonian theory of gravitation, the planet Earth and other planets are taken as a "point" in relation to the Sun. Or, for example, in the "Feynman Lectures on Physics" in § 5 "Universal Gravitation" it is said "One of the most beautiful celestial spectacle — a globular star cluster. Each point is a star." In physics, a material point is a physical concept (model, abstraction) representing a body or region of space whose dimensions (and shape) can be neglected in the conditions of a given problem. Here the word "point" is written in quotation marks and this concept is defined as a small region of space, although this concept is relative and depends on the scale of the problem. In this article, a "point" is a region of 10 Planck units. In mathematics, a point is zero-dimensional, has no size, and therefore is not relevant to the article.

From this, we can assume that the singular "point" from which, according to GR, our Universe arose, was multidimensional.

We can also assume that at collapse of black holes when the black hole matter reaches the Planck density $\rho _{pl}=10^{94}$g/cm.cube. the matter in the black hole singularity is "squeezed" into other (curled into rings) dimensions of space at distances of the order of the Planck length.

Thus, a multidimensional singularity solves the problem of the infinite density of collapsing matter at the center of the black hole.

See also

• Dimension
• Gravitational singularity
• List of unsolved problems in physics