In extreme conditions, when the compression caused by lumping of matter reaches a certain level, obliquity angle of the Estring becomes completely perpendicular to the spatial plane. In other words, the original radial direction that has an oblique angle to the spatial plane (circumference) is re-compressed to become steep again. In such a case, slope of Estring reaches infinity and this is called singularity.

According to our definitions, spatial distance metric and time as quantity of clock-ticks decreases to zero in singularities; hence, they are called black holes. In black holes, the knots untie, and they join to become a single strain formation, which is compressed by gravitational effect only.

Figure 7.17

Geometric Generalization assumes that physical reality is a consequence of mathematical (geometrical) and logical principles. This paper itself is an attempt to describe Nature according to mathematical and logical principles.

On the other hand, it would be a big mistake to get lost in the language of mathematics; we should clearly distinguish the consistency in physical reality and the limits of mathematical language.

Let us discuss a very simple example: Suppose that we have a stick, and we assume that its angle to a plane (such as Earth’s surface) is a property that we use to describe that stick. When the stick becomes completely perpendicular to the plane, tangent of its angle with the plane reaches infinity. However, it would be a mistake to assume that our stick itself has become indefinable; it has just become perpendicular. Surely, there are other parameters describing our stick.

In fact, in previous chapters, we clarified that metrics of distancespatial, timeas clock-ticks, and mass emerge with the formation of knot-like strain structure, and their quantities directly depend on the tightness of the confinement volume alone. Hence, it is completely meaningless to talk about these concepts in the lack of physical knots and vortexes.

The knots or vortexes do not exist in singularities, and the expansion is compressed and restrained by the effect of gravity only. On the other hand, there are still parameters, which describe the physical conditions in singularities. Simply, it is the magnitude of the gravitation, caused by the black hole on space-time. Hence, it is the amount of overall strain in the expanding space-time geometry.

Additionally, even in a black hole, we can talk about a geometric distance towards time dimension (Section 6.2). In a black hole, length of the perpendicular Estring itself becomes the distance towards time dimension, and it represents the difference between radiuses of curvature in spatial dimensions.

Please note that, according to Geometric Generalization, the basic unwrinkled space-time (the inflationary epoch) discussed on Section 5.2 is also a singularity, since radial (time) dimension is completely perpendicular to the spatial plane. Naturally, it is not possible to talk about clock-ticks and mass in the inflationary epoch, because knots or vortexes are not formed yet in this state.