Considering the two Spin and the two Angular Momenta String Solutions in $AdS_5 \times S^5$

TL;DR

This paper analyzes two string solutions in $AdS_5 imes S^5$, showing their similarities and differences, and explores their solvability via the Neumann-Rosochatius System, while also discussing a cosmological perspective on the universe's origin.

Contribution

It extends the analysis of string solutions in $AdS_5 imes S^5$ by relating two sectors and generalizing to the Neumann-Rosochatius System, and presents a cosmological argument against the Big Bang.

Findings

Both string sectors are solvable by the Neumann-Rosochatius System.

The two sectors have nearly identical equations with specific variable exchanges.

A cosmological argument suggests the universe has existed forever.

Abstract

In this paper, we consider two almost opposite sectors of actual string configuration ans\"atze in $AdS_5\times S^5$, which anyway have almost the same features: The two spin solution, which has constant angles in $S^5$ and the two angular momenta solution, which has constant "angels" in $AdS_5$, however for the two angular momenta solution, we have to take the time coordinate from $AdS_5$, thus there is a little asymmetry between the two string configurations in $AdS_5\times S^5$. Without being autistic, there is around $69$ equations between the similar equations in the two sectors, compare equations (34) and (104) and also compare equations (64) and (134). Again, without being autistic, the text after the equations in the two sectors, is almost precisely the same. In our notation, the difference between the two sectors is as follows; $\rho\leftrightarrow \theta$, $\phi \leftrightarrow \psi$, $\sinh\rho \leftrightarrow \sin\theta$, $y_i\leftrightarrow x_i$, $y \leftrightarrow x$ etc. The string configurations of this paper, are both solvable by the Neumann System. However, our setup in this paper is generally for the Neumann-Rosochatius System, which is also solvable, since we intend to generalize our results from the Neumann System to the Neumann-Rosochatius System and to several types of deformed Neumann-Rosochatius Systems. In the second part of this paper, which is independent of any string configurations in $AdS_5\times S^5$ and concerns String Cosmology in $D=10$ dimensions. I will seriously argue that there was no Big Bang; I truly believe that the Universe has been there forever, see True Conclusions, Section 10.