How many speeds of light in the new theory?

Einstein's equation and consequences from it in the new theory.

Ruslan A. Sharipov.

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Einstein's equation was written by him in 1915. A historical essay about it can be found in Wikipedia. It captivated and captivates many people with its simplicity and elegance.

In Wikipedia you will find this equation written with the plus sign in front of the constant Λ. I wrote it with the minus sign. This is my mistake. It was admitted long time ago in 1997, when I was writing my textbook “Classical electrodynamics and  theory of relativity.” The Internet in Russia and in the city of Ufa was much less accessible that time than it is now. Also there was no Wikipedia; it appeared only in 2001. It was necessary to go to libraries and rummage through catalogs in search of even older textbooks. But I was lazy. In order not to contradict my textbook, I now write Einstein’s equation with the minus sign in front of Λ. This is not a big problem. It is enough to take the constant Λ itself with the opposite sign.

Despite its visual simplicity, Einstein's equation is not that simple. Firstly, this is not one equation, but several equations. In it we see two indices i and j, which run through four values 0, 1, 2, 3. Therefore, we get not one equation, but 16. Not all of them are different. In Einstein's equation we see three quantities R, g, and T with subscripts i and j. These are 4x4 matrices. They are symmetric, that is, their elements do not change when the indices i and j are rearranged. Therefore, the number of different Einstein equations is not 16, but 10.

The primary matrix in Einstein's equations is g with indices i and j. It is called the metric or the metric tensor. The matrix R with indices i and j is called the Ricci tensor. The matrix R is calculated through the matrix g according to some certain rules. There is also the non-matrix quantity R in Einstein's equations. This is the scalar curvature. It is also calculated according to some certain rules through the matrices R and g. These rules are part of the content of the mathematical discipline which is called differential geometry. In 1996, I wrote a textbook based on it: “Course of differential geometry.”

But let's not bother studying differential geometry. Instead, let's continue with Einstein's equation. On the right side of the equation we see the matrix T with indices i and j. This matrix is called the energy-momentum tensor. There is matter behind it. Matter is the whole world. It can be light matter, which is familiar to us, and dark matter, which forms the invisible halo of galaxies.

The denominator on the right side of Einstein's equation is usually set to the fourth power of the speed of light. We replaced it with another constant from the list that we discussed in the other post “ How many speeds of light in the new theory?”

Einstein's equation is four-dimensional in its form and its essence. The new theory of gravity, which is called the “3D-brane universe model”, declares a transition from the four-dimensional spacetime back to the three-dimensional space in the form of a 3D brane and to the one-dimensional brane time. The simplest way to make this transition is to replace the 4x4 matrix g with a block-diagonal matrix with two blocks of the sizes 3x3 and 1x1.

Upon substituting such a matrix into Einstein's equations a system of ten equations is obtained. Of these, 7 equations are left, which are divided into two groups: a group of six equations and one separate equation. The group of six equations has the form.

The indices i and j in these equations run through three values 1, 2, 3. The group of the separate seventh equation has the form.

Formally, this equation corresponds to the choice of i=0 and j=0. That's all! The gravity equations of the new theory have been written out. Their discussion is in a separate post.