The field of normals and comoving coordinates

The field of normals and comoving coordinates.

Ruslan A. Sharipov

1 ...  4 5 6 ... ru  

In the new theory, which is called the 3D-brane universe model, there is a special infinite family of 3D-branes (see 3D-branes in thespace-time). It turns spacetime into a 3D brane foliation. Each brane in this foliation corresponds to a specific instant in the evolution of the universe. Only one of the branes corresponds to the current instant, that is, the instant of the present. It is that very brane that has the prototype in the form of the real three-dimensional physical universe in its current state. The remaining branes correspond to past states of the universe that have ceased to exist, and its future states that have not yet formed.

Let us choose at each point of each 3D-brane the vector of unit length perpendicular to the brane and directed towards the future. Such vectors form the vector field of unit normal vectors to the branes.

 

Each vector field has integral curves. This is a mathematical fact. Integral curves of a vector field are curves whose tangent vectors at each point are directed along the vectors of the vector field. In our case the integral curves of the field of unit normal vectors to the branes fill the entire space-time and connect  to each other all of the branes from the 3D-brane foliation.

By choosing x, y, z coordinates on one of the 3D-branes, they can be extended to all other 3D-branes of the foliation along the integral lines of the unit normal vector field. The coordinates x, y, z throughout the space-time obtained in this way are called comoving coordinates. They are usually supplemented with a special fourth coordinate, which is called the brane time t.

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