5D Space,Higher Dimensions etc.

[Anamitra]

Let us consider the metric given by the following line element:

$${ds}^{2}{=}{dr}^{2}{+}{r}^{2}{(}{d}{\theta}^{2}{+}{Sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{)}$$
For a particular value of r we have two independent coordinates in the embedded surface--and the surface has been embedded in a 3D space.For several values of r we have several 2D spheres embedded in the same 3D space.

Now let's take Schwarzschild's Metric:

$${ds}^{2}{=}{(}{1}{-}\frac{2m}{r}{)}{dt}^{2}{-}{(}{1}{-}\frac{2m}{r}{)}^{-1}{dr}^{2}{-}{r}^{2}{(}{d}{\theta}^{2}{+}{Sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{)}$$

For each value of "t" we have a 3D surface embedded in a 4D space having the coordinates(t,r,theta,phi)

For different values of t we have several 3D surfaces [time slices each having a constant value of t] embedded in 4D space. In an interval of time the observer passes though a multitude of such surfaces in the same 4D space.There is no need at all to have a 5D space for the purpose of embedding in General Relativity.

The line element in GR has four coordinates, in general ---one relating to time and three relating to space.
A surface constraint like t= constant or r= const reduces the number of independent variables to three. If we want to embed these surfaces 4 dimensions are enough.

 

[Anamitra]

Let us consider an ant living in a two dimensional[spatial] world [spherical surface] described by:

$${ds}^{2}{=}{r}^{2}{(}{d}{\theta}^{2}{+}{Sin}^{2}{(}{\theta}{)}{d}{\phi}^{2}{)}$$
r=constant.

It sees a flat surface around it just as a human being standing in a vast open field would see. The ant can think of a two dimensional rectangular system(X-Y) in the open space around it.
The horizon is seen as a circular ring at a distance[just like a human being sees, standing in the midst of a large field/meadow extending up to the horizon]. The basic consciousness of the ant is based on flat space ideas . Looking at the horizon the ant feels strange----it must be the end of its universe, it wonders. The ant out of inquisitiveness goes out to investigate horizon--the boundary of the universe. It never reaches the boundary---it gets confused. The ant,being an intelligent one considers this "confusing issue" as a parameter linked to the space it lives in[apart from X and Y]. The ant continues to move away and away from the starting point and to its utter astonishment it discovers that it has come back to the initial point after moving away from it .Now it gets a further conviction that "confusion parameter" may be linked with some coordinate---something perhaps similar to X and Y but definitely "confusing"

It decides to fix up coordinates. Starting from the initial point A it moves and comes back to the same point in the shortest possible time[assume it can move with a constant speed only wrt to the surface]. Starting off in a perpendicular direction , again form A, it moves away and returns , in the shortest possible time moving at the same rate. It considers A to be the origin. For an arbitrary point B it chooses the shortest time route[moving at the same speed], which it considers as a coordinate[the distance along this route is taken to be the coordinate]. The angle it makes with any of the two great circles at the origin[ is the second coordinate.
What about the third coordinate?
The ant says to itself “What would the world be like if it took me a much longer time to come back to the same point[moving at the same rate] along any of the great circles considered previously?”