Spacetime "grids" and Spacetime manifolds

[geordief]

What is the relationship between these two concepts?

 

The former seems to be made up of lines and intervals along the 3 spatial and 1 time axes. When all the lines are drawn in, it gives a 4D ""grid" which can be as finely defined as you like (so long as you stay in the macro level)

 

As I see it the volume demarcated can (should?) be a finite volume .

 

At intersections along the grid there are points where a spacetime event may take place but I think that these points may also be "empty"

 

 

 

The spacetime manifold seem to me to represent the same volume but it only contains the points of intersection of the "grid".

 

It is as if these intersections have been "cast adrift" and "float around " in the volume without the connections that seem to be there in the "grid"

 

Does it look like I have a decent understanding of these two models?

 

Is the manifold a superior model to the "grid" (Minkowski Space Time) ? Does it show things that Minkowski SpaceTime struggles to?

Edited November 2, 2016 by geordief

[MigL]

Locally, they are the same.

Globally, a manifold, a topological space, can be much more complex than a simple grid.

( I believe the simplicity is called 'homeomorphic' )

[geordief]

Locally, they are the same.

Globally, a manifold, a topological space, can be much more complex than a simple grid.

( I believe the simplicity is called 'homeomorphic' )

The global situation is more unfamiliar to me.

 

Locally ,if as you say they are more or less equivalent description , could it be advantageous to connect the points (mathematically) in the manifold so that these connections would be changed by relative proximity to mass-energy sources?

 

Also ,does this manifold model permit of imagining a black hole ,moving through it similarly to how an object might pass through misty air?( (obviously differently but is there an analogy there?)

[studiot]

 

geordief post#1

The former seems to be made up of lines and intervals along the 3 spatial and 1 time axes.

 

Strictly speaking, in spacetime all the axes have the dimensions of distance.

[geordief]

 

Strictly speaking, in spacetime all the axes have the dimensions of distance.

Yes , I know. I just mis-spoke.

 

I am interested in the way Einstein begins (I think ) by talking about measurements in space being represented by a grid or array of physical "scaffold like" metre length rods ,which turn into the purely mathematical [x.y.z] co-ordinates.

 

These are supplemented by the ct axis and then it seems to me that the manifold is a further refinement where the lines between the intersections (points or events) are discarded .

 

What though, is the mathematical connection between these points or events and how does it change when a source of mass-energy is in the vicinity?

Edited November 2, 2016 by geordief