Fourth Dimension

[Appolinaria]

If the fourth dimension is spacetime, is any 3D object that moves through time, now a 4D object?

[ajb]

In a way yes.

 

A particle moving in a d-dimensional space-time will trace out a world-line. That is the "history" of a particle is one dimensional.

 

A string moving in a d-dimensional space-time will sweep out a world-sheet. That is the "history" of the string is two dimensional.

 

And so on for higher dimensional extended objects (branes). They trace out a (higher dimensional) world-volume.

 

It may be very useful to think in terms of these volumes.

[Appolinaria]

Thanks, ajb.

[ajb]

For example, it can be useful to consider string theory in terms of a two dimensional conformal field theory on the world sheet.

[DrRocket]

If the fourth dimension is spacetime, is any 3D object that moves through time, now a 4D object?

 

Spacetime is not a dimension. Spacetime is a 4-dimensional manifold, that is a topological structure in which every point has a neighborhood that "looks like" (i.e. is homeomorphic -- topologically equivalent -- to ) ordinary 4-space.

 

In that local neighborhood one can identify 3 dimensions as spatial and one dimension as timelike. This is due to the nature of the metric imposed on spacetime -- its geometry which is more restrictive than its topology.

 

But there are no global coordinates on spacetime. So there is no universal way to distinguish time from space. It is all just spacetime. This takes some getting used to.

 

Now, as ajb said if you consider the world line of a 3- spatial-dimensional body, that is something that has 3 spatial dimensions in some local coordinate system, then that world lline will be 4-dimensional and will be a basically a tube swept out by a 3-dimensional body along a time-like curve.

 

To make sense of all of this you probably need to study a bit more mathematics and become more familiar with and comfortable with the notion of "dimension". It is not mysterious, but it is a bit more abstract than simply thinking of time as the "fourth dimension". Mathematicians work with spaces of various dimensions, including infinite dimensions, all the time and think nothing of it. Laymen tend to make too much of the mystery of dimensions beyond 3 because they have trouble visualizing it. NOBODY can really visualize higher dimensions, but with some experience one can develop an intuition for higher dimensions based on what we know of dimensions 1-3, which we can visualize. There are some tricks for doing it, but they are not easy to explain until you have more background. Even 2-dimensional manifolds can be very difficult to visualize, because they may be impossible to realize in 3 dimensions (the Klein bottle is a simiple example).

 

It may surprise you to learn that in terms of the topology of manifolds dimensions 3 and 4 are the most difficult. In dimensions 5 and above some things are much easier.

 

http://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture

 

http://en.wikipedia.org/wiki/Classification_of_manifolds

 

http://en.wikipedia.org/wiki/4-manifold

Edited February 11, 2012 by DrRocket

[Appolinaria]

Thanks for the response DrRocket.

 

 

So to add a 3D object here, and it's "history", another space coordinate would have to be added... and so on...