Regarding possibility of infinite Dimensions

Question

Recently i was taking a look at a video explaining the existence of fourth dimension and thereupon that infinite dimensions are possible. Also it showed that what Einstein told about time being a dimension itself was also incorrect as time is always present,even in one dimension.So how's it all possible,is the statement by Einstein not correct ?

Answer 1

I think we'd need to check what is in the video to give a definitive answer. Do you have a link for it?

In relativity the number of dimensions is related to how we specify the position of a spacetime point. If I want to specify my position at the moment I started typing this I have to say where in space I am but also what time I started typing, so I'd specify it as $(t, x, y, z)$, where $t$ is the time I started typing and $x$, $y$ and $z$ give my position in space relative to whatever reference point is convenient. Specifying a point in spacetime takes four numbers, $t$, $x$, $y$ and $z$, so we say spacetime is four dimensional.

You might argue that time isn't really a dimension, but in relativity (both special and general) time is treated like space, and indeed time and space can get mixed together. So time really is a dimension just like length, breadth and width, and you can't separate time out on it's own.

You talk about infinite dimensions, but it's hard to see how spacetime could have infinite dimensions. In String Theory spacetime has ten dimensions, so a point in String Theory spacetime would have to be specified as $(t, x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10})$ i.e. it needs ten numbers. But I don't see how you could have a spacetime that had infinite dimensions.

One possible confusion is that in Mathematics we talk about mathematical objects called spaces e.g. in quantum mechanics we have a Hilbert space and in many areas we have vector spaces. These can be infinite dimensional, but these aren't meant to be physically real: the word "space" is just slightly unfortunate terminology.

General Relativity doesn't actually specify how many dimensions spacetime has. The maths used in it would work just as well with ten dimensions as with four (though I don't think the maths would make any sense with infinite spacetime dimensions). It would also work with more than one dimension of time or even no time dimension, though this would result in a universe that looks nothing like ours. The number of dimensions is an experimental result i.e. GR with one time and three space dimensions predicts a universe that looks just like the one we observe.

Re your comment: I watched the video, and it's a nice explanation of spatial dimensions. However the presenter is confused about the time dimension. No-one since H. G. Wells has described time as the "4th dimension". In relativity we generally label the dimensions by a number, 0, 1, 2, etc, and we normally label time with the dimension zero. So when above I referred to a point in spacetime as $(t, x, y, z)$, we'd actually write this as $(x_0, x_1, x_2, x_3)$, where $x_0$ is the time co-ordinate. So I suppose we should say time is the zeroth dimension, though this is purely a matter of nomenclature and there is no obvious way to order the dimensions.

If you do calculations in relativity you do indeed have to consider time a dimension just like the spatial dimensions. The difference is a property that we call the signature. See http://en.wikipedia.org/wiki/Sign_convention for some info about this, though the article does go on a bit. Time has the opposite signature from the spatial dimensions. Some physicists give time a negative signature and spatial dimensions a positive signature, while others swap this so time is positive and space is negative. It actually makes no difference when you plug the numbers into General Relativity.

To give you a concrete example of this, in special relativity we define an invariant quantity called the line element, $ds$, by:

$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

so we are adding together movements in time to movements in space i.e. we are treating them as the same thing. Note however that the movement in time, $dt$, gets a minus sign while the movements in space get a plus sign. The requirement that the line element, $ds$, be invariant is what causes all the weird effects like time dilation at high speed.