Is time unidirectional because of 4th spatial dimension?

Question

We heard about an expanding universe. Consider an expanding sphere. Consider the surface of the sphere as our 3 dimensional universe. Can time dimension be the radius $R$ of this sphere? And because of its expansion it is one directional?

EDIT: for a moment forget about time. Our universe is expanding. How? It can't expand only in 3 dimentional universe. You can only imagine an expanding 3D universe in a surface of 4D universe like what I explained before. Maybe I was wrong to call this 4th dimention as time. But we must consider another 4th spatial dimention to show that our universe expanding will occur through it. And I think because of this expansion in 4th dimension time is unidirectional. Because expansion through 4th dimensions can't go back so if you put all particles in universe to some previous states it will not means going back through time because there is diffrences in 4th dimention.

I apologize for my poor english.

EDIT:I changed the title: "Is time unidirectional because of 4th spatial dimension?"

Answer 1

I think the idea of spacetime can sometimes be confusing in the sense that people usually imagine time as being another spatial dimension and usually imagine that spacetime is the result of just bringing together time with the spatial coordinates essentially how we make $\mathbb{R}^4$ as $\mathbb{R}\times \mathbb{R}^3$. That is not the point.

I believe we could say that some important things to notice about spacetime are:

1. To label one event (that is, something occurring at a particular instant at a particular place) we need four coordinates $(t,x,y,z)$. The first coordinate, which I called $t$ is what time is, the other three are spatial coordinates.

2. For practical purposes one can think of the difference between $t$ and $(x,y,z)$ being that when you want to register one event, the way you measure $t$ and the way you measure $(x,y,z)$ are fundamentally different. To measure $t$ you can use a clock, to measure $(x,y,z)$ you will need some measuring rods.

3. Finally, if the idea is just that we need to measure time and position, that is $(t,x,y,z)$ to identify an event, why do we need to bring them together? Why can't we just consider three dimensional space by itself and time by itself and stop talking about a four-dimensional thing? The idea is that special relativity showed that space and time make much more sense together. They are not just separate entities, but under the Lorentz transformations, you see that space and time ends up mixing. Since they space and time mix under the Lorentz transformations like different coordinates mix under arbitrary transformations in $\mathbb{R}^3$, it makes sense to model them together as a whole.

The point is that spacetime is a mathematical model. You could consider space and time separately, but the Lorentz transformations are a way to see that they do make sense together.

Another fundamentally difference is the so called spacetime interval. In euclidean three-dimensional space you would consider the distance between two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ to be $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2$.

On spacetime we see that the proper generalization would not be

$$\Delta s^2 = \Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2$$

Rather, it is the so called spacetime interval defined by

$$\Delta s^2 = c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2.$$

This spacetime interval is preserved under the Lorentz transformations and is what really makes sense on the spacetime as a measure of "distance between events".

So I believe the answer is: time is not just another spatial dimension. Is not that we increased the dimensionality of space and time is just like the other spatial coordinates measured along axes with measuring rods. It is that time and space make sense together, bundled in the correct way, this is a mathematical model which agrees with observation, but still, time is measured differently than the position coordinates and more than that, time is not just put together with space to make something exactly like $\mathbb{R}^4$. As stated regarding the Lorentz transformations and the spacetime interval, there are differences.

Answer 2

The first point to make is that the Big Bang didn't happen at a point and the universe isn't like a sphere expanding outwards. Let's get that out of the way so we can get on to the interesting aspects of your question.

Spacetime is a four dimensional structure. One of the dimensions will always be timelike and three will be spacelike, but there is no unique way to divide it into space and time. Phenomena like time dilation arise because different observers disagree about what constitutes the time dimension. So the idea that time can be uniquely defined as a radius of expansion is a non-starter.

However you're very close! When describing any spacetime we have to choose some set of coordinates, one time and three space, and for describing the universe we generally choose coordinates called comoving coordinates. These have the property that there is a well defined time since the Big Bang (the comoving time) that all comoving observers will agree on. Although the universe isn't a sphere as you describe, it can be sliced up into surfaces of constant comoving time. Technically this is a foliation of spacetime into spacelike submanifolds, and for each submanifold the comoving time is indeed the distance to the Big Bang.

In relativity (both special and general) we sort of treat time like a distance, but more precisely we feed time into an expression called the metric that calculates distances. This is discussed further in What is time, does it flow, and if so what defines its direction?. In the case of the comoving coordinates the metric tells us that the distance between some comoving time $t$ and the Big Bang, the proper distance, is equal to $ct$.

Answer 3

Yes it can be. But that's not why time is unidirectional.

There are models of the universe that are consistent with all observations and that have the symmetries we expect where spacetime is $$\mathbb R^4=\{(a,b,c,d):a,b,c,d\in\mathbb R\}$$ and time is equal to $t=\sqrt{a^2+b^2+c^2+d^2}$.

However time is unidirectional for a totally different reason, which is because of the signature of the metric. The metric is what clocks and rulers measure. For instance the thing I called time, $t,$ isn't what clocks measure. It cant be because two clocks that move differently measure differently. So they aren't measuring something about the universe (like where in 4d spacetime you are), they measure something about the particular path they were following in the universe. Rulers do that too. When they move differently they measure differently. So what they measure is something about their path.

What they measure is the metric along their path.

So the metric could allow things to move backwards in time by have two independent time directions (this would give a signature of (2,2) for two independent timelike directions and two independent spacelike directions) so you would have room to do a rotation in time. But the metric in our space doesn't allow you to rotate around all the way, you can only wiggle a bit while you keep moving radially outwards in 4d (in this model, there are other models that are also consistent with observations so far in which time isn't a radial direction in 4d at all).

So the metric is important. For instance it might look like space is getting bigger at a certain rate in this model, after all the 3d set of points with a fixed $t$ seems larger for a larger $t$ but since measurements are based on the metric, the real question is does the metric get larger or smaller on those surfaces of larger $t$ and depending on whether it does the size of the universe (as measured by rulers in the universe that are measuring the metric like all rulers do) could be getting larger, or smaller.

And you could even make the universe collapse in a finite amount of time by making the metric have clocks tick slower and slower for larger $t$ so even though it looks in the picture like you have an infinite amount of time for the universe to live, you might not.

So the details of the metric are super important. And in general relativity the work goes into finding out how that metric changes from point to point and time to time.