How can time be a dimension when it is relative?

Question

I understand that by definition dimension is defined by just another coordinate to pin-point something in space-time. Therefore we need to know not only where but when. At the same time, this somehow over the time changed to imply that there's a "copy" of universe at each time. Meaning if we had a time machine we could go back to in time by just reversing vector's time component.

But how can this be, if time is relative and dependent on speed of reference frame? Does this assume some universal speed the whole space-time is quantified? Is there some constant Planck distance of time?

More over spatial dimensions are constantly exponentially expanding. So if we imagine the time as for example 2D cartoon, it's frame would be ever expanding as the movie goes on. But what effect would relativity/quantum fluctuation have on the frame and it's pixels? A distortion of some kind, surely.

I understand it is just a model and works OK on Earth (meaning locally), and with relativity accounted for in a nearby space. But how about as a whole? Why is this model then expanded to hypothetically allow time-travel? I'm not talking about sci-fi movies, but about scientific papers trying to achieve this (hypothetically).

Is there even a possibility to pin-point a coordinate in this mess? Doesn't that disprove that time is a dimension, not only to travel in, but also just as a coordinate definition? It doesn't have any stable coordinates (except in non-relativistic speeds locally). If time is measurement of the curve an object took in a time dimension, what is this curve tight to? Any spatial coordinate? But those can't be fixed in time, they change all the time (expansion).

If I wanted to go back in time to 1950, I would have to account not for only speed of all the entities (Earth, Sun, Milky Way, Universe?), expansion of the Universe, but also my speed (time velocity) relative to what? Some arbitrary point?

In short: If I have precise coordinates now in 3D [x, y, z, t(now)] and the point exists, how can time be a dimension if this point didn't existed in [x, y, z, t(before)].

Can someone help to explain?

Answer 1

Time is that which the clock shows. Any one clock. Clocks do not all show the same time but their readings are related to each other and that relation is what the theory describes. In non-relativistic theory any two (perfect) clocks can only differ by a constant time difference but they all progress at the same rate. In relativity any two clocks that are in inertial relative motion can differ by a constant time difference and the rate at which they are progressing. That ratio in rates can be calculated from the relative velocity of the clocks.

That time is a dimension is an often misunderstood and, equally often, poorly explained meme. Time is not a dimension, the theory of special relativity merely treats any one clock-time as a dimension of an abstract construct called Minkowski space. Minkowski space is NOT the affine space you live in with an added fourth dimension. It's a local tangent space that describes how relativistic four-vectors and -tensors transform under changes between inertial systems. It is a mathematical construct that makes perfect sense given how clocks and all other physical quantities (!) behave when seen from non-co-moving observers, but it is not a physical space that you can move in.

Answer 2

The universe is a four dimensional object i.e. to locate any point within it you need four numbers. Most commonly we use a coordinate system $(t, x, y, z)$ and the four numbers give location of the spacetime point in this coordinate system.

You ask:

But how can this be, if time is relative and dependent on speed of reference frame?

and the answer is that the speed of the reference frame does not change the universe - it just changes the coordinate system that we use to measure the universe.

In fact this is a key point to understand in relativity (both special and general). If we take two spacetime points and a line connecting them then the separation of the points, i.e. the length of the line between them, is an invariant called the proper length (or proper time - same thing by a different name). It doesn't matter what reference frame you choose, the proper length always has the same value.

All that changes with the reference frame is the coordinate system. Suppose you're travelling at high speed with respect to me. If I use some coordinates $(t, x, y, z)$ and you use your own coordinates $(t', x', y', z')$ then our two coordinate systems won't match up and you and I will disagree about measurements of time and spatial intervals. But it isn't the universe that is changing. The universe is what it is and it's just our measurements of the universe that change depending on the speed.

When you say:

At the same time, this somehow over the time changed to imply that there's a "copy" of universe at each time.

Technically this refers to a foliation. If I choose a time axis then I can divide up the four dimensional universe $(t, x, y, z)$ into a series of three dimensional sub-universes $(x, y, z)$ - one for each value of $t$. But all I'm doing is choosing how I measure out the universe. You, in your reference frame, can also do a foliation of the universe and your foliation won't be the same as mine because your time axis isn't the same as mine. But that doesn't change the universe because the universe doesn't care how you and I measure it out.

Answer 3

You're confusing concepts, as far as I can tell. Being a "dimension" doesn't imply all values of that dimension (or any of them) are arbitrarily reachable or even exist physically at a given time. It doesn't mean that the intuitive sense of all the baggage a more usual dimension comes with, are applicable to time. It doesn't imply time travel or multiple copies. Moving coordinate systems are trivial but that's a tiny part of it.

So we don't know what "actually" exists "now" for "other times" or even if they do, or if they are reachable, to take your last question about t(now) versus t(before). We just don't know.

The thing is, we don't know much about time, except that we seem to agree events happen in it ;-) so finding out what it means, and examining what laws seem to apply to it, and what they might allow, is about the only approach we have, scientifically, for now.