But for all we know, time IS separate from space. Time is not like the 3 spatial coordinates in many physical systems. For example, Entropy increases with time, which doesn't make sense for a space-like property. Also, causality can be used to define a notion of past vs future even in 4D space time.
This makes intuitive sense to me, but isn't the issue that relativity states that time is intrinsically coupled with space? Ones experience of time relative to another's is directly related to the topological features of the space they're in. If it's more curved, time also "curves" and slows down relative to another observer in flatter space.
How does this notion of time become decoupled from space?
One way I imagine it is that "time" as a concept encapsulates at least two properties:
1) The degree of freedom (i.e., dimension) through which things can change; and
2) The unidirectional flow of causal events
Relativity seems more concerned with defining time in terms of casual events (e.g., event horizons) than its dimensionality. If we define "fundamental time" as a dimension that allows for change can it then be decoupled from space?
The core equation of special relativity that couples space with time is (assuming the speed of light is 1):
ds^2 = dx^2 + dy^2 + dz^2 - dt^2
Different observerse disagree on what each individual term of the right hand side of the equation are; but every observers agrees on what ds^2 is.
Thinking in terms of 3 dimensional euclidean space, this makes sense. If you fix your 3 dimensional coordinate system and pick 2 points in space, you can have:
ds^2 = dx^2 + dy^2 + dz^2
Another observer could pick a different orientation for their coordinate system, and arrive at different values for dx, dy, and dz; but they would still have the same ds. This is just the pythagorean theorem. The distance between two points is the same regardless of how you define your axis. This also means that your 3 spatial dimensions are inherently coupled; because there was no particular reason to pick your axis the way you did.
Simmilarly, in the 4 dimensional spacetime defined by the metric:
ds^2 = dx^2 + dy^2 + dz^2 - dt^2
There is no particular reason to pick the particular time axis that you happened to pick. It is coupled with the other 3 dimensions in exactly the same way that the 3 dimensions are coupled in euclidean space.
The only complication here is that rotating your axis under the Lorentzian metric require the Lorentz transform; whereas rotating them under the Euclidean metric requires the Galilean transform.
The coupling described here involves no notion of causality. Nothing in the metric prevents a path from traveling in both directions along the time axis.
Well, even in this metric, there is an obvious separation between the 3 spacial dimensions and time. For example, a negative ds^2 indicates a fundamentally different kind of distance than a positive ds^2. In fact, events separated by positive distances would be considered entirely independent, while those separated by negative distances can have influenced each other; events for which ds^2 = 0 are simultaneous in any frame of reference.
Further, if two events have are separated by a negative ds^2, then all observers will agree on the order in which they happened, though they will not agree on the length of time that passed between them, or the relative positions.
Note that I'm using your version of the equation for the definition of ds^2 > or <0, though in general I've seen it expressed the other way around, ds^2 = dt^2 - (dx^2+dy^2+dz^2).
There is a clear assymetry between the temporal dimension and the spatial dimension; but it is not clear to me that this implies a seperation.
We can distances with ds^2 < 0 timelike, and ds^2 > 0 spacelike. We say that events with a spacelike cannot influence each other; but there is nothing in relativity that requires that (unless you introduce causality as an additional assumption, which we generally do).
This gets more messy in general relativity when you allow for large masses (read. Black holes).
In the case of a non spining black hole, we spacetime is described by the Schwarzschild metric. Expressing this metric in spherical cordinates, you find that when you pass the event horizon the sign of dt^2 and dr^2 flip, where r is distance from the singularity. This means a "timelike" seperation means that events are closer to each other in the time dimension; and events with a "spacelike" seperation are farther from each other in time. That is to say, "time" behaves in the way we think of as "space", and "space" behaves the way we think of as "time".
