Time is the only dimension that has an arrow, and the only dimension which contributes an opposite sign to the metric. Is that just a coincidence? Time is different from space in these two seemingly independent ways.
One of them is generally believed to have to do with special boundary conditions at the beginning of time.
But if you knew nothing about our universe, and you were constructing one from scratch, it seems just as logically possible that you could put the special boundary condition at the edge of any spacial dimension instead.  Where the usually distinction between a spacial and a temporal dimension is which sign it contributes to the metric signature.
I've also wondered about theories with more than one time dimension.  I know these have come up now and then in string theory and other approaches to quantum gravity.  But I've never heard anyone explain whether living in a universe with two time dimensions would mean only one of them has an arrow, or both of them have an arrow... or maybe neither of them necessarily must have an arrow... where I think the last one is probably right.
To put it another way, 3 of the dimensions are unbounded in both the positive and the negative direction... While the 4th is unbounded only in the positive direction, and therefore strictly non-negative if you choose its lower bound as the origin.
Is there anything inconsistent or problematic about a universe whose asymptotic metric signature is (-1, 1, 1, 1) where the -1 direction is unbounded but one or more of the 1 directions are bounded from below (or equivalently, from above) and the metric becomes singular as that special coordinate approaches the origin?  Or is it just random luck, like we had a 1 in 4 shot of the bounded dimension being the -1 and that just happened to be the case?
 A: It is not a coincidence, I think, for the following reason.
First, this question is not just a question about a manifold; it is a question about a manifold and a matter-content together. Considered as a question about a manifold alone I think it cannot be asked in a way that makes sense.
Now the arrow of time is to do with both the behaviour of the manifold and the behaviour of matter, and of course the two are linked via the field equation. But the arrow of time is most easily identified in the increasing entropy of the matter. One could see it in increasingly complex weaving of worldlines of particles (in a classical picture), for example, or in increasingly complex states of fields (in a quantum picture), in such a way that the number of microstates consistent with some given number of macroscopic parameters gets larger as times goes on. The question concerns whether the thing called 'time' here has to be associated with the direction having opposite sign in the metric. It seems to me (I am not quite sure but am proffering arguments) that the answer is 'yes' because the very concept of a worldline is itself already a concept involving time. You can draw spacelike lines in spacetime but they will not be worldlines. The timelike lines we call worldlines are associated with conservation laws, such as conservation of particle number and electric charge in a classical picture. When there are enough conserved quantities along some timelike line then we say the line is a line 'of' something, such as a charged particle, and the something retains an identity along the line. Spacelike lines do not have conservation laws in this way.
By this kind of reasoning I think it is coherent to speak of increase of entropy in the direction associated with opposite sign in the metric, but it is not coherent to do this in other spacetime directions. I have not written nearly enough to make a thorough argument. I have just indicated the sort of observation that I think could be used to make a thorough argument.
A: Minkowski spacetime is a mathematical model constructed to capture aspects of the phenomena we observe. It is a product of the human imagination, like all of our models of physics.
The observed fact that the past is different from the future constrains any model of spacetime. In space, we may rotate ourselves to interchange spatial directions: "forward" may become "backward". Any geometric model of space must accommodate this possibility.
On the other hand, we cannot rotate ourselves to exchange "past" and "future". One consequence of this is that any geometric model that includes time must have only one time dimension. The geometry of Newtonian physics manages this with three spatial dimensions kept completely separate from a single time dimension.
However, this approach struggles with electromagnetic phenomena. Electrodynamics is theoretically simpler in Minkowski's four dimensional spacetime (although we mostly do our approximate practical calculations assuming Newtonian space and time).
Minkowski's spacetime accommodates the arrow of time through its metric signature. While space and time may be partially interchanged through "rotation" (acceleration), timelike intervals may never become spacelike, and vice versa. And then, with only one time dimension, it is impossible to reverse a timelike interval, as the phenomena demand.
If the model had more than one time dimension, it would need some other barrier to rotation in timelike planes. But always remember: this is a theoretical model, an abstraction that exists only in the human mind. It does an excellent job as a model, but it isn't reality. We now understand the Newtonian model as approximate. We may, in the future, understand the Minkowskian model as approximate, and find something else is "better".
A: Remember that the metric signature was discovered in the context of electrodynamics. It is related to the propagation of light and thus, so to speak, a "local" property of space-time. Coordinates being bounded, on the other hand, is a global geometric property of space-time.
In principle, you can thus make the choice of metric signature and coordinate bounds independently. It remains the question of whether the geometry is consistent with equations of GR together with given matter distributions.
And btw. coordinates being bounded does not give them an arrow or an "above"/"below". The arrow of time is a thermodynamic concept as far as I understand.
A: I think time is not a dimension. For Hermann Minkowski, Einstein mathematics teacher and co-founder of spacetime notion, time was an additional dimension (coordinate), probably because it transforms like space coordinates (Lorenz transformation). However, physics is mathematics with units. Time and space have apparently different units. It is more natural to think of 4-dimensonal space with time as an affine parameter along a trajectory there. In that 4-dimensional space all mass points follow their trajectory with velocity c. The infinitesimal length element of that curve is $dl=c\cdot dt$ (see for example  https://physics.stackexchange.com/a/710476/281096). Minkowski metric can be re-written as $$c^2 dt^2=dq^2+dx^2+dy^2+dz^2,$$ where $q$ (quatro) represents the fourth coordinate.
The time as affine parameter on trajectories in the 4-dimensional space have evidently two distinguish directions but it remains parameter and is not an additional dimension.
A: Time can be represented as an imaginary: $it$. This is done to transform the pseudo Euclidean metric to Euclidean.
The fact that entropic time has an arrow associated is admitting that entropic time is unidirectional. This doesn't mean though that the arrow is the cause of the imaginary. A particle can travel on all worldlines on the 4D manifold. A time-like worldline or a light-like one are what we actually encounter. A particle can stay put on one position while traveling in time only but a particle can't stay put in time while traveling to another position instantaneously. This could happen in Newtonian spacetime (instantaneous influence of light or gravity) but it doesn't happen in our universe where the SOL is finite and the same for all.
That being said, tachyons are hypothetical particles always traveling at higher-than-light speed. Which means they always travel backwards in time. They are not able to cross the light-like path of light to change their direction in time. They reside in the two down half-quarter of a space-time diagram cut out by two light-like diagonals and the time and position axes.
A: Whenever you are solving a problem in Mechanics which involves finding time of an interval and if you end up getting time in negative, it doesn't mean that that interval never existed. Yes, the answer may be wrong for that particular question but in mechanics such answers are acceptable.
In mechanics, especially in that of Newtonian, an assumption is made which states that time flow continuously and constantly from infinite past to infinite future. Many questions in mechanics start by stating for considering some event at time t = 0. That means we are assuming some time interval between the time that is flowing from infinite past to infinite future. Now if the time is in negative, it is understood that we are referring to an event that occurred before time t = 0, which we assumed for the sake of that problem.
Hope this help. Incase I am wrong, corrections are welcomed.
A: The reason for these particularities is that time and space are completely different things, besides the Lorentz symmetry they have nothing in common. So if time has characteristics that space has not, there is nothing special about it.
A: The metric signature is arbitrary.  Physics is identical between the $(-, +, +, +)$ signuature to the one in the $(+, -, -, -)$ signature.  It's also worth noting that you don't get an arrow of time at all if you had special relativity in a $(-, -, +, +, +)$ signature
A: The minkowskian representation of space-time is 'not complete', the euclidean metric also has a role to play: either an observer who approaches an event orthogonally, i.e. $s^{2}=c^{2}t^{2}+x^{2}$ , (in 2D to simplify), a simple calculation gives
$\tau=\frac{s}{c}=\sqrt{1+\frac{v^{2}}{c^{2}}}\;t$

It's the same thing to calculate the aberration of light, it is assumed that the light arrives orthogonally to the plane of rotation of the earth...
In general, we can write :  $t = k(\theta,v)\,\tau=\kappa(\alpha,v)\,\tau$, where
$  k(\theta,v)=\frac{1}{\sqrt{1+\frac{v^{2}}{c^{2}}-2\,\frac{v}{c}\,\cos(\theta)}}
$
and
$\kappa(\alpha,v)=\frac{1}{\frac{v}{c}\cos(\alpha)+\sqrt{1-\frac{v^{2}}{c^{2}}\sin^{2}(\alpha)}}$
For: $\cos(\theta)=\frac{v}{c} $ and $\alpha=\pi/2$, we find the Lorentz factor.
