# 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?

• "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." That is a pure consequence of your choice of coordinates. Two of the spherical coordinates in Minkowski spacetime are bounded and one is always positive. May 29 at 7:24
• In our universe we think the time dimension has a lower bound at the Big Bang, but a universe with no matter is a perfectly valid solution in GR and in that universe time is unbounded in both directions. So time being bounded is not fundamental in GR. May 29 at 7:26
• Australian sci-fi author / mathematician / programmer Greg Egan has written stories exploring alternative metrics, both ++++ and --++. He explores the mathematics & physics on his site, illustrated with apps, some interactive. See gregegan.net/ORTHOGONAL/ORTHOGONAL.html & gregegan.net/DICHRONAUTS/DICHRONAUTS.html May 29 at 10:57
• You might be interested: Why does a sign difference between space and time lead to time that only flows forward? (in particular the proof cited by Chiral Anomaly.) May 30 at 9:32
• In a spacetime with to timelike dimensions (for example ++--), it is possible to have closed timelike curves, because the causal "lightcone" is a connected region [in our universe lightcone consists in two separate pieces: future lightcone and past lightcone]. So it is not only the sign associated with a direction in the metric that is important, but also the number of spatial and temporal dimensions. In spacetime (++--) it is possible to move back and forth, both in time and space. Jun 16 at 14:49

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".

• +1 for pointing out that there's no continuous rotation in (3,1) Minkowski spacetime that can exchange past with future, whereas any spacelike vector can be continuously rotated (and scaled) to any other. This is a direct geometric consequence of there being only one time dimension (and more than one spatial dimension), so that the light cone separates spacetime into two disjoint timelike regions but only one connected spacelike region. May 29 at 19:47
• (1) You cannot rotate instantly in space, e.g. from left to right, but only with a time delay. So your rotation is not a circle, but a spiral. One can construct a similar rotation in time with no moving backwards. (2) The Minkowski spacetime has no preferred time direction. One can say positrons are electrons moving back in time and where would be no contradiction (other than the experimental entropy). (3) GR allows closed timelike loops, such as inside the Cauchy horizon of a black hole. So a nice theoretical argument, but it fails. May 30 at 3:23
• (1) In Minkowski geometry, you cannot exchange t for x and then x for -t the way you can exchange x foy y and then y for -x. (2) The spacetime model need not have a preferred time direction. The requirement is that it must not contradict the observation that there is a preferred time direction. The arrow of time is additional physics. (3) Models are not required to support the extrapolation of empirical results beyond their domain. May 30 at 12:31
• The "it's an observed fact" response can be given to just about any question asked in physics, I see it more as a way of avoiding a question than answering it. But I think @IlmariKaronen has done a great job of summing up the useful part of this answer. That makes a lot of sense! The spacial directions can't have any arrow without breaking the isotropy of rotational invariance. Whereas an arrow of time is possible without breaking any continuous symmetry, you only have to break the macro equivalent of T symmetry. I suppose chirality is the closest spacial analog of the arrow of time. Jun 1 at 19:46
• @reductionista Real physics questions are more of the form "how does model X reflect observation Y?" Your argument is essentially that you can start with axioms and use math to get a result. But where do your axioms come from? How do you know your axioms are right? Jun 1 at 19:51

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.

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.

• I agree that being bounded isn't a sufficient condition to imply an arrow. But I do see it as a necessary condition. Statistically, for most of eternity the universe must be in thermodynamic equilibrium... the state of maximum entropy. The only way it can be out of equilibrium is if you are near a boundary where there are special correlations rather than pure random motion. If time were unbounded in both directions, there is no boundary you can be near so the entropy must be constant, ie there is no arrow of time. Jun 1 at 19:15
• @reductionista How would you test this hypothesis? Jun 1 at 22:57
• @JohnDoty In the same way you would test the "hypothesis" you made in your answer when you wrote "If the model had more than one time dimension, it would need some other barrier to rotation in timelike planes." These are not hypotheses, they are mathematical conjectures. You test whether they are true by either proving them or finding a counterexample. Jun 1 at 23:14
• @reductionista That's a statement about getting the math to reflect the physics. But you assume that your math reflects the physics. Jun 1 at 23:19
• The only assumption I'm making is that the basic postulates of statistical mechanics like ergodicity hold. If you're suggesting there might be a loophole in my conjecture due to them not holding, fine... I can't prove you wrong. But so far as I'm aware, nobody has observed a violation of that yet so it seems like a plausible assumption if you want to reflect the physics. Jun 1 at 23:42

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.

• I don't think it is necessary to not call time a dimension. But it can indeed be good to distinguish it from the other dimensions due to its dimension (pun intended, the latter in the sense of units, of course). See for example the book "Formal structure of electromagnetics" by Post, who calls time an "essentially distinguishable coordinate" due to its different physical units. Jun 1 at 6:45
• @kricheli. I am not dogmatic about it. I wanted just to point out that one should not mystified time as the fourth dimension. See my reference in physics.stackexchange.com/a/710504/281096 . Jun 1 at 10:24

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.

• The imaginary time coordinate is either an aid to calculation or an old-fashioned (outdated?) self-deception to make Minkowski space "Euclidean". But what does it contribute to the question? May 29 at 17:18
• @kricheli I'm not sure I understand the question well, maybe. The arrow of time and the imaginary i dont have anything to do with each other. Time can run in both directions. And particles can go back in time or run faster than light. Positrons are electrons traveling backwards in time. Tachyons just go forward in time nut faster than light. In the upper part of a spacetime diagram. May 29 at 18:24
• There are many problems I see with your answer. The most glaring problem? Tachyons are science fiction, they can't exist in our universe for obvious reasons. Jun 22 at 6:20

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.

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.