# What grounds the difference between space and time?

We experience space and time very differently. From the point of view of physics, what fundamentally grounds this difference?

Dimensionality (the fact that there are three spatial dimensions but only one temporal) surely cannot be sufficient, as there are tentative proposals among string theories for models with multiple spatial dimensions, and two time dimensions.

One of the most lauded answers in the philosophy of spacetime has to do with the fact that our laws predict temporally, rather than spatially. That is to say, if we are given enough information about the state of the world at one moment, we can predict (quantum considerations aside) the future state of the world. However if we reverse the roles of time and space here, and instead give information about a single point of space for all of time, it seems we cannot predict spatially. Are there equations in physics that can be considered to predict across space (for a given time)?

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I think the main point IS dimensionality. Let's assume time is still 1D. Then if we have information about the entire extent of space at one point, we can only travel in one direction, so we need no more information. If we have 1 point in (let's say 3D) space, there's an infinite number of directions to consider. We can shrink this down to three independent directions, so I would think we should be perfectly capable of making predictions if we had information about the entire extent of time in 3 non-collinear spatial points. But this is by no means a rigorous argument, could be wrong. – Wouter Feb 5 at 12:54

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Regarding just your last sentence: You might be interested to read up on IVPs (and the associated notion of hyperbolic equations) vs. BVPs (and elliptic equations). – Chris White Feb 6 at 0:40
One important difference is that the time coordinate has order (causality), which is absent in the spatial coordinates. – chaohuang Feb 6 at 0:44
Can you give specific references to the assertion of the "most lauded answers in philosophy", I am not sure I follow the philosophical assertion, nor agree with the way its phrased. – Hal Swyers Feb 9 at 13:19
– Kathryn Boast Mar 3 at 15:09

## 7 Answers

@Kathryn Boast I assume you are looking for an answer that is based on the available experimental evidence we have about nature, not wild new speculations that are not firmly established and supported by experiment. It is very interesting to see how a question as simple as this, has many of us going into a spin… There are several answers one can give, depending on whether your frame of reference is Quantum Mechanics, Special Relativity, General Relativity, Super String Theories, Thermodynamics, Philosophy etc. There is however, one aspect of space and time that is the umbilical cord under all these views. This is the fact that space and time are linked together by the speed of light, c, and form the single geometrical structure we call space-time. If it was not for the invariance and constancy of the speed of light, this space-time structure would be impossible. Our ignorance about this fascinating role played by light in nature, before Einstein taught us, had lead to the notion that space and time are totally different from each other. One can say that the only difference between these two is their functionality. Space offers the ‘room’ for matter to exist and move in, and time offers the facility of keeping track of what matter is doing and in which order. This view has been expressed in various forms by some of the other respondents.

As for the last part of the question, yes, there are many equations in physics that ‘predict’ what is happening all over space, globally, at a particular instant in time. Take for example the electromagnetic scalar potential V(x, y, z, t), chose a particular value for time and you get what is happening to V all over the space (x, y, z), by solving the Poisson differential equation. Another famous example is the Time Dependent Schrodinger Equation. Fixing the time to a particular instant, the solution of the equation will give you what is happening elsewhere in space, in a probabilistic sense of course.

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Before reading my answer, please keep in mind that I am just an undergraduate student, so it might not be entirely correct when it comes to such deep questions.

Time and Space are fundamentally different. In case of quantum mechanics, position is an observable, but time is a parameter. And in field theory even space becomes a parameter along with time. Though lorentz transformation connects spatial dimensions and temporal dimensions, still there is a difference. In a lorentz transformation, no transformation from $t \rightarrow -t$, but rotation can take $x \rightarrow -x$. Physically it means, time is unidirectional, that is in all physical processes one can't go from $t_1 \rightarrow t_2$ such that $t_1 > t_2$, but a particle in space can go both in positive and negative direction.

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Philosophical discussion of this can be started from Zeno's Paradoxes and will go for thousand of pages, but physically speaking we have less choices:

In physics, time goes in one direction, and the most fundamental reason for that is: cause and effect (other reasons can be constructed, but this one is the most intuitive and fundamental), this makes time immediately not totally equivalent to spatial dimensions, and cause and effect forces as to "number" events, such that first event, then second event which is the effect of the first...etc, this numbering creates the "illusion" of the possibility of time measurement, and I say illusion because in this context it seems that this numbering is absolute, but it is not because relativity tells us that information exchange speed is limited, thus this "numbering" is relative.

This possibility of measurement tells us that time is very similar to spatial dimensions, even so cause and effect should not be violated, for that Minkowski space is actually pseudo-Euclidean, and not Euclidean, this gives time it's deserved position: it can be measured as spatial dimensions, but it still not same as spatial dimensions, because it is responsible of "parametrizing" cause and effect.

Regarding

That is to say, if we are given enough information about the state of the world at one moment, we can predict (quantum considerations aside) the future state of the world.

This one can be given a very deep explanation in the framework of Bundle foliation that use in differential geometry, if you familiar with this topic, I can explain it for you.

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My guess would be that time is special for us by definition (assuming the universe is causally connected).

I wouldn't speculate on what would be if there were 3 time dimensions and 1 spatial, let's imagine instead that we live in 1+1 dimensions. Then all the observers/particles/people/cats/forest animals/etc would be divided into three groups: 1) Those, which move with the speed of light with respect to anybody from other groups, 2a) Those which, move slower than the speed of light with respect to each other, 2b) Another group, members of which also move slower than the speed of light with respect to each other, but for whom the members of 2a) move faster than the speed of light. However, 2a and 2b cannot interact, because it would violate causality for each of them.

To make it more illustrative, imagine we belong to a group 2a. Nothing can get accelerated to the speed faster than light - alright (except for light itself, which belongs to group 1). If one imagines something faster than light - it is called tachyons, hence they would belong to group 2b. However, tachyons would violate causality, if they could interact with us (there would be systems of reference where effect precedes the cause). Hence, we do not see tachyons and we have our own time coordinate defined this way.

However, I could have written the same paragraph, replacing 2a with 2b and vice versa. Had we been tachyons, we could also say that we evolve in time, and the laws of physics would be the same for us. Hence the answer to your first questions: We belong to one or another group, and this makes one coordinate special with respect to the other one, and this is why it is called time.

One might argue (see other answers), that the above reasoning contradicts classical quantum mechanics. However, it doesn't, if one considers relativistic quantum mechanics istead of non-relativistic one, unless one considers quantum measurements. Another argument, perhaps, would be, that our consiousness does not need to be time-oriented. Or, in other words, one might ask, why does it take time to think. The answer for this I don't know. Finally, one might refer to irreversible processes, such as enthropy increase in the systems acquiring equilibrium, as another case, where time plays a special role.

This all perhaps has something to do with the fact that our consciousness is aligned with time, hence we tend to write physical laws in time-dependent form and specify initial conditions at a constant time. For reversible processes the laws can be also written in a form, where spatial coordinates are an argument. However, typically, we don't have boundary conditions to provide in order to solve the equations. So, because of consciousness such an approach is not practical. It may be practical, though, for the cases, where we do not have to specify boundary conditions, for example for relativistic wave equations.

Hence, the answer to your second question is yes, there are many such equations (such as wave equation, for example), which allow predictions, considering x as a time coordinate. However, some do not seem to allow such predictions (such as equations dealing with irreversible processes) and for some it is not possible to write practically useful boundary (initial) conditions.

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Are there equations in physics that can be considered to predict across space (for a given time)?

If you take time out of the equation, then you take "change" out of the equation. (And without change, things remain the same "all the time".)

If you take all time out of the equation except for one specific moment, then you take causality out of the equation. Things that happen at the same time in different places, cannot have a causal relationship at that time.

A "prediction across space" for one specific point in time would appear random. Therefore even though I am not a physics student, it would appear to me that logically there can be no such equations, by definition.

We experience space and time very differently. From the point of view of physics, what fundamentally grounds this difference?

From the view of physics, I don't think that there is an accepted answer. I have seen explanations based on "causality" as well as "entropy" as well as "no difference, it's spacetime".

This may be because physics does not necessarily seem to match or explain what "we experience". Many physical laws are reversible in theory, yet we never observe them in reverse. Other laws are merely statistical (like entropy), but we never see nature "roll two sixes" (or when was the last time you saw milk and coffee unmix itself?)

One of the more interesting points about "time" I ran into is the fact that without time there would be no space. If there was no time, but space, then you could go anywhere in no time. This is equivalent to saying that you can be everywhere at the same time, and this is the same as saying that there is no space because there is no difference between different points in time.

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Thanks for this. Do you have any references / links etc. with more detail on the idea in your final paragraph that without time there would be no space? – Kathryn Boast Feb 6 at 9:33
I first ran across the idea in one of the popular science books on time. Unfortunately, I have many and am not sure anymore where I read it first. I think it is “Once before time : a whole story of the universe / by Martin Bojowald - eISBN: 978-0-307-59425-9” but I am not sure. Other good books on time: From Eternity to Here, Sean Carroll, and About Time, Paul Davies. – user1459524 Feb 6 at 13:39

We experience space and time very differently. From the point of view of physics, what fundamentally grounds this difference?

Kathryn, in one sense your question answers itself. It may not be the most satisfying sense, but it's an important one.

The distinction is the one Einstein first made when he proposed special relativity, before his former professor Minkowski re-framed Einstein's theory in more graphical terms: Time is the ticking of a clock, that is, the quantification of measurably cyclic processes. Einstein at first did not attempt to frame this idea into highly graphical terms, but of course did so later after some initial mild grumblings about how Minkowski had made his own theory incomprehensible to him.

It is of course trivially easy to take the concept of regular clock ticks and make that into a concept of distance, but it is not a trivial mapping from the perspective of what must be available to make the mapping. It just seems that way because as organisms capable of existing and surviving in our particular universe, we come pre-equipped with the necessary hardware, and with a situation that makes the idea meaningful.

Does all that sound too complicated for something as simple as counting clock ticks and then representing them along a length-like line? Not really.

You can't access the past as you can a distance, so where does the knowledge of the past reside? In something called a "memory" or a "storage device," which must be independent of the part of the clock that does the ticking. So, you have memory.

You cannot interpret memory without some set of operations that recognize and can act on such a representation of past cycles, treating them within some kind of very different construction as if they existed again. Such operations constitute form of intelligence, including a rather remarkable ability to "simulate" or reproduce the physics of a past event, despite having nothing left of that event except a very wispy pattern of information (itself a very strange concept) about key features of that past event. This collective ability to simulate and operate on wispy, ephemeral, extraordinarily incomplete images of the past, yet somehow manage to achieve a meaningful reproduction of their consequences, we call "intelligence."

But how is that even possible? It seems rather absurd that such slender representations can in fact make meaningful predictions of a much denser and tremendously complex collection of matter an energy.

There the universe itself helps us out both by being based not on total chaos, but on slender, simple, and uniform rules of operation. In the case of time, the universe assists us tremendously by being rich in something called cycles, or almost exact repetitions of patterns. Light is cycle. Electrons going around atoms are cycles. Orbits of planets and bodies are cycles. Vibrations of matter, including the gentle swaying of a pendulum in a grandfather clock, are cycles.

Are cycles simple, then? No, emphatically not! Cycles are walks on the edge of a razor, with chaos on one side and locked-down, frozen simplicity on the other. Planets have cyclic orbits, but add too many bodies or too much time and their simple cycles self-destruct into some form of chaos. But if you go the other way and lock cycles into such extreme sameness that there is no measurable change of any kind from one tick to the next, you achieve not a clock, but perfect oscillator that has no more sense of time than does the world of chaos.

It is only that careful balance of recognizable but slightly different cycles -- that is, of repeated patterns that an intelligence can look at and say "that is still the same light, or that is still the same pendulum, despite the slight changes in position or energy or momentum" -- that makes measurable time possible, and through that allows intelligence -- memory plus meaningful, simulation-like operations that somehow mimic and predict the external world -- to perceive "time." In special relativity this cyclic concept of time is typically represented by the concept of "proper time" $\tau$, which is time as measured by an actual clock.

To achieve length-like time only requires one more comparatively simple step, but that last step also runs the greatest risk of deceiving us. We take our model and out counts of almost-identical cycles, and say "this is like a line, this is like a length. I will represent the progression of cycles as a length, using this distance X I have borrowed from the world I can perceive right now. I will call this axis "time" or $t$, and I will postulate that it exists in addition to the axes of length that I can perceive directly.

It's a very good postulate, and special relativity in particular immediately provides us with some non-trivial substantiation of it by showing us experiments where the easiest way to model the results of velocities near the speed of light as cases where the "time" axis $t$ of the speeding object has been bent and rotated into one of the observer's XYZ axes. But even there, beware! The actual events that get measured are again in terms of cycles -- the cycles or $\tau$ time of the observed object appears (to the observer) to be slowed down. That slowing down can again be mapped into a length-like concept of time, but the mapping still exists. Even there, time as a length-like measurement is indirect in a fashion that should be recognized as part of the process, if you want a more complete picture.

The bottom line of all of this is that if you want to think clearly about complex or advanced concepts of time, don't forget poor old cyclic-only, grandfatherly $\tau$ time as the starting point for all time concepts. It is a deceptively complicated concept, one that says for example that classical physics is just as much "observer dependent" as quantum physics. Why? Because every time you use $t$ in an equation of classical physics, you have implied cycles, and wispy, ephemeral memories of cycles, and remarkable sentient operations that use those misty memories to understand and predict what will happen next, then reason on them.

When you say $t$, you imply $\tau$, and when you imply $\tau$... you imply us.

You have asked more than one question. In your last paragraph, I believe your main question is this:

... if we reverse the roles of time and space here, and instead give information about a single point of space for all of time, it seems we cannot predict spatially. Are there equations in physics that can be considered to predict across space (for a given time)

No.

A space-like slice (subspace) of spacetime stores information, whereas a time-like slice (a worldline) does not. More to the point, the very way a "particle" is defined is an attempt to trim away as much variable information as possible, so that the continuity of conserved quantities such as mass, charge, and spin is emphasized.

In classical physics this focus on reductionism across the time axis leaves you with not much more than the history of how the particle was "bumped," billiard-ball style, as it moves across its path across time. Those deflections provide a small amount of information about the universe as a whole, but the total information encoded is quite trivial compared to that contained in any space-like slice, and is certainly never enough to reconstruct the universe as a whole. The amount of information contained in a single particle path is also highly variable. It approaches zero in the case of a particle that simply sits in a very dark corner of intergalactic space and never interacts with much of anything, so in that case you are left in the dark pretty much about everything else going on in space.

Incidentally, you may wonder why in defining time slices I have focused on the worldlines of individual particles, instead say of a single fixed point in space from the start to the end of the universe. You could use the latter approach, and it gives the same result, since for example a single dark spot in deep intergalactic space has even less information about the rest of the universe than a very bored particle sitting there. However, I don't use that definition because "where" that specific point in space really is quickly becomes entangled with the question of "where it it relative to some set of particle worldlines." Since that is the only way to make such an assertion meaningful, it's easier and more honest simply starting with the particle worldlines.

You also noted in your second paragraph:

... Dimensionality (the fact that there are three spatial dimensions but only one temporal) surely cannot be sufficient ...

Yes. In fact, if you look at what I just wrote, it applies just as well to a 1-to-1 ratio of space axes to time axes as it does to a 3-to-1 ratio. Time is simply the axis of quantity (e.g. mass) conservation, while space is the axis on which the relationships (information) that capture the variable relative configurations of those conserved quantities is expressed.

So what does any of this have to do with my earlier answer about time operating first as cyclic rather than length-like? Quite a bit, actually. The cycles are just repeating patterns of relationships between the conserved, particle-like conserved quantities. So, the conserved-over-time mass of a planet orbits the similarly conserved mass of the sun, and from that detectably similar pattern we define as the year.

It's not that you cannot have change without cycles. It's just that without the concept of some patterns "repeating," you cannot create a truly metric concept of time that like space includes definite lengths and distances. That makes the time version of "distance" rather odd, and a lot more complicated than the space-like version.

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Even beautiful and brilliant discussions typically posess clear underlying ideas beneath. I see two in your answer: 1) cyclic proper time and 2) memory (and correct me if I'm wrong). As to 1) there is nothing special about cycles. I use digital watch, some other person could just have a continuous variable in her pocket. Concerning proper time and memory - what about them after all? Why do we actually store memory temporally and experience $\tau$ as we live rather than some $x$? I don't find a clear answer to this in your discussion. – Alexey Bobrick Feb 5 at 21:37
Alexey, oops, I noticed your question but then forgot about it. Hmm. I... don't know an easy way to answer it. My day job requires examining "obvious" assumptions about how intelligence works in a very skeptical fashion, because computers are exceedingly unforgiving in ways that human intelligences are not. I think the issue is this: The consistent, recognizable persistence of anything is one of those deep and distressingly anthropic mysteries of our universe, because many other options (e.g. a hot ball of gas that never condenses) don't even support "objects," let along "cycles" in them. – Terry Bollinger Feb 12 at 3:57
You asked: "Why do we actually store memory temporally and experience $\tau$ as we live rather than some $x$?" Length-like $t$ time is an exceedingly stringy and fibrous thing, an axis whose very direction is defined by those persistences (local conservation laws) I just mentioned. Persistences are what form the fibers and worldlines of length-like $t$, and through those its overall direction. But precisely because such $t$ fibers define "sameness," it is only their braiding and mutual relations in the much richer and infinitely complex axes of space that one can define memories and meanings. – Terry Bollinger Feb 12 at 4:07
Thank you! I see a very good point here. If you let me rephrase it, it would go as: Conserning conciousness and mind, it is dimensionality, that matters. 3 dimensions provide much broader range for possible structures, than 1 dimension. Hence, space is more suitable for defining a state, than time. – Alexey Bobrick Feb 14 at 21:24
You make a good point your self, but alas, I can't claim that one. 3D is indeed close to optimal for connectivity without degenerating into isolationism, but my point above was that particles in spacetime are represented as world lines, with the line part following the "stringy" time axis. So, just at a thread is boring if you only look along its length and ask "did anything change?", a particle in spacetime is boring along $t$ for much the same reason. But if you have additional axes where the particle can be at just one location, you can e.g. tie knots that can become very complex. – Terry Bollinger Feb 16 at 20:05

i think time and space are substancial equivalent but differ by the scale of their cycle. when you move in space, you can turn around go back to the start or go on too ifinity and you will end up at the starting point, this is at least for me the meanning of the determistic view. with time it's the same but causality will prevent you from just turning around and going back in time,the starting point has changed too many aspect ( caused by the other actors) and you wan't recognize him , but this is true aswell if you move around in space. a nother point is , i think nobody of us toke in acount the axiom of choice.this means the futur will never be determinate by the present but the past is unchangable and fixed. for exsample take the conway's game of life: personly iwould say the game has to be played backwards , because in the game each position has one and only one futur followers, but a infity many of past precurser. in reality it seems to me excactly the opesite. But now theirs my point: if we go beyound the horizont of event and we aloud us to go as long as we need to go to find a constelation where in some futur the game will lead up to take us back to the starting point. it is clear that this constelation must be possible and if it is possible, it means that it has to excist. still a nother point : our memory can stock past events in mind and retrive some simulation and remind us of the past. there must be some material way to save or enclose past events in our mind and if it was just one photon we received by our eyes. now in the moment of remembering, we have to build up some kind of connection to this past and what is it else then to travel in time.here it was maybe just one electron, but what was possible for this one electron must have a basic thruth in this world. in the end, the axiom of choice can't be ignored and has to be taken in acount by some mathematical serius atempt to define this world as we observe her . no body can just turn around and go back to start. neither in space neither in time ; it's just not how i percive this world and don't forget mathematics are maid out of the humin minds and reflect only a part of the world.

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