# What is Meant by Time & Space Being Continuous ?

What is Meant by Time & Space Being "Continuous"?

Hi All,

Can someone tell me what is meant by time & space being "continuous" as opposed to "discontinuous"? What exactly does this mean in laymen terms and is time & space being "continuous" a widely-accepted "theory" or is this what we may call a "scientific fact"? Any examples and references will be much appreciated.

Thanks!

continuous as opposed to discrete (as in quantized).

Time being continuous means that, if one person sets his alarm to go off at x, and another person sets his alarm to go off at y, then I can always pick a z such that x < z < y for my alarm to go off at, of course assuming perfect alarms =) If time were discrete, then there would be two closest values of x and y, where x and y are distinct (i.e., one follows the other), but there would be no in-between time. You can imagine 2 dimensional spacetime being discrete as a chessboard where rows are space and columns are time. You're either on one square, or another, but there's no in between (as far as chess rules are concerned).

Maybe I'm bad analogy guy =)

It appears to be approximately continuous, meaning that we haven't found evidence that it's not. But we have strong reasons to believe that it isn't.

Space and time continuous means that you can continue to divide the two linked variables as much as you want and you can't find the unit - base of both quantity.

If the space and time would be discrete (quantized), we would expect to find the unit - base the we couldn't divide again.

But it's a theoretical question, no experimental evidences again.

I have, notwithstanding, the impression that the space could be discrete. This could solve many many problems concernig our travel in the space, (if we should be able to handle with it).

"The bridging of the chasm between the domains of the discrete and the continuous,or between arithmetic and geometry, is one of the most important - may, the most important - problem of the foundations of mathematics....Of course, the character of reasoning has changed, but,as always, the difficulties are due to the chasm between the discrete and the continuous - that permanent stumbling block which also plays an extremly important role in mathematics, philosophy,and even physics." (P.Fraenkel, Yehoshua Bar-Hillel, FOUNDATIONS OF SET THEORY, North-Holland, 1958)

Space and time continuous means that you can continue to divide the two linked variables as much as you want and you can't find the unit - base of both quantity.

If the space and time would be discrete (quantized), we would expect to find the unit - base the we couldn't divide again.

But it's a theoretical question, no experimental evidences again.

I have, notwithstanding, the impression that the space could be discrete. This could solve many many problems concernig our travel in the space, (if we should be able to handle with it).

Chatman, thank you for your comments and thank you to everyone who replied to my initial question.
I would like to ask why this theory of time & space being continuous has been so widely accepted as almost as a "fact" of science when there is no experimental evidences for it? Can anyone give me a summary as to why or direct me to some literature regarding this?

Thanks!

Why do you place the burden of proof on those who believe it is continuous, and not those who think it is discrete?

In order to find experimental evidence either way, you would need some sort of model of how the discretization happens. For instance, a lattice does not preserve Lorentz invariance, and leads to a speed of light that is frequency dependent at short distances. On the other hand, string theory allows for a "minimum distance" while preserving these symmetries.

robphy
Homework Helper
Gold Member
Here is an interesting quote by Penrose:
Penrose, R., “On the Nature of Quantum Geometry,” in Magic Without Magic,
Klauder, J. (ed.), (Freeman, San Francisco, 1972), pp. 333-354.

"Let me say at the outset that I am not happy with this state of affairs in physical theory. The mathematical continuum has always seemed to me to contain many features which are really very foreign to physics. This point has been argued forcefully, particularly by Schrodinger and also by a number of other physicists and philosophers.
...
I think it must be the case that the all-pervading use of the continuum in physics stems from its mathematical utility rather than from any essential physical reality that it may possess. However, it is not even quite clear that such use of the continuum is not, to some extent, a historical accident.
...
I wish merely to point out the lack of ﬁrm foundation for assigning any physical reality to the conventional continuum concept."

Why do you place the burden of proof on those who believe it is continuous, and not those who think it is discrete?<SNIP>.

Fair enough Ibrits. I've decided to take upon the task of why I have a problem with those who believe it to be continuous. I have taken the time out to write this so I hope you and those who find this thread interesting will take the time to read it.

Ok, let me start by stating that I am no master of philosophy or mathematics or physics. But I am a fan of all three and do have an interest in these core sciences.

Since we're discussing time and space let me begin by discussing my issue with the prevalent beliefs surrounding the former first.

So as for time, we can come up with many definitions for it but the crux of all the definitions is the same in that it is only through the movement of physical bodies relative to a reference point that we can understand the concept of time. All "time-keepers" ancient and modern work on this principle:
(1) the hourglass uses the movement of sand grains, (2) the water clock, the movement of water, (3) the dial-clock, the movement of the gears that move the dials, (4) the solar clock, the rotation of the earth, (5) digital wristwatches, the movement of electrical
impulses through wire, semiconductors, and crystals, (6) the most advanced clock (the cesium clock), the orbit of electrons around the cesium atom, (7) it has been also recently discovered that the human body has its own time-clock (which resembles an hourglass)
whose flow changes every twenty-four hours, etc.

Thus when the human experiences the phenomenon of time, his mind is actually just measuring the number of fluctuations (e.g., ticks in his internal clock) of some sensory phenomenon.
i.) I believe it’s possible to say that the human mind/body is a discontinuous (opposite to ‘continuous’) instrument which calculates and recalculates its current state
at a fixed interval. [as the human mind sends electrical impulses to its neurons which after gaining enough information reconstructs/refreshes its present state similar to how a finite state machine, such as a computer works]. This is why if a phenomenon happens too quick, the mind will not perceive it.
ii.) IF indeed the human mind is discontinuous, then it cannot be easily proven that the
physical world (which consists of space reconstructed through time) is continuous. [As a side note, another proof that the world may NOT be continuous is the Uncertainty Principle in Physics which states that we cannot accurately describe both the location and time of a particle due the delays in receiving the information of its location
at a particular time.]
iii.) If we move a body from point A to point B (and there is one meter between point A and point B), we can count a great number of states of motion, but cannot count an infinite number of states of motion(again due to the delays in receiving the information about the body's movements). Thus, we cannot prove that the body actually went through an infinite number of states.
iv.) It has been said by some physicists over the past 100 years that all matter is made up of distinct and finite building blocks. For example, a one foot iron rod is made up of a finite number of Fe atoms. It is not valid to state any longer that one can divide up a one foot rod an infinite number of times. Now, of course, we can go further and count the protons, neutrons, and electrons in the one foot iron rod, but we will still end up with a finite number at a particular time. Now, of course, we could go even further and count the quarks (e.g., the one's with up spin, down spin, etc.) in the iron rod, but again we in the end would end up with a finite number. It may happen that we find even smaller particles in the future which make up the smallest known particle now; but again in the end we will end up with a finite number of particles no matter how deep we go in this nested scheme.
v.) Thus, we conclude that space which is made up of matter and matter-voids is discontinuous and not continuous.
vi.) The matter-void becomes the domain in which matter rests and we know that the space in matter-voids can only fit a finite number of pieces of matter. Therefore, we conclude that these voids must also be of finite dimensions (otherwise, they would be able to fit an infinite number of pieces of matter).
When we move a body from point A to point B as is noted in section (iii), the body successively comes closer to its destination. For example, when at point A, the body is one meter away from point B. If we move it move it midway between point A and point B, it is now half a meter away. Therefore, we conclude that the matter void between point A and point B is divisible - as the distance remaining can be obtained by dividing the original distance by some factor (in this case '2').
vii.) Anything that is finite in dimensions and is divisible cannot itself be continuous.
This is because if we state that a divisible realm of finite dimension were continuous that translates into the claim that we could break it into an infinite number of pieces and that
the sum of this infinite series would total its finite dimensions. But, if we divide any finite number by infinity, we will get zero. Thus, that would mean that each of the infinite number of pieces which make up the realm of finite dimension would be of zero
size. And if we sum these pieces of zero size, we will get zero whereas we already know that the finite dimension of the matter-void is greater than zero. And this leads to a contradiction which forces us to reject the proposition that a divisible realm of
finite dimension could actually be continuous.
Now returning to my example about moving a body from point A to point B, it is obvious from our above discussion that the body can only have a finite number of movement states between point A and point B since the matter-void through which the body moves is discontinuous.
viii) Now returning to our definition of time, if the number of states of a moving body from point A to point B is always finite, it cannot be proven that time itself is continuous as time is only measurable by the movement/fluctuation of finite sensory phenomena.
ix) Rather, I would state that it is provable that time is discontinuous. Time here
is analogous to the matter-void point stated above and the events which take place at a particular instance in time are analogous to the contained matter in the points stated above. If time were continuous, then that would force us to claim that we could divide a fixed time interval an infinite number of times. But again, each time piece would be of zero length and all of them together would sum to zero. But, as I stated above that since a body moving from point A to point B requires a fixed time interval to be at each location between the two points, time intervals are of sizes greater than zero. And again this would lead to a contradiction would it not?
x.) Next, I would state that it is provable that time (as conceived by ancient and modern man) does not in actuality exist. And what it really is - is an imaginative metaphor created in the human mind to explain the differences in states which the human being experiences at disparate intervals. And the only metaphor of time which conforms to reality (as summarized above) is that of linear (non-circular but not necessarily non-multidimensional (e.g., time may branch out like a tree) forward-progressing time.
xi.) The human being can only experience one physical state at a time. It is only because of the human's memory that he can experience the concept of the past. This is because if the human did not have a memory, then he would only know the state which he is experiencing currently. Thus, in such a case, he would be unable to see things as progressing from his previous experiences to his current experiences. It is only because of the human's imaginative faculties that he can experience the concept of future. This is
because if the human did not have the ability to imagine other than what he currently sees at present, he would be unable to expect another state in the future.
xii) So if the concepts of past and future rest on the human's memory and imaginative faculties (which are internal to him), then it cannot be proven that time as conceived
by the common man actually exists. Rather, I would say that it is provable that time (which is considered by the common man to be a smoothly flowing domain in which events take place) in actuality does not exist.
xiii.) If we state that time is a freely flowing domain, that would force us to claim that time can exist without events to hold; otherwise, it would not be freely flowing but be tied to disparate events (as explained above).
If we propose that time can exist without events to hold, then we would state that it cannot be flowing, but must be stable.
The reason for this is that if time were independently flowing, it would mean time itself could experience change (as it will keep adding discontinuous time intervals to its length as its flow continues). And anything that can experience change needs another "time-like" dimension to quantify its change. We will call this other proposed "time-like" dimension of time "time-2". Now the same thing would apply to "time-2" in that it could either be tied to time or be an independent flowing domain which holds time. And the same argument about the flow of time-2 would apply in that if it could exist and flow independently, then it itself would need a time-like dimension to quantify it. If we propose that time-2 is not an independent flowing domain which holds time, but is tied to time, then that proves that time itself could not have a directional flow (as it has no independent quantifiable domain in which its directional flow can be measured (this is also because time-1 and time-2 are similar in their characteristics and purpose; thus, saying that they are fixed/tied to each other is the same as saying that only time-1 exists; but if an independent time-2 does not exist, then time-1 cannot experience flow/change)).
However, if we propose that time-2 is an independent domain which can experience flow, then we would need yet another time-like dimension which we will call time-3 to quantify time-2's change. And thus, we could continue on like this forever. If we propose at any iteration that time-x is not an independent flowing domain, then that will mean in sum total that time-1 could not experience change or flow (as each level will keep collapsing until we reach the original time-1).
And if we keep stating at each iteration that time-x is an independent flowing domain, we will end up with an infinite series which never ends. This would mean that the sum of the
discontinuous time intervals of each time line at each level at any particular time (in accordance to the measurement of time-1) would neither be odd nor even (as the infinite series of time lines would lead to an infinite number of discontinuous time intervals). However, we know from the laws of mathematics and counting that all discontinuous phenomena must add up to either an odd or even number at a fixed point in time. Thus,
we conclude that such an infinite series of time-lines is impossible and at least one time line at some iteration must not be independent and flowing. But as we stated before, as soon as we conclude that a higher iteration time-line is not independent and flowing, this will cause all of the levels below to collapse until we reach time-1 forcing us to accept that time-1 cannot experience change and thus cannot have a directional flow.
xiv) Thus if time does not consist of an independent flow, it is useless arguing about the direction of its progress (either forward, backward, or both forward and backward simultaneously). Rather, the concept of flow can only be understood as a metaphor for the human's previous memories and future imaginative expectations. And this metaphor can only lead one to consider time to be linear and forward-progressing as one frame is shown to the human at a time which his memory recalls.
Those that hold the possibility of backward flowing or circular time seem to have made the mistake of considering time an independently flowing domain in which events can take place or not take place.

So in conclusion, it seems to me that:
a) Space can most certainly be discontinuous.
b) Time can most certainly be discontinuous.
c) Time, in reality, has no directional flow (but one can metaphorically understand time to be linear and forward-progressing by using human experience as a base for building this metaphor).
Thoughts, comments, criticisms, complaints, compliments?? Be kind now! D H
Staff Emeritus
Thoughts, comments, criticisms, complaints, compliments?? Be kind now! My first thought: Why are you trying so hard to conserve whitespace? A few blank lines would make this much more readable. My second thought, gibberish.

Ok, let me start by stating that I am no master of philosophy or mathematics or physics. But I am a fan of all three and do have an interest in these core sciences.
Philosophy and mathematics are not science.

So as for time, we can come up with many definitions for it but the crux of all the definitions is the same in that it is only through the movement of physical bodies relative to a reference point that we can understand the concept of time.
The things we measure and the ways in which we measure them are completely distinct concepts.

IF indeed the human mind is discontinuous, then it cannot be easily proven that the
physical world (which consists of space reconstructed through time) is continuous.
The universe existed before humanity and will continue to exist long after we are gone. Anthropocentric arguments don't work.

As a side note, another proof that the world may NOT be continuous is the Uncertainty Principle in Physics which states that we cannot accurately describe both the location and time of a particle due the delays in receiving the information of its location at a particular time.
The uncertainty principle does not result from delays in receiving information. If that were the case, we could take these delays into account. The uncertainty principle is about measurements of canonical pairs of variables. Theoretically, I can accurately measure the position of a particle to any degree of accuracy. (That means I have to give up on knowing the particles momentum, of course.) The uncertainty principle has nothing to do with the continuous/discontinuous nature of the universe.
It has been said by some physicists over the past 100 years that all matter is made up of distinct and finite building blocks.
That is the quantum theory of physics. Quantum physics is one of the most rigorously tested theories in science. You don't have to resort to philosophical circumlocutions ("it has been said by some physicists") at this cite. The quantum nature of matter has nothing to do with the quantum nature of space and time (at least as far as we know).

v.) Thus, we conclude that space which is made up of matter and matter-voids is discontinuous and not continuous.
No, you cannot conclude this.

Nature of continuums

As quoted above:

Penrose, R., “On the Nature of Quantum Geometry,” in Magic Without Magic,
Klauder, J. (ed.), (Freeman, San Francisco, 1972), pp. 333-354.

"Let me say at the outset that I am not happy with this state of affairs in physical theory. The mathematical continuum has always seemed to me to contain many features which are really very foreign to physics. This point has been argued forcefully, particularly by Schrodinger and also by a number of other physicists and philosophers.
...
I think it must be the case that the all-pervading use of the continuum in physics stems from its mathematical utility rather than from any essential physical reality that it may possess. However, it is not even quite clear that such use of the continuum is not, to some extent, a historical accident.
...
I wish merely to point out the lack of ﬁrm foundation for assigning any physical reality to the conventional continuum concept." bold fonts added for emphasis

Down to Planck scale of pseudo-Riemannian spacetime manifold (continuum), is finite. So how does one mathematically establish a continuum? By matching, mapping sets; thus a function. Can the real numbers, the rationals, or the integers be matched to spacetime manifold on a coarser to finer scale? No. Rather only a finite set can be matched to spacetime manifold on a coarser to finer scale. Also discreteness depends upon which continuum one is working with. Something simple to remember as more or less synonymous is: manifold i.e. continuum i.e inbetweeness quality.

Chatman, thank you for your comments and thank you to everyone who replied to my initial question.
I would like to ask why this theory of time & space being continuous has been so widely accepted as almost as a "fact" of science when there is no experimental evidences for it? Can anyone give me a summary as to why or direct me to some literature regarding this?
Becasue virtually all of the equations and models of physics are built upon continuous spacetime. Physicists hate changing their mathematical models,so the evidence of discrete spacetime would have to be overwhelming for it to be accepted.

This, despite the fact that at least one theory (scale relativity or as some call it extra-special relativity), which assumes non-discrete spacetime, actually leads to all of the equation sof SR and QM based solely on the addition of that one extra postulate, and predicts more accurate values for the fundamental constants than currently experimentally known. It's got a lot of promise, but has been utterly ignored as "numerology" and "poetry." Pity.

First of all, the continuous model of space-time is just a model? of course it is! as everything in physics is just a model. In our BEST current model, that is classical GR, for space-time, it's a pseudo-Riemanninan differentiable manifold (except at a separable finite set of points), end of discussion.

Are there any REAL and CONCRETE, i.e. EXPERIMENTAL, evidence for thinking space-time may not be continuous? NONE, as far as we can probe it at some 100~200Gevs, NOTHING appears to be discrete.

Are there some GOOD theoretical reasons to believe its discrete at some fundamental level? Ok, this is a matter of discussion but there is a lot of misunderstanding here:

I) For example, in LQG calculations that can ONLY be done at kinematics level, space-time appears to be discrete at Planck-scale. Does it means something? It's well know that for highly non-linear theories as GR, the picture we have at the kinematics level may have nothing to do with the real thing. Before one can solve the full dynamics of this theory, NOTHING of concrete can be said. Also, the area and volume operators which 'are' discrete in this formalism are not even observables.

II) There are some indications - and probably the most close we could get of a 'concrete theoretical evidence' - that at the Planck-level, one should consider the contribution to the Path-integral of GR of non trivial topologies (some hawking works from the 80's, for example). However, non trivial topologies have NOTHING to do with discrete space-time, even if some of these non trivial topologies are related to discrete symmetry groups.

III) Quantization DO NOT means discretization. Is this clear?

IV) There are some evidences space-time cannot be discrete? as far as I know, don't. But this means nothing. It's easy to dream about a situation far (VERY far) away from our current experimental knowledge.