# Does taking the derivative of time really make sense?

#### zeromodz

Think about it, say we have dt. dt is an infinitesimal amount of time that elapses with respect to another quantity. However, physics sets standards for how we define time. The shortest length of time possible is planck time. Planck time is 10^-43 seconds in magnitude and is defined as the shortest meaningful time in the universe. Any time smaller than this makes no physical sense, so there is no such thing as an infinitesimal amount of time. So does dt automatically equal 10^-43 seconds or what?

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#### DragonPetter

I think mathematically, dt can still be considered continuous in many ways if the order of magnitude of change of the quantity is very large, but I wonder if processes actually can occur in this planck time scale to where it would be meaningful.

Say a large star fuses, on average, X amount of atoms over the planck time dt. If X is a large number, then it would imply that, continuously, a smaller number of atoms were burned at a shorter increment of time between this smallest increment limit. Even if the actual process is limited by the planck scale, a continuous differential equation could still have meaning macroscopically.

Either way, we have difference equations and ways to represent discrete time systems. If the fabric of the universe is truly discrete, then of course continuous calculus would break down at some point. I don't know much beyond laymen's physics when talking about planck scales and extra dimensions, so I'd be interested how more knowledgeable people here will interpret your question.

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#### DragonPetter

Think about it, say we have dt. dt is an infinitesimal amount of time that elapses with respect to another quantity. However, physics sets standards for how we define time. The shortest length of time possible is planck time. Planck time is 10^-43 seconds in magnitude and is defined as the shortest meaningful time in the universe. Any time smaller than this makes no physical sense, so there is no such thing as an infinitesimal amount of time. So does dt automatically equal 10^-43 seconds or what?
Also, to answer your question point-blankly, no, taking the derivative of time does not make sense. You are not taking a derivative of time ever, you are taking the derivative of a function with respect to time, and so that distinction I would say means your question doesn't make sense; ignoring the rest of your post of course.

#### Pengwuino

Gold Member
Calculus involves limiting processes. A good example in my mind is how mortgage payments and quantities are determined. You can model the amount of money paid or how much is left or whatever you want using an exponential function. However, it is done in a limiting procedure that dictates that payments are continuous. In reality, you are paying every month, not on a continuous basis. If you look at something where the number of payments is some massive number, say you're paying over 10,000 years and you're paying once a day, you can pretty much say that the payments are a continuous thing. The math allows for arbitrarily small "time between payments" and can take the limit as this $\Delta t \to 0$ so that you get a continuous model. However, the mortgage is in finite steps, 1 every day. So on one hand, you have the math allowing you to make arbitrarily small intervals in such a way that you can pretty much say it's a continuous thing, whereas the mortgage payments are in reality small one-at-a-time things. The math will model the reality stupendously.

The way this relates to physics is kind of the same thing. The plank time is so small that the math correctly models the physics assuming continuous time intervals.

#### DragonPetter

That's not true and doesn't get at the heart of the matter (think relativity). Any parameter, be it time, space, temperature, can be questioned in this same manner.
But you can't take the derivative of a parameter. You must take the derivative of a function with respect to a parameter.

#### vanesch

Staff Emeritus
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Think about it, say we have dt. dt is an infinitesimal amount of time that elapses with respect to another quantity. However, physics sets standards for how we define time. The shortest length of time possible is planck time. Planck time is 10^-43 seconds in magnitude and is defined as the shortest meaningful time in the universe. Any time smaller than this makes no physical sense, so there is no such thing as an infinitesimal amount of time. So does dt automatically equal 10^-43 seconds or what?
The point is that in those situations where you are supposed to take a derivative wrt time, you are working with a *theoretical model* that assumes that time is a *real number* (that is, an element of R). In other words, you are working within the frame of a theoretical model, and within that model, the derivative makes sense. Whether this corresponds to reality or not is a different matter. In fact, you can always assume that a useful model *approaches* reality sufficiently to be useful for the purpose you are using it (otherwise you are just using the wrong model).

Your question is interesting, because it should make you think about the distinction between the mathematical models you're using (Newtonian mechanics, classical electrodynamics, quantum physics.... + approximations and idealisations of the problem at hand, like a "point mass", or a "uniform E field" or "a spherical electrode" to geometrical perfection) and the reality you are trying to model with it, which, you should assume, is always "somewhat" different.

When you're calculating the velocity of a car, you don't give any damn about accuracies beyond 40 decimals, and in any case your idealisations will give you problems before that (did you take into account the thermal vibrations of the car ?). So you can use Newton's model, that there are material points that are represented by a point in Real Euclidean space as a continuous function of a real number, time. And then the derivative IS defined. Even though it will of course not make *physical sense* at the 40th decimal or something.

#### Pengwuino

Gold Member
But you can't take the derivative of a parameter. You must take the derivative of a function with respect to a parameter.
Yes, I shouldn't have written that. I deleted it.

I guess what I really meant to say was that, for example in special relativity, you can find ${{dt}\over{d\tau}}$ if you want to find a relationship between the proper time and coordinate time. You can differentiate time, but actually time is no longer the parameter; the proper time becomes the parameter of interest.

#### arildno

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The point is that in those situations where you are supposed to take a derivative wrt time, you are working with a *theoretical model* that assumes that time is a *real number* (that is, an element of R). In other words, you are working within the frame of a theoretical model, and within that model, the derivative makes sense. Whether this corresponds to reality or not is a different matter. In fact, you can always assume that a useful model *approaches* reality sufficiently to be useful for the purpose you are using it (otherwise you are just using the wrong model).

Your question is interesting, because it should make you think about the distinction between the mathematical models you're using (Newtonian mechanics, classical electrodynamics, quantum physics.... + approximations and idealisations of the problem at hand, like a "point mass", or a "uniform E field" or "a spherical electrode" to geometrical perfection) and the reality you are trying to model with it, which, you should assume, is always "somewhat" different.

When you're calculating the velocity of a car, you don't give any damn about accuracies beyond 40 decimals, and in any case your idealisations will give you problems before that (did you take into account the thermal vibrations of the car ?). So you can use Newton's model, that there are material points that are represented by a point in Real Euclidean space as a continuous function of a real number, time. And then the derivative IS defined. Even though it will of course not make *physical sense* at the 40th decimal or something.
It isn't just that simple. is it vanesch?
After all, there are numerous examples where the (naively) associated difference equation shows quite a different behaviour than the differential equation.

Furthermore, might it be that the divergent integrals appearing within quantum mechanics are symptoms that using continuous models for essentially discrete (and finite) phenomena generates a number of "problems" precisely because the modelling of the world as a continuum (or inter-related continua) is actually physically dead wrong?

Staff Emeritus
However, physics sets standards for how we define time. The shortest length of time possible is planck time.
This is not true, and since the premise of your argument is not true, the conclusions do not follow.

#### zeromodz

This is not true, and since the premise of your argument is not true, the conclusions do not follow.
I thought there was a consensus amoung physicists that planck time is the shortest possible time interval in our universe. When we model the big bang, we use units of planck time to show how fast everything happened. If planck time is not the shortest amount of time, what is?

#### vanesch

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Gold Member
It isn't just that simple. is it vanesch?
After all, there are numerous examples where the (naively) associated difference equation shows quite a different behaviour than the differential equation.
Well, that would only indicate that the "continuous" model is a bad model, but it wouldn't make the taking of the derivative within the continuous model problematic.

BTW, if we could NEVER approach any discrete system more or less reliably with a continuous model, then most fluid dynamics, and most continuum mechanics, most elasticity calculations would be wrong etc... because we all know that matter is not a continuum, and consists of atoms on this level.

Of course, you are right that *sometimes* there's a problem in the continuum model that arises from the lower-lying discreteness. But there's no *mathematical* error in the continuum model because it doesn't describe correctly, in all circumstances, a discrete nature. Within a continuum model, the stress tensor is defined mathematically on scales far below the atomic dimensions for instance. That doesn't make physical sense because we know that the model isn't applicable there, but it is mathematically perfectly defined.

Furthermore, might it be that the divergent integrals appearing within quantum mechanics are symptoms that using continuous models for essentially discrete (and finite) phenomena generates a number of "problems" precisely because the modelling of the world as a continuum (or inter-related continua) is actually physically dead wrong?
You might ALSO have a mathematically problematic "continuum" model of course. But the question of the OP was, as I understood it: "how can we take a time derivative (in, say, Newtonian mechanics), if we somehow "know" (we don't, actually, but suppose we did) that time is discrete ? "
My answer is simply: WITHIN the model of Newtonian mechanics, dynamical variables are real functions of t in R, and you can take the derivative in that model. Whether or not this makes physical sense has nothing to do with the question of whether the mathematical operation is well-defined or not.
And in as much as the model is "applicable", the derivative is also "applicable" (up to the accuracy of the model in this case, and with the caveat you pointed out, that *sometimes* a microscopic discrete behavior is different from the smoothed-out continuum model that one uses for it).

Staff Emeritus
I thought there was a consensus amoung physicists that planck time is the shortest possible time interval in our universe. When we model the big bang, we use units of planck time to show how fast everything happened. If planck time is not the shortest amount of time, what is?
No, there is no such consensus and no evidence that there is a "shortest amount of time".

#### arildno

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Sure enough, vanesch.

I just wished to highlight the distortion that continuum models might do to discrete phenomena, because it seemed to me that OP had been mulling over such issues.

That, surely, does not make continuous modelling mathematically inconsistent, or without predictive value (the ones in use have enormous value).

Furthermore, the best way to find out what flaws our tool might possess is to poke around with it (i.e, do research) and find out where it has tried to do something..unnatural.
Nature will protest, eventually.

#### vanesch

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Gold Member
Sure enough, vanesch.

I just wished to highlight the distortion that continuum models might do to discrete phenomena, because it seemed to me that OP had been mulling over such issues.

That, surely, does not make continuous modelling mathematically inconsistent, or without predictive value (the ones in use have enormous value).

Furthermore, the best way to find out what flaws our tool might possess is to poke around with it (i.e, do research) and find out where it has tried to do something..unnatural.
Nature will protest, eventually.
Yes, you're right of course. My aim was rather pedagogical, because it seemed to me that the OP hadn't grasped the difference between the (Platonic ? ) physical concept of "time" which is not really well-defined outside of a given theoretical frame, and which *might* turn out to be discrete (or not to exist as we think of it, after all), and the mathematical concept that corresponds to it *in a given theoretical frame*, which is of course clearly defined mathematically.

I think it is a very important distinction to make for a physicist: to distinguish between the hard-to-grasp *physical* reality on one hand, and the attempts to *model* them with mathematical concepts (which we call physical theories) on the other hand, and to keep that distinction clearly in one's head. Which doesn't mean of course that we can't loosely talk about "time" in Newtonian mechanics, or about "time" in relativity and take for granted that "real, physical time" (in as far as it exists) IS exactly behaving as the mathematical model we use of it within a certain frame. In other words, when talking about the motion of planets, we usually loosely *take it for granted* that the real parameter t that we use in Newtonian mechanics DOES represent accurately the elusive physical concept of time.

It is also useful to ponder about the "craziness" of the "pretention" of certain physical theories such as Newtonian mechanics.

After all, in order to make sense of "derivative" (which is central to Newtonian physics - the concept has been *invented* by Newton to be able to formulate Newtonian mechanics), you need *real* numbers to represent position and time. Rationals will not do. The standard derivative is not defined over functions from Q to Q.
This means that we couldn't do with any FINITE decimal development of those quantities. Time, specified only to 7 billion decimals, wouldn't do for Newton. We need *infinite* decimals. Here, the OP was talking of about 40 decimals! But for Newton, we needed infinite decimals in time and position. Clearly, the *pretention* of describing nature up to infinite precision (or even postulating the *existence* of the concepts of time and space in nature up to that precision) is totally crazy. There are all chances in the world that time doesn't make sense to infinite precision, that space doesn't make sense up to infinite precision. (as distinguished from *arbitrary* precision which would be modeled by Q)

So we could guess from the start that the "time" concept in Newtonian physics, or "position" in Newtonian physics (and in relativity...) is not representing exactly the elusive *physical* concept we understand (or think to understand) by time and space. But doing as if allows us to solve practical problems using Newtonian physics, or for that matter, relativity. But we should keep the distinction between both (the mathematical concept *in a certain model* and the elusive physical quantity) in the back of our head.

#### arildno

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There is, clearly, no particular reason why the human being is evolved to grasp "reality as it is" fully.

That isn't really so much of a problem as long as you say, we keep reminding us of the distinction between a model of reality and reality in itself.

#### Naty1

.... The shortest length of time possible is planck time.
I think that IS a general consensus......but of course such have been wrong repeatedly thru history. When such physicsts as Brian Greene, Michio Kaku, Kip Thorne, Leonard Susskind and others accept quantum foam, that is quantum jitters,at the tiniest of scales, I'll take that as a "consensus".

Greene says this in FABRIC OF THE COSMOS:

...on scales shorter than Planck distances and durations quantum uncertainty renders the fabric of the cosmos so twisted and distorted that the usual conceptions of space and time are no longer applicable.
But there is no theory that covers just what is going on at those scales....QM and GR seem to diverge, string theory is incomplete as is quantum gravity....so maybe new math is needed.

But certain derivatives MAY still be applicable....a unit step function, for example, represents an instantaneous change and traditional calculus provides a defined derivative.