# Upper limits for jerk and higher derivatives in physics

Is there an upper limit for jerk in physics? What about higher derivatives?

A consequence of special relativity is that no material body can reach or exceed the speed of light in vacuum (due to the relativistic mass increase, or the Lorentz contraction).

I am not interested in obtaining these upper limits based on postulates (and following mathematical implications), like the postulate of the constancy of speed of light in special relativity.

I would like to know whether there are experiments that would lead to these upper limits, as a matter of consistency with empirical evidence. For the speed of light in vacuum we have Maxwell's equations. As far as I know, it was established experimentally that the constant that appears in Maxwell's equations is the speed of light in vacuum.

For acceleration, there is a previous question on StackExchange, Is there a maximum acceleration?, where in an answer it is discussed Caianiello's maximum acceleration (the result is linked to Heisenberg's uncertainty principle, that's the type of answer that I am interested in, as an example).

So for speed and acceleration, I understand that there is experimental evidence that tells us that in order to have consistency (with the empirical evidence), then we must accept that there is a maximum speed and acceleration in physics.

I am not interested in arguments involving the Planck system of units, since these arguments would be sensitive to the choice of normalization.

What about jerk in physics, and higher order derivatives. Is there any experimental data that would force us to accept upper limits for these quantities? Would it be useful to imagine experimental scenarios that would force us to accept these upper limits?

I will reformulate the question, for clarity. I will not focus on the chronological emergence of the mathematical models that explain the experimental data.

In physics, maximum speed can be linked to Maxwell's equations. Maximum acceleration can be linked to Heisenberg's uncertainty principle. Does this trend continue, for jerk and higher derivatives? Because intuition tells me that in physics there must be upper limits/bounds for all of them, and there is a countable infinity of these quantities.

## 2 Answers

I am not sure about this answer, but it may answer the question.

A general equation of motion where the $$n^{th}$$ derivative is a constant can be written as $$x^{\mu} = \sum^{n}_{0} \frac{1}{j!}\frac{d^j x^{\mu}}{d\tau^j}\tau^j$$ The maximum (say) acceleration will be the acceleration that accelerates the body to $$0.999c$$ in a time period of Planck's time. This acceleration can be calculated using the above formula by substituting $$n=2$$ and $$\frac{dx^{m}}{d\tau}=0$$. You will get two equations (assuming the body to move only along one axis), with index $$0$$ and index $$1$$.

In a similar way the jerk can be calculated by setting the first two derivatives ($$\frac{d^jx^{m}}{d\tau^j}$$) to zero.

This is obviously an approximation.

Hope this helps.

• Good effort. Unfortunately, the choice of normalization for the Planck units is an issue. You basically say, for constant acceleration I take the first 3 terms in the Maclaurin series (the rest 0) , for constant jerk I take the first 4 and so on. And then you impose the condition that your system will reach the speed of light in Planck time, and see what information you extract from that. Good effort. First, the choice of normalization for Planck units is an issue. Second, it is only believed (not experimentally proved) that at Planck scale quantum gravitational effect start to matter. – Cristian Dumitrescu Apr 28 '19 at 16:35
• So at Planck scale the domain of applicability of our current mathematical models (in physics) starts to break down (a theory of quantum gravity is necessary). But that does not mean that nature forbids the jerk (and higher derivatives ) to take any higher values, it just says that our mathematical models break down. For speed and acceleration, nature tells us that you would get into contradiction with experimental data (actual observation), and current mathematical models, if you exceed a certain threshold ( for speed and acceleration). – Cristian Dumitrescu Apr 28 '19 at 16:44
• And for a general approach, you have to consider the Lagrangian and Hamiltonian mathematical framework. Once again good effort though. – Cristian Dumitrescu Apr 28 '19 at 16:50
• @CristianDumitrescu Could give me a hint of how to apply Lagrangian approach to this problem? Thanks by the way. – Manvendra Somvanshi Apr 28 '19 at 16:58
• Every domain of physics can be cast within the Lagrangian mathematical framework. You have Lagrangian electrodynamics., Lagrangian quantum mechanics, Lagrangian special/general relativity , and so on. That's why Lagrangian and Hamiltonian formalism are valuable , it's a unifying mathematical approach. It looks like an upper limit for speed comes from experimental data from electromagnetism. An upper limit for acceleration comes from experimental data in quantum mechanics . I don't know what domain of physics will provide the experimental data, in order to set an upper limit for jerk. – Cristian Dumitrescu Apr 28 '19 at 17:35

This answer is a little late, but here are a few thoughts. In the Newtonian world jerk could theoretically be infinite. Hold an object stationary just above the surface of the Earth and drop it. The acceleration instantaneously goes from $$0$$ to $$g$$. But since duration of the infinite jerk is $$0$$, the velocity is still $$0$$ after the jerk ends and then changes only due to the acceleration $$g$$. So there are no infinite velocities or forces or accelerations due to this infinite jerk.

In the quantum world, the uncertainty principle makes time, energy, momentum, and position somewhat fuzzy, which basically stops anything from being instantaneous. This has the effect of keeping the jerk finite, although it still could be quite large.

• The colission time (duration of the colission ) is not zero , consider the time-energy uncertainty relations, or see this link: physics.stackexchange.com/q/53802 . Nothing happens instantaneously, you just have to consider different time scales. – Cristian Dumitrescu Oct 24 '19 at 15:58
• @Cristian Dumitrescu I mentioned nothing about a collision. Read my whole answer. I mention uncertainty in the quantum world but in the Newtonian world, there are not restrictions. – Bill Watts Oct 24 '19 at 21:51