## Main Question or Discussion Point

If this is in the wrong section, I apologise, and would appreciate it if a forum veteran would show me the wrongs of my ways.

## Homework Statement

In one of Zeno's paradoxes, it is argued that travel over any finite distance is impossible, because the distance can be divided an infinite number of times. Therefore there is no first distance to run, because any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin.
I understand though, that there is an indivisible unit of length- the Planck length- which I would have thought makes Zeno's paradox obselete.
However, to me, this presents a new problem.

## Homework Equations

The smallest unit of distance = the speed of light/the smallest unit of time

Or

Planck length = c/Planck time

So how far does anything slower than light travel in the same distance?

## The Attempt at a Solution

Simple logic would suggest that:

½c/Planck time = Planck length/2

Or- something traveling at half the speed of light travels half a Planck length in one Planck time.

Related Beyond the Standard Model News on Phys.org
In the standard model of physics spacetime is not discrete, not in terms of Planck units or anything else.

In string theory, which is the mainstream extension of the standard model, spacetime is not discrete, not in terms of Planck units or anything else.

So what use are Planck units?

Staff Emeritus
2019 Award
What use are meters and seconds? They are convenient for certain measurements and/or calculations.

D H
Staff Emeritus
So what use are Planck units?
They certainly were not created to prove the ludicrous notion that travel is impossible. Physics, unlike philosophy, is grounded in reality. Things move.

The value of Planck units is essentially the same as the value of metric units. Metric units get rid of some conversion factors. Newton's second law does not say F=ma. It says force is proportional to the product of mass and acceleration: F=kma. Choose your units for mass, length, and force carefully and the constant of proportionality becomes one. Moreover, setting that conversion factor to one explicitly shows that there are only two rather than three independent units involved.

Natural units (Planck units are one example of natural units) carries this simplification/consolidation concept several steps further.

Sorry, perhaps "what use are Planck units" was rather crass. What if I rephrase the question "what is the particular function of Planck units?"

"what is the particular function of Planck units?"
The planck units are appropriate for measuring distances, durations, energies etc that are on the scale where we expect string theory to be necessary for predicting the outcomes of experiments.

[Planck units] certainly were not created to prove the ludicrous notion that travel is impossible. Physics, unlike philosophy, is grounded in reality. Things move.
Well naturally, common sense tells us that travel being impossible is a ludacris notion. But where exactly is the flaw in Zeno's argument?

Well naturally, common sense tells us that travel being impossible is a ludacris notion. But where exactly is the flaw in Zeno's argument?
The flaw is in the assumption that the sum of an infinite series is an infinite number.

What CAN you get when you sum an infinite series?
(a) An infinite number
(b) A finite number
(c) Any number you like, finite or infinite

The general answer is (c), in the sense that I can show you the sum of an infinite series that results in any finite number or infinite value, you like.

A. Distance: the infinite sum of all those steps is actually a finite number.
B. Time: the infinite sum of the time spent taking all those steps is a finite number.

The fallacy in Zeno's paradox is in the assumption that an infinite sum always results in an infinite number (a), so would take an infinite amount of time to complete (a).

Given a finite distance and a finite speed, it can be shown that the time to completion is finite, even within the framework of Zeno's paradox.

In other words, you can cut those steps up any way you like, into an infinite number of pieces, but the journey is completed in finite time.

apeiron
Gold Member
All these replies seem off the mark.

The planck scale is indeed a grain, an observed limit on smallness spatially, and also heat or energy density. This does not mean it is actually discrete - broken rather than part of a continuity. As said, it is a limit to how closely you can approach some ultimate smallness - and to reach it would be to break the continuity from which you are attempting this feat.

Your specific question was about imagining being a slower than light object crossing a planckscale space. Well because the scale is about energy density as well as everything else, it would seem to me you would have to double your energy density - to twice what the planck scale permits. So you could not actually go slower than light.

Another way of looking at this is that the planck scale is the smallest possible wavelength - a single oscillation that fits the smallest space, and which thus also defines the energy scale. So if you tried to fit two oscillations into that space, you would be doubling the energy density.

Zeno's paradox itself is not so easily dismissed. Certainly not by stating that an infinite sum of discrete steps can add up to a finite continuous path. This is OK as a maths model - operationally useful - but it buries the wider (meta)physical issue.

Again, the limits story provides an alternative view of what is going on. The paradox can instead be taken as a proof that reality cannot actually be broken into the discrete and the continuous as we so conveniently like to model.

In reality, the world lies between these two extremes. At the lower bound, it is almost broken up into the discrete (QM atoms of spacetime). At the upper bound, it is almost relativistically smooth and continuous. So it is almost dualistic broken. But not quite. Instead it actually exist just within the bounds, the event horizons indeed, of these modelled extremes.

So Zeno's paradox is answered by treating both finite and infinite, the discrete and the continuous, as the limits of observational processes.

BTW, it is because of considerations like these that the quantum gravity issues seems rather clouded. The anguish lies in not being able to collapse the model of the continuous to the model of the discrete - relativity to QM. But why would you expect to do that when the reality you actually want to capture in the models is the meat in the sandwich, the stuff caught in between these two extremes?

Last edited:
^^A fantastically clear and informative reply from apeiron. Thanks very much.

One remaining query though:
you could not actually go slower than light.
Tell me if I'm missing some glaring point here, but this seems to go against my (limited) knowledge of physics. I can understand the wavelength issue- that's fine, but what about everyday physical objects. Presumably a ball thrown 100m through space at 30m/s would travel quite a few Planck lengths on its journey. But I wouldn't have thought that any of the numerous Planck lengths it travels over would be crossed at the speed of light. Surely it would travel one Planck length at 30m/s.

All these replies seem off the mark.

The planck scale is indeed a grain, an observed limit on smallness spatially, and also heat or energy density
This is misinformation, provide a citation!

apeiron
Gold Member
Presumably a ball thrown 100m through space at 30m/s would travel quite a few Planck lengths on its journey. But I wouldn't have thought that any of the numerous Planck lengths it travels over would be crossed at the speed of light. Surely it would travel one Planck length at 30m/s.
The planck scale here would be about your ability to determine whether the edge of you macro-object was on one side or the other of a planck distance. (Or even where the edge of the object is located).

So from a distance, you can see that a mass is moving slowly against some background set of co-ordinates. Zoom in close and you can no longer be sure of anything. Uncertainty kicks in. And the energy requirements to probe this scale go exponential.

Gold Member
[bSo how far does anything slower than light travel in the same distance?

## The Attempt at a Solution

Simple logic would suggest that:

½c/Planck time = Planck length/2

Or- something traveling at half the speed of light travels half a Planck length in one Planck time.