# Universal minimum speed

I had been reading "The Ancestor's Tale" by Richard Dawkins before going to sleep, and woke up the following morning with a curious thought: "Is there a minimum speed?" Now, I have to admit this has very little to do with the evolutionary zoology the book is about ^{*} , but it does raise an interesting series of thoughts.

For one, this question is not "can there be no speed", or more succinctly, "can there be a no-velocity state", like the "velocity" 0 m/s. If I had asked this, I would have greatly disappointed myself; a zero velocity is not actually a velocity, due to the meaning of "zero". While we use 0 as if it were a number in the mathematical language, it isn't actually one - it is a conceptual symbol much like infinity or the complex number, allowing us to perform calculations that make sense, but having no "real world" equivalent. When asking about the existence of a minimal velocity obviously it stands to reason that this phrasing discounts, exactly because of the presense of the word velocity, the situation where no velocity is manifest. I digress a little, but that was important. Anyway, is there some kind of universal minimal velocity?

Possibly.

There are a few things that make a minimal velocity possible: discrete spacetime or discrete energy. The second is the simplest solution, so let's look at that one. Quantum mechanics shows us that if anything, energy is actually quantised - that is, it comes in discrete quantities, rather than in any possible quantity - but it is not entirely clear why, and whether this is universal. The energy released by electrons upon excitation is highly quantised, with an electron requiring a minimal amount of energy to be added to it, in order to release another energy packet, conveniently called a photon.

larger sub-molecular particles do roughly the same thing - all energy conversions (that we know of that is) happen through discrete valued particles, and this leads us to believe that based on the evidence we have so far energy is in fact universally quantised at the quantum level of reality. So this leads us to a certain line of reasoning. If energy is quantised, and velocity is a direct consequence of the addition of energy to a system (which it is), then there has to be a minimal velocity, some velocity value "v", such that a velocity lower than v but higher than zero is universally impossible.

But what if spacetime, too, is discrete? If this is the case, we would suddenly have an interesting problem: velocity is measured in distance units per time unit. Meters per second, miles per hour, parsecs per year, all of these indicate some discrete distance, over some discrete time - but all these discrete values like meters or miles or parsecs, are plucked from a continuous "space", and seconds, hours and years are plucked from continuous "time". What if space and time are not continuous at all?

In fact, let's rephrase: is spacetime discrete?

Well, it might be. We just reached a cutting edge question in science, which is not experimentally answerable yet; a slight problem. However, it is becoming more likely that it actually is discrete at the scale lower than the "Planck length", which is a remarkably small length of roughly 0.000000000000000000000000000000000016 meters. This means that even at the molecular level this scale is simply invisible. To make this invisibily a bit more clear, the estimated "size" (if you can call it that) of a proton, one of the two submolecular particles that form a atomic nucleus (the other being the neutron) is estimated at 0.000000000000001 meters. Even if we got that value wrong by, say, a margin on a thousand-fold (which is quite a ridiculous margin, the real error margin is at most a factor 10), this means that the discretisation of space is still 160 quintillion times as small. That's a million, *times* a million, times *another* million. Times 160.

This is phenomenally small. So small in fact, your human imagination cannot envisage it, because the scale difference is unknown to you - nothing in the world that you experience has that kind of size contrasts. In comparison, the diameter of our sun is roughly 1.4 million kilometers, and the distance from the sun to the edge of our solar system is less than 14 thousand times that. In fact, the distance from one end of our galaxy rather than just the solar system is only 675.752.029.000 times as big as the diameter of the sun. To get a feeling for the size difference between a proton and (possibly) discretised space, you would need to get something that is at least another million times as large. Let's for the sake of argument assume that the current estimate for the size of the entire universe is accurate at 156 billion lightyears. That would be roughly 1.475.842 quintillion kilometers... The scale difference between a proton and discrete spacetime is roughly comparable to the scale between the *entire universe* and just our solar system.

That's, realistically speaking, a scale difference that can drive a person insane trying to envision it.

But then, we're thinking scientifically and zeroes don't phase us one bit. If discretised spacetime is true, then this gives us a "lower bound" on what the minimum velocity is: it cannot be less than 1 discrete space unit per ... per what? if spacetime is discrete then it basically means that not just space, but spacetime is discrete, so our upper limit is actually "one spacetime unit", which already carries in it both possible distance, and possible length of time. Hurray! But what are they?

And that, we just don't know. But that's okay, because the question doesn't need to actually know what it is, though it is fun to think about it - the question was "is there a minimum velocity, below which there is no velocity".

In case spacetime is continuous, the answer seems to be "yes, there is, but it's hard to say how the value should be expressed": the minimal velocity will be whatever the minimal energy packet transmittable by the smallest particles in the universe is, when translated into particlewave motion (which is pretty small, but definite). However, expressing this in for instance miles per hour or meters per seconds will be too suggestive, because even the tiny scale "nanometer" will probably already be too big to represent the velocity involved. In short the velocity unit would be too suggestive in secretly claiming that with this velocity a particlewave could traverse a nanometer, or a meter, or a mile, or a parsec, or any other distance we picked as unit measure. Units are tricky that way.

In case spacetime is discrete, the answer seems to be that "yes, there is a minimal velocity because of the fact that there is a discrete minimal energy value, it lies between 1 space unit per time unit, and a number of space units that can be covered by the aforementioned minimal energy per time unit... and we will know how to write this in normal numbers once we discover what the heck these units are". Except the very fact that we ourselves are embedded in spacetime will probably make it impossible to directly measure what the discretisation factor of spacetime is. Clever physical (in the sense of physics) math might help us, but it'll be daunting nontheless.

So there we have it: "Is there a minimum velocity?" - "yes"

Another burning question answered thanks to a bit of thinking and the progress of science.

^{*)}* I recommend that in addition to reading this book, because it is quite good, you read "The Selfish Gene" by Dawkins, particularly before or after reading works by Stephen J. Gould on the subject of evolutionary "law" (yes, every evolutionary biologist knows it's no more law than a macro-physics law, which is to say a descriptive one), so that you may notice that both extremes these men advocate(d) for evolutionary biology are right, but operate on different levels of abstraction. Once you become accustomed to the multiple perspective interpretation of the world, inherently caused by our own abstraction of it, it becomes blatantly obvious that many contrapositions are simply void because their arguments pivot around different levels.*