This is my second post on the subject of why “the speed of light (in empty space)”, more accurately referred to as “the cosmic speed limit”, is so fast. This speed, denoted c, is about 186,000 miles (300,000 km) per second, which does indeed seem quick.
But as I pointed out in my first post on this subject, this isn’t really the right question, because it implicitly views humans as centrally important and asks why the cosmos as strange. That’s backward. We should instead ask why we ourselves are so slow. Not only does this honor the cosmos properly, making it clear that it is humans that are the oddballs here, this way of asking the question leads us to the answer.
And the answer is this: ordinary atomic material, from which we are made, is fragile. If a living creature were to move (relative to the objects around it) at speeds anywhere close to c, it couldn’t possibly survive its first slip and fall, or its first absent-minded collision with a door frame.
Today I’ll use a principled argument, founded on basic particle physics and its implications for atomic physics, to show that any living creature made from atoms will inevitably view the cosmic speed limit as extremely fast compared to the speeds that it ordinarily experiences.
In the next post I’ll try to go further, and suggest that we must travel at even slower speeds. However, doing so becomes increasingly difficult. We potentially have to get into details of materials, of chemistry, of biology real and imagined, and of the specifics of the Earth. Once we enter this territory, the issues become complex, the logic and conclusions may be debatable, and I’m no longer expert enough to be sure I know what I’m doing. Perhaps readers can help me with that stage when we get to it.
By contrast, the logic presented here today is straightforward and impossible to evade. Along the way it will also teach us why nuclear weapons are so terrifying — why the energy hidden inside atomic nuclei is so much larger than the energies we encounter in our daily activities.
There are a lot of elements to the reasoning, so what I’ll give here is an overall sketch. But you’ll find many more details in the hyperlinks to other pages, some on this website and some beyond.
All Atomic Creatures Will View “c” As Fast
The argument is based on two simple observations about atoms, which I’ll combine in two ways to derive facts about energy and speed. All numbers given below will be very approximate, because we don’t need precision to draw the general conclusion, and trying to be more precise would make the argument longer without making it clearer. (See chapter 6.1 of my book for a discussion of physicists’ “rules of precision”.)
1. The Electromagnetic Force Makes Atoms’ Outer Electrons Slow
The electromagnetic force is a moderately weak force by particle physics standards (despite being much stronger than gravity). It has only about 1% of the strength of what we’d consider a reasonably strong force. As a direct result, the electron in a hydrogen atom moves around the atom at a speed v that is far below c — approximately 1% of c. (See also this article and this article.)
The electrons on the very outer edge of any atom (the “valence electrons”) are affected by an attractive electric force from the positively charged nucleus, but are also repelled by the electric force from all the inner electrons, which have negative charge. The net effect is that positive charge of the nucleus is substantially shielded by the negative charge of the electrons, and the resulting force on the valence electrons is not dramatically different from that found in a hydrogen atom. As a result, these valence electrons are somewhat similar to the one electron in hydrogen, and so they too move around at about 1% of c, typically a bit slower. (This “1%” is very imprecise, but it’s never as small as 0.1%.)
Thus for any atom, the outer electrons have
Here the “~” sign means “is very roughly equal to,” indicating purposeful imprecision. From this we can estimate the typical energy required to disrupt the atom — the “first ionization energy“. That energy is the combination of
- the valence electron’s motion energy 1/2 mv2 and
- the “binding energy” that holds that electron to the atom.
The motion energy and binding energy are similar in size. (This is always true for any force that decreases as 1/(distance-squared), including electric and gravitational forces.) As a result, the first ionization energy is close to the valence electron’s motion energy 1/2 mv2. Just as for a hydrogen atom, this energy is always about 1/10000th, or 0.01%, of the internal E=mc2 energy of an electron. In math terms, the fact that v/c ~ 0.01 for a typical valence electron implies that the ionization energy divided by the electron’s internal energy is roughly
Let me just emphasize again that I’m not being precise because precision isn’t needed for this argument.
Aside: Those of you who know a little quantum physics know that v isn’t really defined for an electron, because it’s a wavicle, not a particle, and that electrons don’t really go around their nuclei in orbits, despite the picture of an atom that’s so often drawn. I’m using Bohr’s cartoon of an atom here, which is enough to get the right estimates for what is going on. But we could do things correctly, and avoid ever writing “v,” by just using the ionization energy all the way through the argument. We’d get the same answer in the end.
2. The Strong Nuclear Force Gives Atoms a Big Mass
An atom has a nucleus at its center, consisting of between 1 and about 300 protons and neutrons. The strong nuclear force forms the protons and neutrons by tightly trapping quarks, anti-quarks and gluons. It then subsequently binds protons and neutrons, somewhat more loosely, into nuclei.
The most common materials, which are forged in the early universe and in ordinary stars, run mainly from hydrogen (1 electron and 1 proton) to iron (26 electrons, 26 protons, and typically 30 neutrons). Since neither hydrogen nor helium is suitable on their own for making life forms, as their chemistry is too simple, it’s reasonable to define a “typical atom” as one with roughly 10 or so protons and neutrons in its nucleus. (Carbon usually has 12, oxygen 16, etc.)
Now we need to take note of three different and unrelated aspects of particle physics.
- The strong nuclear force becomes extremely strong at a distance D that is about 1/100000th of an atom’s diameter. This sets the size of protons and neutrons, and assures they have almost the same mass Mp. That mass, about equal to the mass of a single hydrogen atom, is related to D by the formula Mp ~ h/(cD), where c is the cosmic speed limit and h is Planck’s constant, the mascot of quantum physics.
- The value <H> of the Higgs field corresponds to a mass about 250 times larger than that of a proton,
- The interaction of the Higgs field with electrons is tiny, and characterized by a number called “ye” which is about 1/400000.
Combining the last two tells us the electron’s mass, m ~ ye<H>, is very small compared to the majority of known elementary particles.
Taking all three facts together (which are independent as far as we know [none of them can yet be predicted from first principles]), it turns out that
(More precisely, the proton’s mass is nearly that of 1836 electrons, and the neutron’s 1838; but 2000 is close enough for our current purposes.)
Therefore, for a typical atom with mass M ~ 10 Mp , the ratio of the atom’s mass to the electron’s mass m, which is also the ratio of the energy stored inside that atom to the energy stored inside a single electron, is
Again, this all follows from known facts about subatomic particles. It isn’t something that anyone can explain from scratch, but for our purposes, it’s enough to know that it is true. Let’s see what the consequences are.
3. Atoms with Slow Electrons and a Large Mass are Fragile
Combining sections 1. and 2., let’s compare amount of energy that it takes to pull an outer electron off a typical atom (its first ionization energy), which is approximately equal to the motion-energy 1/2 mv2 of the electron, to the energy Mc2 stored inside that atom:
Thus the amount of energy needed to disrupt an atom is a minuscule fraction of the energy that it carries inside it. Atoms are very fragile indeed!
This now explains why the energies of ordinary life must be so small compared to the energy stored in objects, and why nuclear weapons can draw on so much more energy than we are used to. If the energy per atom involved in ordinary activities, such as catching a ball or jumping up and down, were any more than a tiny fraction of the energy stored inside an atom, many of our atoms would lose one or more electrons right away, instantly ruining our biochemistry and destroying our internal structure. We can only survive because the energy of ordinary life is tiny compared to what the cosmos considers normal.
4. Fragile Atoms Can’t Survive Fast Collisions
A head-on collision between two typical atoms with mass M, each with speed V, will involve energy 1/2 MV2 for each atom. If this collision energy is comparable to or exceeds the energy needed to disrupt either atom, which is about 1/2 mv2 (where as before m is the mass of an electron and v is the speed of a valence electron in the atom) then at least one of the atoms will probably lose an electron. So to avoid this, it must be that
where in the second line I used the result from section 3. Taking the square root of this line, we find
Thus atoms cannot survive intact in any collision whose relative speed is comparable to or faster than 0.00005 c — about 10 miles (15 km) per second. In any collision of ordinary objects at such a speed, the collisions of their individual atoms will lead to widespread atomic disruption, leaving the objects seriously damaged.
Aside: as noted, I have been imprecise all throughout this argument. The true maximum speed for the survival of typical atomic materials may be somewhat slower than 0.00005 c — but not too much so. Perhaps a reader can suggest a more precise estimate?
5. Life (and Anything Else) Made From Atoms Must Move Gingerly
Therefore, for living creatures to avoid injuring themselves irreparably at the atomic level every time they stub their toe or accidentally bump in to one another, they must travel slowly. Relative to objects in their environment, they dare not travel faster than 10 miles (15 km) per second at all times — much less than a 1/10000th of the cosmic speed limit.
And therefore, when they first discover the existence of c, they will all, without exception, express surprise and amazement at how fast it is — more than 10000 times faster than the motions of their ordinary existence.
Humans and Tardigrades
Now, of course, you and I are restricted to much slower speeds than 10 miles (15 km) per second! Even a collision at 10 feet (3 meters) per second, about 7 miles (11 km) per hour, will hurt a lot, and speeds ten times that would surely be fatal. For us, c isn’t just 10,000 faster than what we’re used to — it’s about 100,000,000 times faster than jogging speed!
So clearly this argument gives an overestimate of how fast we can go. There must be additional issues that force us to move even more slowly than the speeds that would disrupt atoms. This is true, but those constraints are much more complicated. That’s why I decided to begin with this relatively simple and very general argument.
What’s good about this argument is that it applies to all atomic objects. It restricts not only natural life on Earth but all imaginable atomic life anywhere — even artificial life that we or some other species might potentially create.
Many organisms are far stronger than we are, tardigrades most famously among them. (Note that tardigrades are small, but not microscopic.) We’ve been making robotic machines for quite some time that can survive and thrive in environments that would kill us instantly. Current technology can already create simple artificial life forms.
Nevertheless, no matter how good our technology, and no matter how intricate the unconscious process of evolution, we will never encounter or construct complex objects that can remain intact in environments where relative speeds are as high as 1/10000th of c, unless
- they remain forever isolated in deep space, where collisions with any large objects are extremely unlikely, and collisions with atoms are infrequent enough that the resulting surface damage is manageable [which is why our spacecraft, not to mention asteroids and planets, are able to travel safely much faster than Earthly objects do], or
- we someday find a building block stronger than atoms from which a robot or artificial life form can be built [but given what we know about the universe so far, this may forever remain science fiction.]
Here’s an interesting and relevant piece of information: it has been shown that frozen tardigrades can survive collisions with sand at tremendously higher speeds than we humans can handle, but only up to about 0.6 miles (0.9 km) per second — about 1/300000th of c. (This makes it challenging for them to travel successfully between planets and moons across the solar system, where typical relative speeds of large objects and meteors are in the 10 km/second range.) On the one hand, this shows that at least some life forms can survive much more rapid collisions than we can. On the other, they have limitations that are consistent with today’s reasoning.
Could one could create a intelligent creature of a larger size that could match the durability of a tardigrade? That is perhaps doubtful, but we can discuss that after the next post.
Slow is Better
Thus by combining basic knowledge concerning our universe —
- the moderate weakness of the electromagnetic force,
- the great strength and trapping ability of the strong nuclear force,
- the substantial value of the Higgs field,
- the tiny interaction of the Higgs field with electrons,
- the nature of typical atomic nuclei, and
- the behavior of valence electrons in typical atoms —
we learn that any object made from atoms cannot endure collisions at speeds anywhere close to c. And now we know the reason why the cosmic speed limit seems so fast — and why nuclear weapons seem so violent.
The reason is simple: in this universe, only the slow survive.
19 Responses
This is about particles, wavicles. Perhaps the true consideration should relate to the single thing which is currently being made or outputted by structures created by these wavicles. If the speed of light, the speed of gravity, were any different, how would it affect the production of empty space.
After all, Empty Space (i.e., the quantum vacuum containing all the fields from which particles and forces are made) is being created. This is not happening by some accident nor by some “design”. The quantity of empty space being produced is a product of the structure of the components in the universe. If the speed of light was any different, it would likely change the production output of the quantum vacuum.
Thanks for the great article. I thought from the title that you might have a theory about the actual speed of light in vacuum and why does it have the value that it has. Why 300000 km/s.
For me to do that, I’d have to have a theory about why humans defined seconds and kilometers when they decided how to measure things.
Take humans out of the equation; imagine kilometers and seconds had never been invented. What should my theory say? What should I be proving the speed of light has to be equal to?
One might suggest that I prove that light can cover the average distance from the Earth to the Moon in about 1.3 seconds. But that still requires a definition of “second”; and it would require I have a theory of why the Earth and the Moon are as far apart as they are right now — which is different from their distance a billion years ago. Same for the Earth and Sun, or the distance across the galaxy.
A better idea would be to argue why light can travel the distance across a hydrogen atom in the time it takes for an electron to vibrate about 2600 times. (I think I did the math right.) That’s a much better idea, because atoms are the same everywhere and have been since they first formed when the universe became cool enough.
But if you think about it, that’s exactly what I did. Except I did it the other way. I argued why the distance across a hydrogen atom is the distance light travels in 2600 electron vibrations, by looking at the nature of electrons and the electric force.
So the mistake is thinking that what is needed is to explain light’s speed in terms of atoms, as though atoms are more fundamental than the cosmic speed limit. That is backwards. The cosmic speed limit is fundamental: it makes sense to define it as equal to 1, and define all speeds as fractions of c. That’s what I did in showing the speed of atomic collisions that is necessary to break an atom apart; it is about 1/10000 of c.
More precisely, we can choose to define time in terms of any unit we like — let’s take seconds (where 1 second is something like a billion trillion electron vibrations) and then define distance as measured in light-seconds (= the distance light travels in one second). When we do that, the cosmic speed limit is exactly one light-second per second.
I assume it’s not particularly useful to ask, “why is c small compared to distances”, because that implies some natural spatial or temporal scale. Is it any more interesting to observe that very little ordinary matter reaches relativistic speeds and wonder why that should be so?
Have only very recently stumbled upon your fascinating blog here. I am absolutely not a physicist, nor have any deep learning in such areas, so please forgive me if I ask a daft question…
You say above “we will never encounter or construct complex objects that can remain intact in environments where relative speeds are as high as 1/10000th of c, unless
we someday find a building block stronger than atoms from which a robot or artificial life form can be built [but given what we know about the universe so far, this may forever remain science fiction.]”
Is there in fact anything within physics as it’s presently understood that indeed prevents such a thing i.e. there existing something other than atoms as some basic “unit”?
I’ve long wondered why it is that despite the seeming brilliance of humans at building such complex understanding, we are still pushing at such limits as the time it would take to fly a space ship to another galaxy. Is it really true that nothing could ever exceed ‘c’ and thus we are indeed doomed to take lifetimes to travel beyond our solar system? Or is it because we have not yet discovered something much more fundamental about the universe, such as an ‘alternative to’ the atom?
Nothing the slightest bit daft about these questions! In fact, they’re such good questions I’d like to answer them in a blog post. Would you mind? (I can either credit you for the questions or state them as anonymous, as you prefer.)
yes, please go ahead. Apologies for a late reply, there was no notifcation system so I only just saw it 🙂
The wave package tracking within the photon moves faster than c.
The previous part had an odd issue, the page was of a fixed length, meaning that later comments and the reply box were cut off. I had to zoom out to 33% to see them. I hope that won’t be the same here.
I thinkt he atom collision argument sets a maximum temperature for atomic matter, past that point molecules will break down.
A fact about water bears that people oft overlook is that their tun stage, resistant as it is, is designed to let them survive when their environment dries out. To actually survive they need damp moss. This somehwat diminishes their chances on another world. Survival is different from proliferation.
Something clearly went wrong at some point with that post, something I’ve never seen before. I don’t see anything obviously wrong on my end, so I don’t know what happened or how to fix it. I’m hoping to get IT advice on Monday.
The atomic collision argument does set a limit for molecules, too, but it is far more complicated. I’ll come back to this over the coming week. A perfect crystal can survive to thousands of degrees, and so the speed limits are still pretty high. So you can’t improve the limit very much at the molecular level unless you can that life can’t be made robust while housed inside a large hollow perfect crystal. If you really want to do better, I think you need to focus on bulk matter, but this is complex.
Water bears/tardigrades: point taken. That said, they’re amazingly tough, and so they raise all sorts of questions about how tough intelligent life forms could be if they evolved under, or were explicitly designed for, extreme environmental circumstances.
Does this reasoning also explain why humans live a long time but are small, relative to the universe?
By my rough tally, humans are about 10^-27 as long as the universe, and 10^-52 as massive as the universe, but live 10^-9 as long as the age of the universe, making us relative giants in the time dimension measured in universes. Is there an intuitive reason why it must be this way? Could we imagine a stable universe where sentient life is 10^-9 universes tall, but only lives 10^-52 times the age of the universe?
Not quite, although your question is quite interesting to think about.
I think it’s first useful to start with the statement that humans are about 6×10^-9 light-seconds tall (that’s 6 feet = 2 meters) and live about 10^9 seconds. That doesn’t refer to the universe, it just says that we live far longer than our size, as measured using a cosmic perspective. This isn’t related to our speeds relative to our surroundings, though, so this isn’t the same issue as the one discussed today.
We can also say that a typical boulder, moon or star lives far longer than its size too. This is even more striking for atoms, of course; they can live vastly longer than their size. Also true for galaxies. It’s true for any stable object in the universe — in fact, that’s almost the definition of (quasi)-stability.
In fact, any life form whose age and size were comparable wouldn’t have time to take a single breath; the nerve signals from brain to lung, and the response of the lung, can’t travel faster than the cosmic speed limit. I think the very definition of life requires it have a very long age-to-size ratio in cosmic units.
What about the universe? We don’t really know how big it is or how old it is, so that makes it hard to say. If we assume that the universe is pretty much only as big as the part that we can see, and that it was born at the time of the Big Bang or just before, then its size and its age are, in cosmic units, about the same. But this may not be the case. Inflation could make it much larger than its age. A history of Big Bangs into the distant past could make it far older than its size. We really have no idea.
So I think the issues here are unrelated. The question of what in the universe is stable and what is not, and why, is a very complex one, and very different for elementary particles, for non-organic clumps of matter, for self-organizing engines, such as hurricanes, and for life forms such as ourselves. These questions of stability occur independent of the larger scale structure of the universe, both in space and in time, and so I don’t see any way to connect them.
There’s more to mine here in your question; I’ll think about it some more. I don’t think life and the global properties of the universe can be connected. But there may be some minimum age-to-size ratio in cosmic units that any lifeform made from atoms must satisfy, and I’ll try to see if I can find it.
As a experimental check, I suspect graphene is the molecule with the highest melting/sublimation point per atomic mass, about 4000K, which if I’ve done my math correctly works out to a rms speed in an ideal gas of roughly 3 km/s. Given the shape of the Boltzmann Distribution, this definitely seems to match your calculations!
Indeed, it’s no accident; binding energy of molecules in the best crystals is naturally related to the same physical scales as first ionization energy. 4000K is also about 0.35 eV in energy, which is a bit below carbon’s ionization energy of 11 eV.
There are actually some substances that go higher than graphene, though. But nothing is solid above 1 eV = 11600 K , and so that really is a strong bound… somewhat stronger than the one I argued for.
Material science probably brings 15 km/s down to about a few km/s. That’s an interesting discussion for the next post. But then you get into biology; can you build life from graphene? If you build it from water and want to keep the water from boiling you need to go down further in speed… etc. It gets messy.
Not an alternative estimate, but a confirmatory one: What is the velocity of a 10 amu atom to carry 1 Rydberg of energy. (Of course being in an environment that can reliably completely ionize Hydrogen is especially hostile to life — we don’t expect to find much non-plasma chemistry around type O stars.)
I note that the maximum speed you have is a bit faster than escape velocity (11-ish km/s) and is comparable to meteoroids entering the atmosphere (10-75 km/s). The latter is well known for destructively heating the front face of the rocks and ice entering the atmosphere. Very little life chemistry seems to occur at “flash vaporizing rocks” temperatures.
On your last paragraph: it’s worth noting that unlike the atomic/particle physics argument that I’ve given here, the fact that escape velocity and meteroid speeds are comparable to the limit I obtained is an accident. There are surely other solar systems and planets where these speeds are significantly different. It’s worth keeping track of those arguments that are universal (literally) and those that would vary from place to place. Our own more extreme speed limitations have a lot to do with specific features of the Earth, its atmosphere, and its existing life forms.
The first post didn’t allow for comments (I had a comment loading problem), but the answer was very much as I expected. Of course, it isn’t primarily the ionization energy that is interesting for evolved organisms but the slightly lower bonding energies of metabolism, but that is detail.
The topic lends itself to questions such as:
– How much stronger could the electromagnetic force be and we still had (slightly less perturbative) atoms?
– Why are orbital speeds roughly hypervelocity (launch shock waves in rocks) but not orders of magnitude more? Only about 2-5 % of hypervelocity ejecta make it back out of Earth’s gravity well in models of when, say, the Chicxulub impactor hit. As far as local panspermia of tardigrades goes it is a very inefficient business.
I forgot to add that the article is well written and engaging. The number of details that goes into the answer is surprising!
I don’t understand what happened to the comments on the first post. People left comments for a while and then were unable to. I don’t see a setting that’s been flipped, but I’ll have to check if somehow something was changed in WordPress defaults. [Although I just got an email that suggests it was a bug.]
On your potential topics; let’s come back to the second one, since that’s more tangential. But when you ask, “How much stronger could the electromagnetic force be and we still had (slightly less perturbative) atoms?”, the answer is that while we don’t really know precisely how to phrase this question so that the answer is meaningful, the force could certainly be 100 times stronger and perhaps 300 or 1000. (What is called for historical reasons the fine-structure constant α is about 1/137 at atomic and larger distances. The question is whether the maximum effective value for α is 1, 4π, or the square root of 4π. Probably it is 1.) If you try to make α larger than that, there are probably some effects that prevent the force from actually increasing in effect. In such a situation, you will still have atoms, but the velocities of the electrons will be close to c, the atom will be a hundred times smaller, and the binding energy will be as large as the electron mass — probably leading to strange behavior. The electron in a hydrogen atom at α ~ 1 would in some ways resemble an innermost-shell electron in uranium or some other element with very high atomic number — relativistic in speed and energy. But that’s only an approximate analogy. To my knowledge, no one has been able to figure out precisely what would happen in a hydrogen atom with α equal to or greater than 1.
We do know an example of an atom-like system with a pretty strong interaction — the charmonium system is bound by a powerful force induced by the strong nuclear force — but it isn’t identical to the case of electric forces because of the long-range trapping (or “confinement”) of flux, which changes the force law. The case of the bound states of up, down and strange quarks is even less similar to the case of the electric force.
In short, the force can presumably be about 100 times stronger before saturating; and atoms can probably exist all the way up to that point, becoming highly relativistic with high speeds and deep binding; but their precise features are not known, at least not to me.
More problematic is that nuclei of high atomic number, which are held together by the strong nuclear force despite the repulsive electrical forces among their protons, will be blown apart as α increases — the increasing electrical forces will overwhelm the strong nuclear force long before α gets to 1. By the time α is 1, you’ll probably have a universe with only hydrogen (and maybe helium), making life impossible.