Why is the Speed of Light So Fast? (Part 2)

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 M_p. That mass, about equal to the mass of a single hydrogen atom, is related to D by the formula M_p ~ 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 “y_e” which is about 1/400000.

Combining the last two tells us the electron’s mass, m ~ y_e<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 M_p , 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.