I’m often asked two very natural and related questions.
- Why is the speed of light, usually denoted c, so astonishingly fast?
- Why, in Einstein’s famous equation relating energy and mass — E=mc2 — does c2, a gargantuan number, appear?
It’s true that the speed of light does seem fast — light can travel from your cell phone to your eyes in a billionth of a second, and in a full second and a half it can travel from the Earth to the Moon.
And indeed the energy stored in your body is comparable to the Earth’s most explosive volcanic eruptions and to the most violent nuclear bombs ever tested — enormously greater than the energy you use to walk across the room or even to lift a heavy suitcase.
What in the name of physics — and chemistry and biology — is responsible for these bewildering features of reality? The answer is fascinating, and originates in particle physics and the resulting structure of matter. It is surprisingly intricate, though, so I’m going to approach this step-by-step over three blog posts. Here’s the first.
Refining and Rephrasing the Questions
Let’s start by being clearer than I was in my first version of this post! The quantity that I’ll be referring to as “c” is the speed of light in empty space. In materials such as air or water or glass, light travels more slowly than “c“. The precise form of question 1, then, is
- 1. Why is the speed of light in empty space, usually denoted c, so astonishingly fast?
We should also recognize that the second question has two sub-questions, one qualitative and one precise:
- 2a. Why, in Einstein’s famous relation between energy E and mass m, does a gargantuan number appear?
- 2b. Why is that gargantuan number equal to the speed of light (in empty space) squared, i.e. c2?
We’ll see that questions 1 and 2a are almost the same question, and have largely the same answer. But as we’ll see, they aren’t phrased well yet.
The problem is that “fast” and “gargantuan” are relative notions. I can run much faster than a slug but much slower than a cheetah. I am huge compared to a bacterium, but not compared to a star. So we ought to start by restating these questions in relative terms; that will help us think them through.
- 1. Why is the speed of light in empty space so much faster than the speeds that we humans ordinarily experience?
- 2a. Why is the energy stored in the masses of ordinary objects (via E=mc2) so much larger than the energy of the ordinary processes experienced in daily life?
The Cosmos Has a Viewpoint
To get us warmed up, I’ll start with a brief quote from my book, chapter 2.
“It’s well-known that light has a characteristic speed, which scientists call c ; this is the speed at which each individual photon travels, too. As scientists discovered centuries ago, c is about 186,000 miles per second. That’s fast, in a way. Our fastest spaceships don’t come anywhere close to that speed. Though my last car was with me for fifteen years, I drove it less than 186,000 miles. At the speed c , you could circle the Earth in a blink of an eye (literally) and travel from my head to my toe in a few billionths of a second.
“And yet c is also slow. It takes light more than one second to travel to the Moon, over eight minutes to reach the Sun, and over four years to reach the next-nearest star. If we sent off a robot spaceship at nearly c to explore the Milky Way, it could visit only a few dozen nearby stars during our lifetimes.
“You and I are small, so we think light runs like a rabbit. But the universe is vast, and from its perspective, light creeps like a turtle.”
The point of this quote is to remind us that we’re not the center of the universe. We are not annointed creatures relative to whom all cosmic facts should be measured. There’s nothing unique or special about the Earth or its size, mass or temperature — nothing materially unique about animals, about mammals more specifically, or about us. The way the cosmos works is not influenced by the objects of our ordinary lives. So our own perspectives are not privileged, and we should be aware that there are other perspectives, ones from which light’s speed (in empty space) is slow and/or from which the energy stored in a human is tiny.
To make our questions really meaningful, then, we ought to step back and ask not just how we view the cosmos but how the cosmos views us. From the universe’s perspective, the questions really are these:
- 1. Why are humans so astonishingly slow compared to the natural speeds of the cosmos?
- 2a. Why are the energies involved in ordinary human affairs so insanely tiny compared to the natural energies one would expect from objects of human scale?
For us to understand how the universe would answer these questions, we have to understand what “natural speed” and “natural energy” might mean from a cosmic perspective. So let’s start there.
The Natural Speed
It’s actually best not to call c “the speed of light”, as it leads to confusion. As noted above, light travels more slowly inside ordinary materials than it does in empty space; by contrast, c does not change inside materials. Meanwhile, c is not just the speed of light in empty space; it also the speed of gravitational waves. Even more important, c is the universal limit on the relative speed of all physical objects. That’s why I and many others often refer to c as “the cosmic speed limit” — because that makes it clear that rather than being a property of light, it is a property of the universe. (Caution: there are lots of conceptual traps and subtleties here, some of which I’ve written about.)
This cosmic speed limit seems to be the same everywhere across the universe (based on our observations of almost unimaginably distant and ancient objects), and so every living intelligent creature in the cosmos can measure it. No other speeds are fixed and reliable in the same way. Compare it, for instance, with the speed of sound. Sound speed varies with temperature and with the material through which it travels, and so this speed is completely different in other planets’ atmospheres and oceans. It could never be used as a cosmic measure of speed that all intelligent species could agree on.
Note, in answer to a commenter’s question: like sound’s speed, though for somewhat different reasons, light’s speed also varies with the material it travels through, and with temperature. But the cosmic speed limit does not. This has observable consequences that particle physicists use in their experiments.
Nor should we think of human speeds of about 1 meter per second (about one yard per second) as “normal speed”. First, if we were peregrine falcons or sloths, we’d view human speed very differently. Second, the now-standard choice of “meter” to measure length and “second” to measure time is arbitrary. A blue whale is many meters long, very big in this sense. But a sufficiently intelligent species of whale wouldn’t use “meter” as their yardstick, and would instead likely define length using a “whaler”, comparable in size to a whale. We’d be a fraction of a whaler tall, and thus seem diminutive by that measure. Similarly, a sequoia tree would probably not want to use “second” as a time-frame; “hour” would be more characteristic.
So the precise way one defines distances and times and speeds, and what makes a length or a duration or a velocity large or small, are all species-dependent, planet-dependent, and perspective-dependent unless you use facts about the cosmos that everyone can agree on. And when it comes to speed, the cosmos has a view on this matter. It says:
“c is normal speed, because that’s the maximum rate at which information can travel from one place to another. No two objects can move relative to each other faster than that. No knowledge can be sent faster than that. There’s no other speed of comparable stability or of comparable importance. So typical objects should always pass each other at a speed that is a reasonable fraction of c.
“But, WOW… you Earth-creatures are absurdly, ridiculously slow! Look at how you crawl around your planet!”
The Appearance of c2
Setting aside the issue of whether c should be viewed as large, small or normal, why is it “natural” that the energy E stored inside an object should be related to its mass m by c2? My answer follows the logic of this post, which goes into more detail about the methods of “dimensional analysis”, one of physicists’ most important tools. You may want to read it if my explanation here seems too sketchy to you.
Einstein’s basic claim was that even a stationary object has energy stored inside it. The amount of that energy, he suggested, is reflected in its mass — specifically its “rest mass” m, which is the mass as measured by an observer who is stationary relative to the object. (For more details on rest mass and on various forms of energy, see chapters 5-8 of my book.)
Any relation between energy and mass must involve the square of a speed (or the product of two speeds.) We find this already in first-year physics. In pre-Einsteinian days, the motion energy (i.e. “kinetic energy”) of a moving object was understood to be equal to the object’s mass m times its speed v squared:
- Newtonian-era motion energy = 1/2 mv2
If you tried to replace v2 with v3 or v99 , the equation would become nonsensical. (As a physicist would say it: the units on the two sides of the equation don’t match.) It would be like claiming that the height of a tree is equal to the color of its leaves — two things of completely different character can’t generally be equal.
But back to Einstein’s claim that a stationary object has energy too. The corresponding formula can’t contain v, since a stationary object has v=0. Some other speed or speeds must appear instead.
Why should that speed be c? Well, it wouldn’t make much sense for an object’s energy/mass relation to depend on the speed of some other object. Imagine if the energy in my body were my mass times the square of the speed of some ultra-distant star. Not only would this be bizarre (and inconsistent even with Galileo’s relativity), what would the formula have meant before the star was born?
No, the relationship between energy and mass for stationary objects must be universal — cosmic — and so it can only depend on speeds that are properties of the universe itself. As far as we know, the universe has only one inherent speed: c. (In fact you can prove that Einstein’s conception of relativity would be inconsistent if there were more than one basic speed.) Therefore any relation between energy and mass must be of the form E = #mc2, where # is a fixed number that someone has to figure out. There’s no other equation that could logically make any sense.
Einstein knew this, of course, even before he wrote his relativity papers. So did all his colleagues.
The fact that the # is equal to 1 is partly a historical accident of definitions, and partly, given this accident, a matter of brilliant deduction and imagination. Click here for some details.
Regarding the question as to whether E = 1/2 mc2 or E = 2 mc2 or E = 4/3 mc2, here physicists got a little lucky historically. The definition of mass was given in Newton’s day, and energy was defined later in just such as a way that, for pre-Einsteinian physics, the motion energy of a moving object is 1/2 mv2. There are sensible reasons for that definition. It is directly related to the definition of momentum as mv, mass times velocity, with no 1/2 or 2 in front. The definition of momentum was in turn was motivated by Newton’s equation F=ma, which defines what we mean by mass. If Newton had put a 1/2 in that equation, defining mass differently, then there’d be a 1/2 in Einstein’s formula too. But with the definitions that Newton and his followers used, the correct equation that matches nature is E=mc2 , with no numerical factor. That’s a nice historical accident; any change in the definition of energy or mass would have affected the sleek appearance of Einstein’s formula.
Now, why was Einstein the one to figure out that, with these definitions, the correct number in the equation is 1, when his colleagues had been trying so hard and getting so close for a couple of decades? He asked the right question, while his colleagues did not. More about that here.
So 2b is answered: in our universe, the only possible relation between E and m for a stationary object is E=#mc2, where # may depend on how one’s culture exactly defines energy and mass, but which happens, with our historical definitions of energy and mass, to be 1.
The Natural Energy
And so, from the universe’s perspective,
“The natural energy for an object with a rest mass m is something like mc2 . When the object is stationary, that’s exactly how much energy it has, and when it’s moving, it has more. And if it’s moving at a natural speed — some moderate fraction of c — then we already know from pre-Einstein physics that its motion energy will be something like 1/2 mv2 , which will be a substantial fraction of mc2 . In short, typical objects in the universe will be seen to carry internal energy mc2 and motion energy which is not so far from mc2.
“But you Earth-creatures … you are like frightened mice, keeping all your activities down to a tiptoe and a whisper! Are you trying to avoid being noticed? Are you cowards, afraid of any drama?”
The answer to the last question is “yes, absolutely”. But more on that in the next post.
Why the Energy Question is a Speed Question
I’ve already now hinted at why the energy question 2a is the same as the speed question 1. The reason the energy stored in ordinary objects seems so large in human terms is that c, the speed of light (in empty space), seems so fast in human terms.
Again,
- Einstein claimed (and it was soon confirmed in experiments) that a stationary object has internal energy mc2, where m is called the “rest mass” of the object.
- For slow objects like us, with v much less than c, we can use Newton’s approximations to Einstein’s equations, for which an object of rest mass m moving at speed v has motion energy 1/2 mv2.
This means that the ratio of an object’s motion energy, which is easily observed in ordinary life, to its internal energy, which is hidden in ordinary life, is
- (motion energy)/(internal energy) = (1/2 mv2 ) / (mc2) = 1/2 (v/c)2
This is extremely tiny if (v/c) itself is very small. And therefore, if we understand why v is so much less than c in daily life, then we will simultaneously understand why the energies of ordinary human affairs are so small compared to the internal energies of typical objects around us.
So when I return to this topic in an upcoming blog post, we’ll explore why particle physics itself assures that the speeds of daily life must be slow.
Here’s the next post in this series!
21 Responses
If a single photon is travelling in an otherwise empty universe isn’t its speed more of a fundamental than distance or time?. Is there any way to measure distance without using speed in this example?
You’re focused on measuring distance in this example. But how do you want to measure time?
I’ve always liked Matt O’Dowd’s use of ‘the speed of causality’ for c; the fastest speed one event may influence another. It’s unrelated to any physical thing or phenomena; light may, under certain circumstances, match it, but it is not intrinsically related to photons in any way.
I’m not sure trees would like to have the sky flick black and blue every 12 seconds. They’re made of cells same as us and can respond to changing light levels or insect attack in minutes. The world around them moves too fast for them to have the luxury of dawdling and some live a shorter time than humans do.
“Sound speed varies with temperature and with the material through which it travels, and so this speed is completely different in other planets’ atmospheres and oceans. It could never be used as a cosmic measure of speed that all intelligent species could agree on.”
As far as I know speed of light also varies with the medium. c is the speed of light in vacuum. What am I missing?
Your question is absolutely spot on, and thank you for asking it. I have rewritten parts of the post to make it clear.
The point is that c is the cosmic speed limit and the speed of light in empty space. Inside a material, c remains the cosmic speed limit, but light slows down and travels at a speed below c, as you noted. That’s why referring to c as “the cosmic speed limit” is so much better than calling it “the speed of light”. If you do the latter, you have to always remember to call it “the speed of light (in empty space)”. Otherwise confusions like yours are immediately generated.
Thanks again for the question; it improved the post.
The speed of light, “the cosmic speed limit” is c because the universe “chose” the fine structured constant = 0.007 297 352 5693.
1. So, why did the universe chose this particular value?
2. If, the universe is cyclic and does collapse, will the next Big Bang necessarily have the same fine structure constant.
3. Here’s a real interesting one, is the fine structure constant associated with the natural resonance of the Higgs field?
4. Finally, is it all about an ideal fuild, the quantum foam, and the geometry of the spacetime just outside a singularity event horizon whether we are talking about black hole, the Big Bang and/or the nucleaus of an atom?
How could Einstein define energy on an absolute scale ? Kinetic energy is the extra energy with respect to rest. But the overall energy in E=mc^2 is with respect to what ?
Can it be defined better than ‘up to a constant’ ? The only thing I can think of is that it is defined via the total conversion of mass into massless energy, that is, into radiation. That, however, is only possible if anti-matter is involved (right?) and anti-matter was unknown of in 1905. So how did Einstein manage to give an empirical meaning to E=mc^2, as opposed to dE = dm c^2 ?
The question that is defined is this: how much energy must be added to a system, at a minimum, to create a new particle from scratch? That doesn’t depend on how you set the zero of energy of the whole system, since it’s the energy of a change in the system that doesn’t alter how you set the zero. For instance, to create an electron and a positron from scratch takes at least 2 m c^2 of energy, where m is the mass of the electron (= the mass of the positron). That is true no matter how you set the zero of the energy, which is the same before and after the process.
Einstein didn’t bother to say this explicitly, because it’s clear to any professional without having to say it out loud.
You could have asked Newton’s compatriots the same question: how did they know that the energy of a moving object is 1/2 m v^2? Why couldn’t one add a constant? And indeed, one can: mc^2. But that constant doesn’t come up if you’re not creating particles from scratch. Before Einstein, it didn’t occur to anyone that you might be able to do that. In any Newtonian process, the mc^2 factor is the same before and after the process, and so it is indeed a constant that doesn’t matter. Only once you have radioactive decay and particle production do you realize that something is happening that requires you to rethink your Newtonian views.
So the question you should really ask is: might there be some other processes, beyond producing particles or particle decays, that could pull energy out of a constant shift in the energy? At that point we have to look to general relativity, where an overall constant shift affects gravity and is no longer unobservable. At that point it becomes absolutely clear that there are no missing constants of energy.
OK. Thanks a lot for your reply. Can we rephrase it as:
– in 1905, with SR, Einstein ‘only’ shows that dE = dm c^2 (where m is the inertial mass)
– in 1915, with GR, Einstein further shows that E = m c^2 (also explaining why inertial and gravitational mass are one and the same)
It’s actually a lot more complicated than that. For one thing, whether m is inertial mass or m is rest mass (and whether E is total energy or just the internal energy of the object) engendered a lot of debate at Einstein’s time. If you take it as rest mass — as particle physicists do, and as Einstein did in his later years — then your last statement isn’t true. More generally, the statements that are needed in general relativity are a lot more complicated than just E=mc^2. So while I understand what you’re trying to do here, I think you aren’t really gaining much by it. The fact is that any equation used in physics needs to be properly interpreted, and never contains all the information that you need to know all by itself. Trying to rewrite it in some new way doesn’t actually make it any clearer. To really understand what E=mc^2 means (both in 1905 and 1915) you need the whole structure of Lorentz transformations and their mathematics, as well as a proper definition of energy and momentum.
I have never managed to write a simple discussion of this, but my latest attempt (which explains why particle physicists, and Einstein in his late years, view m as rest mass) is here: https://profmattstrassler.com/waves-in-an-impossible-sea/waves-in-an-impossible-sea-commentary-and-discussion/chapter-7/chapter-7-endnote-7/
Not material to the article, but I always think the speed of light is suprisingly low (when you consider the intution that it is instance). It takes a second for light to come from the moon.
That said, I asked a load of friends a bit ago if they shared the thought, and no, everybody thinks it’s suprisingly fast. But I am going to stand my ground
It is indeed a matter of perspective. But the relative form of the question still stands: why is c so fast compared with typical human speeds? That still deserves an answer. And it has one, as you’ll see.
Great article! Looking forward to reading the next part. As you have said, the speed of light in the vacuum is a constant of our universe, with a value that is a function of two other properties of the vacuum, the permeability of the vacuum, and the permittivity of the vacuum.
If the speed of light of the vacuum, or of any given electromagnetic wave, were to have a different value, like for instance, faster, that would affect the values of the aforementioned properties of the vacuum, which would imply very stark consequences for the general physical behaviour of our universe, like for instance, how “transparent” or “opaque” would be the vacuum to the propagation of electromagnetic waves (light), or how strong is a magnetic field in the vacuum as induced by a given electric current.
Kind regards, GEN
I have described the properties in the inverse order of the names that I mentioned, my bad
I would strongly urge you not to think this way. The cosmic speed limit would be the same even if light did not exist in our universe, in which case ideas like “permeability” and “permittivity” of the vacuum, derived from imagining space as being like an ordinary material, would not make sense. All you need for the cosmic speed limit is space-time and Einstein’s relativity, and you’d discover this limit using gravity or using electrons and neutrinos even if photons did not exist.
Instead, you should start from a universe with a cosmic speed limit, then put electromagnetic waves in it, and only then define, if you choose, terms like vacuum permittivity and vacuum permeability. When you do so, you will find that there is a relation between them that is fixed by the cosmic speed limit. They cannot separately be varied, and your decision to define two terms, rather than just one, introduced some redundancy.
One should always be careful to derive less fundamental things from more fundamental ones, and not the other way around.
Matt, when you say ” All you need for the cosmic speed limit is space-time and Einstein’s relativity”, I understand that you mean (x, y, z, -ct) as space-time, and indeed, from said space-time you can derive the cosmic speed limit … is that what you mean?
x,yz,-ict, I meant.
No, spacetime may be (x,y,z,t), but special relativity requires that the space-time distance s between two events at time t1 and t2 and positions x1, x2 (setting y and z to zero for brevity) satisfies s^2 = (t1-t2)^2-#(x1-x2)^2 in every inertial frame of reference. We then define # to be c^2; that’s the definition of c, it’s the number which appears in the equation above for s. We then discover — not by experiment, but by simply working out the equations that such a particle must satisfy — that according to any inertial observer, any massless particle in empty space has v=c, and any particle with non-zero mass has v
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The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time, however it further states that energy can neither be created nor destroyed – only converted from one form of energy to another. In other words Energy and mass have always existed, so yo cannot create something out of nothing but from the available mass and energy that has always existed. To my thinking and agreeable to Principles of Relativity, that even the so called vacuum of space or the so called ether is a form or energy or mass (so far unexplained as dark matter) as it has always existed, even if in another form (call it dark matter). As a theologian, I see the Wavicles as (organized power) from an intelligent being, that passes through all energy and mass under two laws. 1) To GOVERN His creatures, according to immutable laws (elements that are given shape, image and mass and which eventually will be converted to another (energy/mass) within a higher sphere/dominion/principality/dimension or realm. 2) To CONTROL the lower (non intelligent elements that make up mass as dust, plants, gas, planets, stars quasars, Pulsars etc, according to laws.
There is no place where there is no law, neither is there no place where there is no mass or energy.
There is a place currently unknown to most intelligent beings which is indicated as the ‘Nucleus of all Nuclei” of which I will defer to legitimise until a later time. In every discourse you make either in your book , podcast, Youtube or any medium, I see all you work as easily converted into theology. I certainly lack the intelligence as far as the mathematical equations that you have attained, never theless, your presentations make perfect sense to me in another bandwidth. Thanking you for your energy in helping many see beyond their limited horizons. Kind regards Joseph
On this: “even the so called vacuum of space or the so called ether is a form or energy or mass (so far unexplained as dark matter) as it has always existed, even if in another form (call it dark matter).” I think you might want to read the book’s Chapters 1-8 carefully. Things — objects — are not forms of energy. Energy is carried by things. The same is true of mass. If you imagine making things out of energy or out of mass, you are misunderstanding what energy and mass are. They are properties of substances, not the ingredients for substances. Do not let loose language lead you astray on this.
Also, I’d like to ask you again to please hold your religious theorizing from this blog, particularly when it veers into scientific speculation with no experimental basis (as in “within a higher sphere/dominion…” or “Nucleus of all nuclei”.) I’d encourage you to write about these ideas on a blog devoted to the larger questions of truth, philosophy, religion, metaphysics, and the like. This is a site devoted to the very limited methods and lessons of science.
In science, we assume there are laws of nature and do experiments to understand what follows from those assumptions. Remarkably, what we learn proves to be very powerful in the practical, material world. But we cannot use these methods to definitively address larger questions of truth, religion, fundamental origins, etc.
You can try to obtain answers for such questions as you see fit, but these are not scientific answers, as they require assumptions beyond those that are necessary to do science. You’re welcome to make those assumptions, of course. But the consequences of those assumptions are not scientific, and belong on a different blog than this one. Here we walk a narrow, straight path, knowing that sticking to that path means that we can answer a few questions with great clarity, while leaving other important questions, including ones essential to being human, completely unaddressed.
In GR, conservation laws are a little bit more complex.
At first, it was not clear how conservation of energy applied to GR, and that gave way to the discussions and debate of Hilbert, Klein, Einstein and Noether about this topic.
It was Noether who made this topic crystal-clear.