Why is the speed of light a random finite number?

Category: Physics
Published: February 7, 2024

By: Christopher S. Baird, author of The Top 50 Science Questions with Surprising Answers and physics professor at West Texas A&M University

The speed of light in vacuum is the universal speed limit, meaning that nothing can ever travel faster than this speed. If the speed of light is a universal and fundamental limit, you may wonder why it has a seemingly random value of 299,792,458 meters per second. Why not 2 m/s more? or 5 m/s less? or a nice round number? What magical thing happens right at that speed?

Public Domain Image
Source: Christopher S. Baird

The first thing to note is that the speed of light in vacuum only has the value of exactly 299,792,458 if you are using meters to measure lengths and seconds to measure time (using the post-1983 definition of the meter). If you instead use miles to measure lengths and hours to measure time, the speed of light in vacuum is approximately 670,616,629 mph. If you instead use miles per second, it is approximately 186,282 mi/s.

As should be obvious at this point, the actual number that describes the speed of light depends on which units you use. There is nothing special about these numbers themselves. These numbers have scientific meaning and can be useful, but only insofar as you have defined a unit system and measured things using that unit system. You can, in principle, define your unit system however you want and get the correct value for the speed of light to be whatever number you want (as long as it's greater than zero). For instance, if I define an "opti-oddimeter" as a distance that is exactly equivalent to 23,060,958 ⁴/₁₃ meters, then the correct value for the speed of light is exactly 13 opti-oddimeters per second, which might make some people uneasy!

The seemingly random nature of the numeric value of the speed of light is an artifact of choosing a non-ideal unit system. Meters, miles, seconds, and hours were defined long before much was known about the speed of light. The original definitions of these units have nothing to do with light, so it should be no surprise that the value for the speed of light using such units is such a seemingly random number.

Can we define a different unit system that is more relevant to the basic physical nature of light and therefore end up with a number for the speed of light that makes more sense? Yes. In fact, this has already been done by physicists. The unit system called the natural units of particle physics has its units defined such that the numeric value for the speed of light is exactly 1.

As you can see, this choice of units leads to a much more natural value for the speed of light in vacuum. The speed of light having a value of exactly 1 makes sense because no proper fraction is greater than 1, just like no speed can be greater than the speed of light. The number 1 is the same as 100%, and 100% often means the largest amount possible. Therefore the value of a speed in natural units gives you an immediate sense of how close that speed is to the highest possible speed. For instance, an object that has a speed of 0.20 in natural units has a speed that is 20% of the speed of light in vacuum, which is the maximum possible speed in the universe. In summary, the seemingly random value of the speed of light in vacuum is an artifact of using a non-ideal unit system. This randomness goes away when you use natural units.

Why don't we use natural units in everday life?

If natural units are so great, why don't we use them in everyday life? First of all, there is historical inertia. People are familiar with using certain units and are resistant to learning something new. However, this problem is bigger than just annoying people. All of the highway speed limit signs would have to be remade using the new units and all packaging would have to be redesigned. All of the rulers, tape measures, speedometers, weather stations, legal documents, and so forth would have to be remade, which would cost trillions of dollars or more. Politicians have tried to get the United States to switch completely to using the metric system for years with only mixed results. It would be even harder to convert the whole world to using the natural units system.

The other problem is that the natural unit system is not very practical in everyday life. It is much easier to fit on a highway speed limit sign the symbols "75 mph" or "120 km/h" than to fit "0.0000002502 natural units" on the sign and still have it be legible from far away. Scientists successfully deal with this sort of problem by using exponential notation, so we could instead write "2.502 × 10^-7 natural units" on the speed limit sign. However, to the average Joe barreling down the highway, this would be even less comprehensible. Even in scientific research, natural units are rarely used because they are not very practical for expressing real-world measurements. Natural units are typically only used by theoretical physicists.

Why is there a universal speed limit?

You now know that the universal speed limit, which is the speed of light, is a nice round number when you use natural units. However, that still leaves the question, Why does it exist at all?

The answer has to do with the fundamental nature of space and time. As an object increases its speed, external observers see its time dimension slow down and its spatial dimension in the direction of motion shrink to a smaller size. The first effect is called time dilation and the second is called length contraction.

These effects sound strange and unbelievable the first time that you hear about them. This is because these effects are so extremely small in everyday life that we don't notice them, meaning that we have no direct experience with them. They only become significant at speeds close to the speed of light. As strange as time dilation and length contraction may sound, they have been proven by a mountain of evidence spanning over a hundred years.

What does this have to do with a universal speed limit? The closer that an object's speed approaches to the universal speed limit, which is the speed of light, the closer its time dimension approaches to being completely stopped and the closer its spatial dimension approaches to being shrunk down to exactly zero size. This means that a valid reference frame can never travel at exactly the speed of light because its space and time would cease to exist, which contradicts the requirements for it to be a valid reference frame. (You may object that light itself manages to travel at exactly the speed of light without any problems. However, light has no mass and therefore does not have its own valid reference frame. So you see, even if we include light in our considerations, we still have the principle that no valid reference frame can travel exactly at the speed of light. This may seem like semantics, but it is actually fundamental to what it means for light to be massless.)

If a valid reference frame can never travel exactly at the speed of light, then a valid reference frame can certainly never travel faster than the speed of light. This makes sense because you can't make time tick slower than stopped and you can't shrink space smaller than zero size, which is what would correspond to traveling faster than light.

What is beyond the speed of light?

The issue is not just that speeds beyond the speed of light are unreachable. Rather, there are no speeds beyond the speed of light. Speed is defined as the distance traveled through space per amount of time, which makes no sense if space and time do not exist. Furthermore, everything needs the background framework of space and time to exist, meaning that nothing exists beyond the speed of light. In fact, there is no "beyond the speed of light." This phrase makes no sense. It's like saying "colder than absolute zero" or "bigger than infinity" or "darker than absolute darkness." You can certainly say these words, but they don't convey anything physically meaningful.

Some people play interesting mental games where they pretend that if time goes slower than stopped, that means that time goes backward, and therefore traveling faster than the speed of light means time traveling to the past. However, they forget the other result that all space would be shrunk to zero size, which is certainly not a situation where time travel, or any kind of travel, is possible.

In summary, the reason that a universal speed limit exists (assuming that we are using the traditional definition of speed) is because time naturally slows toward stopped and space naturally shrinks toward zero size when approaching that speed, so that there is no "beyond the speed of light."

Why can I write down a speed faster than light if it can’t exist?

This is actually a more profound question than you may realize. The answer to this question is commonly misunderstood. The fact that the universal speed limit exists at a particular finite speed is purely an artifact of the unnatural way in which speed has traditionally been defined. If you use a more natural definition of speed called rapidity, then the universal speed limit has a value of infinity. Therefore, you can indeed write down a speed as big as you want and it is still physically possible, when using the rapidity definition of speed. Of course, the universal speed limit being at infinity is not a limit in the sense of a demarcation line between speeds that are allowed and speeds that aren't. It is a limit in the sense that time comes closer to being stopped and space comes closer to being shrunk to zero size the closer that the rapidity gets to infinity.

In other words, the existence of a finite universal speed limit actually arises from the unnatural way in which humans have traditionally defined the concept of speed. The traditional definition of speed works amazingly well at all of the speeds experienced by humans in everyday life. However, the traditional definition of speed becomes unnatural at high speeds. The traditional definition of speed is the distance traveled by an object in a certain amount of time, divided by that amount of time, with both distance and time measured by a standard external observer.

The fact that speed itself makes no sense beyond the speed of light (if using the traditional definition of speed), and yet you can write down a regular number that represents a speed higher than the speed of light, is an indication that our traditional definition of speed is not the best.

Rapidity is a more natural definition of speed when considering time dilation and length contraction. This is reflected by the fact when using the rapidity definition of speed, saying "nothing can travel faster than the speed of light" is equivalent to saying "no number is larger than infinity," which is obviously true. You can write down a number larger than 299,792,458 m/s, but you can't write down a number larger than infinity. (Note that infinity times two is not larger than infinity, it is the same. It is only larger if you are not talking about true infinity, but are instead talking about mathematical limits toward infinity.) To be clear, the fact that light itself has a rapidity value of infinity does not mean that light travels instantaneously or that light experiences no time. It just means that the rapidity definition of speed is not very useful if you are analyzing light itself.

Why don't we use rapidity in everyday life?

The definition of rapidity and the traditional definition of speed with natural units give effectively identical results at all speeds much smaller than the speed of light, which are the only speeds that we ever encounter in everyday life. For this reason, humans can be excused for sticking with the traditional definition of speed. The speeds of birds, cheetahs, rivers, wind, boats, trucks, cars, airplanes, and even the fastest rockets are so far below the speed of light that the traditional definition of speed and the rapidity definition of speed give essentially identical values.

The fastest speed ever achieved by a man-made object—NASA's Parker Solar Probe going 395,000 mph—is only 0.059% the speed of light. This is still small enough that the traditional definition of speed and the rapidity definition of speed give identical results. (Humans have managed to get subatomic particles to travel close to the speed of light, but that is a different story.) This means that for almost all facets of human experience, the traditional definition of speed is just as good as the rapidity definition of speed.

What is rapidity?

Rapidity is defined as the inverse hyperbolic tangent of the distance traveled by an object in one second divided by the distance that light travels in one second. This is equivalent to rapidity being the inverse hyperbolic tangent of the traditional speed of the object divided by the traditional speed of light.

For example, an object that is traveling at 85.0% the speed of light is traveling at 255,000,000 m/s, at 0.850 in natural units, and at a rapidity of 1.26. As another example, an object that is traveling at 99.9% the speed of light is traveling at 299,500,000 m/s, at 0.999 in natural units, and at a rapidity of 3.8. As a last example, an object that is traveling at 99.9999999% the speed of light is traveling at 299,792,457.7 m/s, at 0.999999999 in natural units, and at a rapidity of 10.7. This last example is not purely hypothetical. Researchers at the Large Hadron Collider have accelerated protons to 99.9999991% the speed of light.

"Hold on!" you may say. "Rapidity certainly does not sound like a more natural way to define speed." It's true that the equation defining rapidity is slightly more complicated than the equation defining traditional speed, and that rapidity is less familiar to humans than traditional speed. However, using rapidity instead of traditional speed makes some of the core equations of fundamental physics much simpler. I can demonstrate this by showing you a bunch of equations. Don't feel overwhelmed with trying to understand what each part of these equations mean. Just pay attention to how complicated each equation looks relative to the others.

The image below shows the velocity addition law for colinear motion. This law tells you the resulting velocity when an object is moving relative to one reference frame and that reference frame is moving relative to a second reference frame, and both the object and the moving reference frame are moving in the same direction. For instance, if you are standing in the back of a pickup truck going 30 mph and you throw a baseball directly forward at 50 mph relative to the truck, the velocity addition law tells you how fast the baseball will be going when it hits a tree (assuming air resistance and gravity are negligible).

Public Domain Image, source: Christopher S. Baird.

Our everyday sense is that velocities simply add arithmetically, as shown in the image above in the first equation (and therefore the baseball hits the tree going 80 mph). However, this is technically wrong. Despite being wrong, for almost all speeds that humans encounter in everyday life, simply adding the velocities arithmetically gives you an answer that is indistinguishably close to being correct. At velocities close to the speed of light, the everyday version of this law gives spectacularly wrong answers. Note that the everyday version of this law is technically always wrong, for all speeds. However, it is only detectably wrong for speeds far above the speeds of everyday life. The everyday version of this law, which is perfectly sufficient in everyday life, is called "Galilean velocity addition" or "the velocity addition of classical physics."

The correct expression for the velocity addition law for colinear motion is shown as the second equation in the image above when it is in terms of traditional speed v. This expression is exactly correct for all speeds. As you can see it is quite complicated, which is an artifact of using the traditional definition of speed. In contrast, the third equation in the image above shows the exact same, completely correct expression for this law, but in terms of rapidity w. Note how much simpler the law becomes if you use rapidity. To be clear, the second equation and the third equation are fundamentally exactly the same, but the first equation is not. The first equation and the third equation are fundamentally different, even though they look mathematically similar.

The image below shows the law for the total energy of a moving particle with mass. The first equation in the image below shows the everyday version of this law. As you may guess, this equation is extremely close to giving correct answers for the total energy at speeds that we encounter in everyday life (assuming that you ignore the object's rest energy) but is spectacularly wrong at speeds close to the speed of light.

The second equation in the image below shows the correct version of this law, in terms of the traditional definition of speed v. This second equation is correct at all speeds. The third equation in the image below shows the correct version of this law, but in terms of rapidity w. As you can see, the rapidity version is simpler. (Although, if you don't have much experience with the hyperbolic cosine, it may look more complicated!) Note that if you put in a speed that is greater than the speed of light into the second equation below, v > c, you end up with the square root of a negative number, which gives an error when using real numbers. This indicates that you can't have a speed greater than the speed of light. This is equivalent to putting a number larger than infinity into the third equation below, which obviously can't be done because there are no numbers larger than infinity.

Public Domain Image, source: Christopher S. Baird.

The image below shows the law for the momentum of a moving particle with mass. The first equation in the image below shows the everyday version of this law. Again, as you can probably guess, the everyday version of this law gives answers that are extremely close to correct for everyday speeds but is spectacularly wrong at speeds close to the speed of light. Technically, the first equation is wrong at all speeds.

The second equation in the image below shows the correct version of this law, in terms of the traditional definition of speed v. This second equation is correct at all speeds. The third equation in the image below shows the correct version of this law for all speeds, but in terms of rapidity w. Again, as you can see, the rapidity version is simpler. If you put in a speed that is greater than the speed of light into the second equation below, v > c, you end up with the square root of a negative number, which gives an error when using real numbers. This indicates that you can't have a speed greater than the speed of light. Again, this is equivalent to putting a number larger than infinity into the third equation below, which obviously can't be done because there are no numbers larger than infinity.

Public Domain Image, source: Christopher S. Baird.

The image below shows the law for the blueshift of light. This law is also called the Doppler shift of the wavelength of light when the source of the light is moving directly away from the receiver. Note that the law for the redshift of light (when the source of the light is moving directly toward from the receiver) is the same as the law for the blueshift, except you replace each v with (-v) in the second equation below and replace w with (-w) in the third equation below.

The first equation below shows the everyday form of the blueshift law, which is technically never exactly correct. However, for very low speeds of the object emitting the light, this first equation is extremely close to being correct. For instance, a police officer's speed radar gun uses the Doppler effect to measure the speed of vehicles traveling on roads. However, the speeds of vehicles on the road are very slow compared to the speed of light, meaning that radar guns can be programmed to use the everyday form of the Doppler shift law and still get accurate results.

The second equation below shows the correct version of the blueshift law in terms of the traditional definition of speed v. This second equation is correct at all speeds. The third equation below shows the correct blueshift law in terms of rapidity w. Again, as you can see, the rapidity version is simpler. As with energy and momentum, if you put in a speed that is greater than the speed of light into the second equation below, v > c, then you end up with the square root of a negative number, which gives an error when using real numbers. This again indicates that you can't have a speed greater than the speed of light. This is again equivalent to putting a number larger than infinity into the third equation below, which obviously can't be done because there are no numbers larger than infinity.

Public Domain Image, source: Christopher S. Baird.

In summary, the fact that there is a universal speed limit with a finite value is an artifact of the unnatural way in which speed has been traditionally defined. If you instead define speed in terms of rapidity, the upper limit (if you could call it that) is naturally at infinity. To be absolutely clear, I am not saying that the existence of a universal speed limit for traditional speed is incorrect or inaccurate or a flaw or a hoax. If you choose to use the traditional definition of speed, the fundamental nature of space and time requires there to be an absolute, universal speed limit at exactly c = 299,792,458 m/s.

If you instead define speed in terms of rapidity, then all speeds are possible. However, this statement could be misleading. A rapidity value approaching infinity is equivalent to a traditional speed approaching c = 299,792,458 m/s. In other words, the rapidity being allowed to have any value between zero and infinity is equivalent to the traditional speed being allowed to have any value between zero and c = 299,792,458 m/s. Using the rapidity does not magically make the universal speed limit go away. It just moves the value for the universal speed limit to infinity, as is shown in the image below.

The image below shows the plots of the traditional speed and the rapidity of a moving object as a function of the distance traveled by the object in a second. In this image, the distance traveled by the object in a second has been divided by the distance traveled by light in a second, in order to make it not depend on the choice of unit system. Also, the traditional speed and rapidity were plotted using natural units.

Public Domain Image, source: Christopher S. Baird.

The purple region in the image above shows the values that do not exist physically and can never exist physically. From a physical standpoint, the purple region does not exist at all. As this image shows, using the traditional definition of speed means that the range of possible speeds (the blue line) seems to abruptly stop at a random point that we call the universal speed limit. If we instead use the rapidity definition of speed (the orange line), the fundamental limitations of space and time are still respected (i.e. the orange line does not enter the purple region), but the rapidity can have any value between zero and infinity, meaning that the orange line never stops anywhere.

Also note in the image above that for all speeds ever experienced by humans, by man-made machines, and by earth-bound macroscopic objects (v ≪ 0.1), the rapidity definition of speed and the traditional definition of speed give effectively identical results, like we have been talking about. The distinction between rapidity and traditional speed therefore only matters if you are studying the fundamental nature of spacetime, as we are doing here, or if you are dealing with exotic astronomical phenomena or extremely fast atoms or particles.

With all of that said, the rapidity definition of speed has its limitations. For instance, the rapidity value of light itself has the value "infinity", which is not very helpful if you are trying to do calculations involving only light, such as calculating how long it takes light to travel from the sun to the earth. It can also be misleading because light does not travel instantaneously, i.e. light does not cover an infinite distance in a second. That is not what the rapidity definition is saying. Lastly, to be clear, all of the equations in this article apply only to objects with mass. Objects with no mass, such as light itself, obey different laws. However, even objects with no mass cannot travel beyond the speed of light, because it does not exist.

Topics: length contraction, natural units, rapidity, relativity, speed of light, time dilation