# How can any speed be defined as a constant? [duplicate]

We know that the speed of light is a constant, and can therefore be used to calculate many other relative values, but I'm having difficulty understanding how speed can be a constant, seeing as it's dependent on other variables.

For example, take a universe in which there is nothing other than Earth and the Sun. No other stars, galaxies, planets. In this example, when a rocket leaves Earth and heads towards the Sun, its speed is measurable, according to its varying distance from the Earth and the Sun.

Now delete the Sun from the image. The rocket hasn't changed; it's still heading away from the Earth at the same speed. But what if the Earth itself now starts moving? Now the distance is still changing, but how the individual speeds be calculated?

Now delete the Earth, and we have only a moving rocket in the entire universe. How can its speed be calculated? How do we even know it's still moving? Does the concept of 'speed' still exist?

My question is: If speed is not an entity in itself, but only dependent on other constant factors, how can the speed of anything (let alone light) be a constant? Am I completely missing something here?

Excuse the lack of scientific terms; I'm not a scientist; rather a software developer with an interest in relativity and quantum physics.

Edit: The question is not "Is it possible to have a constant of speed?", in which case it would be possible to answer with, "Yes, and the proof is that there is a constant of speed, here's how to prove it...", which is what most of the answers seem to be doing.

The question is rather "How can there be a constant of speed?" - emphasis on the 'constant' - in which case saying "Here's the proof that there is one" is not an answer. I know there is one, otherwise I wouldn't be asking the question...

• I've deleted some comments that were attempting to answer the question... If you have an answer, please post it as one! Comments are to seek clarification or suggest improvements only. Thanks! Mar 31 '20 at 1:34
• This is quite clearly a duplicate of every "please explain relativity to me" question ever, of which there are very, very, very, very, very many. I'm voting to close as a duplicate as the first decent one that appeared in the linked questions. Apr 2 '20 at 19:40
• Comments are not for extended discussion; this conversation has been moved to chat. Apr 3 '20 at 0:20
• I've just deleted another round of answers in the comments. Please use these only to seek clarification or suggest improvements. Anything else should be an answer (if appropriate) or in the chat room if it isn't an answer. Apr 3 '20 at 0:21
• For your updated clarifications, see my simple answer. The only logical possibility seems to be a maximum relative speed.
– user257090
Apr 6 '20 at 23:38

Speed of light is actually a pretty special case compared to how we typically think of speed (as far as I understand it).

Movement is always relative to some frame of reference. In the case of a single isolated object, it's hard to really think about how you could have any frame of reference without at least a second object to measure the speed relative to. With multiple objects, the relative velocity obviously depends on the movement of each object relative to each other.

Light is a special case because it has the same speed relative to any inertial reference frame. If I'm moving 3 km/s compared to Earth, for example, and someone else were moving 30,000 km/s compared to Earth, we would still both measure light moving at the speed of light relative to each of our reference frames. It's not exactly intuitive, but it's how we've found reality actually works.

It also leads to other weird consequences, like length contraction and time dilation.

• When you say "it's how we've found reality actually works", can you provide any links where I can read up on these examples? Mar 31 '20 at 0:45
• @Still_learning en.wikipedia.org/wiki/Tests_of_special_relativity Seems to have plenty of examples. To be honest, it's not really my area of specialty. There's quite a bit of history behind it.
– JMac
Mar 31 '20 at 0:56
• The most famous by far is en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment Mar 31 '20 at 2:32
• When you understand the "any inertial frame" part, you can appreciate how revolutionary special relativity is. Apr 1 '20 at 11:39
• @Still_learning Einstein's short book, "Relativity: The Special and General Theory". Part 1 essentially consists of him asking the same question you are: If all speed is relative to a reference frame, but the speed of light in a vacuum is constant, then what does that mean? And then working out the implications of that through a series of thought experiments and some basic algebra. (And then having learned what a constant speed of light implies, experiments were run to see if those things were true in reality. q.v. JMac's link)
– Ray
Apr 2 '20 at 14:28

My question is: If speed is not an entity in itself, but only dependent on other constant factors, how can the speed of anything (let alone light) be a constant? Am I completely missing something here?

The question you are really asking is which is the more fundamental, speed or distance? Think about the distances in space. How are we to measure them if we do not measure the time taken for something travelling at known speed? Think too about the structures of matter, how does a ruler maintain its size? It is because of the forces in bondings between atoms or molecules. Those forces are electromagnetic. The are transmitted by photons, in effect by particles of light. It is the speed of the forces (i.e. the speed of light) which create structure and give length to the ruler. Ultimately distance depends on speed, not speed on distance.

Either there is, or there is not, an absolute maximum speed in nature. If there were not, the laws of physics would be different from those we observe. So we know that there is a maximum speed. It happens that light travels at the maximum speed. Because distance depends on speed, all other speeds are fractions of the maximum speed. The speed of light is a constant because all speed are relative to the speed of light.

• I don't really see how the electromagnetic interaction in the ruler being transmitted by photons adds anything to your argument Apr 2 '20 at 17:56

It is counter intuitive, this question or variants of it get asked a lot. Surely if you're travelling towards a beam of light it will appear to be travelling faster? The answer is it won't, every inertial observer measures the speed of light to be exactly $$c$$, regardless of their velocity. It is for this reason that the Galilean transformations break down and in their place we use the Lorentz transformations:

$$t'=\gamma\left(t-\frac{vx}{c}\right)\tag{1},$$ $$x'=\gamma(x-vt)\tag{2},$$ $$y'=y\tag{3},$$ $$z'=z\tag{4}.$$

It is also from this seemingly counter intuitive fact that length contraction and time dilation in special relativity appear.

how speed can be a constant

It's not, it depends on selected reference frame. The only exception is light speed in vacuum, which in any reference frame is $$c$$.

But what if the Earth itself now starts moving? Now the distance is still changing, but how the individual speeds be calculated?

In a pre-relativity times there was a Galilean speed addition rule :

$$\mathbf {u} =\mathbf {v} +\mathbf {u'}$$

So if train goes at $$100 \,\text{km/h}$$ speed relative to ground, and you as passenger are walking inside a train wagons towards a driver with a $$5\, \text{km/h}$$ speed, then you may think that your speed relative to approaching crossroad is $$100 + 5 = 105 \,\text{km/h} ?$$

MAYBE.

In this particular case 105 is good approximation because your and train relative speeds is $$\ll c$$. But let's return back to your rocket-earth example.

Suppose that you want to return back to earth, so you reverse your rocket direction and start moving towards earth with $$0.5c$$ speed. Now, like you said some cosmic cataclysm has happened, near super-nova has exploded and explosion wave pushed earth towards your rocket also at $$0.5c$$ speed. So you may think that now your rocket speed towards earth will be $$0.5c + 0.5c = 1c$$ ?

NO

Simply because only light in vacuum and other massless particles can achieve light speed. Other objects with rest mass $$> 0$$ can neither reach $$c$$ nor exceed it. So to be able to correctly calculate your rocket speed towards earth you need to use special relativity speed addition formula, which is : $$u={v+u' \over 1+(vu'/c^{2})}$$

This is also called a composition law for velocities. Now we plug your and earth relative velocities to CMB and get :

$$\large{u={0.5c+0.5c \over 1+\left(\frac{0.5c \,\, 0.5c}{c^{2}}\right)}} = 0.8c$$

So your rocket moves towards accelerated earth only with $$0.8$$ of light speed. If you find it interesting, try to raise relative speeds to CMB of earth,rocket and see what happens. (Hint: Can you reach $$c$$ ?)

Now delete the Earth, and we have only a moving rocket in the entire universe. How can its speed be calculated? How do we even know it's still moving?

Indeed, if there are no external objects to compare with, it's hard to say a rocket is moving at all. However to remove all external reference frames would be impossible to do. At least you can measure rocket speed with respect to it's exhaust. The same as you see in a moving car like it leaves dirty fog behind. Or you can measure how cosmic microwave background radiation wavelength changes due to Doppler blue/red shift, which happens due to your rocket movement and etc.

You are already right on track to follow Einstein's thoughts. Just follow through with the logic and you will find:

There is only one way that a speed can be constant: It has to be a maximum speed!

If there is exists maximum speed, the usual law of adding speeds must be altered!

And that is what he proposed - and it turned out to be spot on.

If speed is not an entity in itself, but only dependent on other constant factors, how can the speed of anything (let alone light) be a constant? Am I completely missing something here?

What you are missing is that distance and time themselves are not constant. Both distance and time depend in part on the velocity of the observer.

1. You are on Earth, and determine that the distance to the Sun is 93 million miles.
2. You get on a rocket and start flying towards the Sun. You check again and find that the distance, not just between you and the Sun, but between Earth and the Sun is now only 90 million miles even though neither of them has moved towards the other.
3. You turn on your radio and call your friend back on Earth. He says the Earth-Sun distance is still 93 million miles.
4. You turn your rocket around and stop, now motionless between Earth and the Sun. You confirm that the Earth-Sun distance is 93 million miles.
5. You turn the rocket back on to head home, and find that the Earth-Sun distance has somehow shrunk to 90 million miles again.
6. You arrive at home and check the clock. You've been gone 30 minutes.
7. You check your phone that you brought with you on the trip, which is still in airplane mode and hasn't synced yet, and it says you left only about 29 minutes ago.

The thing about the speed of light is that it's not actually light that's special. I've seen the speed of light referred to as the speed of causality, the speed of information, the universal speed limit, the cosmic speed limit, and probably a few more terms I haven't thought of right now.

The speed of light is the fastest that anything can move, and all things that have no mass will always travel at that speed. This speed is more fundamental than distance or time, and anything moving at that speed will always be measured as having that speed regardless of the measurer's movement.

This necessarily requires that people moving at different velocities will see the same interval as having different lengths, in both distance and time. This means that space and time contort in some strange and difficult to understand ways based on your movement, and Relativity is essentially an extended exercise in analyzing exactly what contortions are needed to produce an invariant speed. In any case, the effects of this for moving at any speed you've ever experienced are far too small to notice.

• Do photons actually travel at the speed of causality, or at a value which is a very tiny bit slower than the speed of causality as a consequence of their small but non-zero mass? Mar 31 '20 at 19:44
• @supercat Photons have zero mass. You might be thinking of momentum. Mar 31 '20 at 19:52
• +1 for the sound advice of putting your phone in flight mode during interplanetary space-flight. Apr 2 '20 at 12:24

One has to distinguish between, on the one hand, the physical reality that exists outside of our minds and independently of how we think about it, and, on the other hand, human-made ways to describe this physical reality.

Specifically, how speed is dependent on other quantities, how we measure it, etc. are reflections of how we think of it and describe it. The Universe however does not have to obey our conception of it, and this independence is manifested in experimental measurements, when they come in conflict with our existing notions and theories. This is how the relativity theory came to existence: as an attempt to adapt our description of reality to the experimental fact - the constancy of the speed of electromagnetic waves. Same could be said about quantum mechanics.

[Disclaimer: Throughout this post I will be bracketing the ideas of general relativity because I think they needlessly complicate the story.]

The invariance of the speed of the light is more of a statement about the geometry of the universe than it is a statement about light.

Suppose I am in an inertial frame $$S$$ with coordinates $$(t,x,y,z)$$. If one event (call it event A) takes place at location $$(x_A, y_A, z_A)$$ and time $$t_A$$ in $$S$$ and another event (event B) takes place at location $$(x_B, y_B, z_B)$$ and time $$t_B$$ in $$S$$, we can consider the quantity $$(\Delta s_\alpha)^2 \equiv \alpha (t_A - t_B)^2 - (x_A - x_B)^2 - (y_A - y_B)^2 - (z_A - z_B)^2 \tag{1}$$ which I will abbreviate as $$(\Delta s_\alpha)^2 \equiv \alpha (\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2 \tag{2}$$ Note that the $$\alpha$$ prefactor must be there because time and position do not have the same units. It is our conversion factor between time and space.

If I shift to another inertial frame $$S'$$ with coordinates $$(t', x', y', z')$$ where event $$A$$ has coordinates $$(t'_A, x'_A, y'_A, z'_A)$$ and event $$B$$ has coordinates $$(t'_B, x'_B, y'_B, z'_B)$$, I can compute $$(\Delta s'_\alpha)^2$$ just like I computed $$(\Delta s_\alpha)^2$$ in (1).

How are $$(\Delta s'_\alpha)^2$$ and $$(\Delta s_\alpha )^2$$ related? We have $$(\Delta s_\alpha)^2 - (\Delta s_\alpha ')^2 = \alpha \left[(\Delta t)^2 - (\Delta t ')^2 \right] - (\Delta x - \Delta x')^2 - (\Delta y - \Delta y')^2 - (\Delta z- \Delta z')^2 \tag{3}$$ It is an experimental fact that $$(\Delta t)^2 - (\Delta t')^2$$ is not always zero. In other words, the time difference I measure between two events is not necessarily the same in all inertial reference frames. Suppose we are in a situation where $$(\Delta t)^2 \neq (\Delta t')^2$$. We see that there is a unique value for $$\alpha$$, call it $$\alpha_0 (A,B,S,S')$$, such that the LHS of (3) is $$0$$, i.e. $$(\Delta s_{\alpha_0(A,B,S,S')})^2 = (\Delta s_{\alpha_0(A,B,S,S')}')^2$$ This notation is chosen to remind us that $$\alpha_0$$ could depend on our choice of event $$A$$, event $$B$$, inertial frame $$S$$, or inertial frame $$S'$$.

However, it is an experimental fact that $$\alpha_0$$ does not depend on our choices of events or inertial reference frames. In other words, there is a dimensionful quantity $$\alpha_0$$ such that, for any two events, $$(\Delta s_{\alpha_0})^2$$ is the same in every inertial reference frame.

I want to pause here and emphasize that we have only asked that observers can agree about what constitutes an inertial reference frame and are capable of measuring position and time in their own reference frames. The experimentally verified existence of an $$\alpha_0$$ with the aforementioned properties is a purely geometrical fact. It tells us how space and time relate in inertial reference frames. We haven't said anything about speed or light. In principle (although this is emphatically not how it happened in actual physics history) we could have observed the existence of this $$\alpha_0$$ with nothing more than stopwatches and meter sticks.

Now, you are probably wondering what this $$\alpha_0$$ is. How should we interpret it? To aid in our discussion, we'll switch to writing $$(\Delta s)^2$$ instead of the nonstandard and unwieldy $$(\Delta s_{\alpha_0})^2$$. In physics, $$(\Delta s)^2$$ is known as the "spacetime interval" or simply the "interval" between two events.

To start our investigation, let's consider a pair of events A and B with $$(\Delta s)^2 = 0$$ (Such a pair exists. Consider A $$= (0, 0, 0, 0)$$ and B $$= (1, \sqrt{|\alpha_0|}, 0, 0)$$). This means $$0 = \alpha_0(\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2$$ Rearranging: $$\alpha_0 = \frac{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}{(\Delta t)^2}$$ The top of this fraction we recognize as the squared distance between events A and B, which we will write as $$d_{AB}^2$$. The bottom of this fraction is, of course, the square of the time that passes between event A and event B, for which we'll write $$t_{AB}^2$$. Then $$\alpha_0 = \frac{d_{AB}^2}{t_{AB}^2} = \left(\frac{d_{AB}}{t_{AB}} \right)^2$$ From this we see that $$\alpha_0$$ is nonnegative. Thus, we are free to pick a nonnegative $$c$$ such that $$\alpha_0 = c^2$$. It immediately follows that $$c = \frac{d_{AB}}{t_{AB}}$$ for events A and B with interval $$0$$. The interpretation of $$c$$ is clear. An object which moves uniformly from an event A to an event B separated by a spacetime interval of $$0$$ travels at speed $$c$$. In other words, $$c$$ is how fast you have to move to get between events that have $$0$$ spacetime interval. Moreover, because of how it is defined, $$c$$ is invariant: every observer in every reference frame agrees on how fast $$c$$ is.

You have probably guessed that $$c$$ is the speed of light. But, again, notice that we defined $$c$$ completely in terms of the geometry of spacetime. It just so happens that the speed of light, i.e. how fast photons move through a vacuum, is equal to this geometric constant which describes how fast you have to move to get between events separated by a spacetime interval of $$0$$.

To demonstrate that this is the $$c$$ we all know and love, I'll argue on geometric grounds that $$c$$ is as fast as ordinary matter can go. Suppose I want to get from event A to event B. Let's say that in frame $$S$$ they are time $$t > 0$$ (I can't teleport, and I certainly can't move backward through time) and distance $$d$$ apart, so that I must travel at speed $$v = d/t$$ in $$S$$ to complete my trip.

In the inertial frame $$S'$$ whose origin moves with the same velocity as me, we'll measure $$x_A' = x_B'$$, $$y_A' = y_B'$$, and $$z_A' = z_B'$$. In other words, in $$S'$$ I appear to get from event $$A$$ to event $$B$$ by standing still. What can we say about the interval between $$A$$ and $$B$$? Calculating in $$S'$$, we find $$(\Delta s)^2 = c^2 (\Delta t')^2 - 0^2 - 0^2 - 0^2 = c^2 (\Delta t')^2 > 0$$ But the interval is invariant, so in $$S$$ we must also find $$(\Delta s)^2 > 0$$. On the other hand, in $$S$$ we calculate $$(\Delta s)^2 = c^2 t^2 - d^2$$ Thus $$c^2 t^2 - d^2 > 0$$ which means $$v^2 = \frac{d^2}{t^2} < c^2$$ Therefore $$v < c$$. I can move no faster than $$c$$.

In conclusion, experiments tell us the geometry of spacetime. If our universe has the geometry that our experiments indicate, then there must be an invariant cosmic speed limit $$c$$. Based on experimental evidence and theoretical developments, we think this speed limit happens to be precisely the speed at which light propagates.

In special relativity speed of light is constant with respect to any inertial frame of reference. That is an axiom of special relativity. Having well defined frame of reference is all you need. It makes no difference if in your particular frame of reference there are other objects like Earth, Sun or some other thing.

when a rocket leaves Earth and heads towards the Sun, its speed is measurable, according to its varying distance from the Earth and the Sun

You do not need two objects like Earth and Sun to measure the speed, because in your frame of reference you actually have a ruler (ruler has a reference point on it - the point labeled "zero". Every coordinate that we talk about is a distance with respect to this point ) and a clock. That is sufficient to measure the speed of an object (speed is easily calculated if you know the coordinates of infinitely close starting and ending points and the time object needs to cover the distance between these points). In practice, you certainly can measure the speed of light using $$c= \frac{d_{ES}}{t_{ES}}$$, where $$d_{ES}$$ is the distance between Earth and Sun and $$t_{ES}$$ is the time light needs in order to cover the distance $$d_{ES}$$ (keep in mind that you always have to take into account length contraction and time dilation). However I am trying to emphasize that even though this approach is valid it is not necessary to measure the speed in this way. Actually you do not need to have any objects ( like Earth and Sun)! You only need ruler(with a reference "zero") and clock - that means you only need frame of reference.

But what if the Earth itself now starts moving?

It doesn't matter. As I mentioned previously, in order to measure the speed you need the coordinates of an object (in starting position and ending position) that are defined with respect to the "zero" of a ruler. The Earth does not play any role here. You can, if you like, define the Earth itself as a "zero" (assuming that this is an inertial frame... ). But that means that by definition the Earth is not moving.

The reason why you find this so hard to imagine is because you always assume (implicitly in your head) that there is and absolute space. And that is something that is simply not true in the special theory of relativity. That is the reason why I keep emphasizing the term frame of reference.

You certainly noticed that I used the term inertial frame of reference. That is something that needs to be defined. We can just define that the system is inertial if it has constant speed with respect to distant stars. Actually some books use different definition in order to make things more rigorous, but as it is usually the case, that definition is much less practical.

Let me add one more thing. It is possible to define the speed without the use of coordinates and without the inertial reference frame. But that approach uses the language of differential geometry ...

Speed of light is constant wrt to reference inertial frame

You always need two things to define physical quantities like distance and speed.

Like in your example you have three objects, the sun, the earth and the rocket.

If you delete the Sun you still have two observers for these physical quantities to make sense aka measured.

If you delete earth the rocket alone can’t be used to define distance and speed. Think of questions like distance of rocket from ‘what’? Speed aka change in distance per unit time from ‘what’?

For our universe its just that speed of light aka photons etc measured in any frame of reference is always constant.

Meaning the photons are like rockets(to use your analogy) that always leave a frame at c. Seem to approach a frame at c. Regardless of the motion of the frame itself.

Here is the explanation I received on presenting this question to a maths and physics professor. I think it clarifies the 'soul' of the question (ie the point about the conceptual status of speed rather than proving the speed of light which is what most of the other answers seem to be doing):

All speeds are relative and without reference to another object there is no way (even in theory) to determine its speed – if that has meaning at all.

There is one exception and that is light, which presents itself to an observer with the same speed C (by convention) as viewed from any and every other object in the universe, whether moving or stationary (relative to other objects) even if those objects are moving towards or away from the beam of light. Another way of saying that is that the speed of light is constant. In other words the speed of light (in vacuo) will always appear to be moving at at the speed of light C. This is one of the foundations of Einstein’s relativity. But once again if there are no other objects in the universe (and that means no observers either) then a beam of light might traverse the universe but to talk of its its speed would be meaningless.