# If $c$ is constant to every observer, how can be the speed of an electron moving at $0.9c$ is not constant to every observer?

How can the speed of light, c be the same to an observer who is moving in the same direction with 0.9c and to another observer who isn't moving?

If there is an observer who is following (moving in the same direction) a photon with 0.9c speed, how can that observer see the photon moving still at the same c speed? And if somebody is moving in the opposite direction with 0.9c speed, how can that observer see the same photon (that is moving in opposite direction) moving with the same constant speed c?

Extension: I understand that SR's time dilation makes to any observer c as a constant speed. But my question is: let's say there is something moving at a little bit slower speed, lets say an electron moving at 0.9c. Is this also true for the electron, so that any observer moving at any speed, any direction would see the electron moving at constant 0.9c?

If not (and that is what I am guessing) then I dont get it. Why is it true for EM waves propagating at speed c only? Anything moving even a little bit slower then c will not be seemed to any observer moving in any direction to be moving at the same fast constant speed? Why not?

• The speed of light is a constant in all frames of reference, that is simply a law of nature, but it's direction is not constrained in any way in empty space. – user108787 Oct 7 '16 at 20:00
• Using correctly capitals is important to an usable question quality. Thus, start your senteces with capitals, but don't use all-capital words. I think your question would be a commonly asked naive layman question, but an answerable and on-topic one, b+! You only have to formulate it on a polite and correct way. Maybe it would be closed as a duplicate (questions like this were asked thousands time already here), but in the dupe you could find your answer. – peterh - Reinstate Monica Oct 7 '16 at 20:37

You are mixing two different things:

1. The Special Relativity, it is dealing with the time dilatation, etc.
2. The General Relativity, it is about the spacetime curvature.

The two theories could be used from eachother independently, but although GR without SR is highly uncommon.

Both theories have a very clear math. Similar naive questions are mainly about a seemingly problem about the terminology, and not about their math. The math of (1) is not really above the high school level. The math of the GR is harder.

The answers to your questions are the time dilation. The time of the moving observers slows down (in the reference frame of the "standing" observer).

To your extension: only $c$ is invariant to Lorentz-transformations (Lorentz-transformation means if you move to another frame of reference). If an electron moves with $0.9c$, its speed will depend on where do you see that.

In an 1D linear system, you can use the relativistic velocity addition formula:

$$v_1+v_2 \rightarrow \frac{v_1+v_2}{1+\frac{{v_1}{v_2}}{c^2}}$$

Also in this formula you can see, if one of the velicities is $c$, also the result will be $c$, independently from the other velocity.

EM waves move always with $c$ is coming out from the Maxwell-equations. The proof is shorter as a page and it is only a little bit over the high school level math.

As @CountTo10 (notifiable nick) mentioned in his comment, the direction of the movement can change.

Also the wavelength of the EM waves can change for Lorentz-transformations.

• I wouldn't say that GR without SR is uncommon, I would say that it is impossible. In GR you consider diff-invariance of the metric and in SR you only consider the subset of diffs that are linear transformations. – dpravos Oct 7 '16 at 20:08
• @DavidPravos Ok, but all of the theories, incl. the Newtonian mechanics, is diff-invariant. Why would it make impossible to have a Newtonian-like mechanics with a non-trivial metric? (Note: unfortunately I can't find my source about this, but I am sure I've read about this, although it was mentioned as a highly exotic theory) – peterh - Reinstate Monica Oct 7 '16 at 20:16

Velocity isn't fixed, it's relative, so you have to say what the Electron is moving relative too.

If we set up an example, lets say, you are on Earth and your friend is on Mars and he has an Electron gun and he's shooting electrons at your electron detector, the electrons leave his gun at .9c but relative to you, you have to add the relative velocities of Mars relative to Earth. That's not too different than playing catch while driving, the relative velocity of the cars is added to the velocity of the balls you're playing catch with.

Light is different, you can't add or subtract velocity to light by shooting it off a moving planet because all particles without rest-mass move at the speed of light. Each photon of light has a specific energy and wavelength. What changes is the clocks of the individual who see that light. To them, the light might appear somewhat redder or somewhat bluer depending on their relative motion towards or away from the light source and their relative clock-speed.