# Relatively faster that light? (question)

1. Aug 30, 2015

### Moldj

hey. it's my very first post in here.
i couldn't find any answer to my question so i decided to post it here
so my question is:

if someone moves towards the moon at 51% the speed of light ( relative to moon (or earth)) and another person on moon moved towards the earth at 51 % the speed of light (relative to moon (or earth)) then they must be moving at 102 % the speed of light towards each other. no?

so basically if i go near the speed of light in one direction and something else was going near the speed of light in the exact opposite direction and we passed each other what would we see? or feel?

tnx :) (not sure if i posted it in the right place. and also not sure which prefix i should choose)

Last edited by a moderator: Aug 30, 2015
2. Aug 30, 2015

### phinds

No. From the point of view of a 3rd observer between the two, they have an approach speed to each other greater than c, but this is not a proper motion. In THEIR frames, which is what counts as proper motion, they each see the other as moving at less than c. Google "relativistic velocity addition" for the formula, or do a forum search since this question is answered here with great regularity.

EDIT: I tend to forget how confusing this can be at first, so let me expand on it just slightly.

Where ever I happen to be, if I see you coming at me at .51c from one direction and I see John coming at me at .51c from the other direction, then I can compute the time when you guys will pass each other based on a merge speed of 1.02c. But this is not proper motion. What YOU see is me coming at you at .51c and John coming at you at something like .99c (again, Google for the formula and its derivation).

Last edited: Aug 30, 2015
3. Aug 30, 2015

### Moldj

thanks for the answer :). it's still confusing to me but i will google it and hopefully i will understand :)

Last edited: Aug 30, 2015
4. Aug 30, 2015

### harrylin

Hi Moldj, welcome to physicsforums!

As the max. speed is c, the maximum speed difference between two entities (also called relative speed) is 2c in general. However if you assume to be in rest so that your speed is 0, then that difference cannot be more than c.
https://www.physicsforums.com/threa...trains-speed-relative-to.828679/#post-5205138

And here's an old thread in which your question was answered with more elaboration:

5. Aug 30, 2015

### HallsofIvy

Combining speeds as "u+ v" is Gallilean relativity, not Einsteinian. In Einstein's theory of relativity, if person A is moving toward you at speed v and person B is moving toward you from exactly the opposite direction, then A and B each see the other coming toward them at speed $\frac{u+ v}{1+ \frac{uv}{c^2}}$. In particular, if u and v are both 0.51c then they each see the other coming toward them with speed $\frac{1.2c}{1.26}= 0.95c$ which is still less than c.

6. Sep 2, 2015

### Moldj

however i can't visually understand it yet :)

7. Sep 2, 2015

### PAllen

I don't doubt some books call closing speed relative speed, but it is very non-standard, confusing, terminology nowadays. Relative speed in modern usage refers specifically, as the name implies, to the speed of one body relative to another - i.e. as each 'body' would measure the other's speed. Other speeds are closing or separation speed (for diference in speed of two objects in a given coordinate system), and celerity for distance measured by one observer divided by proper time of a moving body. In SR, in standard coordinates, separation/closing speed is up to 2c, celerity has no upper bound, while relative speed is < c for bodies with mass and c between a body with mass and light or any massless particle. Relative velocity is not really defined between two light signals; celerity is not defined at all for light. However, closing/separation speed is well defined between light signals.

8. Sep 4, 2015

### harrylin

It's also a source of confusion to propose that any arbitrary body or particle constitutes a valid or usable reference system for measurements in SR; evidently we come from different schools! and in this thread as well as in other recent threads we use "speed difference" to avoid misunderstandings, as I even emphasized here.

9. Sep 4, 2015

### Stephanus

Hi Moldj, welcome to PF Forum,
Okay, can you visualize this?
A rocket travels 51%c relative to the earth. And while it travels, the rocket fires a missile 51%c relative to the rocket. So the missle travels... (as HallsOfIvy says) 95% c relative to earth?
Perhaps if you can visualize that, it would be easier to visualize your original question.
Try to rearrange your question as this.
So the velocity of that person toward earth is 95% relative to the speed of light. You can think of the moon as the rocket which fires a missile (the person) toward the earth.
And frankly I haven't calculated u+v/(1+uv), as HallsOfIvy writes, but his/her velocity addition formula is correct.
Hope this will help.

10. Sep 4, 2015

### PAllen

The point is that relative speed is a coordinate invariant quantity in the language of 4 vectors. All other type of speed (coordinate speed, closing speed, celerity) are coordinate dependent. In 4-vectors, the scalar product of two 4-velocities is gamma of the relative speed. Thus, even evaluated in coordinates adapted to a spinning, accelerating rocket, it comes at the same as computed in the rest frame of one of the bodies. IMO, it is very important to use relative velocity only this sense. Of course, the coordinate independent computation is defined such that it is always the same as the coordinate speed in standard coordinates such that one of the bodies is at rest. Using it to mean a speed difference in some generic frame is inconsistent with modern usage. That is a closing or separation speed (or a speed difference, if you prefer), but NOT a relative speed.

Just to be clear, this is the only part of what you said that I object to:

"the maximum speed difference between two entities (also called relative speed) is 2c in general."

NOT also called relative speed, without causing great confusion.

Last edited: Sep 4, 2015