TITLE: Understanding speed of light as an absolute measure
QUESTION [1 upvotes]: Speed of light is absolute, unlike other speeds which are relative to each other, right? Below is an image of a scenario that illustrates my question. If Alice is moving at $c/2$ in one direction and Bob is moving at $c/2$ in the opposite direction, then from Alice point of view Bob would be moving at the speed of light, no? So Photon 1 and Bob would be moving at the same speed according to Alice. Since they are also moving in the same direction, after some time $t$ has passed they should have traveled the same distance $d$ away from her. So is Bob moving at the speed of light? But then again, the speed of light is absolute and Bob should also be moving at the speed of light from our perspective (standing on the ground watching all of this), no?
On the other hand to Bob Photon 1 is still moving at $c$, so they are not moving at the same speed according to Bob. So after time $t$ Photon 1 will have traveled distance $d$ away from Bob. But isn't this a contradiction?
It might well be that a very similar question has been asked before, but since I do not know the answer I do not know what to look for.

REPLY [1 votes]: If Alice is moving at c/2 in one direction and Bob is moving at c/2 in the opposite direction, then from Alice point of view Bob would be moving at the speed of light, no?

This is not how velocity addition works in special relativity. The correct way to add velocities is via the equation $$w' = \frac{v + w}{1 + vw / c^2}$$
where $w$ is the object's velocity in one frame, and $w'$ is the same object's velocity in a second frame moving at speed $v$ relative to the first frame (and $c$ is the speed of light). In your example, $$w' = \frac{0.5c + 0.5c}{1 + \frac{0.5c\times 0.5c}{c^2}}=\left( \frac{1}{1.25}\right) c = 0.8 0c$$ or in other words, according to Alice, Bob is moving at 80% the speed of light.

So Photon 1 and Bob would be moving at the same speed according to Alice

You have to be careful when talking about light. One of the fundamental postulates of relativity is that the speed of light is the same in all inertial frames.

So is Bob moving at the speed of light?

No. Nothing with mass can move at the speed of light. Nothing moves at, or greater than, the speed of light, except light itself.

REPLY [1 votes]: The main factor that seems to be missing in your attempt to follow it.. is that time passes at different rates for the different observers. A second for one observer might be 1.4 seconds for another observer. This means masses can change too. If Bob thinks he has mass but ends up moving relative to me at c, then I will observe him as having no mass, as being light, in order to travel at c. The closer you get to c the weirder it gets to make it work. Id suggest starting with lower speeds.
Two helpful rules: 1. all observers always see all light as moving at c, even if to others they are moving partially along with or away from some of the light. 2. Any two observers agree on their relative speed, so like if Bob observes he’s moving away from me at .5c then I observe that too. A third party may disagree.
But let’s assume the speeds you wrote are 0.3c from the point of view of the stationary “you” in the model, and see what the Bob part of it works out to:
Bob observes that he and I are moving apart at 0.3c, and that the light is moving away from him at c (always true). I observe that the light is moving at 0.7c from him and moving at c from me. When he and I are three miles apart (from his perspective), the light is 10 miles from him (from his perspective) and 13 miles from me. The amount of time needed for light to go TEN miles has passed for him. We disagree on how much time has passed, but we are both right, because time is not absolute, c is absolute. From my perspective, the time passed is different, per time dilation equation, https://en.m.wikipedia.org/wiki/Time_dilation
1/ [ 1 - 0.7^2 ] ^ (0.5) times as much time has passed, 40 percent more time, enough for light to go 14 miles away from me, and he and I are 1.4*3 = 4.2 miles apart according to me, and the light is 14-4.2= 9.8 miles from him, still moving away from him at 0.7c.
If we continue this way, we can keep c constant from everyone’s perspective and each pair agrees on relative speed.
{"set_name": "stack_exchange", "score": 1, "question_id": 652871}