# Relativity and speed of light again - two opposite light sources [duplicate]

Two light sources emit light at the same moment but in opposite directions. At what speed the distance between two light fronts is increasing? c or c * 2?

Note, that there is only one coordinate system here - a system, where these two light sources are placed and they don't move.

## marked as duplicate by Kyle Kanos, Emilio Pisanty, Brandon Enright, Abhimanyu Pallavi Sudhir, Waffle's Crazy PeanutJan 22 '14 at 15:29

• @Kyle Kanos - I'm not talking about a coordinate system related to one of the fronts. That's the difference between this question and one you are referring to. – HEKTO Jan 20 '14 at 16:08
• @Kyle Kanos (and others) - I'm surprised my question is marked as duplicate of "Double light speed". Do you consider two particles, facing each other, and two opposite light sources as basically the same thing? Look, these two questions have different answers - I think one question can't have two different answers – HEKTO Feb 3 '14 at 0:22

I think you are asking: If I turn on a lightbulb, I can imagine a sphere of light spreading radially outward from the bulb at the speed of light. How fast is the diameter of the sphere increasing in the lightbulb's frame? The answer is 2*c.

• So, relativity isn't involved here? – HEKTO Jan 20 '14 at 16:09
• Relativity is not involved because there is no particle moving anywhere near the speed of light (besides, of course, the photon). The distance between two objects is not a reference frame or a particle or any type of quantity which can interact. This is similar to how a shadow can move faster than the speed of light. Or, how you can shine a laser beam at the moon and make the dot move faster than the speed of light. – mcFreid Jan 20 '14 at 16:14
• mcFreid: "Relativity is not involved because [...]" -- Relativity is involved already due to your (correct) characterization of the setup as "a sphere of light spreading radially outward from the bulb" which requires determinations concerning simultaneity. – user12262 Jan 21 '14 at 6:00
• @mcFreid - so, there exist pretty normal physical processes, where some derived parameter can grow with speed > speed of light. For example: the surface of the sphere above grows with speed 4*$\pi$*c^2*t. All this is beyond relativity theory, right? – HEKTO Jan 21 '14 at 22:04
• Yes. Special relativity applies to objects (call it particles, matter, waves, etc...) and inertial reference frames (non-accelerating coordinate systems). The diameter of a sphere is neither of these. – mcFreid Jan 21 '14 at 22:32

Two light sources emit light at the same moment but in opposite directions.

For the question under consideration it is suitable (and also in accordance to the answer already submitted by mcFreid) to specify only one source ("light bulb $L$") having stated a particular signal ($L_{\mathscr E \ast}$, e.g. "the light bulb turning on");
and let's also consider additional participants ($A$, $B$ ... on the one hand, and $P$, $Q$ ... on the other) which are all at rest wrt. $L$ and (pairwise) wrt. each other, where their distance ratios wrt. $L$ satisfy
$\frac{AL}{AP} + \frac{LP}{AP} = 1, \, \frac{AL}{AQ} + \frac{LQ}{AQ} = 1, \, \frac{BL}{BP} + \frac{LP}{BP} = 1$ and so on
(as a specific setup implementation of $P$, $Q$, etc. having been "in opposite direction from $L$" wrt. $A$, $B$, ...).

Finally, set distance ratios $AL / PL = 1$, $BL / QL = 1$, and so on.
Correspondingly, $A$ and $P$ observed the signal event $\mathscr E \! \ast$ simultaneously, $B$ and $Q$ observed the signal event $\mathscr E \! \ast$ simultaneously, and so on.

At what speed the distance between two light fronts is increasing?

I believe that "distance between two light fronts [propagating] in opposite directions" is supposed to mean more correctly the "distance between participants (located in opposite directions from $L$) who observed signal event $\mathscr E \! \ast$ simultaneously".

The values of those distances (as a function of $L$'s duration "$\tau_L( X )$" from having stated $L_{\mathscr E \ast}$ until indication $L_{\circledS}^{X \circledR \mathscr E \ast}$ of $L$ simultaneous to $X$'s indication of having received the signal $\mathscr E \! \ast$) are
$AP = AL + LP = 2 AL = 2 \, c \, \tau_L( A )$,
$BQ = BL + LQ = 2 BL = 2 \, c \, \tau_L( B )$,
and so on.
However: the quantity $2 \, c$ is some "speed" only in a rather lose sense, similar to "phase speed".
It is obviously the quotient of "some distance" divided by "some duration", but not speed (such as: the speed of the signal having been transmitted) in the strict sense of the quotient

• distance between source and receiver; divided by
• duration of the sender from stating the signal until the sender's indication simultaneous to the receicer's indication of having received the signal.