# Interpretations of Second Postulate of Special Relativity

(a) Relative to any two frames of reference $S$ and $S'$, the speed of light is the same regardless of their relative motion.

(b) Relative to a frame $S$, the speed of light is independent of the motion of the source relative to $S$.

The statement (b) can be valid without (a) being valid because (b) was valid in non-relativistic physics. So, I guess (a) does not cover (b), but I am not sure. How can it be shown that (b) is or is not a special case of (a)? Please use 'relative to' when talking about speeds, I get really confused figuring this out.

## 2 Answers

(a) merely says that the speed of light both frames $S$ and $S'$ would report they measured is always the same. The reason this does not automatically cover (b) is that this doesn't necessitate there is only one speed at which light can travel.

Let me show two scenarios: I'll make a hypothetical universe that only has to comply with one of the postulates and we can see what can happen.

First, let's say there was an absolute reference frame and that, relative to this frame, we have two sources of light; a stationary source that emits light that travels at $c$ and a source moving at some velocity $\vec v$ that emits light at $c+\vec v$. The implications of (a) are that every other frame, no matter their relative motion, would report measuring the speed of light from each source as $c$ and $c+\vec v$ respectively. (a) insists that any two observers will always agree on the measured speed of any one beam of light. It does not insist that every beam of light has the same speed.

Now, you might want to say "But if the moving source is effectively just another frame then shouldn't it be generating light at $c$ in its frame, which we should measure the same in every other frame?" The answer is no. Recall I set up a hypothetical situation with an absolute reference frame. A privileged frame that is basically the correct frame. The second source was not in this frame, so to determine what speed the light is at, you have to refer back to the absolute frame; can't just call it $c$. Notice that this situation still obeys (a). The motion of the source in the absolute frame determines the speed of light, but the measured speed of light from any source is the same for all frames of reference. However, this scenario doesn't obey (b), which means (a) is not sufficient to cover (b).

Now, let's set up a second scenario; a universe with an absolute frame of reference where the speed of light from any source is always measured as $c$. In this scenario, we can say that relative to a frame travelling a $\vec v$, they might measure the speed of light as $c+\vec v$. Notice that this scenario complies with (b); in any given frame of reference, you will always measure the same value for the speed of light, regardless the relative motion of the source. However, this scenario doesn't comply with (a) because different frames would disagree on the measured value. Clearly, (b) is not sufficient to cover (a).

If we then try to create a situation where both conditions (a) and (b) are met, we find that our absolute frame of reference is one that always measures the speed of light as $c$ regardless the relative motion of the source AND that every other frame of reference must also measure the same value. This means that there is not any difference between what we called our absolute reference frame and any other reference frame. It also means that there can only ever be one speed of light in all situations.

So we see that not only do we need both postulates to properly specify special relativity, we also find by having both that, in SR, there does not exist a privileged reference frame.

In other words, the first postulate states that all physical laws are equal in all reference frames, while the 2nd states that the speed of light is the same in all directions in all inertial systems. Next you have Lorentz transformations
$x' = \gamma(x-Vt)$ and $t'=\gamma(t-(\beta/c) * x)$.
Imagine the following experiment. We place a single dot light source in the origin of both system $S$ and $S'$ and observe the expansion of the light front. From the first postulate the same spherical wave front must appear the same in both $S$ and $S'$.
In system $S$, light emitted at $t=0$ travels distance $r$ in time $t$ so the following is true
$r=ct$, or,
$x^2 + y^2 + z^2 = c^2t^2$
The wave front must have the same form in system $S'$
$\gamma^2(x-Vt)^2 + y^2 + z^2 = c^2\gamma^2(t- (\beta/c)*x)$
Now you sort the equation out and get $\gamma^2(1-\beta^2)x^2 + y^2 + z^2 = \gamma^2(c^2 - V^2)t^2$
We know that $\gamma^2(1-\beta^2) = 1$ and
$\gamma^2(c^2 - V^2) = c^2\gamma^2(1-\beta^2) = c^2$
And we are back where we started $x^2 + y^2 + z^2 = c^2t^2$ This is the proof that Lorentz transformations, which go hand in hand with Einsteins postulates are valid (so far).
In the proof $V$ is the velocity of the system $S$ relative to system $S'$
$\beta$ is equal to $V/c$
and $\gamma = 1/\surd[(1-V^2/c^2)]$
$c$ is the speed of light and $t$ is time.