Dear Daryl,
You wrote:
1 "Draw a vertical line and a horizontal line and call them t and x, respectively."
OK, in this x(t) plane I consider an antenna A that emits at t=t_1 a signal for synchronization with antenna B which will be arrive there at t_2. Only the distance matters between the position of A at the moment of emission and the position of B at the moment of arrival.
2 "Now draw the lines t=x and t=-x."
I would like you to first clarify the meaning of x. Well, you meant a line x=ct. I know, you did set c=1. This is not my point. I have objections.
The left part in Fig. 1 of your essay shows a vertical line illustrating the position of A as reference. The distance between A and B is shown to increase with growing time. I reiterate, the absolute positions of A and B do not matter. Only the difference counts. Can we consider the right part of your Fig. 1 equivalent? Does it also show a difference that increases with growing time? You didn't yet specify the point x=0. If x=0 in the right part refers to B, then the negative difference grows.
In the case you referred to below, the line t=-x does not belong to light from A to B.
In general, the line t=-x may be confusing in so far it gives rise to not consequently distinguish between past and future. While the distance in 3D corresponds to the always positive radius, positive and negative x have different meanings.
3 "Now draw another line passing through the origin, rotated 30 degrees to the right of the t-axis. Call that t'. If t=x and t=-x represent the paths of photons through x in t, which both recede from an observer who sits at x=0 with unit velocity, and if the line t' is the world-line of another observer who moves through x in t, then clearly the photon that moves along t=x isn't moving away from this observer as quickly as the one that moves along t=-x. "
Sorry, you did not at all justify the introduction of t'. So far you seem the only one who distinguishes between simultaneity and synchronicity. I tried to show in a previous post that Einstein's synchronization has been based on unjustified application of Poincaré synchronization on the case of relative motion between A and B.
I understand that the line t=x refers to a time axis that is orthogonal to the x-axis. 30 degrees to the right of that axis then means x=0.577c.
You introduced a configuration that is different from what Einstein, your Fig. 1 and I referred to. Now you consider a common emitter (you called it observer) at x=0 and light signals propagating to both sides in vacuum i.e. with c. Your "other observer" who is thought to move with only 0.577c to the right is neither an emitter nor a receiver of a light signal. It does not take part in the transmission of energy.
Wouldn't it be pointless to ascribe a fictitious speed with respect to the two light signals to it? Likewise you could argue that the sum of the velocities of both "photons" equals to 2c.
Unfortunately, I see you not right when you wrote "Time, in the sense of an all-pervading "now" does not exist", even if this is Einstein's doctrine: relativity of simultaneity.
Regards,
Eckard