[geordief (OP)]

https://arxiv.org/pdf/physics/0302045.pdf

I have been working my way through this approach for a while now (have come back to it 3 or 4 times ) and am getting stuck at this point

Using Eqs. (9) and (10), we obtain
X(x2 + h, t, v) − X(x2, t, v) = X(x1 + h, t, v) − X(x1, t, v). (11)

Dividing both sides by h and taking the limit h → 0, we obtain ∂X /∂x ¦ x2 = ∂X/ ∂x¦ x1   (12)

Apologies for the poor formatting ,copy and paste  comes up short here.


Can anyone help me through this tricky bit?
The author says he is dividing by h  but the result of the partial differentiation (that is what he is doing?) seems to be a division by x .

My partial differentiation skills are rusty and it would take me a lot of work to bring them up to scratch...

What about the article as a whole?Does anyone find it interesting?

He goes on to say that   "Thus, the function X(x, t, v) must be a linear function of x"

Can anyone clarify that particular point as well?

[Halc]

I cannot follow the notation used to get (12).

Following what the author is trying to do, we're basically assuming our universe but without light or any other thing (gravity waves for instance) that moves at some frame independent constant speed.
That's fine, but there are several sets of physical laws (including Newtonian ones) that work just fine using just Galilean relativity.  To actually derive the laws of relativity from it all, one must distinguish between various sets of rules by there empirical differences.  In other words, if we can't use constant light speed as an empirical premise, we need to use some other observation to derive these laws.  I don't see any at all, and hence SR cannot follow from just a premise of Galilean relativity.

That said, I followed along at least as far as you did, and equation 9 is wrong, which would show up if they actually did an empirical test for it (hard to do without using light).

Working backwards, equations 1,2 are also wrong, or at least uses incorrect terminology.  They seem to be describing an event at x and t, but they don't call it an event.

Let us consider two inertial frames S and S′, where the second one moves with a speed v, along the x-axis, with respect to the first one. The co-ordinates and time in the S-frame will be denoted by x and t, and in the frame S′, they will be denoted with a prime. The space-time transformation equations have the form
x′ = X(x, t, v), (1)
t′ = T(x, t, v), (2)

A frame does not have coordinates.  If it gets assigned an origin (a designated event where x and t are both zero), then an event can be described with (x,t) in that frame.  If no origin is specified, only relations like speeds and such can be specified in the selected frame, but not coordinates.
So the second frame is indeed specified with the v above, but not the origin of this frame relative to the origin of the first frame.  So we need not only x and t of the event we want to transform, but also x₀ and t₀, the coordinates of the origin of S' in S.
I am assuming both frames use the same origin event, a simplification which allows us to move on.  The author seems to assume this without saying it.

Back to equation 9.  I notice that the equations all came in pairs up until now, but with 9, they drop 10: the T computation.  This is a mistake.

Suppose there is a rod placed along the x-axis such that its ends are at points x1 and x2 in the frame
S, with x2 > x1. In the frame S′, the ends will be at the points X(x1, t, v) and
X(x2, t, v), so that the length would be
l′ = X(x2, t, v) − X(x1, t, v)   (9)

They didn't compute t2' = T(x2, t, v) and t1' = T(x1, t, v).  The above length computation is valid only if t1' and t2' are the same time, but they're not.  An empirical test would verify this, but they don't seem to reference any empirical facts, so this is actually just not demonstrated at this point.  Equation 9 is not valid until it has been demonstrated, and in reality, it isn't the case.

[geordief (OP)]

Thanks,you have given me food for thought.

Perhaps I will get back again when I have digested what you said.

[jeffreyH]

To add to what Halc has said. You need some way of verifying both time and distance. Even in Galilean relativity. The time at two remote points within a coordinate system is a huge issue. The point itself does not have a time associated with it. An object sitting at or passing through that point can have an associated time. It is the remote location with respect to an 'observer' that matters.

This paper ignores the issue altogether. Which is the wrong way to proceed.