What was wrong with Dingle?

Comment on "What GPS tells us about the Twin Paradox":

If the variables v, t, l, and x are of the rocket ship frame, and the variables v, T, L, and X are of the earth frame; then, vt is the distance the earth travels within the rocket ship frame, and vT is the distance the rocket ship travels within the earth frame. The variable l is the length of the rocket ship within the rocket ship's frame, and the variable L is the same length of an identical rocket ship at rest upon earth. The variable x is the total distance of L/g (whereby g is gamma) plus -vt within the rocket ship frame, and X is the total distance of l/g plus vt within the earth's frame.

Correctly, the rocket ship's motion as observed within earth's frame is given by:
l=g(X-vT), and solved for X equals (l/g)+(vT)=X
Furthermore, earth's motion as observed within the rocket frame is given by:
L=g(x+vt), and solved for x equals (L/g)-(vt)=x

Notice, the variables T and x are not paired within a single equation, nor are the variables t and X paired within a single equation. Why? Because only clocks at rest within a particular reference frame can give the speed of an object's motion as observed from within that particular reference frame. Only clocks at rest within the earth's reference frame can give the speed of the rocket ship's motion within the earth's reference frame, and only clocks at rest within the rocket ship's reference frame can give the speed of the earth's motion within the rocket ship's reference frame. However, in equation (1) of "what GPS tells us about the Twin Paradox", this is not apparent.

More importantly, variables X and x never exist within any single equation. However, in equation (1) of "What GPS tells us of the Twin Paradox", this is not true.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by ebg</i>
<br />Correctly, the rocket ship's motion as observed within earth's frame is given by: l=g(X-vT)... More importantly, variables X and x never exist within any single equation. However, in equation (1) of "What GPS tells us of the Twin Paradox", this is not true.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Equation (1) is the standard Lorentz transformations, the second of which reads: x=g(X-vT). How does x in eq. (1) differ from l in your equation? I see the two as identical.

BTW, I have about two weeks of travel starting Wednesday, half of which will be in Europe, so we're probably not going to be able to finish this discussion until after I return. -|Tom|-

[ebg] "If the variables v, t, l, and x are of the rocket ship frame, and the variables v, T, L, and X are of the earth frame ... The variable l is the length of the rocket ship within the rocket ship's frame ... "

My understanding of the Lorentz transform variables is a little different. For example -

T and X are time and distance from origin in the Earth frame.
t and x are time and distance from origin in the rocket frame **as seen from Earth**.
v is the speed difference between these frames.

???,
LB

By this interpretation, l is the (shortened) length of the rocket seen from Earth.

l = x1 - x2

where:
x1 = x coordinate of nose
x2 = x coordinate of tail

???,
LB

In the rocket's frame l is not shortened.

[ebg] "Notice, the variables T and x are not paired within a single equation, nor are the variables t and X paired within a single equation. Why? ... More importantly, variables X and x never exist within any single equation."

These variable pairings were there all along, but hidden by your variable substitution. If you make the inverse substitution I mentioned above (replacing your variable l with the origiinal variable x, and of course L with X for the Earth frame) then you see why the variable parings you were looking for did not appear to exist in your equations.

LB