Re: A Geometrical Proof of the Non-invariance of the Spacetime Interval

[David Rutherford <drutherford@SOFTCOM.NET>]

Bob LaMontagne wrote:
> David Rutherford wrote:
>
> > In response to a question asking if it's a fair statement that I expect
> > a line drawn from E1 to E2 to "coincide" in different frames, I replied:
> >
> >         No, I don't expect the line to coincide in different reference
> >         frames. In the case I gave, the lines don't coincide in F and
> >         F', but the point I'm trying to make is that they don't coincide
> >         in spacetime, either. When I say they don't coincide in
> >         spacetime, I mean, they don't coincide independently and apart
> >         from either reference frame.
>
> If E1 is a an event in spacetime, and E2 is a separate event in spacetime,
> how is there more than one unique "line" connecting them? F and F' may have
> different descriptions for that line, but there's nothing to "coincide
> independently" since there's only one line being referred to.
>
> You certainly wouldn't ask if E1 "coincides independently" in F and F'
> because it's only one event. Likewise for E2. Why then ask that question of
> the line from E1 to E2?

Maybe I can show the way I think it should be by using Galilean
relativity as an analogy. I know that the Galilean transformations
aren't used to find the spacetime interval between events, but I think
they can be used to make my point.

Say the situation is the same as before, so that event E1 occurs at the
coincidence of the origins of F and F'. Then at time t = t, event E2
occurs at x = vt. The Galilean transformations give the coordinates of
E2 in F', in this situation, as

x' = x - vt = vt - vt = 0
y' = y = 0
z' = z = 0
t' = t

Now suppose that observers in F' are asked to give the distance between
E1 and E2. Should they say that x' = 0 (which is what the analogous
Lorentz transformations say), or should they say x' = x = vt = vt'? I
say they should say the latter.

For example, think of E1 as being the driving of a stake into the ground
at x = y = z = t = 0 in F, and E2 as the driving of a second stake at
x = vt, y = 0, z = 0, t = t. I say that a valid set of transformation
equations should give a comparison of the measurements of the distance
between those stakes at the time of E2 (that is, at t' = t), and the
time between the driving of the stakes at the time of E2 (that is, at
t' = t) in F and F'.

--
Dave Rutherford
"New Transformation Equations and the Electric Field Four-vector"
http://www.softcom.net/users/der555/newtransform.pdf

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