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?

I guess I should have included the rest of my reply to the person who
emailed me. It addresses this issue. Here it is:

The spatial coordinates of E1 are fixed at x = 0 in F and x' = 0 in F',
"for all time" [in SR]. But at the occurrence of E2, the assigned
spatial position of E1 in F' (x' = 0) is the same as the spatial
position of E2 in F' (x' = 0) [otherwise x' would not be zero at
t = t]. The problem is that the spatial position of E1 in F (x = 0) is
_not_ the same as the spatial position of E2 in F (x = vt). The lines
from E1 to E2 in F and F' both begin at _times_ t = t' = 0, but they
begin at different positions _in_space_. In F, the line begins at x = 0,
but in F' the line begins at the position _in_space_ of E2, that is
x = vt. If they begin at different spatial positions in the same frame
(F), then they begin at different positions in spacetime. Therefore,
lines drawn between E1 and E2 in each frame don't coincide in spacetime
[since they begin at different points in space].

The occurrence of E1 in both frames is at [times] t = t' = 0 (for all
times :-)). But the _spatial_ position of E1 in F' should be the spatial
position of the origin of F' at t' = 0, not the spatial position of the
origin of F' at t' = t'.

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

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