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



At 23:04 -0800 1/20/03, David Rutherford wrote:

As I said very early on, I'm not referring to the invariance of the
_magnitude_ of the spacetime interval. I'm referring to the lines drawn
between the coordinates of the events in each frame. I claim that they
don't coincide in spacetime, therefore, they don't describe an invariant
entity (the spacetime four-vector between the events).

That the spacetime vector may not point in the same "direction" is
not of concern. The only thing invariant in the Lorentz
transformation is its *magnitude.* It may well "point" in some
different direction, depending on whether one interval may be
spacelike and the other timelike, or whatever.

It seems to me that you have raised a non-issue here.

Imagine a reference frame that can be described by a horizontal axis
labeled "space," or "x", and a vertical axis labeled "time," or
"ict." Then construct a spacetime vector from the origin to some
point in this two-dimensional representation. This then, will be the
vector representing, say, the motion of a particle in that reference
frame. From any other reference frame, moving parallel to the spacial
axis, when the coordinate origins coincide at the beginning of the
motion at issue, the vector will point in a different direction, but
will still lie along the circular arc between the space and time axes
that includes the tip of the vector. In other words, the magnitude
will still be x^2-c^2*t^2, even though x and t are different in the
two reference frames. You can see the sketch that I have described in
the book "Relativity Visualized," by Lewis Epstein (San Francisco:
Insight Press, 1985), pp. 78ff.

Hugh
--

Hugh Haskell
<mailto:haskell@ncssm.edu>
<mailto:hhaskell@mindspring.com>

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