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



David Rutherford writes:

I'm not claiming that the magnitude of the spacetime interval is not
invariant.

It seems to me that that was almost exactly what you were claiming.
In your first message you wrote

I think I can show that the spacetime interval ... is not invariant ...

Except for the addition of the phrase "magnitude of the" in your
latest message, these two statements would seem to be explicitly
contradictory. If you really *are* trying to draw a distinction
between the spacetime interval and its magnitude, then I would remind
you that the spacetime interval *is* a scalar quantity--it is the
magnitude of a spacetime four vector that "locates" one event
relative to another.

I'm claiming that, since the assigned positions of E1 in F
and F' don't coincide in space (meaning they are not independent of any
reference frame), at times later than t = 0, lines drawn between E1 and
E2 representing the spacetime interval between E1 and E2 in F and F'
don't coincide in spacetime (meaning they are not independent of any
reference frame). In other words, they aren't the same line.

I can't make sense of this. I think you may be suffering from some
fundamental confusion about the geometry of spacetime, the nature of
spacetime four-vectors, and, particularly the distinctions between
*vectors*, *components* of vectors in particular coordinate systems,
and *magnitudes* of vectors.

The spacetime interval is the magnitude of a four-vector that
"locates" event E2 relative to event E1 in spacetime. That four
vector exists independently and apart from coordinate systems; it can
not (and does not) "depend" on the choice of a coordinate system. Of
course the *description* of that four vector (in the form of its
components) will be different in different coordinate systems in
analogy to the more familiar case of vectors in three dimensional
Euclidean space.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm

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