On 23 Aug 2005, at 06:14, Maru Dubshinki wrote:
A minor point: why are you representing the cartesian distance formula
in squared form? I've always elsewhere seen it as sqrt(x^2+y^2+z^2).
If you look at the relativistic formula
d^2 = t^2 - x^2 - y^2 - z^2
(which uses my preferred sign convention, which differs from Dan's)
then you'll see that depending on the relative sizes of the time and
space components the d^2 can be positive, zero or negative. Somewhat
confusingly, d^2 is called the "squared interval", as in the case of
positive d^2, sqrt(d^2) is the elapsed proper time. However, in the
case of negative d^2, sqrt(-d^2) is the proper distance. The formula
itself - more or less - is called the metric. (Strictly speaking, the
metric requires infinitesimal changes in coordinates rather than the
coordinates themselves.)
I suppose there are two reasons for this squared convention. Firstly,
because it's cleaner to present it in terms of a real quantity d^2
than in terms of a possibly imaginary quantity d that must then be
manipulated in various confusing ways to make it real. Secondly,
because in general relativity the metric is more complicated and can
include cross-terms in the coordinates and coefficients that vary
from place to place or with time, but the terms are always quadratic
in coordinates.
How exactly does that work for space-like relationships? Is this
potential to mix up ordering of A and B what allows reverse time
travel?
Yes, that's exactly right. If you have a faster than light drive, you
can exploit the ambiguity in ordering of events with spacelike
separations in such a way that you can travel backwards in time. I
explain how in more detail in an article on my weblog:
http://www.theculture.org/rich/sharpblue/archives/000089.html
(This isn't the only way to get time travel out of relativity. In
general relativity it's possible to produce metrics that allow time
travel without moving faster than light, the paths that allow this
being called "closed timelike curves".)
Rich
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