Science > Physics > Is there a "law of cosines" for Minkowski Flat Space?
| Topic: |
Science > Physics |
| User: |
"Robert J. Kolker" |
| Date: |
06 Dec 2003 03:27:01 PM |
| Object: |
Is there a "law of cosines" for Minkowski Flat Space? |
Given three events P_1 = (x_1, y_i, z_i, t_1) i = 0,1,2
Is there a formulate that will give the interval from P_1 to P_2 given
the intervals from P_0 to P_1 and P_0 to P_2? I am looking for something
like the law cosines for triangles in a Euclidean flat space.
TIA
Bob Kolker
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| User: "Edward Green" |
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| Title: Re: Is there a "law of cosines" for Minkowski Flat Space? |
07 Dec 2003 02:16:22 PM |
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"Robert J. Kolker" <bobkolker@attbi.com> wrote in message news:<bqthj8$25lsai$1@ID-76471.news.uni-berlin.de>...
Given three events P_1 = (x_1, y_i, z_i, t_1) i = 0,1,2
Is there a formulate that will give the interval from P_1 to P_2 given
the intervals from P_0 to P_1 and P_0 to P_2? I am looking for something
like the law cosines for triangles in a Euclidean flat space.
Correct me if I'm wrong, but doens't the Minkowski metric allow
vectors other than the zero vector to have zero length?
I think this is going to be a problem. In fact, using what I know
empirically about the metric, without calculation: send two light
beams out from the origin at t=0 in your rest frame, simultaneously
reaching points R1,R2 at time t1. The interval between R1,t1 and
R2,t1 is just the Euclidean distance, is it not? And yet the
intervals from the spacetime origin to these events are both zero.
Now just how are you going to get this non-zero Euclidean distance out
of two zero intervals and something like a cosine of an included
angle? Especially since we would have the very same inputs to your
putative "law of consines" at any other time t2, but some different
Euclidean distance in the output?
They don't call it an "indefinite metric" for nothing.
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