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4-D Minkowski vector space

by Maepez
Tags: distance, minkowski space, vectors
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Oct9-09, 05:42 PM
P: 2
1. The problem statement, all variables and given/known data
In the 4-D Minkowski vector space [you can think of this as the locations of events in space-time given by (t, x, y, z)] consider the vectors pointing to the following events: (4ns, -1m, 2, 7) and (2ns, 3m, 1m, 9m)
(a) Find the distance between the events.
(b) Find the innerproduct between the two events.

2. Relevant equations
Distance Formula

3. The attempt at a solution
My initial thoughts are to "subtract" one point from another to get a vector from the first point to the second. Then take the square root of the sum of each component squared. The only thing that bothers me is the different units (ns and m).

For part (b), the inner product is just a dot product of the two original vectors. Is that a correct assumption? (I've never studied anything like this before, and it seems important.)

Thanks in advance for any advice!
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Oct9-09, 06:19 PM
HW Helper
gabbagabbahey's Avatar
P: 5,003
I'm pretty sure that by "distance between the two events", they mean [itex]d=(\Delta x^2+\Delta y^2+\Delta z^2)^{1/2}[/itex]...If they asked for the space-time separation (or interval) between the events, then you would calculate [itex]\Delta s=(c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2)^{1/2}[/itex] (Using the convention that [itex]s^2=c^2t^2-x^2-y^2-z^2[/itex] )...However, different authors will use different terminology, so you should check with your Professor. Alternatively, the definition of "distance between events", that you are expected to use might be found in your text, or at least your notes.

As for the inner product, are the 4-vectors you posted contravariant or covariant? What is the definition for the inner product between two 4-vectors that is given in your text/notes?
Oct9-09, 06:33 PM
P: 2
Thanks very much for your reply, gabba. It seems my intuition was leading me in the right direction on the distance. Unfortunately, the only information I have was posted in the question. I'll consult the text, too, with the ideas you have given me. As far as note are concerned, I don't have anything covering Mindowski vector space. We're discussing Eigenvectors in class and something completely different in the homework. :\

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