Vector Space Algebra of Minkowski Space

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  • #1
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Consider the Minkowski space of 4 dimensions with signature (- + + +). How does the vector space algebra work here? More specifically given 3 space like orthonormal vectors how do we define fourth vector orthogonal to these vectors? I am looking for an appropriate vector product like it is in the case of 3-dimesnsions: i ^ j = k etc.
 

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  • #2
HallsofIvy
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Three "space like" vectors? Obviously, any vector orthogonal to all three would be time like: <a, 0, 0, 0>.

The nearest you can come to the cross product in 4 dimensions is the "alternating product" [itex]z_i= \epsilon_{ijkl}u_jv_kw_l}[/itex] where [itex]\epsilon[/itex] is defined by [itex]\epsilon_{ijkl}= 1[/itex] if ijkl is an even permutation of 1234, [itex]\epsilon_{ijkl}= -1[/itex] if ijkl is an odd permutation of 1234, [itex]\epsilon_{ijkl}= 0[/itex] if ijkl is not a permutation of 1234 (i.e. at least two indices are the same). Notice that that involves the product of 3 vectors (which is what you want).
 
  • #3
robphy
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HallsofIvy's construction is essentially the computation of the [4-]volume determined by four vectors (as edges of a parallelepiped).

To complete your problem, you could do this...
given an orthonormal set [tex]x^a[/tex], [tex]y^a[/tex], and [tex]z^a[/tex],
choose any vector [tex]u^a[/tex] so that [tex]e_{abcd}u^a x^b y^c z^d <>0[/tex] (so that [tex]u^a[/tex] is linearly independent of the set you have).
With this [tex]u^a[/tex], subtract out all of the components parallel to the orthonormal set... [tex]t^a=u^a- (g_{bc}u^b \hat x^c) \hat x^a - (g_{bc}u^b \hat y^c) \hat y^a - (g_{bc}u^b \hat z^c) \hat z^a[/tex].

Check the signs with your metric signature conventions.
 
  • #4
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Spacetime Algebra

For an excellent and thorough formulation of the algebra, go to the link
http://modelingnts.la.asu.edu/html/STC.html [Broken]
and click on the link at the very top of the page, Spacetime Calculus, to download a pdf file.
 
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  • #5
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pkleinod, the link you provided is really an excellent source and is proving to very useful to me. Thanks and a Happy New Year to all the blokes who responded to this thread !
 

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