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Vector Space Algebra of Minkowski Space

  1. Dec 14, 2007 #1
    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.
     
  2. jcsd
  3. Dec 14, 2007 #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).
     
  4. Dec 15, 2007 #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.
     
  5. Jan 1, 2008 #4
    Spacetime Algebra

    For an excellent and thorough formulation of the algebra, go to the link
    http://modelingnts.la.asu.edu/html/STC.html
    and click on the link at the very top of the page, Spacetime Calculus, to download a pdf file.
     
  6. Jan 2, 2008 #5
    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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