Vector Space Algebra of Minkowski Space

[thermobum]

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.

 

[HallsofIvy]

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" $z_i= \epsilon_{ijkl}u_jv_kw_l}$ where $\epsilon$ is defined by $\epsilon_{ijkl}= 1$ if ijkl is an even permutation of 1234, $\epsilon_{ijkl}= -1$ if ijkl is an odd permutation of 1234, $\epsilon_{ijkl}= 0$ 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).

 

[robphy]

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 $$x^a$$, $$y^a$$, and $$z^a$$,
choose any vector $$u^a$$ so that $$e_{abcd}u^a x^b y^c z^d <>0$$ (so that $$u^a$$ is linearly independent of the set you have).
With this $$u^a$$, subtract out all of the components parallel to the orthonormal set... $$t^a=u^a- (g_{bc}u^b \hat x^c) \hat x^a - (g_{bc}u^b \hat y^c) \hat y^a - (g_{bc}u^b \hat z^c) \hat z^a$$.

Check the signs with your metric signature conventions.

 

[pkleinod]

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.

 

[thermobum]

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 !