Xref: utzoo sci.math:17854 sci.physics:19954 comp.theory.dynamic-sys:265
Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!usc!wuarchive!uunet!munnari.oz.au!bruce!dbrmelb!johnm
From: johnm@dbrmelb.dbrhi.oz (John Mashford)
Newsgroups: sci.math,sci.physics,comp.theory.dynamic-sys
Subject: tensor problem
Keywords: tensors, Minkowski space
Message-ID: <908@dbrmelb.dbrhi.oz>
Date: 4 Jun 91 07:46:08 GMT
Followup-To: poster
Organization: CSIRO, Div. Building Constr. and Eng'ing, Melb., Australia
Lines: 36

In article
<Date: 31 May 91 07:56:26 GMT
Xref: dbrmelb sci.math:10713 sci.physics:7218 comp.theory.dynamic-sys:278>
johnm@mel.dbce.csiro.au (John Mashford) writes

>I have the following problem with tensors:

>Let eta = diag(1,-1,-1,-1) be the standard metric tensor for Minkowski space
>and let < , > be the associated inner product.

>By taking {e(i) : i = 0,...,3} to be the standard basis for Minkowski space we
>can satisfy the following equations
    <e(i),e(j)> = eta(i,j), for i,j = 0,...,3.

>My problem is to find 16 vectors {mu(i,j) : i,j = 0,...3} which satisfy
    <mu(i,r),mu(j,s)> = eta(i,j)eta(r,s).

>If {mu(i,j)} is a solution and L is a Lorentz transformation then {L(mu(i,j))}
>is also a solution.

Since there can be no more than 4 orthonormal vectors in Minkowski space the 
problem has no solution. What my problem really is is to find 16 vectors which
satisfy
    <mu(i,r),mu(j,s)> + <mu(j,r),mu(i,s)> = 2eta(i,j)eta(r,s).

I can exhibit a solution when < , > is the standard inner product for real 4
dimensional space and eta is the Kronecker delta.

Any help at all towards finding a solution or proving that no solution exists
would be very gratefully received.
___

| John Mashford   Commonwealth Scientific and Industrial Research Organization |
|                 Post Office Box 56, Highett, Victoria, Australia   3190      |
| Internet: johnm@mel.dbce.csiro.au  Tel: +61 3 556 2211  Fax: +61 3 556 2819  |
|______________________________________________________________________________|