johnm@dbrmelb.dbrhi.oz (John Mashford) (06/04/91)
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 | |______________________________________________________________________________|