johnm@dbrmelb.dbrhi.oz (John Mashford) (05/31/91)
Hello networld,
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.
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 |
|______________________________________________________________________________|
erudite_signature();mcanally@kurims.kyoto-u.ac.jp (David Scott McAnally) (06/01/91)
In article <907@dbrmelb.dbrhi.oz> johnm@dbrmelb.dbrhi.oz (John Mashford) writes: >Hello networld, > >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. > >Any help at all towards finding a solution or proving that no solution exists >would be very gratefully received. The answer is that 16 such _vectors_ can't exist, as they would have to be mutually orthogonal, requiring then that they be linearly independent. This is shown by <sum a(i,r)mu(i,r),mu(j,s)> = a(j,s)eta(j,j)eta(s,s). On the other hand, it is possible to define such an inner product on rank 2 tensors, and to find 16 such tensors mu(i,r), and to have the Lorentz invariance mentioned above. David McAnally kurims.kurims.kyoto-u.ac.jp "Well, Ah'll be hornswoggled!" "Look, your personal habits are of no interest to us." US Major and Graeme Garden: The Goodies (Clown Virus)