- #1
graupner1000
- 16
- 1
Hi all,
once again I'm stuck on something I am quite certain is silly, but here it goes. My confusion pertains to the equation
[itex]Ric=R^{a}\otimes e_{a}[/itex]
where [itex]Ric[/itex] is the Ricci tensor, [itex]R^{a}[/itex] is the Ricci 1-form and [itex]e_{a}[/itex] are the elements of an orthonormal basis.
Now, let's say for arguments sake that [itex]a=0,1,2[/itex] and I have a Ricci 1-form that looks something like this (What I'm actually trying to work out is a lot larger but follows a similar pattern)
[itex]R^{a}=\left[ \begin{array}{c} Ae_{0} + Be_{1} \\ Be_{0} - Ae_{1} \\ e_{2} \end{array} \right][/itex]
where [itex]A[/itex] and [itex]B[/itex] are constants. The next step would be to take the tensor product of [itex]R^{a}[/itex] and [itex]e_{a}[/itex] and this is where the problem lies. My instinct would be to treat this as an outer product so you end up with something like
[itex]R^{a}\otimes e_{a}=\left[ \begin{array}{ccc} (Ae_{0} + Be_{1})e_{0} & (Ae_{0} + Be_{1})e_{1} & (Ae_{0} + Be_{1})e_{2} \\ (Ae_{0} - Be_{1})e_{0} & (Ae_{0} - Be_{1})e_{1} & (Ae_{0} - Be_{1})e_{2} \\ e_{2}e_{0} & e_{2}e_{1} & e_{2}e_{2} \end{array} \right][/itex]
But that seems to be ignoring the sum over [itex]a[/itex] (or is this the operation it implies?) and more importantly, I really doubt there should be multiplication between the elements, i.e does
[itex](Ae_{0} + Be_{1})e_{0}[/itex]
imply
[itex](Ae_{0} + Be_{1})\otimes e_{0}[/itex]
or
[itex](Ae_{0} + Be_{1})\wedge e_{0}[/itex]
As said, this is a really silly thing to be stuck with and probably means that I've missed(read not paid attention to) something really basic so any help would be very much appreciated.
once again I'm stuck on something I am quite certain is silly, but here it goes. My confusion pertains to the equation
[itex]Ric=R^{a}\otimes e_{a}[/itex]
where [itex]Ric[/itex] is the Ricci tensor, [itex]R^{a}[/itex] is the Ricci 1-form and [itex]e_{a}[/itex] are the elements of an orthonormal basis.
Now, let's say for arguments sake that [itex]a=0,1,2[/itex] and I have a Ricci 1-form that looks something like this (What I'm actually trying to work out is a lot larger but follows a similar pattern)
[itex]R^{a}=\left[ \begin{array}{c} Ae_{0} + Be_{1} \\ Be_{0} - Ae_{1} \\ e_{2} \end{array} \right][/itex]
where [itex]A[/itex] and [itex]B[/itex] are constants. The next step would be to take the tensor product of [itex]R^{a}[/itex] and [itex]e_{a}[/itex] and this is where the problem lies. My instinct would be to treat this as an outer product so you end up with something like
[itex]R^{a}\otimes e_{a}=\left[ \begin{array}{ccc} (Ae_{0} + Be_{1})e_{0} & (Ae_{0} + Be_{1})e_{1} & (Ae_{0} + Be_{1})e_{2} \\ (Ae_{0} - Be_{1})e_{0} & (Ae_{0} - Be_{1})e_{1} & (Ae_{0} - Be_{1})e_{2} \\ e_{2}e_{0} & e_{2}e_{1} & e_{2}e_{2} \end{array} \right][/itex]
But that seems to be ignoring the sum over [itex]a[/itex] (or is this the operation it implies?) and more importantly, I really doubt there should be multiplication between the elements, i.e does
[itex](Ae_{0} + Be_{1})e_{0}[/itex]
imply
[itex](Ae_{0} + Be_{1})\otimes e_{0}[/itex]
or
[itex](Ae_{0} + Be_{1})\wedge e_{0}[/itex]
As said, this is a really silly thing to be stuck with and probably means that I've missed(read not paid attention to) something really basic so any help would be very much appreciated.