- #1
emma83
- 33
- 0
Hello,
I am still having a hard time with tensors...
The answer is probably obvious, but is it always the case (for an arbitrary metric tensor [tex]g_{\mu \nu}[/tex] that [tex]g_{ab}g_{cd}=g_{cd}g_{ab}[/tex] ?
I was trying to find a formal proof for that, and was wondering if we could use the relations:
(1) [tex]g_{ab}=\frac{\partial x^{c}}{\partial x^{a}} \frac{\partial x^{d}}{\partial x^{b}} g_{cd}[/tex]
(2) [tex]g_{cd}=\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{b}}{\partial x^{d}} g_{ab}[/tex]
And then multiply the left-hand side of (1) and (2) together and use the fact that the fractions of partial derivatives commute with the metric tensor and cancel each other to get:
[tex]g_{ab}g_{cd}
=(\frac{\partial x^{c}}{\partial x^{a}} \frac{\partial x^{d}}{\partial x^{b}} g_{cd}) (\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{b}}{\partial x^{d}} g_{ab})
= \frac{\partial x^{c}}{\partial x^{a}}\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{d}}{\partial x^{b}} \frac{\partial x^{b}}{\partial x^{d}} g_{cd} g_{ab}
= g_{cd}g_{ab}[/tex]
Does that make sense ?!
Thanks for your help...
I am still having a hard time with tensors...
The answer is probably obvious, but is it always the case (for an arbitrary metric tensor [tex]g_{\mu \nu}[/tex] that [tex]g_{ab}g_{cd}=g_{cd}g_{ab}[/tex] ?
I was trying to find a formal proof for that, and was wondering if we could use the relations:
(1) [tex]g_{ab}=\frac{\partial x^{c}}{\partial x^{a}} \frac{\partial x^{d}}{\partial x^{b}} g_{cd}[/tex]
(2) [tex]g_{cd}=\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{b}}{\partial x^{d}} g_{ab}[/tex]
And then multiply the left-hand side of (1) and (2) together and use the fact that the fractions of partial derivatives commute with the metric tensor and cancel each other to get:
[tex]g_{ab}g_{cd}
=(\frac{\partial x^{c}}{\partial x^{a}} \frac{\partial x^{d}}{\partial x^{b}} g_{cd}) (\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{b}}{\partial x^{d}} g_{ab})
= \frac{\partial x^{c}}{\partial x^{a}}\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{d}}{\partial x^{b}} \frac{\partial x^{b}}{\partial x^{d}} g_{cd} g_{ab}
= g_{cd}g_{ab}[/tex]
Does that make sense ?!
Thanks for your help...