I'm working on the electromagnetic stress-energy tensor and I've found this in a book by Landau-Lifshitz:
[itex]
T^{i}_{k} = -\frac{1}{4\pi} \frac{\partial A_{\ell}}{\partial x^{i}} F^{k\ell}+\frac{1}{16\pi}\delta^{k}_{i} F_{\ell m} F^{\ell m}
[/itex]
Becomes:
[itex]
T^{ik} = -\frac{1}{4\pi} \frac{\partial A^{\ell}}{\partial x_{i}} F^{k}_{\ell}+\frac{1}{16\pi}g^{ik} F_{\ell m} F^{\ell m}
[/itex]
I was wondering how this work? [itex]F^{ik}[/itex] is the electromagnetic field tensor, [itex]A_{\ell}[/itex] is the potential of the field.
[itex]
T^{i}_{k} = -\frac{1}{4\pi} \frac{\partial A_{\ell}}{\partial x^{i}} F^{k\ell}+\frac{1}{16\pi}\delta^{k}_{i} F_{\ell m} F^{\ell m}
[/itex]
Becomes:
[itex]
T^{ik} = -\frac{1}{4\pi} \frac{\partial A^{\ell}}{\partial x_{i}} F^{k}_{\ell}+\frac{1}{16\pi}g^{ik} F_{\ell m} F^{\ell m}
[/itex]
I was wondering how this work? [itex]F^{ik}[/itex] is the electromagnetic field tensor, [itex]A_{\ell}[/itex] is the potential of the field.