- #1
ProfDawgstein
- 80
- 1
[This is mostly about notation]
I was working on a problem where I had to prove that [itex] div(B) [/itex] remains invariant under lorentz transformations. That was not too hard, so I came up with
[itex] div(B) = \partial_{\mu} B^{\mu} [/itex]
must equal
[itex] div(B) = \partial'_{\mu} B'^{\mu} [/itex]
so I did a lorentz transformation on both [itex]B[/itex] and [itex] \partial [/itex].
[itex] \partial'_{\mu} = \widetilde{L}^{\nu}_{\ \mu} \partial_{\nu} [/itex]
and
[itex] B'^{\mu} = L^{\mu}_{\ \nu} B^{\nu} [/itex]
where [itex]L[/itex] is the lorentz transformation matrix and [itex] \widetilde{L} [/itex] its inverse.
Now I simply plug in both of them into the first equation (changing [itex]\nu[/itex] to [itex]\alpha[/itex])
[itex] div(B) = \widetilde{L}^{\nu}_{\ \mu} \partial_{\nu} L^{\mu}_{\ \alpha} B^{\alpha} [/itex].
Because [itex] \partial_{\nu} [/itex] applies to [itex] B^{\nu} [/itex] I can change the order.
Then I get [itex] div(B) = \widetilde{L}^{\nu}_{\ \mu} L^{\mu}_{\ \alpha} \partial_{\nu} B^{\alpha} [/itex].
Now I use [itex] \widetilde{L}^{\nu}_{\ \mu} L^{\mu}_{\ \alpha} = \delta^{\nu}_{\alpha} [/itex] which changes the previous equation to [itex] \delta^{\nu}_{\alpha} \partial_{\nu} B^{\alpha} [/itex] and then I get [itex] div(B) = \partial_{\alpha} B^{\alpha} [/itex].
This is the same equation as on the top -> QED.
1) Is this okay (notationally)?
2) I tried the same for the curl of E
[itex] curl(E)_i = \varepsilon_{ijk} \partial_{j} E_{k} [/itex] ( in old-school notation )
But the problem is that I do not really know how to write it down correctly.
Is it [itex] \varepsilon_{ijk} [/itex] or [itex] \varepsilon^{ijk} [/itex]?
I am also not sure about if the curl of E (or E itself) is contravariant or covariant and how to transform them correctly to get the same I got for div(B).
The problem was from Cheng's "Relativity, Gravitation and Cosmology" (problem 2.4).
He does not use this type of notation in the solutions.
Thank you in advance.
First post :)
I was working on a problem where I had to prove that [itex] div(B) [/itex] remains invariant under lorentz transformations. That was not too hard, so I came up with
[itex] div(B) = \partial_{\mu} B^{\mu} [/itex]
must equal
[itex] div(B) = \partial'_{\mu} B'^{\mu} [/itex]
so I did a lorentz transformation on both [itex]B[/itex] and [itex] \partial [/itex].
[itex] \partial'_{\mu} = \widetilde{L}^{\nu}_{\ \mu} \partial_{\nu} [/itex]
and
[itex] B'^{\mu} = L^{\mu}_{\ \nu} B^{\nu} [/itex]
where [itex]L[/itex] is the lorentz transformation matrix and [itex] \widetilde{L} [/itex] its inverse.
Now I simply plug in both of them into the first equation (changing [itex]\nu[/itex] to [itex]\alpha[/itex])
[itex] div(B) = \widetilde{L}^{\nu}_{\ \mu} \partial_{\nu} L^{\mu}_{\ \alpha} B^{\alpha} [/itex].
Because [itex] \partial_{\nu} [/itex] applies to [itex] B^{\nu} [/itex] I can change the order.
Then I get [itex] div(B) = \widetilde{L}^{\nu}_{\ \mu} L^{\mu}_{\ \alpha} \partial_{\nu} B^{\alpha} [/itex].
Now I use [itex] \widetilde{L}^{\nu}_{\ \mu} L^{\mu}_{\ \alpha} = \delta^{\nu}_{\alpha} [/itex] which changes the previous equation to [itex] \delta^{\nu}_{\alpha} \partial_{\nu} B^{\alpha} [/itex] and then I get [itex] div(B) = \partial_{\alpha} B^{\alpha} [/itex].
This is the same equation as on the top -> QED.
1) Is this okay (notationally)?
2) I tried the same for the curl of E
[itex] curl(E)_i = \varepsilon_{ijk} \partial_{j} E_{k} [/itex] ( in old-school notation )
But the problem is that I do not really know how to write it down correctly.
Is it [itex] \varepsilon_{ijk} [/itex] or [itex] \varepsilon^{ijk} [/itex]?
I am also not sure about if the curl of E (or E itself) is contravariant or covariant and how to transform them correctly to get the same I got for div(B).
The problem was from Cheng's "Relativity, Gravitation and Cosmology" (problem 2.4).
He does not use this type of notation in the solutions.
Thank you in advance.
First post :)