Monday, May 14, 2018

general relativity - Equivalence of two formulation of Maxwell equations on manifolds


I have read about a generalization of Maxwell equation on manifolds that employs differential forms and Hodge duality that goes as follow: dF=0anddF=J.

As I understand exterior derivative is defined to be the differential on 0-forms and to be an antiderivation (roughly speaking). The Hodge dual of a p-form w is defined as the unique (np)-form w that satisfy ηw=η,wvol for all p-forms η. We can express the Faraday tensor in coordinates such that the metric tensor is trivialized at one point i.e. gμν=ημν. Then one should compute the hodge dual using the general formula in a basis: α=1k!(nk)!ϵi1,,in|det(g)|αj1,,jkgi1,j1gik,jkeik+1ein
That formula can be used to obtain F in a single point if one put gμν=ημν. But due to the vanishing of first derivative of the metric tensor once differentiate with the exterior derivative one gets the correct expression in the point where the metric was trivialized. By decomposing the effect of the derivative as spatial and temporal parts one gets Maxwell equation. Here I'm following John Baez and Javier P. Muniain "Gauge fields, knots and gravity". Thus one sees that in an local inertial coordinate reference usual Maxwell equation are obtained. This formulation uses the fact that there is a Lorenzian metric and that the manifold is orientable, but doesn't rely on any addiction structure like a connection.


However I am aware of another way to generalize maxwell equation: the minimal coupling which is the substitution of usual partial derivative with the covariant derivative leading to aFab=Jband[aFbc]=0.

With the Levi-Civita connection.


I don't understand how this two are connected. Are those two generalization the same? How can it be possible if the differential form formulation is not aware of Levi-Civita connection? It will be remarkable if the "right" connection popped out from the differential forms version of the Maxwell equation!



Answer



Maxwell's equations in differential form notation reads dF=0 ,dF=J .

We now show that these equations are equivalent to [aFbc]=0, aFab=Jb.


First, by definition [aFbc]=[aFbc]+Γd[abFc]dΓd[acFb]d

If the connection is torsion-free (not necessarily the Levi-Civita connection) then Γa[bc]=0 so that [aFbc]=[aFbc]=13(dF)abc
The last equality is true by definition. Thus, [aFbc]dF=0.


Next, consider the second equation aFab=aFab+ΓaacFcb+ΓbacFac

Again, if the connection is torsion free, then the last term is zero. To simplify the second term, we have to assume that Γ is the Levi-Civita connection so that Γaac=12gab(agcb+cgabbgac)=12gabcgab=12clogdetg=1detgcdetg
Then, we have aFab=aFab+1detgcdetgFcb=1detga(detgFab) .
So we can write Maxwell's equation as e(detgFed)=detgJd
Now, contract both sides with the Levi-Civita symbol (not tensor), ˜εabcd to get e(detg˜εabcdFed)=detg˜εabcdJd
Now, recall the the Levi-Civita tensor is εabcd=detg˜εabcd e(εabcdFed)=εabcdJd=(J)abc
The last equality is true by definition. Finally, we wish to write the LHS in terms of F. To do this, we write Fed=12εedmn(F)mn
Then, (J)abc=12e(εabcdεdemn(F)mn)
Then, we use the property εabcdεdemn=6δe[aδmbδnc]
Finally, (J)abc=3δe[aδmbδnc]e(F)mn=3[a(F)bc]=(dF)abc
where again, the last equility is the definition of d. Thus, we see that aFab=JbdF=J.


QED.



PS - To answer your last question. The differential form notation is aware of the connection through the Hodge dual in which detg enters. Note also that in the divergence of any p-form, only the following component of the connection appears - Γaab which depends entirely on detg. Other components never appear, i.e. in full generality aT[abc]=1detga(detgT[abc]) .


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