Hello there! I have a more or less silly and possibly obvious question that's been bugging me a lot. Here's the premise:(adsbygoogle = window.adsbygoogle || []).push({});

Take a space-time ##(M,g_{ab})## that admits a covariantly constant (unit) time-like vector field ##\xi^a## i.e. ##\nabla_a \xi^b = 0## (such a space-time must necessarily have a degenerate Ricci tensor). Given an electromagnetic field ##F_{ab}##, we can decompose it relative to ##\xi^a## into an electric and magnetic field as per ##E^a = F^{a}{}{}_{b}\xi^b## and ##B^a = \frac{1}{2}\epsilon^{abcd}F_{cd}\xi_b##.

It can be shown as a consequence of ##\nabla_a \xi^b = 0## that ##\nabla^a F_{ab} = J_b## and ##\nabla_{[a}F_{bc]} = 0## if and only if ##D_a B^a =0##, ##\epsilon^{abc}D_b E_c = -\xi^b \nabla_b B^a##, ##D_a E^a = \rho##, and ##\epsilon^{abc}D_b B_c = j^a + \xi^b \nabla_b E^a## where ##D_a## is the induced derivative operator on the one-parameter family of space-like hypersurfaces orthogonal to ##\xi^a## (such a family exists because ##\xi_{[a}\nabla_{b}\xi_{c]} = 0## trivially), ##J^a## is the 4-current density, ##j^a## is the 3-current density, ##\rho## is the charge density, and ##\epsilon_{abc} = \xi^d \epsilon_{dabc}## is the induced volume element on the space-like hypersurfaces.

If you consider the special case where ##g_{ab} = \eta_{ab}## and ##M = \mathbb{R}^{4}## (Minkowski space-time) then the integral curves of ##\xi^a## are just worldlines of inertial observers and the 4 equations above for the electric and magnetic fields just take on the usual form of Maxwell's equations in global inertial frames: ##\vec{\nabla}\cdot \vec{E} = \rho##, ##\vec{\nabla}\cdot \vec{B} = 0##, ##\vec{\nabla}\times \vec{E} = -\partial_t \vec{B}##, and ##\vec{\nabla}\times \vec{B} = \vec{j} + \partial_{t}\vec{E}##.

But what if we aren't in Minkowski space-time? What physical significance/interpretation (if any) does the family of observers following orbits of ##\xi^a## have then? What bothered me is that instead of simply requiringanyfamily of locally inertial observers i.e. simply having ##\xi^a \nabla_a \xi^b = 0##, we required a family of observers having ##\nabla_a \xi^b = 0## and such observers are not only locally inertial but also following orbits of a time-like killing field that is also twist-free.

But why physically do we need this (much) stronger restriction? I would imagine that if we wanted the covariant form of Maxwell's equations to decompose into a curved space-time version of the canonical vector calculus form of Maxwell's equations in a global inertial frame in flat space-time then we should be able to get that usinganyfamily of locally inertial observers.

Thanks in advance!

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# Constant time-like vector fields-Maxwell's equations

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