Maxwell's equations remains invariant under lorentz transformation proof
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In Peskin and Schroeder page 37, it is written that
Using vector and tensor fields, we can write a variety of Lorentz-invariant equations.
Criteria for Lorentz invariance: In general, any equation in which each term has the same set of uncontracted Lorentz indices will naturally be invariant under Lorentz transformations.
I would like to explicitly show that the above criteria is valid for Maxwell's equations ∂μFμν=0∂μFμν=0or ∂2Aν−∂ν∂μAμ=0∂2Aν−∂ν∂μAμ=0.
Solution 1: Maxwell's equations follow from the Lagrangian
LMAXWELL=−14(Fμν)2=−14(∂μAν−∂νAμ)2LMAXWELL=−14(Fμν)2=−14(∂μAν−∂νAμ)2
which is a Lorentz scalar, so this means that the equation of motion is Lorentz-invariant as well. That's one way to convince yourself that the above Maxwell's equations are, in fact, Lorentz invariant. Is this correct?
Solution 2: I would like to actively transform the electromagnetic field strength tensor FμνFμνand show that the Maxwell's equations ∂μFμν=0∂μFμν=0 or ∂2Aν−∂ν∂μAμ=0∂2Aν−∂ν∂μAμ=0remain Lorentz invariant.
I can see that ∂2∂2 and ∂μAμ∂μAμ will not Lorentz transform as they are Lorentz scalars.
Under an active Lorentz transformation, Vμ(x)→ΛμνVν(Λ−1x)Vμ(x)→ΛνμVν(Λ−1x). So, will AνAν and ∂ν∂νLorentz transform in the same way?
Using vector and tensor fields, we can write a variety of Lorentz-invariant equations.
Criteria for Lorentz invariance: In general, any equation in which each term has the same set of uncontracted Lorentz indices will naturally be invariant under Lorentz transformations.
I would like to explicitly show that the above criteria is valid for Maxwell's equations ∂μFμν=0∂μFμν=0or ∂2Aν−∂ν∂μAμ=0∂2Aν−∂ν∂μAμ=0.
Solution 1: Maxwell's equations follow from the Lagrangian
LMAXWELL=−14(Fμν)2=−14(∂μAν−∂νAμ)2LMAXWELL=−14(Fμν)2=−14(∂μAν−∂νAμ)2
which is a Lorentz scalar, so this means that the equation of motion is Lorentz-invariant as well. That's one way to convince yourself that the above Maxwell's equations are, in fact, Lorentz invariant. Is this correct?
Solution 2: I would like to actively transform the electromagnetic field strength tensor FμνFμνand show that the Maxwell's equations ∂μFμν=0∂μFμν=0 or ∂2Aν−∂ν∂μAμ=0∂2Aν−∂ν∂μAμ=0remain Lorentz invariant.
I can see that ∂2∂2 and ∂μAμ∂μAμ will not Lorentz transform as they are Lorentz scalars.
Under an active Lorentz transformation, Vμ(x)→ΛμνVν(Λ−1x)Vμ(x)→ΛνμVν(Λ−1x). So, will AνAν and ∂ν∂νLorentz transform in the same way?
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