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Is gauge invariant?

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… the defining property of a Lorentz transformation, :

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… 4-vectors and (Lorentz)-tensors are transformed like this:

and

where we have used the conventional notation

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Let us then take your equation and apply on both sides (recall this Lorentz transformation does not depend on ), and try rewriting everything in terms of prime quantities:

This “game” can always be done with contracted indices, …

— answered Jul 7, 2020 at 15:06

— ohneVal

— Lorentz invariance of the Lorentz force law

— Physics StackExchange

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How come ?

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Note that

does not mean that

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Also

does not mean that

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At the first glance, it seems to be unlikely that

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because while in , for any , ‘s have visible negative terms; in , only ‘s do.

Without additional mathematical properties among those physical quantities, the identity is impossible to prove.

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Just for future reference:

From the invariance of the spacetime interval it follows

— Wikipedia on

Lorentz transformation.

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Inserting the identity into the expression:

This path does not work. Also, the formula is plain wrong!

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Inserting the identity into the expression *before* (actually *without*) lowering the index:

— Me@2022-08-23 12:03:54 PM

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