.
Is gauge invariant?
.
~~~
… 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
.
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
.
Also
does not mean that
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At the first glance, it seems to be unlikely that
,
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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