New Transformation Equations and the Electric Field Four-vector

The Invariance of Charge Density

Because of the invariance of the Euclidean inner product, we can show that the charge density, , is the same in all inertial reference frames. First, we transform the gradient of the electric field

Then, by setting , we have

Now, due to the orthogonality of , we get

And, finally, since and , we conclude that

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