New Transformation Equations and the Electric Field Four-vector

16. The Spacetime Derivative

We would like to introduce an analogy to the element of spacetime interval,

involving the second partial derivative of an arbitrary scalar function of the coordinates, ,

which we call the spacetime derivative. Since the spacetime interval, , and the scalar function, , are invariant under transformation between arbitrary inertial reference frames, we can say that and , therefore,

In the primed frame

so that, from (16.2), (16.3), and (16.4), we can say

Now, in the case that

in (16.4), we have

In this case, , where is the invariant proper time and, since and , we can say that

Therefore, in this case, (16.5) becomes

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