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I am aware that acceleration is not invariant under lorenrz transformations, but I was sure that the first postulate of special relativity implied that newton’s second law in its original form, F=dp/dt, where p is the relativistic momentum, was invariant. However, the following 2 questions imply otherwise:

Relativistic electromagnetism and electromagnetic forces on 2 protons

Magnetic force between 2 moving charges

It seems that the magnetic force between 2 moving charges depends on the frame of reference, and even goes to 0 in the relativistic limit. Doesn’t this violate the first postulate of special relativity?

Vulgar Mechanick
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1 Answers1

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The fact that Newton’s Second Law in the form

$$\mathbf F=\frac{d\mathbf p}{dt}$$

is relativistically form-invariant, meaning that in another inertial frame

$$\mathbf F’=\frac{d\mathbf p’}{dt’},$$

does not mean that force doesn’t transform under a Lorentz boost. It does transform, in the same way as the time derivative of relativistic momentum does. $\mathbf F’\ne\mathbf F$.

See this paper for a discussion of the Lorentz transformation of a three-force $\mathbf F$.

G. Smith
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