Newton’s Third Law and Self-Forces

Newton’s third law tells us that, for every action there is an equal and oppositely directed reaction. Since this law’s “action” and “reaction” have both magnitude and direction, they qualify as vectors. In modern terms we think of Newton’s actions and reactions as being forces, and reserve the word action for a scalar quantity¹.

Now as every student is taught early on, Newton 3 implies two entities: (1) an agent (or actor), and (2) an object (or reactor). But one can interpret Newton as referring to a single physical entity, provided one allows for the existence of “self-forces”. In this view every elementary particle, subjected to an “external” field, is also subject to an equal and oppositely directed self-field, and the following law is suggested:

Every elementary particle, when subjected to a conservative “external” electric or gravitational field, engenders an equal and oppositely directed self-field. The external and self-fields always sum to zero.

To the extent this law is true, Newton 2 can be summarized by the following three equations:

mg_external + qE_external = dp/dt,          (1)

mg_self + mg_external = 0,                    (2)

qE_self + qE_external = 0.                     (3)

Let us for now focus on the electric field (Eq. 3). According to it, E_self always equals –E_external. Assuming that Maxwell’s equations are physical laws, what is the Maxwellian paradigm for E_self? In answering this question let us imagine an elementary particle initially at rest at the Origin of a rectangular coordinate system. Let there be an ambient external, uniform electric field E_external pointing toward -x. According to Maxwell (and Newton):

• The subject charge experiences a negative acceleration, a_x.
• There is accordingly a nonzero B field at points on a circle lying in the yz-plane and with radius dr.
• The curl of this circulating B field equates to an E_self field, right at the charge, where E_self points toward positive x.
• According to Eq. 3, E_self equals –E_external.

The same reasoning applies to gravitation, but according to gravitomagnetic theory m and g are mathematically imaginary quantities, and mg points opposite to g, etc.

In summary, conventional wisdom is that Newton was referring to two discrete entities when he wrote his third law. He didn’t know about self-forces.

Or did he???

1. For a modern view of “action” see The Feynman Lectures on Physics, volume 1, chapter 19.