# Relativity and Newton's Third Law

*Most recent answer: 06/23/2012*

Q:

What precisely is Newton's third law? In particular, does it apply for objects separated in space? Or does the law only work for "contact forces"? The reason I ask is because some people say relativity breaks Newton's third law, while others say it doesn't, but it appears to all come down to disagreement on the details of Newton's third law.

- Sam (age 17)

Springfield, IL

- Sam (age 17)

Springfield, IL

A:

Newton's third law simply says, in its modern form, that the momentum of any isolated collection of things is conserved. Someone might have gotten the impression that for certain long-range forces (electromagnetism) momentum was not conserved, but that's only because they forgot to include the momentum of the electromagnetic field itself.

Relativity is not only compatible with the third law, it's very closely related. The overall symmetry of special relativity, in which all places are fundamentally the same and any uniformly moving object can be treated as being at rest, actually implies Newton's third law.

Perhaps what people said was that relativity violates Newton's second law. This states, in Newton's form or in a modern version, that the time rate of change of the momentum equals the outside force. This remains true in special relativity. However, this law is often given as F=ma, which implicitly assumes that the inertial mass m doesn't change as the object accelerates at rate a under force F. The inertial mass does, however, change as an object's speed changes, so in that standard form the law is wrong. Although Newton's statement of the law remains true, it doesn't mean quite what he thought since he didn't realize that the inertial mass is velocity-dependent.

Mike W.

Relativity is not only compatible with the third law, it's very closely related. The overall symmetry of special relativity, in which all places are fundamentally the same and any uniformly moving object can be treated as being at rest, actually implies Newton's third law.

Perhaps what people said was that relativity violates Newton's second law. This states, in Newton's form or in a modern version, that the time rate of change of the momentum equals the outside force. This remains true in special relativity. However, this law is often given as F=ma, which implicitly assumes that the inertial mass m doesn't change as the object accelerates at rate a under force F. The inertial mass does, however, change as an object's speed changes, so in that standard form the law is wrong. Although Newton's statement of the law remains true, it doesn't mean quite what he thought since he didn't realize that the inertial mass is velocity-dependent.

Mike W.

*(published on 06/23/2012)*