good - newtons third law

7/30/2019 Good - Newtons Third Law

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Gary L. Mathis All rights reserved

Newton's Third Law.doc Page 1 of 3

Newtons Third Law in Special Relativity

Introduction

Does Newtons Third Law hold up in Special Relativity? Well, this essay shows

that depends.

Analysis

In Special Relativity there are three forces to deal with. These are:

1. The Minkowski force, defined byd

d=

p

K

where p is the 4-momentum

and is the proper time;

2. The ordinary force, defined byd

dt=

pF where m= p u is the relativistic

momentum and tis the laboratory time;

3. The Newtonian force, defined by NN

ddt

= pF where N m=p u is the

classical momentum. The Newtonian force is the force of classical

physics.

The validity of Newtons Third Law in SR depends on which of these forces areinvolved.

Minkowski Force

SupposeCM

V is the 4-velocity of the CM of a system of particles moving through

spacetime. By analogy with nonrelativistic particles, we define P , the total 4-

momentum for the system, to beCM

M=P V (1)whereMis the total mass for all the particles in the system. The above equation meansthe total momentum of the system is represented by a single point of mass Mthat is

moving with velocityCM

V . Let us now restrict ourselves to a system that contains only

two particles. Then

( )1 2 1 2 CMm m= + = +P p p V (2)

where 1 2,p p are the individual 4-momenta of the two particles whose masses are 1 2,m m .

We now define the following Minkowski forces for this system:

11

1

22

2

, force associated with the CM

, force on particle 1

, force on particle 2

CM

CM

d

d

d

d

d

d

=

=

=

P

K

pK

pK

(3)


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1 2, are the proper times associated with the particles, CM is the proper time at the

center of mass.CM

K is the Minkowski force associated with the CM and 1 2,K K are the

Minkowski forces acting on the individual particles in the system. Recall, in the

nonrelativistic theory of system of particles the Newtonian force associated with the CMgives the total force acting on the system. However, the statement below shows this is

not the case for the Minkowski force.

Statement 1:: the Minkowski force associated with the CM of a system does not represent

the total force acting on the system.

-Check:

( ) 1 21 2

1 21 2

1 2

1 1 2 2

1 2

but

because

and

CM

CM CM CM CM

CM

CM CM

d dd d

d d d d

d d

d d

d d d d

d d d d

= = + = +

+ = +

p pPK p p

p pK K K

p p p p

(4)

The failure of the Minkowski force associated with the CM to represent the totalMinkowski force for all the particles is due to the fact that there does not exist a universal

time for a system of particles; i.e., in general 1 2CMd d d because each particle hasits own proper time interval as does the CM.

In nonrelativistic theory, if the force associated with the center of mass is zero then the

forces that act between the particles obeys Newtons third law. As we see this is not thecase in Special Relativity.

Statement 2:: the Minkowski force does not obey Newtons third law of motion.

-Check: show what happens ifCM

P is constant.

1 2

1 2

1 2

if is constant

but , from above

if

CM

CM

CM

CM CM

CM

CM

d

d=

=

+

+ =

PK

K 0 P

K K K

K K 0 K 0K K

(5)

Again, the failure of Newtons third law vis--vis the Minkowski force is due to theabsence of a universal proper time that applies the same to all the particles in a system.


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Ordinary Force

Statement 3:: Newtons third law is obeyed by the ordinary force of Special Relativity,

d

dt=

pF , where m= p u is the spatial part of the 4-momentum vector (u is the

nonrelativistic particle velocity).

-Check: to prove this we use the conservation of relativistic momentum for two particlesand set the total ordinary force to zero so the only forces acting are those between theparticles.

1 2

1 2

1 2

1 2

1 2

, conservation of relativistic momentum

, total ordinary force on the system

if

d dd

dt dt dt

= +

= +

= + =

+ = =

=

P p p

p pP

F F F F

F F 0 F 0

F F

Note the relativistic momentum is based on the laboratory time, t, which is the same for

all the particles as observed from S; however, the relativistic force is not Lorentz

invariant as is the Minkowski force; i.e., we cannot compute the relativistic force in S byperforming a Lorentz transform on the force in S.

Newtonian Force

Statement 4:: Newons third law is not obeyed by the classical Newtonian force.

-Check: let F be the ordinary force of special relativity andN

F be the classical

Newtonian force.

( )

( ) ( )

1 2

3

2

3 3

1 21 1 1 1 1 2 2 2 2 22 2

1 2

1 2

, from above

but "ordinary" force in SR

because, in general,

N N

N N N N

N N

c

c c

=

= +

+ = +

F F

F F u u F

F u u F F u u F

F F

u u

(6)

Note: if 1 2,u u c

good - newtons third law

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