Notes on Relativity and Cosmology - Physics Department, UCSB

144 CHAPTER 6. DYNAMICS: ENERGY AND ...
The third law of Newtonian Physics: When two objects (A and B) exert
forces F A on B and F B on A on each other, these forces have the same size but
act in opposite directions.
To understand why this is a problem, let’s think about the gravitational forces
between the Sun and the Earth.
Sun
Earth
Recall that Newton said that the force between the earth and the sun is given
by an inverse square law: F = GM earthM sun
d
where d is the distance between
2
them. In particular, the force between the earth and sun decreases if they move
farther apart. Let’s draw a spacetime diagram showing the two objects moving
apart.
F 2
F2
t 2
F
1
F 1
t 1
At some time t 1 when they are close together, there is some strong force F 1
acting on each object. Then, later, when they are farther apart, there is some
weaker force F 2 acting on each object.
However, what happens if we consider this diagram in a moving reference frame?
I have drawn in a line of simultaneity (the dashed line) for a different reference
frame above, and we can see that it passes through one point marked F 1 and one
point marked F 2 ! This shows that Newton’s third law as stated above cannot
possibly hold 1 in all reference frames.
So, Newton’s third law has to go. But of course, Newton’s third law is not
completely wrong – it worked very well for several hundred years! So, as
with the law of composition of velocities and Newton’s second law, we may
expect that it is an approximation to some other (more correct) law, with this
approximation being valid only for velocities that are very small compared to c.
It turns out that this was not such a shock to Einstein, as there had been a bit
of trouble with Newton’s third law even before relativity itself was understood.
Again, the culprit was electromagnetism.
1 You might wonder if you could somehow save the third law by having the concept of force
depend on which inertial frame you use to describe the system. Then in the moving frame, the
forces would not be F 1 and F 2 . However, the forces in the moving frame must still somehow
be determined by F 1 and F 2 . Thus, if F 1 and F 2 do not agree, neither can the forces in the
moving frame.