Notes on Relativity and Cosmology - Physics Department, UCSB

7.1. THE GRAVITATIONAL FIELD 169
directed toward each other, where G = 6.67 × 10 −11 Nm 2 /kg 2 is called
“Newton’s Gravitational Constant.” G is a kind of intrinsic measure of
how strong the gravitational force is.
It turns out that this rule is not compatible with special relativity. In particular,
having learned relativity we now believe that it should not be possible to send
messages faster than the speed of light. However, Newton’s rule above would
allow us to do so using gravity. The point is that Newton said that the force
depends on the separation between the objects at this instant 2 .
Example: The earth is about eight light-minutes from the sun. This means that,
at the speed of light, a message would take eight minutes to travel from the sun
to the earth. However, suppose that, unbeknownst to us, some aliens are about
to move the sun.
Then, based on our understanding of relativity, we would expect it to take eight
minutes for us to find out! But Newton would have expected us to find out
instantly because the force on the earth would shift (changing the tides and
other things.....)
Force before
Force after
7.1.2 The importance of the field
Now, it is important to understand how Maxwell’s equations get around this
sort of problem. That is to say, what if the Sun were a positive electric charge,
the earth were a big negative electric charge, and they were held together by an
Electro-Magnetic field? We said that Maxwell’s equations are consistent with
relativity – so how what would they tell us happens when the aliens move the
sun?
The point is that the positive charge does not act directly on the negative charge.
Instead, the positive charge sets up an electric field which tells the negative
charge how to move.
+ -
When the positive charge is moved, the electric field around it must change,
but it turns out that the field does not change everywhere at the same time.
2 Note that there is also an issue of simultaneity here. Which events on the two separated
worldlines should one compare to compute the distance? Which notion of ‘this instant’ would
one use to pick out these events?