Notes on Relativity and Cosmology - Physics Department, UCSB

7.6. HOMEWORK PROBLEMS 189
of acceleration is now relative as well – it is equivalent to being in a gravitational
field.
While this point is related to why the study of gravity historically acquired the
name “General Relativity,” it is not clear that this is an especially useful way
to think about things. In particular, I want to stress that one can still measure
one’s proper acceleration as the acceleration relative to a nearby (i.e., local!)
freely falling frame.
⋆⋆ Thus, there is an absolute distinction between freely falling and not freely
falling. Whether you wish to identify these terms with non-accelerating and
accelerating is just a question of semantics – though most modern relativists find
it convenient to do so. As a result, I will use a language in which acceleration is
not a relative concept but in which it implicitly means “acceleration measured
locally with respect to freely falling frames.”
7.5.2 And what about the speed of light?
There is a question that you probably wanted to ask a few paragraphs back,
but then I went on to other things.... I said that in a general gravitational field
there are no frames of reference in which light rays always travel in straight lines
at constant speed. So, after all of our struggles, have we finally thrown out the
constancy of the speed of light?
No, not completely. There is one very important statement left. Suppose that
we measure the speed of light at some event (E) in a frame of reference that
falls freely at event E. Then, near event E things in this frame work just like
they do in inertial frames – so, light moves at speed c and in a straight line.
Said in our new language:
As measured locally in a freely falling frame, light always moves in straight
lines at speed c.
7.6 Homework Problems
7-1. This first problem is an exercise in thinking about things from the perspective
of a freely falling object. Remember that Newton’s law of gravity
tells us that (i) objects near the earth are attracted to the center of the
earth and (ii) objects closer to the earth are attracted more strongly than
are objects farther away. You should include both of these effects
as you work this problem.
(a) Suppose that three small stones are released from high above the
earth as shown below. Sketch their worldlines on a spacetime diagram
drawn in the reference frame of the middle stone. Use the
x direction (shown below) for the spatial direction of your diagram.
(That is, sketch the motion in the x, t plane.) [Hint: In the case
shown below, the three stones are nearly the same distance