Michael Fowler
University of Virginia
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In Einstein's little book Relativity: the Special and the General
Theory, he introduces general relativity with a parable. He
imagines going into deep space, far away from gravitational fields,
where any body moving at steady speed in a straight line will
continue in that state for a very long time. He imagines building
a space station out there - in his words, "a spacious chest
resembling a room with an observer inside who is equipped with
apparatus." Einstein points out that there will be no gravity,
the observer will tend to float around inside the room.
But now a rope is attached to a hook in the middle of the lid
of this "chest" and an unspecified "being"
pulls on the rope with a constant force. The chest and its contents,
including the observer, accelerate "upwards" at a constant
rate. How does all this look to the man in the room? He finds
himself moving towards what is now the "floor" and needs
to use his leg muscles to stand. If he releases anything, it accelerates
towards the floor, and in fact all bodies accelerate at the same
rate. If he were a normal human being, he would assume the room
to be in a gravitational field, and might wonder why the room
itself didn't fall. Just then he would discover the hook and
rope, and conclude that the room was suspended by the rope.
Einstein asks: should we just smile at this misguided soul? His
answer is no - the observer in the chest's point of view is just
as valid as an outsider's. In other words, being inside the
(from an outside perspective) uniformly accelerating room
is physically equivalent to being in a uniform gravitational field.
This is the basic postulate of general relativity. Special relativity
said that all inertial frames were equivalent. General relativity
extends this to accelerating frames, and states their equivalence
to frames in which there is a gravitational field. This is called
the Equivalence Principle.
The acceleration could also be used to cancel an existing gravitational
field - for example, inside a freely falling elevator passengers
are weightless, conditions are equivalent to those in the unaccelerated
space station in outer space.
It is important to realize that this equivalence between a gravitational
field and acceleration is only possible because the gravitational
mass is exactly equal to the inertial mass. There is no way to
cancel out electric fields, for example, by going to an accelerated
frame, since many different charge to mass ratios are possible.
As physics has developed, the concept of fields has been very
valuable in understanding how bodies interact with each other.
We visualize the electric field lines coming out from a charge,
and know that something is there in the space around the charge
which exerts a force on another charge coming into the neighborhood.
We can even compute the energy density stored in the electric
field, locally proportional to the square of the electric field
intensity. It is tempting to think that the gravitational field
is quite similar - after all, it's another inverse square field.
Evidently, though, this is not the case. If by going to an accelerated
frame the gravitational field can be made to vanish, at least
locally, it cannot be that it stores energy in a simply defined
local way like the electric field.
We should emphasize that going to an accelerating frame can only cancel a constant gravitational field, of course, so there is no accelerating frame in which the whole gravitational field of, say, a massive body is zero, since the field necessarily points in different directions in different regions of the space surrounding the body.
Consider a freely falling elevator near the surface of the earth,
and suppose a laser fixed in one wall of the elevator sends a
pulse of light horizontally across to the corresponding point
on the opposite wall of the elevator. Inside the elevator, where
there are no fields present, the environment is that of an inertial
frame, and the light will certainly be observed to proceed directly
across the elevator. Imagine now that the elevator has windows,
and an outsider at rest relative to the earth observes the light.
As the light crosses the elevator, the elevator is of course accelerating
downwards at g, so since the flash of light will hit the
opposite elevator wall at precisely the height relative to the
elevator at which it began, the outside observer will conclude
that the flash of light also accelerates downwards at g.
In fact, the light could have been emitted at the instant the
elevator was released from rest, so we must conclude that light
falls in an initially parabolic path in a constant gravitational
field. Of course, the light is traveling very fast, so the curvature
of the path is small! Nevertheless, the Equivalence Principle
forces us to the conclusion that the path of a light beam is bent
by a gravitational field.
The curvature of the path of light in a gravitational field was
first detected in 1919, by observing stars very near to the sun
during a solar eclipse. The deflection for stars observed very
close to the sun was 1.7 seconds of arc, which meant measuring
image positions on a photograph to an accuracy of hundredths of
a millimeter, quite an achievement at the time.
One might conclude from the brief discussion above that a light
beam in a gravitational field follows the same path a Newtonian
particle would if it moved at the speed of light. This is true
in the limit of small deviations from a straight line in a constant
field, but is not true even for small deviations for a spatially
varying field, such as the field near the sun the starlight travels
through in the eclipse experiment mentioned above. We could try
to construct the path by having the light pass through a series
of freely falling (fireproof!) elevators, all falling towards
the center of the sun, but then the elevators are accelerating
relative to each other (since they are all falling along radii),
and matching up the path of the light beam through the series
is tricky. If it is done correctly (as Einstein did) it turns
out that the angle the light beam is bent through is twice that
predicted by a naïve Newtonian theory.
What happens if we shine the pulse of light vertically down
inside a freely falling elevator, from a laser in the center of
the ceiling to a point in the center of the floor? Let us suppose
the flash of light leaves the ceiling at the instant the elevator
is released into free fall. If the elevator has height h,
it takes time h/c to reach the floor. This means
the floor is moving downwards at speed gh/c when
the light hits.
Question: Will an observer on the floor of the elevator
see the light as Doppler shifted?
The answer has to be no, because inside the elevator, by the Equivalence
Principle, conditions are identical to those in an inertial frame
with no fields present. There is nothing to change the frequency
of the light. This implies, however, that to an outside observer,
stationary in the earth's gravitational field, the frequency of
the light will change. This is because he will agree with
the elevator observer on what was the initial frequency f of
the light as it left the laser in the ceiling (the elevator was
at rest relative to the earth at that moment) so if the elevator
operator maintains the light had the same frequency f as
it hit the elevator floor, which is moving at gh/c
relative to the earth at that instant, the earth observer will
say the light has frequency f(1 + v/c) =
f(1+gh/c2), using the Doppler formula
for very low speeds.
We conclude from this that light shining downwards in a gravitational field is shifted to a higher frequency. Putting the laser in the elevator floor, it is clear that light shining upwards in a gravitational field is red-shifted to lower frequency. Einstein suggested that this prediction could be checked by looking at characteristic spectral lines of atoms near the surfaces of very dense stars, which should be red-shifted compared with the same atoms observed on earth, and this was confirmed. This has since been observed much more accurately. An amusing consequence, since the atomic oscillations which emit the radiation are after all just accurate clocks, is that time passes at different rates at different altitudes. The US atomic standard clock, kept at 5400 feet in Boulder, gains 5 microseconds per year over an identical clock almost at sea level in the Royal Observatory at Greenwich, England. Both clocks are accurate to one microsecond per year.
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Copyright ©1997 Michael Fowler