*Michael Fowler, University of Virginia*

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 classical Newtonian particle would,
if one could assume 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 $f\left(1+v/c\right)=f\left(1+gh/{c}^{2}\right)$ 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. This means you would age more slowly if you lived on the surface of
a planet with a large gravitational field. Of course, it might not be
very comfortable.

Despite what you might suspect, the fact that time passes at different rates
at different altitudes has significant practical consequences. An
important *everyday *application of general relativity is the Global
Positioning System. A GPS unit finds out where it is by detecting signals
sent from orbiting satellites at precisely timed intervals. If all the
satellites emit signals simultaneously, and the GPS unit detects signals from
four different satellites, there will be three relative time delays between the
signals it detects. The signals themselves are encoded to give the GPS
unit the precise position of the satellite they came from at the time of transmission.
With this information, the GPS unit can use the speed of light to translate the
detected time delays into distances, and therefore compute its own position on
earth by triangulation.

But this has to be done very precisely! Bearing in mind that the speed
of light is about one foot per nanosecond, an error of 100 nanoseconds or so
could, for example, put an airplane off the runway in a blind landing.
This means the clocks in the satellites timing when the signals are sent out
must certainly be accurate to 100 nanoseconds a day. That is one part in
10^{12}. It is easy to check that both the special relativistic
time dilation correction from the speed of the satellite, and the general
relativistic gravitational potential correction are much greater than that, so
the clocks in the satellites must be corrected appropriately. (The
satellites go around the earth once every twelve hours, which puts them at a
distance of about four earth radii. The calculations of time dilation
from the speed of the satellite, and the clock rate change from the
gravitational potential, are left as exercises for the student.) For more
details, see the lecture by Neil Ashby here.

In fact, Ashby reports that when the first Cesium clock was put in orbit in 1977, those involved were sufficiently skeptical of general relativity that the clock was not corrected for the gravitational redshift effect. But—just in case Einstein turned out to be right—the satellite was equipped with a synthesizer that could be switched on if necessary to add the appropriate relativistic corrections. After letting the clock run for three weeks with the synthesizer turned off, it was found to differ from an identical clock at ground level by precisely the amount predicted by special plus general relativity, limited only by the accuracy of the clock. This simple experiment verified the predicted gravitational redshift to about one percent accuracy! The synthesizer was turned on and left on.