## Remarks on General Relativity

*Michael**
Fowler*

*University*

*of*
Virginia

### Einstein’s Parable

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.

### Some Consequences of the Equivalence Principle

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*/*c*^{2}), 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

### General Relativity and the Global Positioning System

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.