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Introduction to GR

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/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.  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.

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 1012.  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.