The Theory of General Relativity
The Riddle of Gravitation
Even after the realization that gravity and accelerated motion are indistinguishable and equivalent, Einstein was still troubled by a problem with the law of universal gravitation that plagued Newton and all scientists afterwards—just what was the nature of gravitation. In his Principia, Newton admitted that he could not "feign hypothesis" and that he would leave it to his successors to decipher the nature of gravitation. As envisioned by Newton and the scientists of the Enlightenment who followed, gravitation was a force that acted instantly over an distance. That lead to a troubling paradox, however, after the speed of light had been measured to be extremely fast, but finite.
Newton's Dirty Little Secret
Consider a troubling
hypothetical situation in which the Sun instantly disappears. According
to Newton's law of gravitation, the absence of the Sun's mass would instantly
result in Earth and the other planets flying off into space because the
Sun's gravitational influence on them was gone. The last rays of light leaving
the Sun before it disappeared, however, would continue traveling outward
and would reach Earth's orbit 500 seconds (8 minutes 20 seconds) later. In
other words, we would still see the Sun after it disappeared, but the change
in Earth's orbit would tell us the Sun was no longer there! How could
it be there and NOT be there at the same time?
Einstein knew the answer had to be connected to the equivalence principle. Using the mathematics and postulates of special relativity he was able to pull all the pieces together.
Riding Along on a Carousel
Imagine that we have a very special carousel—one
that can rotate at speeds very near the speed of light—and wish to
determine its circumference. First, while the carousel is stationary,
Ginger (being lazy and taking advantage of her lab partner's skills
a bit too much) just watches and lets Marianne measure the distance from
the center to the edge of the carousel with a meter stick. At the edge,
Marianne announces the carousel's radius is 10 meters. Hence, its circumference
is simply 62.8 meters (from C = 2πr).
Being a thoughtful astronomy student, Marianne knows it's important to
check "their" work
by actually measuring the circumference of the carousel while Ginger,
sitting at the center, watches her move around the edge with the meter
stick. As expected, the measured circumference is indeed 62.8 meters,
confirming the accuracy of the initial measurement and calculation.
Marianne now states that they—or rather, she—must repeat the measurements in a second trial, but by this time the carousel has started and is rotating at 0.87c (87% of the speed of light!). Not wanting to carry her lab partner through the entire experiment, Marianne puts Ginger to work measuring the radius while she takes a break and sits at the center. When Ginger reaches the edge, she agrees that the radius of the carousel is indeed 10 meters. So far so good—the calculated circumference (62.8m) agrees with the first trial, but Marianne figures she better have Ginger measure the circumference directly as a final check. As Ginger sets out to measure the circumference by moving around the edge, Marianne watches carefully and soon begins to doubt her partner's ability to perform even this simple measurement. To Marianne's disbelief, Ginger confidently announces the circumference it 125.6 meters! Twice as large as the measurement Marianne made just a few minutes before.
How can that be? Even Ginger isn't so incompetent that she would get the circumference wrong by so much! Marianne thinks about what could be wrong while telling Ginger to try measuring the circumference again. As Ginger sets out to conduct the measurement of the circumference a second time, Marianne begins to realize the result is again as it should be—125.6 meters.
"Don't worry, I've got it. The number is right,"Marianne exclaims confidently.
Ginger is baffled. How can both numbers be right?
The answer, Marianne explains, came to her when she noticed the background was moving. The carousel had been started after she sat down at the center and made Ginger start measuring. The measurement of the radius was unaffected since that measurement was perpendicular to the direction of the carousel's motion. Only when Ginger physically measured the circumference was she laying the meter stick in the direction of the rotation. Moving at 0.87c, the meter stick was contracted by 50% as predicted by the Lorentz transformations of special relativity! Thus, the number of lengths of the meter stick required to complete one circumference was twice as big as it was for the stationary carousel, or 125.6 meters. There is no error, just length contraction. Ginger didn't notice the length contraction since she was stationary with respect to the meterstick. Only Marianne, watching from the center of the rotating carousel, noticed the meterstick looked short.
Acceleration as Warpage
If we were to envision this rotating carousel and plot its shape, the fact
that the calculated circumference is 62.8 m and the measured circumference
is 125.6 m would result in a "warped"
carousel that is not flat like it was when it was stationary.
The rotating carousel is, in fact, accelerating since any point other than the center is constantly changing direction. Acceleration distorts measurements of space! Applying the Principle of Equivalence, we are forced to conclude that gravitational fields must also distort, or warp, space.
This is the solution that eluded Newton—gravity is not a spooky force that acts instantly over distances in a static spatial playing field. Gravity is a manifestation of the curvature of spacetime produced by the presence of mass. A star (see figure at right) has a gravitational influence on its surroundings because its mass distorts the shape of the spacetime in the vicinity of the star. The greater the mass, the greater the curvature of spacetime.
This curvature is also responsible for the influence gravity has on orbital dynamics. A planet orbits a star because the planet moves through the curved spacetime of a star (see figure at right.) In the mathematical representation of spacetime, the planet moves along the "straightest" (shortest) path in curved spacetime. In another sense, Newton's force of gravity on the macroscopic scale is a fictitious force—it is not some spooky "action" between to objects of mass, but rather an interaction produced by movement through curved spacetime.
The Speed of Gravity
If the Sun were to suddenly disappear, its last light rays would race outward
through space at exactly the same speed that the gravitational
"rebound" of spacetime would. That rebound would be perceived
as a graviational wave rippling through spacetime much like a ripple
races outward from the point where a rock is dropped into a pond.
Hence, the visual disappearance of the Sun 500 seconds after the
fact would coincide exactly with the "felt" disappearance
of the Sun's gravitational influence.