General theory of relativity
See Theory of Relativity for a comprehensive treatment of this topic.
The General Theory of Relativity is an extension of special relativity, dealing with curved coordinate systems, accelerating frames of reference, curvilinear motion, and curvature of spacetime itself. It could be said that general relativity is to special relativity as vector calculus is to vector algebra. General relativity is best known for its formulation of gravity as a fictitious force arising from the curvature of spacetime. In fact, "general relativity" and "Einstein's formulation of gravity" are nearly synonymous in many people's minds.
The general theory of relativity was first published by Marcel Grossman in 1913 and David Hilbert and Albert Einstein in 1916. However, "even though General Relativity has passed many tests, most physicists don’t believe it is ultimately correct because it conflicts with quantum mechanics." 
General relativity, like quantum mechanics (the other of the two theories comprising "modern physics") both have reputations for being notoriously complicated and difficult to understand. In fact, in the early decades of the 20th century, general relativity had a sort of cult status in this regard. General relativity and quantum mechanics are both advanced college-level and postgraduate level topics. Hence this article can't possibly give a comprehensive explation of general relativity at the expert level. But we will attempt to give a rough outline, for lay people, of the general relativistic formulation of gravity.
In the weak field approximation, where velocities of moving objects are low and gravitational fields are not very severe, the theory of general relativity is said to reduce to the law of universal gravitation. That is to say, under those circumstances the equations of general relativity are mathematically equivalent to the equations of Newtonian gravitation.
Modern science does not say that Newtonian (classical) gravity is wrong. It is obviously very very nearly correct. In the weak field approximation, such as one finds in our solar system, the differences between general relativity and Newtonian gravity are miniscule. It takes very sensitive tests to show the difference. The history of those tests is a fascinating subject, and will be covered near the end of this article. But in all tests conducted so far, where there are discrepancies between the predictions of general relativity and Newtonian gravity (or other competing theories for that matter), experimental results have shown general relativity to be a better description.
Outside of the solar system, one can find stronger gravitational fields, and other phenomena, such as quasars and neutron stars, that permit even more definitive tests. General relativity appears to pass those tests as well.
This is not to say, by any means, that general relativity is the ultimate, perfect theory. It has never been unified with modern formulations of quantum mechanics, and it is therefore known to be incorrect at extremely small scales. Just as Newtonian gravity is very nearly correct, and completely correct for its time, general relativity is believed to be very nearly correct, but not completely so. Contemporary speculation on the next step involves extremely esoteric notions such as string theory, gravitons, and "quantum loop gravity".
The theory is sometimes explained with a thought experiment developed by Einstein involving two elevators. The first elevator is stationary on the Earth, while the other is being pulled through space at a constant acceleration of g. Einstein realized that under Newtonian mechanics, any physical experiment carried out in the elevators would give the same result. This realization is known as the equivalence principle and it states that accelerating frames of reference and gravitational fields are indistinguishable. General relativity is the theory of gravity that incorporates special relativity and the equivalence principle.
The general theory of relativity is a metric theory, sometimes also called a geometric theory. Metric theories describe physical phenomena in terms of differential geometry. This stands in contrast to Isaac Newton's Law of Universal Gravitation, which described gravity in terms of a vector field. In the case of general relativity, the theory relates stress-energy — an extension of the concept of mass — and the curvature of spacetime. In the words of physicist John Wheeler, "Space tells matter how to move, matter tells space how to curve."
Coordinate Systems and Spacetime Diagrams
Figure 1 shows a "spacetime diagram", with my house, my neighbor's house, and my neighbor walking from his house to mine.
- This is the same kind of diagram that is used in explanations of special relativity. The "spacetime" is sometimes called "Minkowski space". Spacetime is actually four-dimensional, but we can only show two dimensions, so we leave out y and z. The single x spatial coordinate is good enough for our purposes, so the diagram has x going from left to right, and t (time) going upward. For the purposes of this explanation, don't worry about the considerations of special relativity such as the speed of light, the Lorentz transform, or light cones. None of that is important just now.
The diagram shows the calibration, in space (that is, x) and time. These measurements are made with respect to my (stationary) frame of reference. My house is at x=0, and my neighbor's house is at x=1250 (feet). My neighbor walks at 250 feet per minute.
The diagram shows some "events"—my house, now; my house, 5 minutes from now; and my neighbor's house now and 5 minutes from now. The diagonal line depicts my neighbor walking from his house to mine, arriving 5 minutes from now. That line is called his world line. The line going straight up in my house is my own world line (I'm sitting at home.)
A car is driving down the street, from left to right. Figure 2 shows the same four events and two world lines, but with different calibration—the car's own coordinate system. The car is driving 500 feet per minute, but in the opposite direction. The event of my neighbor's arrival at my house is now at x=-2500. It's way behind the car, though the car was directly in front of my house at t=0.
Because the car's frame of reference is in motion, the calibration lines in figure 2 are not perpendicular. The formerly vertical lines are now slanted. But there is something very important to notice about the two coordinate systems: They are flat. The flatness comes from the fact that the calibration lines are straight and parallel. The boxes created by the lines are parallelograms. But note that the lines don't have to be perpendicular, and the boxes don't have to be rectangles. Straight parallel lines and parallelograms are all that is required.
These two flat coordinate systems have a very important physical property: Neither I, sitting at home, nor a passenger in the car, experiences any "fictitious forces". That is, people in the car don't feel any recoil from acceleration, or centrifugal force, or Coriolis force. These frames of reference are said to be inertial. This leads to an important principle of geometrical physics:
- Inertial frames of reference have flat coordinate systems. Flat coordinate systems lead to an absence of fictitious forces.
Now consider figure 3. The coordinate system is once again that of the car, but the car is accelerating, starting at a standstill in front of my house at t=0. Its world line is curved. Once again, it crosses paths with my neighbor. This case is very different from the other two. The calibration lines are curved, and the boxes that they create are not parallelograms. This coordinate system is curved. Another thing to notice is that people in the car will feel a fictitious force—a "recoil" force agains the back of the seat. This frame of reference is not inertial.
- Accelerating frames of reference have curved coordinate systems. Curved coordinate systems lead to fictitious forces.
- .... !!!! We need these three diagrams, of course. I'll do them, but they will take a lot of work. If anyone else has the tools and expertise to do this, and more skill than I, feel free to make them, or to communicate with me (PatrickD).
... In progress ...
Qualitative Introduction to General Relativity
The relationship between the curvature of spacetime and the motions of freely falling bodies is often explained by an easily imagined analogy: bowling balls and golf balls on a trampoline.
Imagine that we place a golf ball on an ordinary backyard trampoline. If we give the golf ball a slight push, it will roll along in a straight line until friction brings it to a halt. But if we imagine that friction doesn't exist, then the golf ball will roll in a straight line at a constant speed forever — or at least until it reaches the edge of the trampoline and falls off.
Now imagine a bowling ball sitting in the middle of a trampoline. The trampoline isn't a rigid surface, so it deforms where the weight of the bowling ball pushes it down. This causes the surface to be curved downward, toward the ground.
If we place a golf ball near the edge of the trampoline, it will begin to roll toward the bowling ball, because the trampoline is sloped downward in that direction. The golf ball will start off moving very slowly, then pick up speed as it approaches the bowling ball, until finally it collides with the bowling ball and comes to rest.
But if we give the golf ball a slight push in a direction perpendicular to the direction of the bowling ball, then it will move in a curved path. If we only push it a little bit, the golf ball will curve slightly, but still collide with the bowling ball. If we push the golf ball somewhat harder, it will curve toward the bowling ball, pass by it on one side and climb back out of the depression until it reaches the edge and falls off.
But if we're very careful, and give the golf ball just the right push, it will curve completely around the bowling ball and return to our hand.
This is, in a nutshell, how spacetime and matter interact under the theory of general relativity. Massive objects — represented in our analogy by the bowling ball — curve spacetime. Less-massive objects also curve spacetime, but to a lesser extent. If the object is small enough, like our golf ball, the amount of curvature is so slight that we can't even measure it.
The way the golf ball moved in the three scenarios we imagined correspond to conic-section orbits, or Kepler orbits. When we just placed the golf ball and it rolled straight toward the bowling ball, that was a degenerate orbit: a straight line. When we gave it a push and it curved around the bowling ball and off the edge of the trampoline, that was a hyperbolic orbit. And when we gave it just the right push so that it curved around the bowling ball and back to our hand, that was an elliptical orbit.
These are the same orbits that are predicted by Isaac Newton's law of universal gravitation. But in Newton's equations, objects move in conic-section orbits because of a force that accelerates them toward the central mass. In general relativity, objects move in conic-section orbits because spacetime itself is curved, just like our imaginary trampoline was curved by the bowling ball.
Of course, our analogy is far from perfect. Our imaginary trampoline curved downward, toward the ground, pushed down by the weight of the bowling ball. That's not how spacetime behaves in general relativity. It curves, but not toward anything, not in any direction. Spacetime in general relativity is instead said to have intrinsic curvature, which is mathematically quite simple but very difficult to visualize.
And of course there are many, many other aspects of general relativity that our imaginary trampoline didn't model. But the analogy captures the essential nature of the theory: the bowling ball caused the trampoline to be curved, and the curvature of the trampoline caused the golf ball to move in a different way than if the trampoline had been flat. This is the essence of general relativity: matter tells space how to curve, and space tells matter how to move.
Quantitative Introduction to General Relativity
Consequences and Tests of General Relativity
General relativity provides one explanation for the precession of Mercury's perihelion, which was moving at a different speed than that predicted by a simple application of Newton's law of universal gravitation. (Previous scientists had attempted to explain it by the gravitational pull of a hypothetical planet inside Mercury's orbit, which they called Vulcan. This could also be explained by altering the precise inverse-square relation of Newtonian gravity to distance, but that was disfavored by mathematicians due to its inelegance in integrating.)
General relativity predicts that the path of light will be distorted when it passes near a massive object. In 1919, Sir Arthur Eddington, an esteemed English astronomer, used this to test general relativity by observing the bending of starlight around the sun during a total solar eclipse.. (A smaller degree of bending could also be consistent with Newton's theory, if one hypothesized light to consist of particles. However, that particle theory of light had gone out of favor previously.) Eddington detected a bending of light, but his range of error overlapped both Einstein's and Newton's predictions. Upon his return to England, Eddington declared that his observations proven the theory of relativity. His experiment was later confirmed by more rigorous experiments, such as those performed by the Hubble Space Telescope . Lorentz has this to say on the discrepancies between the empirical eclipse data and Einstein's predictions.
- It indeed seems that the discrepancies may be ascribed to faults in observations, which supposition is supported by the fact that the observations at Prince's Island, which, it is true, did not turn out quite as well as those mentioned above, gave the result, of 1.64, somewhat lower than Einstein's figure.
Modern-day astronomers suspect they also see such "gravitational lensing" going on between galaxies, where one galactic cluster distorts the paths of the light passing around it.
Relation to Special Relativity
Special relativity is the limiting case of general relativity where all gravitational fields are weak. Alternatively, special relativity is the limiting case of general relativity when all reference frames are inertial (non-accelerating and without gravity).
- Einstein, Albert (1916), "The Foundation of the General Theory of Relativity" (PDF), Annalen der Physik 49
- A simple explanation of General Relativity