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General theory of relativity (view source)
Revision as of 17:08, November 13, 2009
, 17:08, November 13, 2009Added qualitative intro draft plus some structure, threw up "work in progress" template while I fill in the blanks.
{{Progress}}
The '''general theory of relativity''' is a theory of gravity that is compatible with [[Special theory of relativity|Special Relativity]]. The theory was first published by [[Albert Einstein]] in 1916.
The '''general theory of relativity''' is a theory of gravity that is compatible with [[Special theory of relativity|Special Relativity]]. The theory was first published by [[Albert Einstein]] in 1916.
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|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."<ref>Misner, Thorne & Wheeler. ''Gravitation.'' (1973)</ref> 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.
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|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."<ref>Misner, Thorne & Wheeler. ''Gravitation.'' (1973)</ref>
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.
The theory was inspired by 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 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.
==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|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==
{{Template:Math-h}}
General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional [[manifold]], which looks locally like [[Minkowski space]], and which acquires [[curvature]] due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from [[Euclidean geometry]]: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along [[geodesic]]s in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.
General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional [[manifold]], which looks locally like [[Minkowski space]], and which acquires [[curvature]] due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from [[Euclidean geometry]]: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along [[geodesic]]s in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.
:<math> G_{uv} = 8\pi\, T_{uv} </math>
:<math> G_{uv} = 8\pi\, T_{uv} </math>
where ''G<sub>uv</sub>'' is the [[Einstein curvature tensor]], and ''T<sub>uv</sub>'' is the [[stress-energy tensor]], ''G<sub>uv</sub>'' and ''T<sub>uv</sub>'' are both rank 2 symmetric tensors. The GR field equations is a system of [[partial differential equations]] that relates the curvature of space to the mass occupying the space.
where ''G<sub>uv</sub>'' is the [[Einstein curvature tensor]], and ''T<sub>uv</sub>'' is the [[stress-energy tensor]], ''G<sub>uv</sub>'' and ''T<sub>uv</sub>'' are both rank 2 symmetric tensors. The GR field equations is a system of [[partial differential equations]] that relates the curvature of space to the mass occupying the space.
===The right side of the equation: the stress-energy tensor===
===The left side of the equation: the Einstein curvature tensor===
==Exact Solutions in General Relativity==
==Tests of General Relativity==
General relativity provides one explanation for the seemingly anomalous precession of Mercury's perihelion. There are other explanations based in Newtonian gravity, such as factoring in the pull of the other planets on Mercury's orbit. One Newtonian explanation requires a slight alternation to the precise inverse-square relation of Newtonian gravity to distance, which is disfavored by mathematicians due to its inelegance in integrating.
General relativity provides one explanation for the seemingly anomalous precession of Mercury's perihelion. There are other explanations based in Newtonian gravity, such as factoring in the pull of the other planets on Mercury's orbit. One Newtonian explanation requires a slight alternation to the precise inverse-square relation of Newtonian gravity to distance, which is disfavored by mathematicians due to its inelegance in integrating.