Difference between revisions of "General theory of relativity"
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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. | 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 | + | 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. | 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. | ||
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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. | ||
| − | The | + | The mathematical expression of the theory of general relativity takes the form of the ''Einstein field equations,'' a set of ten nonlinear partial differential equations. While solving these equations is quite difficult, examining them provides valuable insight into the structure and meaning of the theory. |
| − | :<math> G_{ | + | |
| − | + | In their general form, the Einstein field equations are written as a single [[tensor]] equation in [[abstract index notation]] relating the curvature of spacetime to sources of curvature such as energy density and momentum. | |
| + | |||
| + | :<math> G_{\mu\nu} = 8\,\pi\,G\,T_{\mu\nu} </math> | ||
| + | |||
| + | In this form, <math>G_{\mu\nu}</math> represents the ''Einstein tensor,'' <math>G</math> is the same ''gravitational constant'' that appears in the [[Law of Universal Gravitation|law of universal gravitation]], and <math>T_{\mu\nu}</math> is the ''stress-energy tensor'' (sometimes referred to as the ''energy-momentum tensor).'' The indices <math>\mu</math> and <math>\nu</math> range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with [[Special theory of relativity|special relativity]]. | ||
| + | |||
| + | The left side of the equation — the Einstein tensor — describes the curvature of spacetime in the region under examination. The right side of the equation describes everything in that region that affects the curvature of spacetime. | ||
| + | |||
| + | As we can clearly see even in this simplified form, the Einstein field equations can be solved "in either direction." Given a description of the gravitating matter, energy, momentum and fields in a region of spacetime, we can calculate the curvature of spacetime surrounding that region. On the other hand, given a description of the curvature of a region spacetime, we can calculate the motion of a test particle anywhere within that region. | ||
| + | |||
| + | Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it. | ||
===The right side of the equation: the stress-energy tensor=== | ===The right side of the equation: the stress-energy tensor=== | ||
Revision as of 18:42, November 13, 2009
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The general theory of relativity is a theory of gravity that is compatible with 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, 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."[1]
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.
Contents
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
|
This article/section deals with mathematical concepts appropriate for late high school or early college. |
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 geodesics 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.
The mathematical expression of the theory of general relativity takes the form of the Einstein field equations, a set of ten nonlinear partial differential equations. While solving these equations is quite difficult, examining them provides valuable insight into the structure and meaning of the theory.
In their general form, the Einstein field equations are written as a single tensor equation in abstract index notation relating the curvature of spacetime to sources of curvature such as energy density and momentum.
In this form, 
represents the Einstein tensor, 
is the same gravitational constant that appears in the law of universal gravitation, and 
is the stress-energy tensor (sometimes referred to as the energy-momentum tensor). The indices 
and 
range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with special relativity.
The left side of the equation — the Einstein tensor — describes the curvature of spacetime in the region under examination. The right side of the equation describes everything in that region that affects the curvature of spacetime.
As we can clearly see even in this simplified form, the Einstein field equations can be solved "in either direction." Given a description of the gravitating matter, energy, momentum and fields in a region of spacetime, we can calculate the curvature of spacetime surrounding that region. On the other hand, given a description of the curvature of a region spacetime, we can calculate the motion of a test particle anywhere within that region.
Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.
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.
British Historian Paul Johnson declares the turning point in 20th century to have been when fellow Briton Sir Arthur Eddington, an esteemed English astronomer, ventured out on a boat off Africa in 1919 with a local Army unit to observe the bending of starlight around the sun during a total eclipse. Upon his return to England declared that his observations proven the theory of relativity. In fact recent analysis of Eddington's work revealed that he was biased in selecting his data, and that overall his data were inconclusive about the theory of relativity. The prediction was later confirmed by more rigorous experiments, such as those performed by the Hubble Space Telescope [2][3][4]. 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.[5]
The prediction that light is bent by gravity is predicted both by Newtonian physics and relativity, but relativity predicts a larger deflection.
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).
External Links
- Einstein, Albert (1916), "The Foundation of the General Theory of Relativity" (PDF), Annalen der Physik 49
References
- ↑ Misner, Thorne & Wheeler. Gravitation. (1973)
- ↑ Hubble Gravitational Lens Photo
- ↑ Gravitational Lensing
- ↑ [1]
- ↑ Lorentz, H.A. The Einstein Theory of Relativity
