Introduction to general relativity
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- This article is intended as a general, non-technical introduction. For the main encyclopedia article, please see General relativity.
General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1916. It unifies Einstein's earlier special relativity with Sir Isaac Newton's law of universal gravitation. This is done with the insight[1] that gravitation is not due to a force but rather is a manifestation of curved space and time.
| General Relativity |
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| General relativity |
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Introduction to... |
| Fundamental concepts |
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Special relativity · Equivalence principle |
| Phenomena |
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Black hole · Event horizon · Lenses |
| Equations |
| Advanced theories |
| Solutions |
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Schwarzschild · Kasner · Kerr |
| Scientists |
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Einstein · Minkowski · Eddington |
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Contents |
[edit] Einstein's treatment of gravitation
[edit] The equivalence principle
Einstein was uncomfortable with the equations he had for gravitation as a force. Then he noticed that a person free falling in an elevator will experience that he is floating, something very similar to an inertial system in space.
He postulated that a free falling system is a privileged system, similar to the inertial systems of special relativity, but while an inertial system follows a straight line in spacetime (see special relativity), these free-falling systems follow bent lines.
Einstein further postulated that the presence of the earth bends the space-time inertial paths in some way, making the straight lines of the inertial systems into curves. With this in mind, we can explain ordinary things in a new way:
- A satellite circling the earth is following an inertial path, with the path being curved by the presence of the earth.
- If we throw a stone it will make something close to a parabola, but in fact is like an ellipse. The stone is following an inertial path like the one of the satellite.
- And when we are sitting on a chair, we are trying to follow an inertial path or orbit, but the chair does not allow us to do it. Therefore we are accelerated with respect to the inertial path of free fall. This explains the sensation of acceleration or gravity when living on earth.
Therefore, the gravitational field we feel at the surface of the earth is really a fictitious force like those of other non-inertial frames of reference. From this moment, we will use in this article the word "gravitational" to refer to any fictitious field.
[edit] Stretched rubber analogy
The following analogy has been used to explain gravity. This analogy, however, does not explain how space is related to time. It only explains how gravity occurs according to general relativity and is not altogether correct. It is simply an analogy.
The analogy is that an elastic substance such as rubber is stretched over a frame. Imagine that the sun is a heavy ball. This ball will create a depression. Planets such as our Earth are other balls that orbit around the sun. Planets will orbit around the sun because they will orbit in the depression. All of the balls create a depression but the sun creates the largest as it is much heavier. According to general relativity mass affects space. Mass causes distortion of space as does an iron ball on a stretched rubber surface. Therefore, according to general relativity, gravity is a result of the distortion of space and not a force.
Click here for a PBS video on this analogy.
[edit] Extension of special relativity to non-inertial frames of reference
Though initially general relativity was intended as an extension of special relativity to non-inertial frames, nowadays only the theory of gravitation is considered GR, because accelerated frames of reference can be examined in special relativity by considering the instantaneous inertial frame of reference that an accelerated observer is in at a given event.
Therefore relativity in non-inertial frames is outside of GR strictly speaking, but though everything described here belongs to SR, traditionally it was GR, and anyway it is still related to GR in several ways. The important part is that the former consideration (frames are equivalent to an instantaneous inertial FR) leads to two related effects: gravitational time dilation and gravitational red shift. As everything valid for accelerated frames in flat spacetime is equivalent for fixed frames in curved spacetime this effects will be important in GR.
To deduce these two effects, just consider two people in an accelerating rocket ship with one being above the other (in the direction of acceleration) in the rocket ship. If the "lower" person emits a beam of light towards the "upper" person, during the time the light is traveling the upper will accelerate from the inertial frame in which the light was emitted to one which is moving away from the source of the light. As a result, the light will be red-shifted for the upper person. This is the gravitational red shift effect. Similarly, the light emitted by the upper person will be blue shifted for the lower person (since the lower person is being accelerated towards the source of the light).
The red-shifting being continuous reveals another effect for accelerated observers: gravitational time dilation. Since the red-shift means that the light is not vibrating as fast, it must follow that time for the lower person runs slow in the perception of the upper person.
Another effect is the bending of light. For an accelerated observer, a beam of light which is initially traveling horizontally will be bent "downwards" over time as the observers accelerate into "upwards" moving frames of reference.
[edit] Confirming experiments
By virtue of the equivalence principle as described above, all of these effects should be observable in the gravitational fields of the Earth and the stars. For example:
- The gravitational red-shift of light was confirmed by Pound and Rebka in 1959 [2].
- The Hafele-Keating experiment validated gravitational time dilation. An even more rigorous confirmation was done by the GPS system.
- In 1919, Eddington verified the bending of light by the Sun's gravitational field.
[edit] The geometry of gravitation
In general relativity, spacetime is non-Euclidean, or curved. The need for curvature arises from the equivalence principle and a child's simple question: "What keeps the people on the other side of the world from falling off?". In other words, should not the inertial paths on the other side of the Earth take objects away from the planet? Instead, all free-fall trajectories in the vicinity of a massive object will draw objects towards it.
The issue can be better illustrated by considering two balls on opposite sides of the Earth that start at rest with respect to the center of the Earth (and therefore with respect to each other). In the at-rest state, they are taking parallel world lines through spacetime. Now if both balls are allowed to free fall starting at a specified time, they will accelerate towards each other. The resultant movement towards each other means that their world lines are no longer parallel.
Einstein needed a way in which parallel inertial world lines could become non-parallel, something that special relativity does not allow. Einstein found the answer in curvature. An example of curvature is the surface of the Earth. For example, the lines of longitude are locally straight on the surface of the Earth. At the equator, they are parallel, but at the poles they cross. The lines of longitude also describe geodesic paths on the surface of the Earth, and in fact any great circle is a geodesic on the Earth.
With this in mind, Einstein proposed that inertially moving objects follow timelike geodesic world lines through spacetime. Given an appropriate curvature of the spacetime, free fall and orbital motion can now be inertial motion, as described above.
[edit] The Einstein field equations
As orbital motion and free-fall are dependent on the presence of a massive object, it follows in general relativity and related metric theories of gravitation that the presence of mass somehow curves spacetime. Furthermore, mass is a form of energy in relativity (due to E=mc2), and energy and momentum are intertwined in relativity (just as space and time are intertwined). So it follows that presence of mass, energy, and momentum (or "matter") causes spacetime to be curved.
In general relativity, this relationship between matter and curvature is described by the Einstein field equations. These equations were discovered by Einstein in late 1915. The Einstein field equations are expressed using tensor calculus, and are a collection of up to 10 independent simultaneous differential equations. These field equations are solved to create metrics of spacetime. (A metric of spacetime describes the invariant intervals squared between neighboring positions in spacetime whose coordinates differ by an infinitesimal amount. The simplest metric of spacetime is the Minkowski metric.) These metrics described the shape of the spacetime, and the curvature of spacetime and equations of motion for inertially moving objects can be obtained from it.
The actual shapes of spacetime are described by solutions of the Einstein field equations. In particular, the Schwarzschild solution (1916) describes the gravitational field around a spherically symmetric massive object. The geodesics of the Schwarzschild solution describe the observed behavior of objects being acted on gravitationally, including the anomalous perihelion precession of Mercury and the bending of light as it passes the Sun.
[edit] Experimental tests
General Relativity is consistent with all currently available measurements of large-scale phenomena. Arthur Eddington found observational evidence for the bending of light passing the Sun as predicted by general relativity in 1919. Subsequent observations have confirmed Eddington's results, and observations of a pulsar which is occulted by the Sun every year have permitted this confirmation to be done to a high degree of accuracy. There have also in the years since 1919 been numerous other tests of general relativity, all of which have confirmed Einstein's theory. Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the classical tests: the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury.
More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density.
The equivalence principle, the postulate of general relativity that presumes that inertial mass and gravitational mass are the same, is also under test. Past, present, and future tests are discussed in the equivalence principle article.
Even to this day, scientists try to challenge General Relativity with more and more precise direct experiments. The goal of these tests is to shed light on the yet unknown relationship between gravity and quantum mechanics. Space probes are used to either make very sensitive measurements over large distances, or to bring the instruments into an environment that is much more controlled than it could be on Earth. For example, in 2004 a dedicated satellite for gravity experiments, called Gravity Probe B, was launched to test general relativity's predicted frame-dragging effect, among others. Also, land-based experiments like LIGO and a host of "bar detectors" are trying to detect gravitational waves directly. A space-based hunt for gravitational waves, LISA, is in its early stages. It should be sensitive to low frequency gravitational waves from many sources, perhaps including the Big Bang.
Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be the speed of light. In 2002, the Fomalont-Kopeikin experiment produced measurements of the speed of gravity which matched this prediction. However, this experiment has not yet been widely peer-reviewed, and is facing criticism from those who claim that Fomalont-Kopeikin did nothing more than measure the speed of light in a convoluted manner.
The Pioneer anomaly is an empirical observation that the positions of the Pioneer 10 and Pioneer 11 space probes differ very slightly from what would be expected according to known effects (gravitational or otherwise). The possibility of new physics has not been ruled out, despite very thorough investigation in search of a more prosaic explanation.
[edit] See also
[edit] Notes
- ^ In the early 1920s Arthur Eddington claimed that there were only 3 people in the world who understood general relativity.
