General relativity's mathematical foundations go back to the axioms of Euclidean geometry and the many attempts over the centuries to prove Euclid's fifth postulate, that parallel lines remain always equidistant, culminating with the realisation by Lobachevsky, Bolyai and Gauss that this axiom need not be true. The general mathematics of non-Euclidean geometries was developed by Gauss' student, Riemann, but these were thought to be wholly inapplicable to the real world until Einstein developed his theory of relativity.
Gauss had realised that there is no a priori reason that the geometry of space should be Euclidean. What this means is that if a physicist holds up a stick, and a cartographer stands some distance away and measures its length by a triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if the physicist brings the stick to him and he measures its length directly. Of course for a stick he could not in practice measure the difference between the two measurements, but there are equivalent measurements which do detect the non-Euclidean geometry of space-time directly; for example the Pound-Rebka experiment (1959) detected the change in wavelength of light from a cobalt source rising 22.5 meters against gravity in a shaft in the Jefferson Physical Laboratory at Harvard, and the rate of atomic clocks in GPS satellites orbiting the Earth has to be corrected for the effect of gravity.
Newton's theory of gravity had assumed that objects did in fact have absolute velocities: that some things really were at rest while others really were in motion. He realized, and made clear, that there was no way these absolutes could be measured. All the measurements one can make provide only velocities relative to one's own velocity (positions relative to one's own position, and so forth), and all the laws of mechanics would appear to operate identically no matter how one was moving. Newton believed, however, that the theory could not be made sense of without presupposing that there are absolute values, even if they cannot be determined. In fact, Newtonian mechanics can be made to work without this assumption: the outcome is rather innocuous, and should not be confused with Einstein's relativity which further requires the constancy of the speed of light.
In the nineteenth century Maxwell formulated a set of equations—Maxwell's field equations—that demonstrated that light should behave as a wave emitted by electromagnetic fields which would travel at a fixed velocity through space. This appeared to provide a way around Newton's relativity: by comparing one's own speed with the speed of light in one's vicinity, one should be able to measure one's absolute speed--or, what is practically the same, one's speed relative to a frame of reference that would be the same for all observers.
The assumption was whatever medium light was travelling through—whatever it was waves of—could be treated as a background against which to make other measurements. This inspired a search to determine the earth's velocity through this cosmic backdrop or "ether"—the "ether drift." The speed of light measured from the surface of the earth should appear to be greater when the earth was moving against the ether, slower when they were moving in the same direction. (Since the earth was hurtling through space and spinning, there should be at least some regularly changing measurements here.) A test made by Michelson and Morley toward the end of the century had the astonishing result that the speed of light appeared to be the same in every direction.
In his 1905 paper "On the Electrodynamics of Moving Bodies", Einstein explained these results in his theory of special relativity.
Outline of the theory
The fundamental idea in relativity is that we cannot talk of the physical quantities of velocity or acceleration without first defining a reference frame, and that a reference frame is defined by choosing particular matter as the basis for its definition. Thus all motion is defined and quantified relative to other matter. In the special theory of relativity it is assumed that reference frames can be extended indefinitely in all directions in space and time. The theory of special relativity concerns itself with inertial (non-accelerating) frames while general relativity deals with all frames of reference. In the general theory it is recognised that we can only define local frames to given accuracy for finite time periods and finite regions of space (similarly we can draw flat maps of regions of the surface of the earth but we cannot extend them to cover the whole surface without distortion). In general relativity Newton's laws are assumed to hold in local reference frames. In particular free particles travel in straight lines in local inertial (Lorentz) frames. When these lines are extended they do not appear straight, and are known as geodesics. Thus Newton's first law is replaced by the law of geodesic motion.
We distinguish inertial reference frames, in which bodies maintain a uniform state of motion unless acted upon by another body, from non-inertial frames in which freely moving bodies have an acceleration deriving from the reference frame itself. In non-inertial frames there is a perceived force which is accounted for by the acceleration of the frame, not by the direct influence of other matter. Thus we feel g-forces when cornering on the roads when we use a car as the physical base of our reference frame. Similarly there are coriolis and centrifugal forces when we define reference frames based on rotating matter (such as the Earth or a child's roundabout). The principle of equivalence in general relativity states that there is no local experiment to distinguish non-rotating free fall in a gravitational field from uniform motion in the absence of a gravitational field. In short there is no gravity in a reference frame in free fall. From this perspective the observed gravity at the surface of the Earth is the force observed in a reference frame defined from matter at the surface which is not free, but is acted on from below by the matter within the Earth, and is analogous to the g-forces felt in a car.
Mathematically, Einstein models space-time by a four-dimensional pseudo-Riemannian manifold, and his field equation states that the manifold's curvature at a point is directly related to the stress energy tensor at that point; the latter tensor being a measure of the density of matter and energy. Curvature tells matter how to move, and matter tells space how to curve.
The field equation is not uniquely proven, and there is room for other models, provided that they do not contradict observation. General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law. Few physicists doubt that such a theory of everything will give general relativity in the appropriate limit, just as general relativity predicts Newton's law of gravity in the non-relativistic limit.
Einstein's field equation contains a parameter called the "cosmological constant" which was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations by Hubble a decade later confirmed that our universe is in fact not static but expanding. So Λ was abandoned, with Einstein calling it the "biggest blunder [I] ever made". However, quite recently, improved astronomical techniques have found that a non-zero value of Λ is needed to explain some observations.
The field equation reads as follows:
where is the Ricci curvature tensor, is the scalar curvature, is the metric tensor, is the cosmological constant, is the stress-energy tensor, is pi, is the speed of light and is the gravitational constant which also occurs in Newton's law of gravity. describes the metric of the manifold and is a symmetric 4 x 4 tensor, so it has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6.
The study of the solutions of this equation is one of the activities of a branch of astronomy named cosmology. It leads to the prediction of black holes and to the different models of evolution of the universe.
The vierbein formulation of general relativity
This is an alternative equivalent formulation of general relativity using four reference vector fields, called a vierbein or tetrad. We have four vector fields, ea, a=0,1,2,3 such that g(ea,eb)=ηab where
See sign convention. One thing to note is that we can perform an independent proper, orthochronous Lorentz transformation at each point (subject to smoothness, of course) and still get a valid tetrad. So, the tetrad formulation of GR is a gauge theory, but with a noncompact gauge group SO(3,1). It is also invariant under diffeomorphisms.
See vierbein and Palatini action for more details. See Einstein-Cartan theory for an extension of general relativity to include spin.
- The theory appeared to me then, and still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art, to be enjoyed and admired from a distance. —Max Born
- Bondi, Herman, Relativity and Common Sense, Heinemann (1964). A school level introduction to the principle of relativity by a renowned scientist.
- Carroll, Sean M., Spacetime and Geometry: An introduction to general relativity, Addison Wesley, San Francisco (2004). ISBN 0-8053-8732-3. A modern graduate level textbook; notes from an earlier version are available at arXiv:gr-qc/9712019.
- D'Inverno, Ray, Introducing Einstein's Relativity, Oxford University Press (1993). A modern undergraduate level text.
- Einstein, Albert, Relativity: The special and general theory. ISBN 0517884410. The special and general relativity theories in their original form.
- Epstein, Lewis Caroll, Relativity Visualized. ISBN 093521805X. Requires no mathematical background. Actually explains general relativity, rather than merely hinting at it with a few metaphors.
- Misner, Thorne, Wheeler, Gravitation, Freeman (1973). ISBN 0716703440. A classic graduate level text book, which, if somewhat long winded, pays more attention to the geometrical basis and the development of ideas in general relativity than some more modern approaches.
- Perret, W. and G.B. Jeffrey, trans.: The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, New York Dover (1923).
- Thorne, Kip, and Stephen Hawking, Black Holes and Time Warps, Papermac (1995). A recent popular account, by a leading expert.
- J. J. O'Connor and E. F. Robertson, History of General Relativity at the MacTutor History of Mathematics archive.
- Reflections on Relativity A complete online course on Relativity
- MIT 8.962 Course Notes Notes and handouts from the MIT 8.962 course on General Relativity
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