Gauss had realised that there is no prior 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.
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. But 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 geodesic?s. 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 force]?s 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, and 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 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]?" Λ which was originally introduced by Einstein to allow for a static universe. 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, but 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 Rik is the [Ricci curvature tensor]?, R is the [Ricci curvature scalar]?, gik is the [metric tensor]?, Λ is the cosmological constant, Tik is the [stress-energy tensor]?, π is pi, c is the speed of light and G is the gravitational constant which also occurs in Newton's law of gravity. gik 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.
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