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## General Relativity

Special relativity is generalised to accommodate non-inertial reference frames. This is done via the principle of relativity, illustrated in figure 15 and stated below.

It is not possible to distinguish (in a closed system) between the effects produced by a gravitational field and those produced by an acceleration of the closed system This principle allows one to replace the effects of gravity by equivalent effects based on the geometry of space-time. Once gravity is abolished'' in this way, and there is no force of gravity'' then all (gravitating) objects will have motions described by Newton's First Law. That is, those in motion will continue in a straight line at constant velocity. However, straight line'' now means only locally straight (locally parallel to a co-ordinate axis in space). However, the geometry of space is now warped'' (no longer Euclidean) in such a way that the objects actual trajectory is similar'' to that calculated in the classical way. Einstein wrote down a Field Equation which allowed the warping of the geometry of space-time to be calculated given a certain mass distribution.

The trajectory of the moon around the earth is locally straight in a space-time region warped by the presence of the earth's mass. Such straight lines'' are called geodesics, defined as the shortest distance between two points in a curved space. This is illustrated in figure 16. This is not simply an alternative but equivalent way of looking at gravity. It would not be such a disturbing idea if that were so ! It is easy to see that dramatic new gravitational'' effects may be predicted.

• Because mass distributions warp space-time, a photon, which also has to travel along a geodesic (locally straight line in the warped space), will also be affected by the mass distribution. Thus General Relativity predicts that photons are subject to gravitational attraction ! Note that the classical theory (60)

did not predict this for the photon ( ).
• Very dense matter can warp space so much that nothing, no particles (not even light) can ever escape, once they pass closer than a certain distance, known as the event horizon. Such objects are known as black holes. The density of nuclear matter, when aggregated in amounts equivalent to a large star, is sufficient to realise a black hole.

General Relativity is now widely accepted, following three major experimental verifications :

• The perihelion precession of mercury is beyond that expected by classical theories, but exactly that predicated by General Relativity.
• The gravitational red-shift of light (ie loss of energy ( ) by light as it escapes a gravitating body), as predicted by General Relativity has been quantitatively verified.
• The bending of light in a gravitational field has been verified spectacularly during a lunar eclipse of the sun, and more recently by gravitational lensing.

Black holes have not yet been definitely verified, although there are many strong candidates in the cosmos, and further compelling theoretical evidences.   Next: The Global Positioning System Up: Relativistic Mechanics [8 lectures] Previous: Massless particles
Simon Connell 2006-02-21