Tuesday, April 25, 2017
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The mollusc of light

The rigid structure of the space-time of Special Relativity - like that of Newtonian space - is completely destroyed by the impact of gravitation. The space-time continuum is soft, deformed by the matter it contains, and the matter in it moves according to its curvature.

The trajectories of the light rays nevertheless continue to follow the shortest paths. The frame of this space-time ‘mollusc’ is still woven by light, and its representation by light cones still summarises the essence of General Relativity.

Another helpful way of visualising curved space-time and its influence on matter uses a rubber sheet. Imagine a portion of space­time reduced to two dimensions, made of an elastic material. In the absence of any other object, the material remains flat. If a ball is placed on it, the material deforms, making a hollow around the ball that is the deeper the greater the mass of the ball. This type of representation, which appears fanciful, can be made math­ematically rigorous by what are called embedding diagrams.

Einstein's equations

‘The idea that physicists would in future have to study the theory of tensors created real panic amongst them following the firsuannouncement that Einstein’s predictions had been . verified. ’

A. Whitehead (1920)

All theories have their equations. Einstein’s gravitational field equations relate the degree of distortion in space and time to the nature ana motion or tne sources 01 gravitation, matter tells space-time how it must curve; space-time tells matter how it must move.

Einstein’s equations are extremely complex. The quantities involved are no longer just forces and accelerations, but also distances and durations. They are tensors, each a kind of table with several entries containing all the information about geometry and matter.

The action of gravitation on matter is more complicated than that of an electric field, and requires more complex mathematical entities to describe it than numbers and vectors with three components. To be convinced of this we remember that in Newton’s gravitational theory only the gravitational mass of a body is a source of gravitation, this mass being represented by a single number intrinsically associated with the body. In Einstein’s theory, the gravitational mass is only one of the components of the total quantity of gravity associated with a body. Special Relativity (which is always valid in a small region of space-time, where gravitation is uniform) already shows that all forms of energy are equivalent to mass and hence gravitating. Now the energy of a body depends on the relative motion of the observer measuring it. In i stationary body, all the energy is contained in its ‘rest mass’ (E = me2!); but as soon as the body moves its kinetic energy will create mass and thus gravitation. To evaluate the gravitational effect of a body, it is therefore necessary to combine its energy at rest with a ‘momentum vector’ describing its motion. This is why the full description of sources of gravity uses the ‘energy- momentum tensor’.

Furthermore, at each point in space-time 20 numbers are required to describe the curvature. The geometric deformations of space and time therefore require the use of a ‘curvature tensor’ (we remember that the curvature becomes more and more complicated as the number of dimensions increases). Einstein’s equations simply describe the relationship between the curvature tensor and the energy-momentum tensor, placing them on either side of an equality: matter creates curvature and curvature makes matter move. This is not the book to spell out all the richness of Einstein’s equations. The different components of the curvature tensors and the energy-momentum tensors are so tightly interlinked that in general it is not possible to find an exact solution, or even to define globally what is space and what is time. Also, we have to idealise the sources of gravitation in order to calculate ‘something’; to the point that most of the solutions found (describing curved space-times) bear no relation to real space and time. Einstein’s equations are in some sense too prolific, and allow an infinity of theoretical universes with bizarre properties.

This richness has perhaps harmed the credibility of Einstein’s theory. However, we should not get the idea that General Relativity only predicts properties which cannot be observed or are beyond human understanding. On the contrary, Einstein was both a physicist and a philosopher, and accordingly tried to describe the Universe, beginning with the Solar System. Using approximate solutions of his equations, he first calculated three measurable j effects of gravitation in the Solar System not predicted by Newton’s  law of attraction: the deviation of light rays passing close to the Sun, anomalies in the orbit of Mercury, and the lowering of electro­magnetic frequencies in a gravitational field. The next section will discuss the success of these three predictions of General Relativity.

Besides these cases, there are naturally-occurring situations where the simplifications imposed on the sources of gravitation are perfectly justified; the resulting exact solutions to Einstein’s equations give a satisfactory description of some part or other of the Universe. Paradoxically, these simplifications are most fruitful at two extreme distance scales. We can calculate the gravitational field produced by an isolated body in a vacuum (i.e. the space-time distortions around the body). The region around a star - for example the Solar System - or that near a black hole, is well described by this solution, as the matter is effectively concentrated in a small region of space-time surrounded by near-vacuum. At the opposite extreme, we can calculate the average gravitational field of the Universe as a whole (its geometry), because on a very large scale matter is spread more or less evenly and galaxies act like molecules in a homogeneous cosmic gas. General Relativity thus allows us to do cosmology, that is, to study the shape and evolution of the Universe in its entirety. Moreover until the advent of relativistic astrophysics in the 1970s, cosmology was the only real field of application for General Relativity - along with black holes, of course.

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