Einstein's general relativity describes gravitational effects by a set of 'field equations' of the form

                     Geometry(ij) = Matter(ij),

where i, j are indices taking the values 1, 2, 3, 4.

Matter(ij) obeys four identities,

                     Matter(ij);i = 0, 

where the repetition of i means that we take the sum of the terms for i = 1, 2, 3, 4, and the semi-colon (;) denotes 'covariant differentiation'.

Obviously, the geometry tensor Geometry(ij) has to obey the same kind of identity if the field equations are to hold. There are in fact two and only two geometrical tensors with this property. One of them is very simple, being the 'metric tensor' g(ij). The other is formed from g(ij) using a fairly complicated formula. We shall call this second tensor Geom(ij).

Einstein started with the field equations

 
(1)                    Geom(ij) = kappa x Matter(ij),

where kappa is a constant whose value is easily found in the 'Newtonian limit', the case of small velocities of matter. In this case the equations (1) reduce approximately to the single Newtonian equation for the gravitational potential.

When Einstein applied (1) to cosmology, he postulated a static metric with an isotropic and homogeneous distribution of matter, and found a solution in which matter has a constant average density and pressure. The pressure, however, came out negative.

Since Einstein could not give any meaning to the negative pressure, he modified the field equations to read:

(2)         Geom(ij) + lambda x g(ij) = kappa x Matter(ij),

where lambda is the famous 'cosmological constant' (which is small). A suitable solution was then possible.

Einstein regarded the extra 'cosmological term' as having no other justification. Friedman then found that without the cosmological term one can have an expanding universe, and this is taken to be confirmed by Hubble's discovery of the red shift in the spectra of nebulae. For Einstein, the cosmological term became his 'biggest blunder'. In fact, it was not.

Einstein was looking for simplicity and physical justification. But (2) is the most general mathematical form of field equations in which the left hand side is constructed out of the geometry. Our approach to cosmology in general relativity should therefore always be to use (2) and determine the constants kappa and lambda by comparison with observation.

Reference: Albert Einstein, The Meaning of Relativity, Methuen, London, 4th edn, 1950 (or 3rd edn, 1946), Appendix I.

Last updated: 04 January 2005