Solutions of
the Einstein Equations

The quantitative description of the Universe by General Relativity requires obtaining solutions to what are termed the Einstein field equations. These are 10 equations that must be solved simultaneously; they are notoriously difficult, and only a few solutions are known. (Technically, the equations to be solved are known to mathematicians as coupled, non-linear, partial differential equations; we may take that as precise shorthand for "very difficult to solve"!)

Solutions of the Equations

Among these solutions, there are two general classes:
  1. Static Universes
  2. Dynamic Universes
where static universes have distance scales that are constant in time and dynamic universes are either expanding or contracting over time. It may be shown that the cosmological principle (the assumption that there are no preferred positions or directions in the Universe) is inconsistent with a static universe. That is, the cosmological principle requires that the Universe be either expanding or contracting.

The Cosmological Constant

When Einstein first realized that the solution of his equations subject to the constraints of the cosmological principle led to universes that were not static, he was dismayed because at the time (the period between 1915 and 1920) the expansion of the Universe had not yet been discovered by Hubble. This led Einstein to make what he later characterized as the "greatest blunder of his life". To get a static Universe, he added an artificial term to his field equations that stabilized the Universe against expansion or contraction. This term has come to be known as the cosmological constant or the vacuum energy density.

With this new term, Einstein obtained a static solution. Later, when Hubble demonstrated that the Universe was actually not static but expanding, Einstein realized that he had missed a tremendous opportunity. If he had possessed sufficient confidence in his original equations, he would have predicted that either his theory was wrong or the Universe was expanding or contracting, well before there was experimental evidence of the expansion.

Is the Cosmological Constant Zero?

The discovery of the expansion removed the immediate need for the cosmological constant. It is still possible that there is a cosmological constant but it must be very small in the present Universe to be consistent with the data. An important unresolved issue in astronomy is whether the cosmological constant is identically zero, or just very small in the present Universe. Let us also note that an effective cosmological constant plays a role in the inflationary theories that we shall discuss later.

The adjacent image shows one of the most distant type 1a supernovae yet observed, SN1997cj, in the constellation Ursa Major at a distance of about 5 billion light years (Ref). It was discovered in 1997 by the Canada-France-Hawaii Telescope on Mauna Kea, and the Hubble Space Telescope was then used to resolve it from its host galaxy and study the decay of its light curve. By using the standard candle properties of the type 1a lightcurve, the distance to the supernova could then be determined. When this was compared with the redshift of the host galaxy (z = 0.5), it was then possible to estimate the rate of change in the expansion rate of the Universe (because Sn1997cj is 5 billion light years distant, it actually exploded when the Universe was only about 1/3 of its present age and the light is just now reaching us).

Analysis of this and related data have led to the claim (Ref) that the Universe is accelerating (rather than decelerating) because of a finite cosmological constant. If true, this finite cosmological constant would imply that most of the energy of the Universe is neither in matter nor radiation, but tied up in what physicists called the energy density of the vacuum. The acceleration is then a consequence of this energy density that permeates all of "empty" space. However, this result is controversial because a second group analyzing type 1a supernova light curves finds no evidence requiring a finite cosmological constant. The resolution of this controversy is a major issue of current research in astronomy.

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