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The Physics of Dark Energy
An exploration of a recent discovery in cosmology.

— Copyright © 2007, P. Lutus  Message Page

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Relativistic Cosmology

By publishing two key papers, "On the Electrodynamics of Moving Bodies" (1905)1, introducing what we now call Special Relativity, and "The Foundation of the General Theory of Relativity" (1916)2, Albert Einstein almost singlehandedly rebuilt the foundation of physics. Einstein's system did away with a requirement for an ether, explained gravitation as a geometric consequence of spacetime's interplay with matter, and abandoned the notion of absolute space and time.

With respect to the present subject, the new relativity theory introduced some problems as it did away with the ether. While the ether theory prevailed, it was not difficult to explain the apparent stability of the universe — after all, it might be infinite and unchanging in size, and on a large scale the ether might have some frictional properties to keep things in place. But Einstein's mathematical description of the universe didn't possess this static quality. Even though the universe appeared to be steady-state according to the observations of Einstein's time, the equations argued against this — the matter of the universe should either fall inward or outward, depending on the overall distribution and density of matter.

Einstein addressed this apparent conflict between theory and observation by adding a term to his equations, a "cosmological constant" that served to keep widely separated masses apart, while keeping all observed orbital and gravitational behaviors unchanged.

The First Cosmological Constant

Let's look once again at the classical Newtonian gravitational equation, and see how it might be changed to allow a static universe to exist. Here is the original form, introduced on the prior page:

(2)    f = G m1 m2

Remember about this equation that it applies to all masses. In a star cluster that might have 100,000 members, the above equation describes the gravitational acceleration between each possible pairing of stars (for n stars, that's n(n-1)/2 pairings), each with their respective masses and separations.

To produce a mathematical result that agrees with the universe of Einstein's day, we have the following requirements:

  • The universe (of Einstein's time) appears to be static and unchanging, but the new relativity theory argues for instability between masses, possibly a wholesale inward collapse of all the mass in the universe.
  • Any modification of the gravitational relationship between masses at great distances can't measurably change the observed behavior of masses at short ranges.
  • For a reason explained below, a practical solution cannot create a force that declines proportional to the square of distance as gravitation does, instead it must show a different relationship with distance.

Considering these requirements, Einstein added a term to his equations, which had the effect of changing the classical gravitation equation thus (this is greatly simplified):

(3)    f = Λ - G  m1 m2

Where Λ (Greek letter Lambda) represents a small repulsive force that operates over great distances and is unrelated to either mass or distance. It can be seen that, for particular values of Λ, masses would behave as one would expect at relatively short ranges, but at great distances the acceleration produced by Λ would balance the gravitational acceleration of the right-hand terms.

After this modification of Einstein's equations, several things took place to cause the unraveling of this application for a cosmological constant. First, better observation revealed that the universe is not static, it is expanding (ultimately leading to the Big Bang theory). Second, even if the universe had turned out to be static, the application of a single constant term cannot produce the required stability.

Later in this article set, I provide an interactive Java applet that the reader can use to experiment with different values for Λ and see the results on a sample of gravitationally interacting bodies, but as it turns out, in a universe filled with interacting masses, virtually none of them can be balanced by a single cosmological constant — they will either drift toward one another or apart. The only way to assure that the constant has the desired effect is to group it with G, like this:

(4)    f = (Λ - G)  m1 m2

This change assures that a particular value of Λ always cancels G, which would have made the equation agree with the appearance of the large-scale observed universe of Einstein's day, but in order to work, it has the unfortunate side effect of eliminating gravitation altogether.

All these problems eventually occurred to Einstein, who then described the cosmological constant as the biggest blunder of his life.

  1. On the Electrodynamics of Moving Bodies, A. Einstein, 1905
  2. The Foundation of the General Theory of Relativity, A. Einstein, 1916

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