Einstein's Cosmological Constant

Introduction

Einstein called the cosmological constant "the worst mistake of my career". He originally developed the constant to balance his field equation of General Relativity. If the universe were static and not inflationary or deflationary (as believed in his time) then gravity would need to be balanced out to negate the attractive forces of matter and energy. Otherwise, the universe would have no choice but to collapse in on itself.

There are reasons to believe that the cosmological constant is necessary in the study of Relativity. Today's current belief system put forth by Edwin Hubble states the universe is inflationary or expanding, and the overall shape of the universe is flat. However, the total matter and energy of the universe only adds up to 30% of the matter and energy needed to make the universe flat. The cosmological constant adds the extra 30% needed.

In addition, the constant accounts for the fact that if we had an open universe containing the amount of matter we have measured, its age would be younger than that of the oldest stars; not possible.

The constant and General Relativity

With the introduction of General Relativity gravity was now treated as a "dent or "warp" in the fabric of space-time.

This fabric is measured by three-dimensional world lines that bend and stretch with warps formed by gravity. The mathematics used to measure world lines is Riemannian geometry, which traces lines along a three-dimensional surface. The diagram below shows an example of space-time warping.

The world lines of the diagram trace the path of the arrows following the folds. This section of the continuum is referred to as a manifold, or reference frame, as used in Special Relativity. Each reference frame is independent, and can have forces acting on it that do not act on adjacent frames.

Einstein needed to define position in space-time due to the failure of the Michelson-Morley experiment to detect any medium or "ether" that gravity exits in. Time is treated as a fourth coordinate. The field equation that defines General Relativity and the curvature due to gravity is:

R- gR/2+ gG/c^4T

Where R= the Ricci curvature tensor, R= the scalar curvature, g= the metric tensor, G/3c^2(Einstein's cosmological constant),T= the stress-energy tensor non-gravitational matter, energy, and forces at any given point in space-time, c= the speed of light in a vacuum, G= the gravitational constant from Newton's Law of Gravitation, and 3.1415

The basic tenant of the Einstein field equation is that the curvature of space-time is directly affected by the force of gravity applied by the mass of the object. It also takes into account the measurement of the curvature (g), the non-gravitational forces (T), and the interrelations of fields. In an over-simplified way it states, "curvature tells matter how to move, and matter tells space how to curve" (2). 2.

The constant defined

AS stated previously, the cosmological constant is G/3c^2where c= the speed of light in a vacuum, G= the gravitational constant from Newton's Law of Gravitation, 3.1415 and pressure associated with a vacuum. The pressure is negative or repulsive to the positive force of gravity, which it acts to expand the universe. As the universe expands, matter and energy are spread out and gravity diminishes. For vacuum energy, the work done during the expansion provides the amount of energy needed to fill the new volume with the same density. Therefore the constant remains constant, and the universes gravitational repulsion never changes (1).

Observational Data

Age of the universe is one of the main reasons to believe the constant is necessary. If we date the oldest stars mainly in globular clusters they are 15-18 billion years old. If we calculate the universe using detectable matter (x~.3) the age is 10-13 billion years, and is 8-11 billion years using the flat (expanding) model. With the constant in the equation the flat universe is now 12-16 billion years, much realistic.

Red shift, or the bending of light from distant sources, is also affected by the constant. Numerous studies using statistical analysis of red shifted objects show that a flat universe with a cosmological constant is excluded at the 97% confidence level, and is much more likely than a open, matter dominated universe.

The light is bent due to gravitational lensing, an affect that massive objects such as supernovae or quasars have on space-time around them in that they cause light to behave like a river of mud around a person's foot. The mud (light) is forced around the foot (object) and its wavelengths are shortened (moved toward red or red shifted) if the object is moving away from the observer and are lengthed (moved toward blue or blue shifted) if the object is moving toward the observer.

References:

1. http://super.colorado.edu/~michaele/Lambda/lambda.html

2. http://en.wikipedia.org/wiki/General_relativity

3.