Einstein and Gamow Take a Walk
Prediction of the cosmic microwave background was ignored and forgotten for nearly two decades. Robert Dicke and James Peebles of Princeton, unaware of Gamow's work, independently calculated CMB temperature at 10K. Nearby at Bell Labs, physicists Arthur Penzias and Joseph Wilson were tuning a large microwave antenna to communicatewith satellites. Finding a strange background signal, Penzias and Wilson accidentally discovered the CMB. It's temperature would eventually be measured as 2.7K, barely half Gamow's original prediction.
One day in the 1940's Einstein and Gamow were walking through Princeton. Gamow mentioned that one of his students had calculated that it was possible to make a star from nothing; its gravitational energy is equal and opposite to rest energy. Einstein, realising that this could apply to the whole Universe, stopped in the middle of the street. Unfotunately history has not recorded the math behind this calculation or who Gamow's student was.
The gravitational potential energy U of two particles is: U = -GMm/R. The total mechanical energy of a star works out to:
E ~ -(3/10)(GM^2)/R
Where M and R are the star's Mass and Radius. If our Sun's luminosity came from gravitational collapse, this energy would be used up in only 10 million years. Early in the 20th century, this led astronomers to conclude that something beyond gravity powered the stars.
What had Gamow's student found? If we equate an object's total potential energy with rest energy, we get:
GMm/R = mc^2
R = GM/c^2
Where M and R relate to a distant centre of Mass. If G, M and c are fixed this is the spherical Einstein Space. Such a Space would collapse unless it were expanding or supported by some repulsive force.
As nige and others have found, we can insert R = ct. Applied to the Universe, Scale R is distance to the Big Bang, age t multiplied by conversion factor c.
ct = GM/c^2
GM = tc^3
This can also be derived from Relativity, but doing so raises petty mathematical objections. There are multiple ways to derive GM = tc^3, but this is among the simplest and the most difficult to contest.
Thanks to George Gamow, we learned many things about physics and cosmology. His prediction of the CMB, though inaccurate in value, was eventually proved by Penzias and Wilson. We may never know exactly what Einstein and Gamow had hit upon while crossing the street. Perhaps they suspected that total energy of the Universe is zero. Our Universe may be the ultimate free lunch, which has allowed it to grow from a tiny point to the immensity we enjoy today.