Here is a clue to a solution of the Christmas and New Year puzzle in the previous post in the form of the Euler equations for a self-gravitating gas:
God’s Equation: Matter and Antimatter
According to the mathematician Pierre-Simon Laplace, the World is governed by the differential equation:
where is the gravitational potential depending on space-time coordinates , is the 2nd order Laplace differential operator. If you are a fan of relativity theory, you can think if this equation as Einstein’s equation and if you have a religious inclination, as God’s equation. Recall that Laplace upon the question by Napoleon about the absence of God in his mathematics, responded: Sire, I have no need of that hypothesis.
The right hand side of God’s equation expresses the presence of matter/antimatter given as the Laplacian of the gravitational potential :
- where ( convex), there is matter with density
- where ( concave), there is antimatter with density .
- matter around points of local minimum of
- antimatter around points of local maximum of .
Newton’s 2nd Law
Conservation of Matter/Antimatter
Starting from a smooth gravitational potential with a smooth distribution of matter where and antimatter where , we have
- where and
- where and
which combined with the direction of the force shows
- concentration of matter around points of minimum of
- thinning of antimatter around points of maximum of .
We thus identify a tendency of concentration of matter and thinning of antimatter, which we can associate with the formation of galaxies separated by large almost voids.
Dark Energy and Matter
The gravitational potential generates a repulsive force between regions of matter separated by almost voids of antimatter, which can be interpreted as an effect of dark energy. We conclude that God’s equation appears to come out of a truly heavenly inspiration. It is natural to identify real visible matter as delta-functions of connected with singularities of , dark matter with smooth , and antimatter with smooth , see the above figure for a schematic illustration.
Visible matter may develop from dark matter by concentration into delta-functions, while antimatter is subject to thinning rather than concentration.
Sire, I Have No Need of Big Bang
The initial distribution of matter and antimatter can be viewed as resulting from a perturbation of a zero potential, where the Laplacian acts as an amplifier of the perturbation. The Universe with its distribution of matter and antimatter would thus be created by the Laplacian operator acting on a little innocent perturbation of the gravitational potential as expressed by God’s equation . Laplace could thus say about Big Bang: Sire. I have no need of that hypothesis.
Comnpare with illuminating illustration by Joel Primack: