The proportional connection between energy and the space-time continuum is the speed of light such that the space-time continuum (S) is the product of lightspeed (c) and the energy (E): S=cE. The vacuum as the totality of the space-time continuum is capable of infinitesimal matter and energy interactions. These are the quantum vacuum fluctuations. The probability of vacuum direct interaction with energy is v/c where (v) is the speed or rate of space-time displacement of the vacuum. The probability of the vacuum interaction with matter is v/c where the (v) is now the absolute value of the linear velocity of matter. Special relativity had already established the probability of energy interaction with matter as v/c where the (v) is the same absolute value of the linear velocity of matter. There seem to be three distinct types of powers of speed: v, v, and v. However, their distinctions never violate the following three simple inequalities: v<c, v<c, and v<c everywhere and everywhen in the entire universe.
The reciprocals of these probabilities would then define the rate of change of state transformations between vacuum and energy, between vacuum and matter, and between energy and matter. The rate of energy-matter transformation is derived from Einstein’s mass-energy equivalence: E=mc such the rate is given as c=E/m. Given a tiny mass, its energy requirement is also tiny such that the ratio of energy to mass remains a constant c. As the speed of the mass increases, the rate of change decreases, and approaches unity. If the rate of change is unity then E=m and mass and energy exchange has equal probability. The rate of change between vacuum and energy normalized to v is c/v given by S/vE while the rate of change between vacuum and matter normalized to v is c/v given by S/vm. As v approaches c, both c/v and c/v approach unity. But as v approaches zero both c/v and c/v approach infinity. However, c/v indicates a linear variation while c/v indicates a cubic variation. The first is a path integral. The second is a volume integral. The variation between energy and matter c/v is a surface integral. These integrals imply Abelian additive, commutative, and associative properties as sums of infinite series expansions.


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