John Wheeler famously declared that spacetime tells matter how to move; matter tells spacetime how to curve. The mass density operator quantifies the degree to which matter is told to move.
The six Planck units comprising the gravitational constant fall into two categories. The first two units, 𝑙𝑃 / 𝑚𝑃 create the mass density operator from inputs of mass and radius (or distance). This simple operator quantifies the gravitational field using only the density of matter in a region of space.
The remaining units in the gravitational constant, 𝑐2, describe the mechanical properties of a second body. The mass density operator transforms 𝑐2 into the correct proportions of inertial mass and velocity potentials. These units can also be arranged into the unit dimensions of energy, force, and acceleration.
The mass density operator is dimensionless like other operators, but it requires observables in two unit dimensions–length and mass. The operator combines the ratios of Planck length to the distance 𝑟, and the ratio of a system’s mass to the Planck mass. The mass density operator requires both ratios to produce the intensive quantity of mass density.
The New Foundation Model represents mass density in the length and mass unit dimensions. For a given quantity of mass 𝑀, there is a mass density equal to or less than the maximum mass density potential. A single vertical line across both unit dimensions signifies a black hole with mass 𝑀.
The mass-density operator in formulas
The signature characteristic of equations describing gravitational field potentials is a mass input in the numerator and a length input in the denominator. These two inputs create the mass density operator from Planck units embedded in the gravitational constant. The remaining inputs and Planck units are arranged to quantify different physical dynamics, as demonstrated by the gravitational constant and gravitational formulas.
The maximum mass density potential produces an important physical constant in the ratio of Planck length to Planck mass. This potential is equal to 1 in natural units, and 7.43 x 10-28 mkg-1 in SI units.
The significance of this ratio is evident in the properties of black holes. Half the ratio of radius to mass is equal to 7.43 x 10-28 mkg-1 for black holes according to the Schwarzschild radius formula. A limit to the density of mass in a black hole suggests that black holes are not singularities and that a better description of black hole geometry is needed.