Planck mass, hereinafter m_p, is the mass value obtained by appropriately combining constant G, c and hbar as (hbar c / G)^1/2. Used value 2,178 10^-5 g.

Planck length, hereinafter l_p = (hbar G / c³)^1/2, , is the Compton wavelength of the Planck mass particle. The value used in this web is 1,616 10^-33 cm.

Planck time, hereinafter t_p = (hbar G / c⁵)^1/2, is the time that it takes light to travel Planck length. Value used 5,389 10^-44 sg.

An observer's events horizon with rest mass m₀: H_s=2G m₀ / c² o H_s= 2G h / c³ l_c where l_c is the observer Compton wavelength.

A black hole is, by definition, the geometric space in which the escape speed of a gravitational field becomes the same or bigger than c (speed of light), to be exact, if a spherical and homogeneous distribution of mass M is considered, it would be a sphere having a radius r= 2GM/c². Black holes considered here never have any charge or angular momentum.

Applying the Heisenberg uncertainty principle, a particle of mass m should have a minimum angular momentum of 1/2 hbar (mvr =1/2 n h/2p, n>0) with regard to a black hole of mass M. Consequently, it is possible for there to be a minimum speed, v, intrinsic and normal to the axis that joins them. The resulting gravitational force is F= - GMm/r² + mv²/r. Furthermore, the resulting gravitational field may be considered as:

V = -GMm/r + n² hbar²/(8 m r²)    (1.1)
or
V = -GMm/r + m c² n² l²/(8 r²)  where l= hbar / m c    (1.1)

In the paragraph above, it was postulated that the particle trajectories in the gravitational field of a black hole are orbits around the center of the gravitational field at each point with a quantized angular momentum of L = 1/2 n hbar. This postulate is analogous to the one for the atom as given by Bohr.

We can now study how a black hole characterized by radius r= 2GM/c² "grows". By definition, a black hole cannot lose its identity as a black hole while it is growing. What is the minimum amount of energy that can be absorbed without losing that characteristic and what is the minimum time in which this can take place? Since this energy is a characteristic of black holes, it will always be the same.

Let us consider the new radius r₁ = 2G (M+m)/c² consequence of the absorption of m and calculate the difference from the previous radius, r₀:

r₁ - r₀ = 2G (M+m)/c² - 2GM/ c²

r₁ - r₀ = 2Gm/ c²

On the other hand, this increment in energy must comply with the uncertainty principle:

mc² Dt >= hbar/2 and Dt (minimum) = hbar/ 2 mc²

If c is the limit of the expansion speed of the black hole, then we have c = r₁ - r₀ / Dt (for the minimum time) and consequently, m² = c hbar / 4 G. From this equation the value of m that coincides with half the value of the Planck mass can be calculated as m = (c hbar/ G)^1/2 / 2, r₁ - r₀ = l_p and Dt= t_p. Therefore, a black hole may be considered to grow by absorbing quantum energy to half the value of the Planck mass. For any length of time, the black hole accumulates energy on the exterior of its event horizon up to the amount of half the Planck mass, growing a Planck length at a Planck time. The mass value of the black hole would be N m_p/2; N is a whole number that indicates the order of the last absorption process and the radius of the black hole is therefore N l_p.

In the previous paragraph, c is the maximum limit of the expansion speed of a black hole, and, implicitly, it is also the minimum limit of the expansion speed in every process of absorption. This limit is justified when the event horizon is also subject to the condition of the escape speed of a black hole equal to c.

It may be observed that a black hole grows by energy quanta of 1/2 m_p c² each Planck time. However, this event or interaction between the black hole and the rest of the universe don't have reason to be continuous, since the matter contribution around the hole is not continuous (you can have isolated black holes). If the matter contribution is abundant, the growth will go becoming from growth pulsatile to a continuous growth to the light speed like maximum (c = l_p/ t_p). Black hole density may easily be calculated as:

Density = N m_p / (8/3) p N³ l_p³  or  3 c⁵/ 8 p N² h G² g/cc    (1.2)

Keeping in mind that N l_p is the black hole radius, R, we find that the black hole density is given by the equation r = 3 c² / 8 p G R², i.e., a black hole always has critical energy density.

We can also calculate the following spherical cap density that will be absorbed by the black hole:

Superficial density = m_p / (8 p N² l_p ²) l_p g/cc   or   m_p / (8 p N² l_p ³) g /cc    (1.3)

And the potential energy of the particles that enter the black hole. If, in the equation that appears in (1.1), m c² n²l²/(8 r²), we substitute m for 1/2 m_p, we have:

V = - G m_p m_p / 4l_p + 1/4 m_p c² = 0     since G m_p / l_p = c²     (1.4)

All particles enter with the same zero potential energy. If we introduce an integration constant of -1/2 m_p c² (choosing the appropriate zero level gravitational potential) and since all the particles enter with relativistic energy of 1/2 m_p c², we may affirm that all of them enter with zero mechanical energy.

A black hole, therefore, is fully defined by a whole number N.

_{© Jorge Ales, 2002. http://www.livinguniverseweb.com}