A transparent derivation of Painlevé-Gullstrand’s coordinates (used in this paper) is the following: Consider a radially, free-falling observer that starts at rest from infinity. For this observer, the equations for 
as a function of the proper time
, and the equation for the Schwarzschild time, 
, as a function of
, are respectively (cf. [
where 
and 
are two constants. The equations are written in geometrized units. It follows from Equation (1a) that the second term in Equation (1b) is equal to
. This relation suggests the following transformation of coordinates,
the constants in Equations (1a) and (1b) have been ignored because only 
is important. From Equation (2) we obtain,
which can also be derived directly from the equations,
valid for a radial plunge with no kinetic energy at infinity. The new metric can be written,
The radially ingoing and outgoing light rays are found to be (
it follows from the geodesic equations that 
is an affine parameter),
The condition 
requires that,
the value of 
obtained from Equation (6) agrees with Equation (1a).
, in Equation (4) for the metric will be designated hereafter by t, whereas 
will be a particle’s proper time. For radial motions the equations for 
and 
are,
, kv is the Killing vector, namely (1, 0), and k is a constant (the Killing constant). For metrics that are independent of t, an integral of the geodesic equation exists, that is given by the scalar product of (1, 0) with u, which is Equation (7.1), namely the first equation in (7). Equation (7.2) follows then from the normalization condition for u, i.e.,
. In the derivation of this equation, terms in the product 
appear, but they cancel. In Equation (7.1), 
is dimensionless and it is clear that one can assign to 
the dimension of a velocity (if dimensions are introduced the first term would be multiplied by
.) We assume that 
and
, the energy per unit rest mass, are related by, 
, and find that Eq uation (7.2), i.e., the second equation in (7), can be written,
is constant it follows from
, that 
is also a constant, and if
, also
.
, because then (unlike other cases as, e.g., for
, in particular) solutions can easily be found. If
, and for radial motions, the equation for u is found to be,
Equation (9) is valid only below the event horizon and it follows from Equations (7.2) and (8) that the value of 
given by Equation (9) is the minimum fall velocity, and furthermore that
. Therefore inside the event horizon, 
must be larger than
. Below the event horizon, the potential energy decreases, but this decrease is compensated by the increase in the minimum allowed value for
.
. The value
, corresponds to the orbit of a free-falling observer that starts at rest from infinity. The value of 
for all the solutions with 
is unity. It will be shown that
, defines the orbit of a particle, with the same energy per unit mass as the free falling observer, but confined below the event horizon. It is convenient to use an orthogonal system of coordinates associated with the falling observer. It can be readily verified that the vectors,
and the Killing vector, kv, can be written,
, and
, denote vectors in the coordinate and orthonormal basis respectively. It is clear that in Equation (11), 
, and that
.
the equations that need to be satisfied are,
, in Equation (13b), shows that u is the velocity of a particle that is stationary with respect to the falling observer, as expected from our choice of basic vectors in Equation (10). The same conclusion can be reached in the coordinate basis (cf. the first equation in (14a)), because it follows from Equation (1a), that
.
and has no physical meaning above the event horizon because there,
. Conversely, the second equation in (14a) represents an outgoing particle with no physical meaning below the event horizon. Here, the minimum descent velocity is equal to 
from Equation (9), and 
in Equation (14b) must satisfy the inequality
, which is indeed the case.
, is timelike above the event horizon (out), and spacelike below (in). In the quantum creation of a pair of particles, energy conservation requires that [
, 
are the four-velocities of the created pair. Above the event horizon, 
, must be positive because it is proportional to the particle’s energy measured by an observer with velocity kv. Therefore 
must be negative and equal to
. For values of
, it follows from Equation (14b), that the velocity of the created particle below the event horizon must be, 
From the time dependence of the solutions in Equation (14a), it is apparent that 
must then be taken equal to
which is the outgoing solution for
. The particle above the event horizon cannot fall back into the black hole. Because win and wout are both very approximately equal to one, it is straightforward to show that 
requires that Equation (16.1) below, be satisfied. Equation (16.2) follows from the relation, 
,
and 
are the radial coordinates of the particles forming the pair, an intuitively attractive result.
and 
are the energies of the respective particles divided by their mass. Because
, global energy conservation requires that the mass of the particle below the EH be equal to minus the mass of the particle above the EH, i.e., the energy of the particle below the event horizon must be negative in agreement with Schutz’s [
such that 
because then the outside particle cannot escape to infinity.
, where 
is the Schwarzschild time, and 
is the proper time of the particles at the event horizon. From Equation (3b) it follows that 
and then from Equation (14a) we obtain for an outside particle,
. Equation (17) shows that for the outside observer, N particles will have been produced in the incomparably larger time, 
which suggests a weak observed productions of particles. However, the Hawking radiation from a black hole is also very weak. Field theory calculations show that black holes emits as though it were a black body with temperature,
, is truly the physical temperature of the black hole, not merely a quantity paying a role mathematically analogous to temperature in the laws of black hole mechanics (cf. [