In this article we will use the formulas with the velocity of light $c\equiv 1$.

We have shown in [1] that the correct Schwarzschild solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\text{d}{s}^2=\left(1+\frac{2GM}{r_g}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM}{r_g}}\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$ (1)

where M is (a constant, which is equal to) the total mass of the system, and $t_g$, $r_g$, ${\theta }_g$, ${\phi }_g$ are the space-time coordinates measured relatively to a reference frame that is integral with a space-time curved for expressing the gravitational field as the curvature of the space-time. In other words, this equation (1) is expressed in curved coordinates.

We can note that in the (1) the coefficient of $\text{d}{t}_g^2$ is always greater than one (except in the case where there is no gravitational field, in which case this coefficient is equal to one), and the more intense the gravitational field is, the greater this coefficient is. We have already shown in [1] that this is a sign that the coordinates are curved and is closely related to the fact that the clocks in a gravitational field run more slowly.

On the other hand, we have also shown in [1] that the correct Schwarzschild solution expressed as the formally flat metric in the curved coordinates, which metric is expressed as a function of the commonly used coordinates, is:

$\text{d}{s}_g^2=\frac{1}{1+\frac{2GM}{r_g}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}\right)\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$ (2)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates.

As for the relation between $r_g$ and r, we have:

$\text{d}{r}^2=\frac{1}{1+\frac{2GM}{r_g}}\text{d}{r}_g^2$ (3)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM}{r_g}}\text{d}{r}_g^2}$ (4)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g^2\text{d}{r}_g}{r_g+2GM}}$ (5)

From which, we have:

$r^2={\left[r_g^2-4GM{r}_g+8G^2{M}^2\ln \left(2GM+r_g\right)\right]}_0^{r_g}$ (6)

From which, we have:

$r^2=r_g^2-4GM{r}_g+8G^2{M}^2\ln \left(1+\frac{r_g}{2GM}\right)$ (7)

According to the (7) for $r_g\to 0$ also $r\to 0$ and for $r_g\to +\infty$ also $r\to +\infty$. Moreover, as $r_g$ increases, r also increases as can be seen by differentiating the right side of the (7) with respect to $r_g$ or directly from the (5). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as can be seen directly from the (4).

Furthermore, for $r_g\gg 2GM$ we have that $r\cong r_g\left(1-\frac{2GM}{r_g}\right)$ and also $r_g\cong r\left(1+\frac{2GM}{r}\right)$. This implies that for $\frac{2GM}{r}\ll 1$ the presence of $r_g$ instead of r in the (2) implies only corrections to the second order in $\frac{2GM}{r}$.

On the other hand, for $r_g\ll 2GM$ we have $r^2\cong \frac{r_g^3}{3GM}$.

In particular for $r_g=2GM$, $r^2=4G^2{M}^2\left(2\ln 2-1\right)$, and so in this case we have $r=2GM\sqrt{2\ln 2-1}\cong 1.24GM$.

Moreover, for $r_g=4GM$, $r^2=8G^2{M}^2\ln 3$, and so in this case we have $r=2GM\sqrt{2\ln 3}\cong 2.96GM$.

Instead for $r_g=8GM$, $r^2=8G^2{M}^2\left(4+\ln 5\right)$, and so in this case we have $r=2GM\sqrt{2\left(4+\ln 5\right)}\cong 6.70GM$.

Instead, the common erroneous expression for the Schwarzschild solution is [2] :

$\text{d}{s}^2=\left(1-\frac{2GM}{r}\right)\text{d}{t}^2-\frac{1}{1-\frac{2GM}{r}}\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$ (8)

We can note that we have not any singularity in the correct expressions (1) and (2) for any value of $r>0$ and of $r_g>0$. Therefore we can say that there is not any event horizon and therefore that there is not any black hole [2] [3] .

In particular, the coefficients of $\text{d}{t}^2$ in (2) and of $\text{d}{t}_g^2$ in (1) are always >0, while the coefficients of $\text{d}{r}^2$ in (2) and of $\text{d}{r}_g^2$ in (1) are always <0. Therefore since the coefficients of $\text{d}{t}^2$ in (2) and of $\text{d}{t}_g^2$ in (1) are always not negative the time (that is, the temporal coordinate) does not become in any case as a spatial coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] . Moreover, since the coefficients of $\text{d}{r}^2$ in (2) and of $\text{d}{r}_g^2$ in (1) are always not positive, also here the spatial coordinate r (or $r_g$) does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Therefore, the light cones are always orientated in the usual way, in particular there is not any horizontal inclination of the light cones, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

As we have already shown in [1] , the correct formula of the ratio between the times when they are measured in two different positions in this case is:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}}{1+\frac{2GM}{r_g{}_2}}}$ (9)

Therefore in this case we have that in the presence of a gravitational field the clocks go more slowly (that is, that the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a gravitational field flow more slowly). And the more intense is the gravitational field the more slowly the clocks go (that is, the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a more intense gravitational field flow more slowly than the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a less intense gravitational field).

Analogously, as we have already shown in [1] , the correct ratio of the relative frequencies ${\nu }_1=1/\text{d}{t}_g{}_1$ and ${\nu }_2=1/\text{d}{t}_g{}_2$ is:

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}}{1+\frac{2GM}{r_g{}_2}}}$ (10)

This is the correct formula for the gravitational redshift of light in the correct Schwarzschild metric. Therefore, the gravitational redshift is always not infinite for any value of $r>0$ and of $r_g>0$.

We can note that the two formulas (9) and (10) are different from those obtained usually from the incorrect expression of the Schwarzschild solution by

means of an incorrect procedure [1] [2] , but are equal at the first order in $\frac{2GM}{r}$

(to the incorrect formulas). In fact, the formulas commonly used, instead of the two correct formulas (9) and (10), are respectively [1] [2] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM}{r_2}}{1-\frac{2GM}{r_1}}}$ (11)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM}{r_2}}{1-\frac{2GM}{r_1}}}$ (12)

As we have already noted in [1] , the light formally moves in a straight line relative to the reference system curved for expressing the presence of the gravitational field. Therefore we can impose the condition $\text{d}{s}_g^2=0$ for the propagation of light. Therefore from the (2) if we take a radial motion of the light in the commonly used coordinates ( $\text{d}\theta =0$ and $\text{d}\phi =0$) we have:

$\frac{1}{1+\frac{2GM}{r_g}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}\right)\text{d}{r}^2=0$ (13)

From which we have that the radial velocity of light in the commonly used coordinates is equal to [1] [2] :

$v_l=\frac{\text{d}r}{\text{d}t}=\frac{1}{1+\frac{2GM}{r_g}}$ (14)

We can note that the radial velocity of light in the commonly used coordinates is always ≤1, and is equal to 1 only when there is not a gravitational field. Moreover this formula is in agreement with the available experimental data [2] .

For further information on the correct Schwarzschild solution, in particular how it is obtained, please refer to [1] .

As for the Reissner-Nordstrøm solution [2] [3] , that is the solution that represents the curved space-time geometry surrounding an electrically charged mass, we have that the commonly used expressions for the space-time interval and the electric field are respectively [2] :

$\text{d}{s}^2=\left(1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}\right)\text{d}{t}^2-\frac{1}{1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}}\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$ (15)

and

$E\left(r\right)=\frac{Q}{r^2}$ (16)

where M and $Q$ are two constants: M is the total mass of the system and $Q$ is the total electric charge of the system.

In the (15) there are singularities when

$1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}=0$ (17)

that is, when

$r^2-2GMr+G{Q}^2=0$ (18)

or, better yet, when

$r=GM\pm \sqrt{G^2{M}^2-G{Q}^2}$ (19)

Obviously there are real singularities only when $G^2{M}^2-G{Q}^2\ge 0$.

But the expression (15) is erroneous: in fact, by means of the fact that the correct Reissner-Nordstrøm solution must be equal to the correct Schwarzschild solution for $Q=0$, we can say that the correct Reissner-Nordstrøm solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\text{d}{s}^2=\left(1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$ (20)

where M and $Q$ are two constants: M is the total mass of the system and $Q$ is the total electric charge of the system. Moreover $t_g$, $r_g$, ${\theta }_g$, ${\phi }_g$ are the space-time coordinates measured relatively to a reference frame that is integral with a space-time curved for expressing the gravitational field as the curvature of the space-time. In other words, this equation (20) is expressed in curved coordinates.

We can note that in the (20) the coefficient of $\text{d}{t}_g^2$ is always greater than one (except in the case where there is no gravitational field, in which case this coefficient is equal to one), and the more intense the gravitational field is, the greater this coefficient is. We have already shown in [1] that this is a sign that the coordinates are curved and is closely related to the fact that the clocks in a gravitational field run more slowly.

Analogously to the case of the Schwarzschild metric, we can write also in this case the formally flat metric in the curved coordinates, which metric is expressed as a function of the commonly used coordinates. In this case we have:

$\text{d}{s}_g^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}\right)\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$ (21)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates.

As for the relation between $r_g$ and r, we have:

$\text{d}{r}^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2$ (22)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2}$ (23)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g^3\text{d}{r}_g}{r_g^2+2GM{r}_g+G{Q}^2}}$ (24)

From which, we have:

$\begin{array}{l}{r}^2=[r_g^2-4GM{r}_g+G\left(4G{M}^2-Q^2\right)\ln \left(r_g^2+2GM{r}_g+G{Q}^2\right)\begin{array}{c}\stackrel{}{}\\ \end{array}\\ +{\frac{G^{2}M\left(3Q^2-4G{M}^2\right)}{\sqrt{G^2{M}^2-G{Q}^2}}\ln \left(\frac{GM+r_g-\sqrt{G^2{M}^2-G{Q}^2}}{GM+r_g+\sqrt{G^2{M}^2-G{Q}^2}}\right)]}_0^{r_g}\end{array}$ (25)

From which, we have:

$\begin{array}{c}{r}^2=r_g^2-4GM{r}_g+G\left(4G{M}^2-Q^2\right)\ln \left(1+\frac{r_g^2+2GM{r}_g}{G{Q}^2}\right)\\ +\frac{G^{2}M\left(3Q^2-4G{M}^2\right)}{\sqrt{G^2{M}^2-G{Q}^2}}\cdot \ln (\frac{GM+r_g-\sqrt{G^2{M}^2-G{Q}^2}}{GM+r_g+\sqrt{G^2{M}^2-G{Q}^2}}\\ \cdot \frac{GM+\sqrt{G^2{M}^2-G{Q}^2}}{GM-\sqrt{G^2{M}^2-G{Q}^2}})\end{array}$ (26)

According to the (26) for $r_g\to 0$ also $r\to 0$ and for $r_g\to +\infty$ also $r\to +\infty$. Moreover, as $r_g$ increases, r also increases, as we can see directly from the (24). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as we can see directly from the (23).

Furthermore, for $r_g\to +\infty$ we have that $r\cong r_g\left(1-\frac{2GM}{r_g}\right)$ and also $r_g\cong r\left(1+\frac{2GM}{r}\right)$. This implies that, in this case, the presence of $r_g$ instead of r in the (21) implies only corrections to the second order in $\frac{2GM}{r}$.

On the other hand, for $r_g\to 0$ we have $r\cong \frac{r_g^2}{\sqrt{2G{Q}^2}}$, as we can see easily from the (24).

Also in this case as in that of Schwarzschild, we have not any singularity in the correct expressions (20) and (21) for any value of $r>0$ and of $r_g>0$. Therefore we can say that there is not any event horizon and therefore that there is not any black hole [2] [3] .

In particular, also here the coefficients of $\text{d}{t}^2$ in (21) and of $\text{d}{t}_g^2$ in (20) are always >0, while the coefficients of $\text{d}{r}^2$ in (21) and of $\text{d}{r}_g^2$ in (20) are always <0. Therefore, since the coefficients of $\text{d}{t}^2$ in (21) and of $\text{d}{t}_g^2$ in (20) are always not negative, the time (that is, the temporal coordinate) does not become in any case as a spatial coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] . Moreover, since the coefficients of $\text{d}{r}^2$ in (21) and of $\text{d}{r}_g^2$ in (20) are always not positive, also here the spatial coordinate r (or $r_g$) does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Therefore, also here the light cones are always orientated in the usual way, in particular there is not any horizontal inclination of the light cones, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

In this case the correct formulas (9) and (10) become respectively:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{G{Q}^2}{r_g{}_1^2}}{1+\frac{2GM}{r_g{}_2}+\frac{G{Q}^2}{r_g{}_2^2}}}$ (27)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{G{Q}^2}{r_g{}_1^2}}{1+\frac{2GM}{r_g{}_2}+\frac{G{Q}^2}{r_g{}_2^2}}}$ (28)

Therefore also in this case we have that in the presence of a gravitational field the clocks go more slowly (that is, that the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a gravitational field flow more slowly). And the more intense is the gravitational field the more slowly the clocks go (that is, the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a more intense gravitational field flow more slowly than the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a less intense gravitational field).

Furthermore, the gravitational redshift, that is expressed by the formula (28), is always not infinite for any value of $r>0$ and of $r_g>0$, as in the correct Schwarzschild solution [1] .

We can note that the two correct formulas (27) and (28) are different from those obtained usually from the incorrect expression of the Reissner-Nordstrøm solution by means of an incorrect procedure [1] [2] . In fact, the formulas commonly used, instead of the two correct formulas (27) and (28), are respectively [1] [2] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM}{r_2}+\frac{G{Q}^2}{r_2^2}}{1-\frac{2GM}{r_1}+\frac{G{Q}^2}{r_1^2}}}$ (29)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM}{r_2}+\frac{G{Q}^2}{r_2^2}}{1-\frac{2GM}{r_1}+\frac{G{Q}^2}{r_1^2}}}$ (30)

In this case the formula (14) of the radial velocity of light in the commonly used coordinates [1] [2] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}$ (31)

We can note that, also here, the radial velocity of light in the commonly used coordinates is always ≤1, and is equal to 1 only when there is not a gravitational field.

As for the Kerr solution, that is the solution that represents the curved space-time geometry surrounding a rotating mass, we have that the commonly used expression is [2] :

$\text{d}{s}^2=\text{d}{t}^2-\frac{{\rho }^2}{\Delta }\text{d}{r}^2-{\rho }^2\text{d}{\theta }^2-\left(r^2+a^2\right){\sin }^2\theta \text{d}{\phi }^2-\frac{2GMr}{{\rho }^2}{\left(\text{d}t-a{\sin }^2\theta \text{d}\phi \right)}^2$ (32)

where ${\rho }^2$ and Δ are functions of r and $\theta$:

${\rho }^2\equiv r^2+a^2{\cos }^2\theta$ (33)

$\Delta \equiv r^2-2GMr+a^2$ (34)

and M and $a$ are constants: M is the total mass of the system and $a$ is the spin angular momentum of the system per unit mass.

In the (32) there are singularities (event horizons) when

$r^2-2GMr+a^2=0$ (35)

that is, when

$r=GM\pm \sqrt{G^2{M}^2-a^2}$ (36)

Obviously there are real singularities (event horizons) only when $G^2{M}^2-a^2\ge 0$.

Moreover we have that the coefficient $g_{00}$ is equal to zero when

$r^2-2GMr+a^2{\cos }^2\theta =0$ (37)

that is, when

$r=GM\pm \sqrt{G^2{M}^2-a^2{\cos }^2\theta }$ (38)

Therefore there are two distinct surfaces where $g_{00}$ is equal to zero. These surfaces are infinite-redshift surfaces. Obviously such surfaces really exist only where $G^2{M}^2-a^2{\cos }^2\theta \ge 0$.

But the expression (32) is erroneous: in fact, by means of the fact that the correct Kerr solution must be equal to the correct Schwarzschild solution for $a=0$, we can say that the correct Kerr solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\begin{array}{c}\text{d}{s}^2=\text{d}{t}_g^2-\frac{{\rho }^2}{\Delta }\text{d}{r}_g^2-{\rho }^2\text{d}{\theta }_g^2-\left(r_g{}^2+a^2\right){\sin }^2{\theta }_g\text{d}{\phi }_g^2\\ +\frac{2GM{r}_g}{{\rho }^2}{\left(\text{d}{t}_g-a{\sin }^2{\theta }_g\text{d}{\phi }_g\right)}^2\end{array}$ (39)

where ${\rho }^2$ and Δ are functions of $r_g$ and ${\theta }_g$:

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (40)

$\Delta \equiv r_g^2+2GM{r}_g+a^2$ (41)

and M and $a$ are constants: M is the total mass of the system and $a$ is the spin angular momentum of the system per unit mass. Moreover $t_g$, $r_g$, ${\theta }_g$, ${\phi }_g$ are the space-time coordinates measured relatively to a reference frame that is integral with a space-time curved for expressing the gravitational field as the curvature of the space-time. In other words, this equation (39) is expressed in curved coordinates.

We can note that in the (39) the coefficient of $\text{d}{t}_g^2$ is always greater than one (except in the case where there is no gravitational field, in which case this coefficient is equal to one), and the more intense the gravitational field is, the greater this coefficient is. As mentioned before, we have already shown in [1] that this is a sign that the coordinates are curved and is closely related to the fact that the clocks in a gravitational field run more slowly.

Also in this case as in that of Schwarzschild, we have not any singularity in the correct expression (39) for any value of $r_g>0$. Therefore we can say that there is not any event horizon and therefore that there is not any black hole [2] [3] .

Furthermore, since the coefficient of $\text{d}{t}_g^2$ in the (39) is always ≥1, there is not any infinite-redshift surface and the time (that is, the temporal coordinate) does not become in any case as a spatial coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] . Moreover, since the coefficient of $\text{d}{r}_g^2$ in the (39) is always not positive, also here the spatial coordinate $r_g$ does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Therefore also here the light cones are always orientated in the usual way, in particular there is not any horizontal inclination of the light cones, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Now, it is very complicated to express this metric as the formally flat metric in the curved coordinates, which metric is expressed as a function of the commonly used coordinates, but however we can obtain easily a such expression for $\text{d}{\theta }_g=0$ and $\text{d}{\phi }_g=0$. In fact the (39) for $\text{d}{\theta }_g=0$ and $\text{d}{\phi }_g=0$ becomes:

$\text{d}{s}^2=\left(1+\frac{2GM{r}_g}{{\rho }^2}\right)\text{d}{t}_g^2-\frac{{\rho }^2}{\Delta }\text{d}{r}_g^2$ (42)

where ${\rho }^2$ and Δ are functions of $r_g$:

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (43)

$\Delta \equiv r_g^2+2GM{r}_g+a^2$ (44)

Therefore the (42) can be written as:

$\text{d}{s}^2=\left(1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{t}_g^2-\frac{r_g^2+a^2{\cos }^2{\theta }_g}{r_g^2+2GM{r}_g+a^2}\text{d}{r}_g^2$ (45)

or also as:

$\text{d}{s}^2=\left(1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2$ (46)

Now by analogy with the previous cases we can write in this case the formally flat metric in the curved coordinates, which metric is expressed as a function of the commonly used coordinates, as:

$\text{d}{s}_g^2=\frac{1}{1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{t}^2-\left(1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{r}^2$ (47)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates and ${\theta }_g$ is a constant.

As for the relation between $r_g$ and r, we have:

$\text{d}{r}^2=\frac{1}{1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2$ (48)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2}$ (49)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g\left(r_g^2+a^2{\cos }^2{\theta }_g\right)\text{d}{r}_g}{r_g^2+2GM{r}_g+a^2}}$ (50)

From which, we have:

$\begin{array}{l}{r}^2=[r_g^2-4GM{r}_g+\left(-a^2{\sin }^2{\theta }_g+4G^2{M}^2\right)\ln \left(r_g^2+2GM{r}_g+a^2\right)\begin{array}{c}\stackrel{}{}\\ \end{array}\\ +\frac{GM\left(2a^2+a^2{\sin }^2{\theta }_g-4G^2{M}^2\right)}{\sqrt{G^2{M}^2-a^2}}{\cdot \ln \left(\frac{GM+r_g-\sqrt{G^2{M}^2-a^2}}{GM+r_g+\sqrt{G^2{M}^2-a^2}}\right)]}_0^{r_g}\end{array}$ (51)

From which, we have:

$\begin{array}{c}{r}^2=r_g^2-4GM{r}_g+\left(-a^2{\sin }^2{\theta }_g+4G^2{M}^2\right)\ln \left(1+\frac{r_g^2+2GM{r}_g}{a^2}\right)\\ +\frac{GM\left(2a^2+a^2{\sin }^2{\theta }_g-4G^2{M}^2\right)}{\sqrt{G^2{M}^2-a^2}}\\ \cdot \ln \left(\frac{GM+r_g-\sqrt{G^2{M}^2-a^2}}{GM+r_g+\sqrt{G^2{M}^2-a^2}}\cdot \frac{GM+\sqrt{G^2{M}^2-a^2}}{GM-\sqrt{G^2{M}^2-a^2}}\right)\end{array}$ (52)

According to the (52) for $r_g\to 0$ also $r\to 0$ and for $r_g\to +\infty$ also $r\to +\infty$. Moreover, as $r_g$ increases, r also increases, as we can see directly from the (50). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as we can see directly from the (49).

Furthermore, for $r_g\to +\infty$ we have that $r\cong r_g\left(1-\frac{2GM}{r_g}\right)$ and also $r_g\cong r\left(1+\frac{2GM}{r}\right)$. This implies that, in this case, the presence of $r_g$ instead of r in the (47) implies only corrections to the second order in $\frac{2GM}{r}$.

On the other hand, for $r_g\to 0$ in the case of ${\cos }^2{\theta }_g\ne 0$ we have

$r\cong r_g\left|\cos {\theta }_g\right|$, while in the case of ${\cos }^2{\theta }_g=0$ we have $r\cong \frac{r_g^2}{\sqrt{2a^2}}$, as we can see

easily from the (50).

In this case the correct formulas (9) and (10) become respectively:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM{r}_g{}_1}{r_g{}_1^2+a^2{\cos }^2{\theta }_g}}{1+\frac{2GM{r}_g{}_2}{r_g{}_2^2+a^2{\cos }^2{\theta }_g}}}$ (53)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM{r}_g{}_1}{r_g{}_1^2+a^2{\cos }^2{\theta }_g}}{1+\frac{2GM{r}_g{}_2}{r_g{}_2^2+a^2{\cos }^2{\theta }_g}}}$ (54)

Therefore also in this case we have that in the presence of a gravitational field the clocks go more slowly (that is, that the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a gravitational field flow more slowly). And the more intense is the gravitational field the more slowly the clocks go (that is, the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a more intense gravitational field flow more slowly than the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a less intense gravitational field).

Furthermore, the gravitational redshift, that is expressed by the formula (54), is always not infinite for any value of $r>0$ and of $r_g>0$, as in the correct Schwarzschild solution.

We can note that the two correct formulas (53) and (54) are different from those obtained usually from the incorrect expression of the Kerr solution by means of an incorrect procedure [1] [2] . In fact, the formulas commonly used, instead of the two correct formulas (53) and (54), are respectively [1] [2] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM{r}_2}{r_2^2+a^2{\cos }^2\theta }}{1-\frac{2GM{r}_1}{r_1^2+a^2{\cos }^2\theta }}}$ (55)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM{r}_2}{r_2^2+a^2{\cos }^2\theta }}{1-\frac{2GM{r}_1}{r_1^2+a^2{\cos }^2\theta }}}$ (56)

In this case the formula (14) of the radial velocity of light in the commonly used coordinates [1] [2] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\sqrt{\frac{1}{\left(1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\left(1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)}}$ (57)

We can note that, also here, the radial velocity of light in the commonly used coordinates is always ≤1, and is equal to 1 only when there is not a gravitational field.

As for the Kerr-Newman solution [2] [3] , that is the solution that represents the curved space-time geometry surrounding an electrically charged rotating mass, we have that the commonly used expression is [3] :

$\text{d}{s}^2=\frac{\Delta }{{\rho }^2}{\left(\text{d}t-a{\sin }^2\theta \text{d}\phi \right)}^2-\frac{{\sin }^2\theta }{{\rho }^2}{\left[\left(r^2+a^2\right)\text{d}\phi -a\text{d}t\right]}^2-\frac{{\rho }^2}{\Delta }\text{d}{r}^2-{\rho }^2\text{d}{\theta }^2$ (58)

where ${\rho }^2$ and Δ are functions of r and $\theta$:

$\Delta \equiv r^2-2GMr+a^2+G{Q}^2$ (59)

${\rho }^2\equiv r^2+a^2{\cos }^2\theta$ (60)

and M, $Q$ and $a$ are constants: M is the total mass of the system, $Q$ is the total electric charge of the system, and $a$ is the spin angular momentum of the system per unit mass.

In the (58) there are singularities (event horizons) when

$r^2-2GMr+a^2+G{Q}^2=0$ (61)

that is, when

$r=GM\pm \sqrt{G^2{M}^2-a^2-G{Q}^2}$ (62)

Obviously there are real singularities (event horizons) only when $G^2{M}^2-a^2-G{Q}^2\ge 0$.

Moreover we have that the coefficient $g_{00}$ is equal to zero when

$r^2-2GMr+a^2{\cos }^2\theta +G{Q}^2=0$ (63)

that is, when

$r=GM\pm \sqrt{G^2{M}^2-a^2{\cos }^2\theta -G{Q}^2}$ (64)

Therefore there are two distinct surfaces where $g_{00}$ is equal to zero. These surfaces are infinite-redshift surfaces. Obviously such surfaces really exist only where $G^2{M}^2-a^2{\cos }^2\theta -G{Q}^2\ge 0$.

But also here the expression (58) is erroneous: in fact, by means of the fact that the correct Kerr-Newman solution must be equal to the correct Kerr solution for $Q=0$, and must be equal to the correct Reissner-Nordstrøm solution for $a=0$, and must be equal to the correct Schwarzschild solution for $Q=0$ and $a=0$, we can say that the correct Kerr-Newman solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\begin{array}{l}\text{d}{s}^2=\frac{\Delta }{{\rho }^2}{\left(\text{d}{t}_g-a{\sin }^2{\theta }_g\text{d}{\phi }_g\right)}^2-\frac{{\sin }^2{\theta }_g}{{\rho }^2}{\left[\left(r_g^2+a^2\right)\text{d}{\phi }_g-a\text{d}{t}_g\right]}^2\\ -\frac{{\rho }^2}{\Delta }\text{d}{r}_g^2-{\rho }^2\text{d}{\theta }_g^2\end{array}$ (65)

where ${\rho }^2$ and Δ are functions of $r_g$ and ${\theta }_g$:

$\Delta \equiv r_g^2+2GM{r}_g+a^2+G{Q}^2$ (66)

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (67)

and M, $Q$ and $a$ are constants: M is the total mass of the system, $Q$ is the total electric charge of the system, and $a$ is the spin angular momentum of the system per unit mass. Moreover $t_g$, $r_g$, ${\theta }_g$, ${\phi }_g$ are the space-time coordinates measured relatively to a reference frame that is integral with a space-time curved for expressing the gravitational field as the curvature of the space-time. In other words, this equation (65) is expressed in curved coordinates.

As we can see, the difference between the correct formula and the incorrect formula is only in the definition of Δ, in which we have respectively $+2GM{r}_g$ instead of $-2GM{r}_g$.

Moreover, we can note that in the (65) the coefficient of $\text{d}{t}_g^2$ is always greater than one (except in the case where there is no gravitational field, in which case this coefficient is equal to one), and the more intense the gravitational field is, the greater this coefficient is. As mentioned before, we have already shown in [1] that this is a sign that the coordinates are curved and is closely related to the fact that the clocks in a gravitational field run more slowly.

Also in this case as in that of Schwarzschild, we have not any singularity in the correct expression (65) for any value of $r_g>0$. Therefore we can say that there is not any event horizon and therefore that there is not any black hole [2] [3] .

Furthermore, since the coefficient of $\text{d}{t}_g^2$ in the (65) is always ≥1, there is not any infinite-redshift surface and the time (that is, the temporal coordinate) does not become in any case as a spatial coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] . Moreover, since the coefficient of $\text{d}{r}_g^2$ in the (65) is always not positive, also here the spatial coordinate $r_g$ does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Therefore, also here the light cones are always orientated in the usual way, in particular there is not any horizontal inclination of the light cones, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Now, also in this case, it is very complicated to express this metric as the formally flat metric in the curved coordinates, which metric is expressed as a function of the commonly used coordinates, but however we can obtain easily a such expression for $\text{d}{\theta }_g=0$ and $\text{d}{\phi }_g=0$. In fact the (65) for $\text{d}{\theta }_g=0$ and $\text{d}{\phi }_g=0$ becomes:

$\text{d}{s}^2=\frac{\Delta -a^2{\sin }^2{\theta }_g}{{\rho }^2}\text{d}{t}_g^2-\frac{{\rho }^2}{\Delta }\text{d}{r}_g^2$ (68)

where ${\rho }^2$ and Δ are functions of $r_g$:

$\Delta \equiv r_g^2+2GM{r}_g+a^2+G{Q}^2$ (69)

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (70)

Therefore the (68) can be written as:

$\text{d}{s}^2=\left(1+\frac{2GM{r}_g+G{Q}^2}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM{r}_g+G{Q}^2+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2$ (71)

Now by analogy with the previous cases we can write in this case the formally flat metric in the curved coordinates, which metric is expressed as a function of the commonly used coordinates, as:

$\text{d}{s}_g^2=\frac{1}{1+\frac{2GM{r}_g+G{Q}^2}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{t}^2-\left(1+\frac{2GM{r}_g+G{Q}^2+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{r}^2$ (72)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates and ${\theta }_g$ is a constant.

As for the relation between $r_g$ and r, we have:

$\text{d}{r}^2=\frac{1}{1+\frac{2GM{r}_g+G{Q}^2+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2$ (73)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM{r}_g+G{Q}^2+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2}$ (74)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g\left(r_g^2+a^2{\cos }^2{\theta }_g\right)\text{d}{r}_g}{r_g^2+2GM{r}_g+G{Q}^2+a^2}}$ (75)

From which, we have:

$\begin{array}{l}{r}^2=[r_g^2-4GM{r}_g+\left(4G^2{M}^2-a^2{\sin }^2{\theta }_g-G{Q}^2\right)\begin{array}{c}\stackrel{}{}\\ \end{array}\\ \cdot \ln \left(r_g^2+2GM{r}_g+G{Q}^2+a^2\right)\\ +\frac{GM\left(2a^2+a^2{\sin }^2{\theta }_g+3G{Q}^2-4G^2{M}^2\right)}{\sqrt{G^2{M}^2-G{Q}^2-a^2}}\\ {\cdot \ln \left(\frac{GM+r_g-\sqrt{G^2{M}^2-G{Q}^2-a^2}}{GM+r_g+\sqrt{G^2{M}^2-G{Q}^2-a^2}}\right)]}_0^{r_g}\end{array}$ (76)

From which, we have:

$\begin{array}{c}{r}^2=r_g^2-4GM{r}_g+\left(4G^2{M}^2-a^2{\sin }^2{\theta }_g-G{Q}^2\right)\\ \cdot \ln \left(1+\frac{r_g^2+2GM{r}_g}{G{Q}^2+a^2}\right)+\frac{GM\left(2a^2+a^2{\sin }^2{\theta }_g+3G{Q}^2-4G^2{M}^2\right)}{\sqrt{G^2{M}^2-G{Q}^2-a^2}}\\ \cdot \ln \left(\frac{GM+r_g-\sqrt{G^2{M}^2-G{Q}^2-a^2}}{GM+r_g+\sqrt{G^2{M}^2-G{Q}^2-a^2}}\cdot \frac{GM+\sqrt{G^2{M}^2-G{Q}^2-a^2}}{GM-\sqrt{G^2{M}^2-G{Q}^2-a^2}}\right)\end{array}$ (77)

According to the (77) for $r_g\to 0$ also $r\to 0$ and for $r_g\to +\infty$ also $r\to +\infty$. Moreover, as $r_g$ increases, r also increases, as we can see directly from the (75). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as we can see directly from the (74).

Furthermore, for $r_g\to +\infty$ we have that $r\cong r_g\left(1-\frac{2GM}{r_g}\right)$ and also $r_g\cong r\left(1+\frac{2GM}{r}\right)$. This implies that, in this case, the presence of $r_g$ instead of r in the (72) implies only corrections to the second order in $\frac{2GM}{r}$.

On the other hand for $r_g\to 0$ in the case of ${\cos }^2{\theta }_g\ne 0$ we have $r\cong r_g\sqrt{\frac{a^2{\cos }^2{\theta }_g}{G{Q}^2+a^2}}$, while in the case of ${\cos }^2{\theta }_g=0$ we have $r\cong \frac{r_g^2}{\sqrt{2\left(G{Q}^2+a^2\right)}}$, as we can see easily from the (75).

In this case the correct formulas (9) and (10) become respectively:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM{r}_g{}_1+G{Q}^2}{r_g{}_1^2+a^2{\cos }^2{\theta }_g}}{1+\frac{2GM{r}_g{}_2+G{Q}^2}{r_g{}_2^2+a^2{\cos }^2{\theta }_g}}}$ (78)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM{r}_g{}_1+G{Q}^2}{r_g{}_1^2+a^2{\cos }^2{\theta }_g}}{1+\frac{2GM{r}_g{}_2+G{Q}^2}{r_g{}_2^2+a^2{\cos }^2{\theta }_g}}}$ (79)

Therefore also in this case we have that in the presence of a gravitational field the clocks go more slowly (that is, that the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a gravitational field flow more slowly). And the more intense is the gravitational field the more slowly the clocks go (that is, the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a more intense gravitational field flow more slowly than the measurements of time relatively to a reference frame that is integral with a space-time curved for expressing the presence of a less intense gravitational field).

Furthermore, the gravitational redshift, that is expressed by the formula (79), is always not infinite for any value of $r_g>0$ or of $r>0$, as in the correct Schwarzschild solution.

We can note that the two correct formulas (78) and (79) are different from those obtained usually from the incorrect expression of the Kerr-Newman solution by means of an incorrect procedure [1] [2] . In fact, the formulas commonly used, instead of the two correct formulas (78) and (79), are respectively [1] [2] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1+\frac{-2GM{r}_2+G{Q}^2}{r_2^2+a^2{\cos }^2\theta }}{1+\frac{-2GM{r}_1+G{Q}^2}{r_1^2+a^2{\cos }^2\theta }}}$ (80)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{-2GM{r}_2+G{Q}^2}{r_2^2+a^2{\cos }^2\theta }}{1+\frac{-2GM{r}_1+G{Q}^2}{r_1^2+a^2{\cos }^2\theta }}}$ (81)

In this case the formula (14) of the radial velocity of light in the commonly used coordinates [1] [2] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\sqrt{\frac{1}{\left(1+\frac{2GM{r}_g+G{Q}^2}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\left(1+\frac{2GM{r}_g+G{Q}^2+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)}}$ (82)

We can note that, also here, the radial velocity of light in the commonly used coordinates is always ≤1, and is equal to 1 only when there is not a gravitational field.

We have seen that these correct metrics of Schwarzschild, Reissner-Nordstrøm, Kerr and Kerr-Newman do not imply any event horizon and, consequently, these metrics do not imply any black hole.

In fact, in these metrics there is not any singularity in general at any value of $r>0$ or of $r_g>0$.

Moreover, according to these metrics, for any value of $r>0$ or of $r_g>0$ the coefficient of $\text{d}{t}^2$ and the coefficient of $\text{d}{t}_g^2$ always remain positive and the coefficient of $\text{d}{r}^2$ and the coefficient of $\text{d}{r}_g^2$ always remain negative. Therefore, since the coefficient of $\text{d}{t}^2$ and the coefficient of $\text{d}{t}_g^2$ are always not negative the time (that is, the temporal coordinate) does not become in any case as a spatial coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] . On the other hand, since the coefficient of $\text{d}{r}^2$ and the coefficient of $\text{d}{r}_g^2$ are always not positive the spatial coordinate r or $r_g$ does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

Consequently, the light cones are always orientated in the usual way, in particular there is not any horizontal inclination of the light cones, contrary to the common treatment of the space-time inside the event horizon [2] [3] .

As we have seen, the corrections of this article imply that the correct solutions of these metrics do not entail any event horizon and, consequently, any black hole, since there is not any black hole without an event horizon [1] . Therefore, this article confutes all the physics that on the basis of these metrics foresees the possibility of the existence of event horizons and black holes [2] [3] [16] [17] .

On the other hand, we have seen that the alleged proofs in favour of the existence of black holes and event horizons based respectively on x-ray astronomy, on alleged gravitational waves and on alleged photos of black holes are not conclusive and therefore are not sufficient to discriminate between the commonly accepted theory and the same theory corrected according to this article [1] .

Moreover, we have observed that the corrections, that we have proposed here, to the commonly accepted theory are very small in the normal experimental situations, so the fact that so far no difference has been noted between the commonly accepted theory and the experimental results is not strange. In fact, as we have

seen, in the usual case of $\frac{2GM}{r}\ll 1$, $\frac{G{Q}^2}{r^2}$ negligible compared to $\frac{2GM}{r}$ and

$a^2$ negligible compared to $r^2$, we have that the difference between the previsions of these erroneous metrics and the previsions of these correct metrics is

practically only in the second order in $\frac{2GM}{r}$ [1] . And, as we have already noted, all the experiments conducted so far have not had errors so small as to test differences at the second order in $\frac{2GM}{r}$ [2] .

However, as we have already noted, recently atomic clocks have been constructed with a sensitivity such as to test these small differences in experiments that are feasible in the solar system [1] . Therefore, it would be appropriate to try to make a crucial experiment that discriminates between the commonly accepted theory and the same theory corrected according to this article [1] .

Finally, according to this article, all the physics that is based on these incorrect metrics should be modified on the basis of the correct formulas that we have calculated [1] .