In a recent article [1] , we have corrected the traditional derivation of the Schwarzschild metric when the cosmological constant is equal to zero, thus obtaining the formulation of the correct Schwarzschild metric when the cosmological constant is equal to zero, which formulation is different from that of the traditional Schwarzschild metric when the cosmological constant is equal to zero.

Then, in another article [2] by starting from this correct Schwarzschild metric when the cosmological constant is equal to zero, we have corrected also the Reissner-Nordstrøm, Kerr and Kerr-Newman metrics when the cosmological constant is equal to zero.

Now, in this article, by starting from these correct Schwarzschild, Reissner-Nordstrøm, Kerr and Kerr-Newman metrics when the cosmological constant is equal to zero, we want to obtain the formulations of the correct Schwarzschild, Reissner-Nordstrøm, Kerr and Kerr-Newman metrics when the cosmological constant is greater than zero.

Moreover, we will analyse these results and their consequences.

Finally, we will see the experimental prospects, proposing in particular some possible crucial experiments between the commonly accepted theory and the same theory corrected according to this article.

In this article all the formulas are expressed with the velocity of light $c\equiv 1$, unless otherwise indicated.

As for the Schwarzschild solution, that is the solution that represents the curved space-time geometry surrounding a spherically symmetric mass distribution, we have that the commonly used expression when the cosmological constant is equal to zero is [3] :

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

where $M$ is (a constant, which is equal to) the total mass of the system.

In the (1) there is a singularity (event horizon) when $r=2GM$.

But the expression (1) is erroneous: in fact, as we have shown in [1] , when the cosmological constant is equal to zero the correct Schwarzschild solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2\equiv \text{d}{t}^2-\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

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

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 (2) is expressed in curved coordinates.

We can note that in the (2) 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, when the cosmological constant is equal to zero, 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:

$\text{d}{s}_g^2=\frac{1}{1+\frac{2GM}{r_g}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}\right)\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$ (3)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates (that is, $\text{d}{s}_g^2\equiv \text{d}{t}_g^2-\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$).

Obviously, $r$ (or $r_g$) is always greater than zero since the Schwarzschild solution when the cosmological constant is equal to zero is a solution for vacuum region surrounding a spherically symmetric mass distribution.

As for the relation between $r_g$ and $r$, we have [1] :

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

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM}{r_g}}\text{d}{r}_g^2}$ (5)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g^2\text{d}{r}_g}{r_g+2GM}}$ (6)

From which, we have [1] :

$r^2=r_g^2-4GM{r}_g+8G^2{M}^2\ln \left(1+\frac{r_g}{2GM}\right)$ (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 (6). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as can be seen directly from the (5).

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 (3) implies only corrections to the second order in $\frac{2GM}{r}$.

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

We can note that we have not any singularity in the correct expressions (2) and (3) for any value of $r>0$ and of $r_g>0$. Consequently, we can infer that there is not any event horizon and therefore there is not any black hole [3] [4] .

In particular, the coefficients of $\text{d}{t}^2$ in (3) and of $\text{d}{t}_g^2$ in (2) are always >0, while the coefficients of $\text{d}{r}^2$ in (3) and of $\text{d}{r}_g^2$ in (2) are always <0. Therefore since the coefficients of $\text{d}{t}^2$ in (3) and of $\text{d}{t}_g^2$ in (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 [3] [4] . Moreover, since the coefficients of $\text{d}{r}^2$ in (3) and of $\text{d}{r}_g^2$ in (2) 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 [3] [4] .

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 [3] [4] .

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

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}}{1+\frac{2GM}{r_g{}_2}}}$ (8)

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 the gravitational field is, 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:

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}}{1+\frac{2GM}{r_g{}_2}}}$ (9)

This is the correct formula for the gravitational redshift of light in the correct Schwarzschild metric when the cosmological constant is equal to zero. Therefore, in this case the gravitational redshift is always not infinite for any value of $r>0$ and of $r_g>0$.

We can note that the two correct formulas (8) and (9) are different from those that can be obtained from the incorrect expression of the Schwarzschild solution when the cosmological constant is equal to zero by means of an incorrect procedure [1] [3] , but are equal at the first order in $\frac{2GM}{r}$ (to the incorrect formulas). In fact, in this case the incorrect formulas analogous of the two correct formulas (8) and (9) are respectively [1] [3] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM}{r_2}}{1-\frac{2GM}{r_1}}}$ (10)

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

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. Consequently from the (3) 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:

$\frac{1}{1+\frac{2GM}{r_g}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}\right)\text{d}{r}^2=0$ (12)

From which we have that the correct radial velocity of light in the commonly used coordinates is equal to [1] [3] :

$v_l=\frac{\text{d}r}{\text{d}t}=\frac{1}{1+\frac{2GM}{r_g}}$ (13)

We can note that the correct 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 [3] .

On the other hand, by using the incorrect formula (1), we obtain that the incorrect radial velocity of light in the commonly used coordinates is equal to [3] :

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

When the cosmological constant Λ is greater than zero (instead of being equal to zero), the Einstein’s field equation in vacuum, as H. C. Ohanian, R. Ruffini and S. Weinberg state [3] [5] , is:

$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=-\Lambda g_{\mu \nu }$ (15)

instead of:

$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=0$ (16)

Consequently, when the cosmological constant Λ is greater than zero, the commonly used Schwarzschild solution is [3] :

$\begin{array}{c}\text{d}{s}^2=\left(1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}\right)\text{d}{t}^2-\frac{1}{1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}}\text{d}{r}^2\\ -r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2\end{array}$ (17)

where $M$ is (a constant, which is equal to) the total mass of the system.

Because Λ is very small there is a range of radii (in cgs units, $\frac{GM}{c^2}\ll r\ll \frac{c}{\sqrt{\Lambda }}$) in which the space-time geometry is nearly flat. At the lower end of this range, the effect of the mass $M$ dominates; at the upper end of this range, the effect of the cosmological term dominates. Moreover, in the Newtonian approximation the Newtonian potential becomes [3] :

$\Phi =-\frac{GM}{r}-\frac{\Lambda }{6}{r}^2$ (18)

In the (17) there are singularities (event horizons) when:

$1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}=0$ (19)

On the other hand, from the (17), by analogy with the derivation of the correct Schwarzschild solution when $\Lambda =0$ [1] , we can infer that the correct Schwarzschild solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2\equiv \text{d}{t}^2-\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, when $\Lambda >0$ is equal to:

$\begin{array}{c}\text{d}{s}^2=\left(1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}}\text{d}{r}_g^2\\ -r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2\end{array}$ (20)

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 (20) is expressed in curved coordinates.

Therefore, when $\Lambda >0$ we have $\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}$ instead of $\frac{2GM}{r_g}$, in the correct formulas.

Moreover, we can note that in the (20) the coefficient of $d{t}_g^2$ 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 our treatment of the correct Schwarzschild solution when the cosmological constant is equal to zero [1] we can express the correct Schwarzschild solution, when the cosmological constant is greater than zero, as 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:

$\begin{array}{c}\text{d}{s}_g^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}\right)\text{d}{r}^2\\ -r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2\end{array}$ (21)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates (that is, $\text{d}{s}_g^2\equiv \text{d}{t}_g^2-\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$).

Obviously, also here, $r$ (or $r_g$) is always greater than zero since the Schwarzschild solution when the cosmological constant is greater than zero is a solution for vacuum region surrounding a spherically symmetric mass distribution.

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

$\text{d}{r}^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}}\text{d}{r}_g^2$ (22)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}}\text{d}{r}_g^2}$ (23)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{6r_g^2\text{d}{r}_g}{3r_g+6GM+\Lambda r_g^3}}$ (24)

According to the (24) 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).

For $r_g$ not too large we have that the (24) is practically equal to the analogous formula with $\Lambda =0$ [1] , also because we know that the cosmological constant is very small.

Instead for $r_g\to +\infty$ we have that the term with the cosmological constant dominates. In this case from the (24) we have:

$r^2\cong {\displaystyle {\int }_1^{r_g}\frac{6r_g^2\text{d}{r}_g}{\Lambda r_g^3}}=\frac{6}{\Lambda }\text{ln}\left(r_g\right)$ (25)

Also in this case, as in that of the correct Schwarzschild solution when the cosmological constant is equal to zero [1] , we have not any singularity in the correct expressions (20) and (21) for any value of $r>0$ (or of $r_g>0$). Consequently we can infer that also here there is not any event horizon and therefore there is not any black hole [3] [4] .

In particular, also here in the (20) and in the (21) the coefficients of $\text{d}{t}_g^2$ and of $\text{d}{t}^2$ are always >0, therefore since the coefficients of $\text{d}{t}_g^2$ and of $\text{d}{t}^2$ 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 [3] [4] . Moreover, since in the (20) and in the (21) the coefficients of $\text{d}{r}_g^2$ and of $\text{d}{r}^2$ are always not positive, also here the spatial coordinate does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [3] [4] .

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 [3] [4] .

In this case the formula (8) of the correct ratio between the times when they are measured in two different positions becomes:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{\Lambda r_g{}_1^2}{3}}{1+\frac{2GM}{r_g{}_2}+\frac{\Lambda r_g{}_2^2}{3}}}$ (26)

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 the gravitational field is, 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).

While (in this case) the formula (9) of the correct ratio of the relative frequencies ${\nu }_1=1/\text{d}{t}_g{}_1$ and ${\nu }_2=1/\text{d}{t}_g{}_2$ becomes:

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{\Lambda r_g{}_1^2}{3}}{1+\frac{2GM}{r_g{}_2}+\frac{\Lambda r_g{}_2^2}{3}}}$ (27)

This is the correct formula for the gravitational redshift of light in this case. Therefore, also here the gravitational redshift is always not infinite for any value of $r_g>0$ (or of $r>0$), as in the correct Schwarzschild solution when the cosmological constant is equal to zero [1] .

We can note that the two correct formulas (26) and (27) are different from those that can be obtained from the incorrect expression of the Schwarzschild solution when the cosmological constant is greater than zero by means of an incorrect procedure [1] [3] . In particular, in this case the incorrect formulas analogous of the two correct formulas (26) and (27) are respectively:

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM}{r_2}-\frac{\Lambda r_2^2}{3}}{1-\frac{2GM}{r_1}-\frac{\Lambda r_1^2}{3}}}$ (28)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM}{r_2}-\frac{\Lambda r_2^2}{3}}{1-\frac{2GM}{r_1}-\frac{\Lambda r_1^2}{3}}}$ (29)

In this case the formula (13) of the correct radial velocity of light in the commonly used coordinates [1] [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}}$ (30)

We can note that, also here, the correct 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.

Instead in this case the formula (14) of the incorrect radial velocity of light in the commonly used coordinates [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}$ (31)

As for the Reissner-Nordstrøm solution [3] [4] , 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, when the cosmological constant is equal to zero, are respectively [3] :

$\begin{array}{c}\text{d}{s}^2=\left(1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}\right)\text{d}{t}^2-\frac{1}{1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}}\text{d}{r}^2\\ -r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2\end{array}$ (32)

and:

$E\left(r\right)=\frac{Q}{r^2}$ (33)

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 (32) there are singularities (event horizons) when:

$1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}=0$ (34)

that is, when:

$r^2-2GMr+G{Q}^2=0$ (35)

or, better yet, when:

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

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

But the expression (32) is erroneous: in fact, by means of the fact that for $Q=0$ the correct Reissner-Nordstrøm solution when the cosmological constant is equal to zero must be equal to the correct Schwarzschild solution when the cosmological constant is equal to zero, we can infer, as we have already noted in [2] , that, when the cosmological constant is equal to zero, the correct Reissner-Nordstrøm solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2\equiv \text{d}{t}^2-\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\begin{array}{c}\text{d}{s}^2=\left(1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2\\ -r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2\end{array}$ (37)

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 (37) is expressed in curved coordinates.

We can note that in the (37) 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 when the cosmological constant is equal to zero [1] , 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 [2] :

$\begin{array}{c}\text{d}{s}_g^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}\right)\text{d}{r}^2\\ -r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2\end{array}$ (38)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates (that is, $\text{d}{s}_g^2\equiv \text{d}{t}_g^2-\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$).

Obviously, also here, $r$ (or $r_g$) is always greater than zero since the Reissner-Nordstrøm solution when the cosmological constant is equal to zero is a solution for vacuum region surrounding an electrically charged mass.

As for the relation between $r_g$ and $r$, we have [2] :

$\text{d}{r}^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g{}^2}}\text{d}{r}_g^2$ (39)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2}$ (40)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g^3\text{d}{r}_g}{r_g^2+2GM{r}_g+G{Q}^2}}$ (41)

From which, we have [2] :

$\begin{array}{c}{r}^2=r_g^2-4GM{r}_g+G\left(4G{M}^2-Q^2\right)\text{ln}\left(1+\frac{r_g^2+2GM{r}_g}{G{Q}^2}\right)\\ +\frac{G^{2}M\left(3Q^2-4G{M}^2\right)}{\sqrt{G^2{M}^2-G{Q}^2}}\\ \cdot \text{ln}\left(\frac{GM+r_g-\sqrt{G^2{M}^2-G{Q}^2}}{GM+r_g+\sqrt{G^2{M}^2-G{Q}^2}}\cdot \frac{GM+\sqrt{G^2{M}^2-G{Q}^2}}{GM-\sqrt{G^2{M}^2-G{Q}^2}}\right)\end{array}$ (42)

According to the (42) 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 (41). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as we can see directly from the (40).

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 (38) 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 (41).

Also in this case, as in that of the correct Schwarzschild solution when the cosmological constant is equal to zero [1] , we have not any singularity in the correct expressions (37) and (38) for any value of $r>0$ and of $r_g>0$. Consequently we can infer that also here there is not any event horizon and therefore there is not any black hole [3] [4] .

In particular, also here the coefficients of $\text{d}{t}^2$ in (38) and of $\text{d}{t}_g^2$ in (37) are always >0, while the coefficients of $\text{d}{r}^2$ in (38) and of $\text{d}{r}_g^2$ in (37) are always <0. Therefore, since the coefficients of $\text{d}{t}^2$ in (38) and of $\text{d}{t}_g^2$ in (37) 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 [3] [4] . Moreover, since the coefficients of $\text{d}{r}^2$ in (38) and of $\text{d}{r}_g^2$ in (37) 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 [3] [4] .

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 [3] [4] .

In this case the formula (8) of the correct ratio between the times when they are measured in two different positions becomes:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{G{Q}^2}{r_g{}_1^2}}{1+\frac{2GM}{r_g{}_2}+\frac{G{Q}^2}{r_g{}_2^2}}}$ (43)

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 the gravitational field is, 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).

While (in this case) the formula (9) of the correct ratio of the relative frequencies ${\nu }_1=1/\text{d}{t}_g{}_1$ and ${\nu }_2=1/\text{d}{t}_g{}_2$ becomes:

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{G{Q}^2}{r_g{}_1^2}}{1+\frac{2GM}{r_g{}_2}+\frac{G{Q}^2}{r_g{}_2^2}}}$ (44)

This is the correct formula for the gravitational redshift of light in this case. Therefore, also here the gravitational redshift is always not infinite for any value of $r_g>0$ (or of $r>0$), as in the correct Schwarzschild solution when the cosmological constant is equal to zero [1] .

We can note that the two correct formulas (43) and (44) are different from those that can be obtained from the incorrect expression of the Reissner-Nordstrøm solution when the cosmological constant is equal to zero by means of an incorrect procedure [1] - [3] . In fact, in this case the incorrect formulas analogous of the two correct formulas (43) and (44) are respectively [1] - [3] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM}{r_2}+\frac{G{Q}^2}{r_2^2}}{1-\frac{2GM}{r_1}+\frac{G{Q}^2}{r_1^2}}}$ (45)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM}{r_2}+\frac{G{Q}^2}{r_2^2}}{1-\frac{2GM}{r_1}+\frac{G{Q}^2}{r_1^2}}}$ (46)

In this case the formula (13) of the correct radial velocity of light in the commonly used coordinates [1] - [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\frac{1}{1+\frac{2GM}{r_g}+\frac{G{Q}^2}{r_g^2}}$ (47)

We can note that, also here, the correct 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.

On the other hand, in this case the formula (14) of the incorrect radial velocity of light in the commonly used coordinates [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=1-\frac{2GM}{r}+\frac{G{Q}^2}{r^2}$ (48)

By analogy with the (17) and the (32), we can infer that, when the cosmological constant Λ is greater than zero, the commonly accepted Reissner-Nordstrøm solution is:

$\begin{array}{c}\text{d}{s}^2=\left(1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}+\frac{G{Q}^2}{r^2}\right)\text{d}{t}^2-\frac{1}{1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}+\frac{G{Q}^2}{r^2}}\text{d}{r}^2\\ -r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2\end{array}$ (49)

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 (49) there are singularities (event horizons) when:

$1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}+\frac{G{Q}^2}{r^2}=0$ (50)

On the other hand, by analogy with the (20) and the (37), we can infer that, when the cosmological constant Λ is greater than zero, the correct Reissner-Nordstrøm solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2\equiv \text{d}{t}^2-\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\begin{array}{c}\text{d}{s}^2=\left(1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g^2}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2\\ -r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2\end{array}$ (51)

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 (51) is expressed in curved coordinates.

We can note that in the (51) 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 our treatment of the correct Reissner-Nordstrøm solution when the cosmological constant Λ is equal to zero [2] we can express the correct Reissner-Nordstrøm solution, when the cosmological constant Λ is greater than zero, as 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:

$\begin{array}{c}\text{d}{s}_g^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g^2}}\text{d}{t}^2-\left(1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g^2}\right)\text{d}{r}^2\\ -r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2\end{array}$ (52)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates (that is, $\text{d}{s}_g^2\equiv \text{d}{t}_g^2-\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$).

Obviously, also here, $r$ (or $r_g$) is always greater than zero since the Reissner-Nordstrøm solution when the cosmological constant is greater than zero is a solution for vacuum region surrounding an electrically charged mass.

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

$\text{d}{r}^2=\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g^2}}\text{d}{r}_g^2$ (53)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g{}^2}}\text{d}{r}_g^2}$ (54)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{6r_g^3\text{d}{r}_g}{3r_g^2+6GM{r}_g+\Lambda r_g^4+3G{Q}^2}}$ (55)

According to the (55) 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 (55). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as we can see directly from the (54).

For $r_g$ not too large we have that the (55) is practically equal to the analogous formula with $\Lambda =0$ [2] , also because we know that the cosmological constant is very small. In particular, for $r_g\to 0^{+}$ we have $r\cong \frac{r_g^2}{\sqrt{2G{Q}^2}}$, as we can see easily from the (55).

Instead for $r_g\to +\infty$ we have that the term with the cosmological constant dominates. In this case from the (55) we have:

$r^2\cong {\displaystyle {\int }_1^{r_g}\frac{6r_g^3\text{d}{r}_g}{\Lambda r_g^4}}=\frac{6}{\Lambda }\text{ln}\left(r_g\right)$ (56)

Also in this case, as in that of the correct Schwarzschild solution when the cosmological constant is equal to zero [1] , we have not any singularity in the correct expressions (51) and (52) for any value of $r>0$ (or of $r_g>0$). Consequently we can infer that also here there is not any event horizon and therefore there is not any black hole [3] [4] .

In particular, also here in the (51) and in the (52) the coefficients of $\text{d}{t}_g^2$ and of $\text{d}{t}^2$ are always >0, therefore since the coefficients of $\text{d}{t}_g^2$ and of $\text{d}{t}^2$ 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 [3] [4] . Moreover, since in the (51) and in the (52) the coefficients of $\text{d}{r}_g^2$ and of $\text{d}{r}^2$ are always not positive, also here the spatial coordinate does not become in any case as a temporal coordinate, contrary to the common treatment of the space-time inside the event horizon [3] [4] .

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 [3] [4] .

In this case the formula (8) of the correct ratio between the times when they are measured in two different positions becomes:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{\Lambda r_g{}_1^2}{3}+\frac{G{Q}^2}{r_g{}_1^2}}{1+\frac{2GM}{r_g{}_2}+\frac{\Lambda r_g{}_2^2}{3}+\frac{G{Q}^2}{r_g{}_2^2}}}$ (57)

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 the gravitational field is, 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).

While (in this case) the formula (9) of the correct ratio of the relative frequencies ${\nu }_1=1/\text{d}{t}_g{}_1$ and ${\nu }_2=1/\text{d}{t}_g{}_2$ becomes:

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM}{r_g{}_1}+\frac{\Lambda r_g{}_1^2}{3}+\frac{G{Q}^2}{r_g{}_1^2}}{1+\frac{2GM}{r_g{}_2}+\frac{\Lambda r_g{}_2^2}{3}+\frac{G{Q}^2}{r_g{}_2^2}}}$ (58)

This is the correct formula for the gravitational redshift of light in this case. Therefore, also here the gravitational redshift is always not infinite for any value of $r_g>0$ (or of $r>0$), as in the correct Schwarzschild solution when the cosmological constant is equal to zero [1] .

We can note that the two correct formulas (57) and (58) are different from those that can be obtained from the incorrect expression of the Reissner-Nordstrøm solution when the cosmological constant is greater than zero by means of an incorrect procedure [1] - [3] . In particular, in this case the incorrect formulas analogous of the two correct formulas (57) and (58) are respectively:

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM}{r_2}-\frac{\Lambda r_2^2}{3}+\frac{G{Q}^2}{r_2^2}}{1-\frac{2GM}{r_1}-\frac{\Lambda r_1^2}{3}+\frac{G{Q}^2}{r_1^2}}}$ (59)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM}{r_2}-\frac{\Lambda r_2^2}{3}+\frac{G{Q}^2}{r_2^2}}{1-\frac{2GM}{r_1}-\frac{\Lambda r_1^2}{3}+\frac{G{Q}^2}{r_1^2}}}$ (60)

In this case the formula (13) of the correct radial velocity of light in the commonly used coordinates [1] - [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\frac{1}{1+\frac{2GM}{r_g}+\frac{\Lambda r_g^2}{3}+\frac{G{Q}^2}{r_g^2}}$ (61)

We can note that, also here, the correct 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.

On the other hand, in this case the formula (14) of the incorrect radial velocity of light in the commonly used coordinates [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}+\frac{G{Q}^2}{r^2}$ (62)

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 when the cosmological constant is equal to zero is [3] :

$\begin{array}{c}\text{d}{s}^2=\text{d}{t}^2-\frac{{\rho }^2}{\Delta }\text{d}{r}^2-{\rho }^2\text{d}{\theta }^2-\left(r^2+a^2\right){\sin }^2\theta \text{d}{\phi }^2\\ -\frac{2GMr}{{\rho }^2}{\left(\text{d}t-a{\sin }^2\theta \text{d}\phi \right)}^2\end{array}$ (63)

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

${\rho }^2\equiv r^2+a^2{\cos }^2\theta$ (64)

$\Delta \equiv r^2-2GMr+a^2$ (65)

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 (63) there are singularities (event horizons) when:

$r^2-2GMr+a^2=0$ (66)

that is, when:

$r=GM\pm \sqrt{G^2{M}^2-a^2}$ (67)

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:

$r^2-2GMr+a^2{\cos }^2\theta =0$ (68)

that is, when:

$r=GM\pm \sqrt{G^2{M}^2-a^2{\cos }^2\theta }$ (69)

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 (63) is erroneous: in fact, by means of the fact that for $a=0$ the correct Kerr solution when the cosmological constant is equal to zero must be equal to the correct Schwarzschild solution when the cosmological constant is equal to zero, we can infer, as we have already noted in [2] , that, when the cosmological constant is equal to zero, the correct Kerr solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2\equiv \text{d}{t}^2-\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\begin{array}{c}\text{d}{s}^2=\text{d}{t}_g^2-\frac{{\rho }^2}{\Delta }\text{d}{r}_g^2-{\rho }^2\text{d}{\theta }_g^2-\left(r_g^2+a^2\right){\sin }^2{\theta }_g\text{d}{\phi }_g^2\\ +\frac{2GM{r}_g}{{\rho }^2}{\left(\text{d}{t}_g-a{\sin }^2{\theta }_g\text{d}{\phi }_g\right)}^2\end{array}$ (70)

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

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (71)

$\Delta \equiv r_g^2+2GM{r}_g+a^2$ (72)

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 (70) is expressed in curved coordinates.

We can note that in the (70) 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.

Also in this case, as in that of the correct Schwarzschild solution when the cosmological constant is equal to zero [1] , we have not any singularity in the correct expression (70) for any value of $r_g>0$. Consequently we can infer that also here there is not any event horizon and therefore there is not any black hole [3] [4] .

Furthermore, the coefficient of $\text{d}{t}_g^2$ in the (70) is always ≥1, therefore, since the coefficient of $\text{d}{t}_g^2$ is always not zero, there is not any infinite-redshift surface and, since the coefficient of $\text{d}{t}_g^2$ is 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 [3] [4] . Moreover, since the coefficient of $\text{d}{r}_g^2$ in the (70) 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 [3] [4] .

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 [3] [4] .

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 (70) for $\text{d}{\theta }_g=0$ and $\text{d}{\phi }_g=0$ becomes:

$\text{d}{s}^2=\left(1+\frac{2GM{r}_g}{{\rho }^2}\right)\text{d}{t}_g^2-\frac{{\rho }^2}{\Delta }d{r}_g^2$ (73)

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

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (74)

$\Delta \equiv r_g^2+2GM{r}_g+a^2$ (75)

Therefore the (73) can be written as:

$\text{d}{s}^2=\left(1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{t}_g^2-\frac{1}{1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2$ (76)

Now by analogy with the previous cases [1] [2] 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:

$\text{d}{s}_g^2=\frac{1}{1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{t}^2-\left(1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\text{d}{r}^2$ (77)

where $\text{d}{s}_g^2$ is a metric formally flat in the curved coordinates (that is, $\text{d}{s}_g^2\equiv \text{d}{t}_g^2-\text{d}{r}_g^2-r_g^2\text{d}{\theta }_g^2-r_g^2{\sin }^2{\theta }_g\text{d}{\phi }_g^2$) and ${\theta }_g$ is a constant.

Obviously, also here, $r$ (or $r_g$) is always greater than zero since the Kerr solution when the cosmological constant is equal to zero is a solution for vacuum region surrounding a rotating mass.

As for the relation between $r_g$ and $r$, we have [2] :

$\text{d}{r}^2=\frac{1}{1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2$ (78)

From which, we have:

${\displaystyle {\int }_0^{r^2}\text{d}{r}^2}={\displaystyle {\int }_0^{r_g^2}\frac{1}{1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}}\text{d}{r}_g^2}$ (79)

From which, we have:

$r^2={\displaystyle {\int }_0^{r_g}\frac{2r_g\left(r_g^2+a^2{\cos }^2{\theta }_g\right)\text{d}{r}_g}{r_g^2+2GM{r}_g+a^2}}$ (80)

From which, we have [2] :

$\begin{array}{c}{r}^2=r_g^2-4GM{r}_g+\left(-a^2{\sin }^2{\theta }_g+4G^2{M}^2\right)\text{ln}\left(1+\frac{r_g^2+2GM{r}_g}{a^2}\right)\\ +\frac{GM\left(2a^2+a^2{\sin }^2{\theta }_g-4G^2{M}^2\right)}{\sqrt{G^2{M}^2-a^2}}\\ \cdot \text{ln}\left(\frac{GM+r_g-\sqrt{G^2{M}^2-a^2}}{GM+r_g+\sqrt{G^2{M}^2-a^2}}\cdot \frac{GM+\sqrt{G^2{M}^2-a^2}}{GM-\sqrt{G^2{M}^2-a^2}}\right)\end{array}$ (81)

According to the (81) 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 (80). On the other side, for any value of $r_g>0$ we have that $r<r_g$, as we can see directly from the (79).

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 (77) 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 (80).

In this case the formula (8) of the correct ratio between the times when they are measured in two different positions becomes:

$\frac{\text{d}{t}_g{}_2}{\text{d}{t}_g{}_1}=\sqrt{\frac{1+\frac{2GM{r}_g{}_1}{r_g{}_1^2+a^2{\cos }^2{\theta }_g}}{1+\frac{2GM{r}_g{}_2}{r_g{}_2^2+a^2{\cos }^2{\theta }_g}}}$ (82)

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 the gravitational field is, 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).

While (in this case) the formula (9) of the correct ratio of the relative frequencies ${\nu }_1=1/\text{d}{t}_g{}_1$ and ${\nu }_2=1/\text{d}{t}_g{}_2$ becomes:

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1+\frac{2GM{r}_g{}_1}{r_g{}_1^2+a^2{\cos }^2{\theta }_g}}{1+\frac{2GM{r}_g{}_2}{r_g{}_2^2+a^2{\cos }^2{\theta }_g}}}$ (83)

This is the correct formula for the gravitational redshift of light in this case. Therefore, also here the gravitational redshift is always not infinite for any value of $r_g>0$ (or of $r>0$), as in the correct Schwarzschild solution when the cosmological constant is equal to zero [1] .

We can note that the two correct formulas (82) and (83) are different from those that can be obtained from the incorrect expression of the Kerr solution when the cosmological constant is equal to zero by means of an incorrect procedure [1] - [3] . In fact, in this case the incorrect formulas analogous of the two correct formulas (82) and (83) are respectively [1] - [3] :

$\frac{\text{d}{t}_2}{\text{d}{t}_1}=\sqrt{\frac{1-\frac{2GM{r}_2}{r_2^2+a^2{\cos }^2\theta }}{1-\frac{2GM{r}_1}{r_1^2+a^2{\cos }^2\theta }}}$ (84)

$\frac{{\nu }_1}{{\nu }_2}=\sqrt{\frac{1-\frac{2GM{r}_2}{r_2^2+a^2{\cos }^2\theta }}{1-\frac{2GM{r}_1}{r_1^2+a^2{\cos }^2\theta }}}$ (85)

In this case the formula (13) of the correct radial velocity of light in the commonly used coordinates [1] - [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\sqrt{\frac{1}{\left(1+\frac{2GM{r}_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)\left(1+\frac{2GM{r}_g+a^2{\sin }^2{\theta }_g}{r_g^2+a^2{\cos }^2{\theta }_g}\right)}}$ (86)

We can note that, also here, the correct 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.

On the other hand, in this case the formula (14) of the incorrect radial velocity of light in the commonly used coordinates [3] becomes:

$v_l=\frac{\text{d}r}{\text{d}t}=\sqrt{\left(1-\frac{2GMr}{r^2+a^2{\cos }^2\theta }\right)\left(1+\frac{-2GMr+a^2{\sin }^2\theta }{r^2+a^2{\cos }^2\theta }\right)}$ (87)

By analogy with the (17) and the (63), we can infer that, when the cosmological constant $\Lambda$ is greater than zero, the commonly accepted Kerr solution is:

$\begin{array}{c}\text{d}{s}^2=\text{d}{t}^2-\frac{{\rho }^2}{\Delta }\text{d}{r}^2-{\rho }^2\text{d}{\theta }^2-\left(r^2+a^2\right){\sin }^2\theta \text{d}{\phi }^2\\ -\frac{2GMr+\frac{\Lambda }{3}{r}^4}{{\rho }^2}{\left(\text{d}t-a{\sin }^2\theta \text{d}\phi \right)}^2\end{array}$ (88)

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

${\rho }^2\equiv r^2+a^2{\cos }^2\theta$ (89)

$\Delta \equiv r^2-2GMr-\frac{\Lambda }{3}{r}^4+a^2$ (90)

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 (88) there are singularities (event horizons) when:

$r^2-2GMr-\frac{\Lambda }{3}{r}^4+a^2=0$ (91)

Moreover, in the (88) the coefficient $g_{00}$ is equal to zero (and therefore we have infinite redshift) when:

$r^2-2GMr-\frac{\Lambda }{3}{r}^4+a^2{\cos }^2\theta =0$ (92)

On the other hand, by analogy with the (20) and the (70), we can infer that, when the cosmological constant Λ is greater than zero, the correct Kerr solution expressed as the formally flat metric in the commonly used coordinates ( $\text{d}{s}^2\equiv \text{d}{t}^2-\text{d}{r}^2-r^2\text{d}{\theta }^2-r^2{\sin }^2\theta \text{d}{\phi }^2$), which metric is expressed as a function of the coordinates curved for expressing the gravitational field as the curvature of the space-time, is:

$\begin{array}{c}\text{d}{s}^2=\text{d}{t}_g^2-\frac{{\rho }^2}{\Delta }\text{d}{r}_g^2-{\rho }^2\text{d}{\theta }_g^2-\left(r_g^2+a^2\right){\sin }^2{\theta }_g\text{d}{\phi }_g^2\\ +\frac{2GM{r}_g+\frac{\Lambda }{3}{r}_g^4}{{\rho }^2}{\left(\text{d}{t}_g-a{\sin }^2{\theta }_g\text{d}{\phi }_g\right)}^2\end{array}$ (93)

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

${\rho }^2\equiv r_g^2+a^2{\cos }^2{\theta }_g$ (94)

$\Delta \equiv r_g^2+2GM{r}_g+\frac{\Lambda }{3}{r}_g^4+a^2$ (95)