De Sitter-Invariant Black Holes

Abstract

In the de Sitter-invariant approach to gravitation, all solutions to the gravitational field equations are spacetimes that reduce locally to de Sitter. Consequently, besides including a Schwarzschild event horizon, the de Sitter-invariant black hole also has a de Sitter cosmic horizon. Accordingly, it can lodge matter and dark energies. Owing to this additional structure concerning Poincaré-invariant general relativity, such a black hole can establish a link between the black hole dynamics and the universe’s evolution. Possible implications for cosmology are discussed, and a comparison with recent observations indicating the existence of a cosmological coupling of black holes is presented.

1. Introduction

A fundamental issue in spacetime physics is the lack of special relativity governing the Planck scale kinematics. The problem is that the short-distance structure of spacetime at the Planck scale relies on an observer-independent parameter given by the Planck length. Since Poincaré-invariant Einstein’s special relativity does not admit a finite invariant length, it cannot describe Planck-scale kinematics [1]. One then has to look for a special relativity that complies with the presence of a finite invariant length. A possible solution to this puzzle is to assume that, instead of being governed by the Poincaré-invariant Einstein’s special relativity, spacetime’s local kinematics is governed by the de Sitter-invariant special relativity [2]. Since the latter admits a finite invariant length while preserving the Lorentz symmetry, it provides a consistent description of the Planck scale kinematics [3].

If special relativity changes, all relativistic theories undergo concomitant changes. In particular, general relativity changes in the de Sitter-invariant general relativity [4], with the corresponding field equation referred to as the de Sitter-invariant Einstein’s equation. In contrast to the usual Einstein’s equation, whose solutions are spacetimes that reduce locally to Minkowski, all solutions to the de Sitter-invariant Einstein’s equations are spacetimes that reduce locally to de Sitter, where the local kinematics occurs. Since the cosmological term $\Lambda$, defined as the sectional curvature of the background de Sitter spacetime, is encoded in the spacetime’s local kinematics, it is an inherent part of the theory.

De Sitter-invariant general relativity gives rise to deviations from general relativity for energies comparable to Planck energy and at the universe’s large scale [3]. In most cases, therefore, the local cosmological term will be negligible. This is the case with the solar system, where the deviations concerning general relativity are expected to be tiny and virtually undetectable. On the other hand, since black holes involve extremely high energy densities, the dark energy associated with the local cosmological term $\Lambda$ can be relevant enough to produce deviations concerning standard general relativity.

Relying on the de Sitter-invariant approach to gravitation, the primary purpose of this paper is to obtain the de Sitter-invariant Schwarzschild solution and explore its mathematical and physical properties. These results will then be compared with recent observations suggesting the existence of a cosmological coupling of black holes.

2. Minkowski and de Sitter as Quotient Spaces

Spacetimes with constant sectional curvature are maximally symmetric because they carry the maximum number of Killing vectors. Flat Minkowski spacetime M is the simplest one. Its kinematic group is the Poincaré group $\mathcal{P}=\mathcal{L}\oslash \mathcal{T}$, the semi-direct product between Lorentz $\mathcal{L}$ and the translation group $\mathcal{T}$. Algebraically, it is defined as the quotient space [5]:

$M=\mathcal{P}/\mathcal{L}.$

The Lorentz subgroup is responsible for the isotropy around a given point of M, and the translation symmetry enforces this isotropy around all other points. In this case, homogeneity means that all points of Minkowski are equivalent under spacetime translations. One then says that Minkowski is transitive under translations, whose generators are written as:

(1)$P_{\nu }={\delta }_{\nu }^{\alpha }{\partial }_{\alpha },$

with ${\delta }_{\rho }^{\alpha }$ the Killing vectors of spacetime translations.

Another maximally symmetric spacetime is the de Sitter space $dS$, whose kinematic group is the de Sitter group $SO(1,4)$. Algebraically, the de Sitter spacetime is defined as the quotient space [5]:

(2)$dS=SO(1,4)/\mathcal{L}.$

As in Minkowski, the Lorentz subgroup is responsible for the isotropy around a given point of $dS$. To determine the homogeneity, we recall that the de Sitter spacetime can be viewed as a hyperboloid embedded in the $(1+4)$-dimensional pseudo-Euclidean space with Cartesian coordinates ${\chi }^A$ ($A,B,C\cdots =0,\cdots 4$) and metric:

${\eta }_{AB}=\mathrm{diag}(+1,-1,-1,-1,-1),$

inclusion whose points satisfy [6]:

(3)${\eta }_{AB}{\chi }^A{\chi }^B=-l^2.$

In terms of the coordinates ${\chi }^A$, the generators of infinitesimal de Sitter transformations are written as:

(4)$L_{AB}\equiv {\xi }_{AB}^C\frac{\partial }{\partial {\chi }^C}$

where:

(5)${\xi }_{AB}^C={\eta }_{AD}{\chi }^D{\delta }_B^C-{\eta }_{BD}{\chi }^D{\delta }_A^C$

are the associated Killing vectors.

In terms of the four-dimensional stereographic coordinates $\left\{x^{\mu }\right\}$ [7], the ten de Sitter generators (4) are given by:

(6)$L_{\mu \nu }={\xi }_{\mu \nu }^{\alpha }{\partial }_{\alpha }\mathrm{and}{\Pi }_{\nu }={\xi }_{\nu }^{\alpha }{\partial }_{\alpha },$

where $L_{\mu \nu }$ represent the Lorentz generators and ${\Pi }_{\nu }$ stand for the so-called de Sitter “translation” generators, with ${\xi }_{\mu \nu }^{\alpha }$ and ${\xi }_{\nu }^{\alpha }$ the corresponding Killing vectors.¹ In these coordinates, the Killing vectors of the de Sitter “translations” are written in the form:

(7)${\xi }_{\nu }^{\alpha }={\delta }_{\nu }^{\alpha }-\frac{1}{4l^2}{\vartheta }_{\nu }^{\alpha },$

where:

(8)${\delta }_{\nu }^{\alpha }\mathrm{and}{\vartheta }_{\nu }^{\alpha }=2{\eta }_{\nu \rho }{x}^{\rho }{x}^{\alpha }-{\sigma }^2{\delta }_{\nu }^{\alpha }$

are, respectively, the Killing vectors of translations and proper conformal transformations. In this case, the de Sitter “translation” generators can be rewritten as:

(9)${\Pi }_{\nu }=P_{\nu }-\frac{1}{4l^2}{K}_{\nu },$

where:

(10)$P_{\nu }={\delta }_{\nu }^{\alpha }{\partial }_{\alpha }\mathrm{and}{K}_{\nu }={\vartheta }_{\nu }^{\alpha }{\partial }_{\alpha }$

are, respectively, the translation and proper (or local) conformal generators [8]. Equations (1) and (9) show that, whereas Minkowski is transitive under translations, the de Sitter spacetime is transitive under a combination of translations and proper conformal transformations [9].

3. De Sitter-Invariant General Relativity

We review in this section some properties of the solutions to the de Sitter-invariant Einstein’s equation and discuss the main differences regarding the solutions to the standard Einstein’s equation.

3.1. Poincaré-Invariant Einstein’s Equation

As local transformations, diffeomorphisms must comply with the local homogeneity of spacetime. In the standard case of general relativity, where the Poincaré group governs the local kinematics, spacetime reduces locally to Minkowski. Since Minkowski is transitive under translations, a diffeomorphism in these spacetimes is defined as

(11)${\delta }_P{x}^{\mu }={\delta }_{\alpha }^{\mu }{ϵ}^{\alpha }\left(x\right),$

with ${\delta }_{\alpha }^{\mu }$ the Killing vectors of translations. From Noether’s theorem, the invariance of the source Lagrangian under the diffeomorphism (11) yields the energy-momentum conservation law:

(12)${\nabla }_{\nu }{T}^{\mu \nu }=0\mathrm{with}{T}^{\mu \nu }={\delta }_{\alpha }^{\mu }{T}^{\alpha \nu }.$

Consequently, Einstein’s equation is written as follows:

(13)$G^{\mu \nu }\equiv R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R=-\frac{8\pi G}{c^4}{T}^{\mu \nu }.$

3.2. De Sitter-Invariant Einstein’s Equation

In the case of the de Sitter-invariant general relativity, where the de Sitter group governs the local kinematics, spacetime reduces locally to de Sitter. Since de Sitter is transitive under de Sitter “translations”, a diffeomorphism in these spacetimes is defined as [10]

(14)${\delta }_{\Pi }{x}^{\mu }={\xi }_{\alpha }^{\mu }{ϵ}^{\alpha }\left(x\right),$

with ${\xi }_{\alpha }^{\mu }$ the Killing vectors (7) of the de Sitter “translations”. From Noether’s theorem, the invariance of the source Lagrangian under the diffeomorphism (14) yields the covariant conservation law:

(15)${\nabla }_{\nu }{\Pi }^{\mu \nu }=0\mathrm{with}{\Pi }^{\mu \nu }={\xi }_{\alpha }^{\mu }{T}^{\alpha \nu }.$

Consequently, the de Sitter-invariant Einstein’s equation is written as follows:

(16)${\mathcal{G}}^{\mu \nu }\equiv {\mathcal{R}}^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }\mathcal{R}=-\frac{8\pi G}{c^4}{\Pi }^{\mu \nu },$

where the Riemann tensor ${\mathcal{R}}^{\alpha }{}_{\beta \mu \nu }$ represents both the dynamic curvature of general relativity and the kinematic curvature of the background de Sitter spacetime. In this theory, therefore, the cosmological term $\Lambda$, defined as the sectional curvature of the background de Sitter spacetime, is a constitutive part of the theory and does not need to be added by hand to the gravitational field equation. Consequently, the cosmological term $\Lambda$ does not appear explicitly in the field equation, and the second Bianchi identity does not restrict it to be constant.

3.3. The Source of the Cosmological Term

Substituting the Killing vectors (7) in the source current ${\Pi }^{\mu \nu }$, it splits in the form

(17)${\Pi }^{\mu \nu }=T^{\mu \nu }-(1/4l^2)K^{\mu \nu },$

where:

(18)$T^{\mu \nu }={\delta }_{\alpha }^{\mu }{T}^{\alpha \nu }\mathrm{and}{K}^{\mu \nu }={\vartheta }_{\alpha }^{\mu }{T}^{\alpha \nu }$

are, respectively, the energy momentum and the proper conformal currents. In the de Sitter-invariant approach to gravitation, the conserved quantity is not the sum but the difference between the two types of energy. This property stems from the different nature of the dynamic interactions: whereas gravitation is attractive, the $\Lambda$ interaction is repulsive. Consequently, the evolution of any gravitational system depends on the energy difference, not the sum.

Analogously to the source decomposition, the Einstein tensor ${\mathcal{G}}^{\mu \nu }$ splits in the form:

(19)${\mathcal{G}}^{\mu \nu }=G^{\mu \nu }-{\widehat{G}}^{\mu \nu },$

where $G^{\mu \nu }$ is general relativity’s Einstein tensor and ${\widehat{G}}^{\mu \nu }$ is the Einstein tensor of the local de Sitter spacetime. In stereographic coordinates, therefore, the de Sitter-invariant Einstein’s Equation (16) assumes the form:

(20)$\left(R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R\right)-\left({\widehat{R}}^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }\widehat{R}\right)=-\frac{8\pi G}{c^4}\left[T^{\mu \nu }-(1/4l^2)K^{\mu \nu }\right].$

We see from this equation that any matter source in local de Sitter spacetimes gives rise to both an energy momentum and a proper conformal current. The energy-momentum current $T^{\alpha \nu }$ keeps its role as the source of general relativity’s dynamic curvature. In contrast, the proper conformal current $K^{\alpha \nu }$, appears as the source of the kinematic curvature of the local de Sitter spacetime. Taking the trace and identifying ${\widehat{G}}^{\mu }{}_{\mu }\sim -\Lambda$, it becomes

(21)$R-\Lambda =\frac{8\pi G}{c^4}\left[T^{\mu }{}_{\mu }-(1/4l^2)K^{\mu }{}_{\mu }\right].$

In this particular case, the energy-momentum tensor is traceless, the scalar curvature R vanishes, and the above equation reduces to an equation for the cosmological term,

(22)$\Lambda =\frac{8\pi G}{c^4}\frac{K^{\mu }{}_{\mu }}{4l^2}.$

For a more detailed discussion on the source of $\Lambda$, see Section 3 of Ref. [3].

4. The de Sitter-Invariant Black Hole

Following the standard procedure, we now obtain the Schwarzschild solution to the de Sitter-invariant Einstein’s Equation (16). For the sake of comparison, we first obtain the standard Schwarzschild solution.

4.1. Spherically Symmetric Ansatz

Let us consider a centrally symmetric distribution of matter. Since the gravitational field produced by such a source will also have central symmetry, its metric tensor can be written in the form [11]:

(23)$d{s}^2=e^{\kappa }{c}^{2}d{t}^2-e^{\lambda }d{r}^2-r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right),$

with $\kappa (r,t)$ and $\lambda (r,t)$ functions of the coordinates r and t. Denoting by $\{x^0,x^1,x^2,x^3\}$, respectively, the coordinates $\{ct,r,\theta ,\varphi \}$, the non-zero components of the metric tensor are:

(24)$g_{00}=e^{\kappa },g_{11}=-e^{\lambda },g_{22}=-r^2,g_{33}=-r^2{sin}^2\theta .$

The inverse components are:

(25)$g^{00}=e^{-\kappa },g^{11}=-e^{-\lambda },g^{22}=-r^{-2},g^{33}=-r^{-2}{sin}^{-2}\theta .$

Denoting differentiation with respect to r with a ‘prime’, and differentiation with respect to $ct$ with a ‘dot’, the non-vanishing components of the Levi-Civita connection are:

(26)$\begin{array}{ccc}{\Gamma }^1{}_{11}={\lambda }^{\prime }/2\hfill & {\Gamma }^0{}_{10}={\kappa }^{\prime }/2\hfill & {\Gamma }^2{}_{33}=-sin\theta cos\theta \hfill \\ \hfill & \hfill & \\ {\Gamma }^0{}_{11}=(\dot{\lambda }/2)e^{\lambda -\kappa }\hfill & {\Gamma }^1{}_{22}=-r{e}^{-\lambda }\hfill & {\Gamma }^1{}_{00}=({\kappa }^{\prime }/2)e^{\kappa -\lambda }\hfill \\ \hfill & \hfill & \\ {\Gamma }^2{}_{12}={\Gamma }^3{}_{13}=1/r\hfill & {\Gamma }^3{}_{23}=cot\theta \hfill & {\Gamma }^0{}_{00}=\dot{\kappa }/2\hfill \\ \hfill & \hfill & \\ {\Gamma }^1{}_{10}=\dot{\lambda }/2\hfill & {\Gamma }^1{}_{33}=-r{sin}^2\theta e^{-\lambda }\hfill \end{array}$

Computing the Ricci tensor $R^{\mu }{}_{\nu }$ and substituting in Einstein’s equation:

(27)$R^{\mu }{}_{\nu }-\frac{1}{2}{\delta }_{\nu }^{\mu }R=-\frac{8\pi G}{c^4}{T}^{\mu }{}_{\nu },$

a straightforward computation yields the following equations:

(28)$-e^{-\lambda }\left(\frac{{\kappa }^{\prime }}{r}+\frac{1}{r^2}\right)+\frac{1}{r^2}=-\frac{8\pi G}{c^4}{T}^1{}_1$

(29)$\begin{array}{c}\hfill -\frac{1}{2}{e}^{-\lambda }\left({\kappa }^{″}+\frac{{\kappa }^{\prime 2}}{2}+\frac{{\kappa }^{\prime }-{\lambda }^{\prime }}{r}-\frac{{\kappa }^{\prime }{\lambda }^{\prime }}{2}\right)+\frac{1}{2}{e}^{-\kappa }\left(\ddot{\lambda }+\frac{{\dot{\lambda }}^2}{2}-\frac{\dot{\lambda }\dot{\kappa }}{2}\right)=-\frac{8\pi G}{c^4}{T}^2{}_2=-\frac{8\pi G}{c^4}{T}^3{}_3\end{array}$

(30)$-e^{-\lambda }\left(\frac{1}{r^2}-\frac{{\lambda }^{\prime }}{r}\right)+\frac{1}{r^2}=-\frac{8\pi G}{c^4}{T}^0{}_0$

(31)$-e^{-\lambda }\frac{\dot{\lambda }}{r}=-\frac{8\pi G}{c^4}{T}^1{}_0.$

All other components of the field Equation (27) vanish identically.

4.2. The Poincaré-Invariant Schwarzschild Solution

We look for a vacuum solution, i.e., a solution that holds outside the masses producing the gravitational field, where the source current $T^{\alpha }{}_{\nu }$ vanishes. After setting the source current $T^{\alpha }{}_{\nu }$ equal to zero, and considering that Equation (29) is redundant because it follows from the other three equations, we are left with

(32)$e^{-\lambda }\left(\frac{{\kappa }^{\prime }}{r}+\frac{1}{r^2}\right)-\frac{1}{r^2}=0$

(33)$e^{-\lambda }\left(\frac{{\lambda }^{\prime }}{r}-\frac{1}{r^2}\right)+\frac{1}{r^2}=0$

(34)$\dot{\lambda }=0.$

Equation (34) shows that $\lambda$ does not depend on the time. Furthermore, adding Equations (32) and (33) we find ${\lambda }^{\prime }+{\kappa }^{\prime }=0$, whose integration yields

(35)$\lambda +\kappa =a\left(t\right)$

with $a\left(t\right)$ an arbitrary function of the time. However, when we chose the quadratic interval $d{s}^2$ in the form (23), there remained the possibility of an arbitrary transformation of the time of the form $t=a\left(t^{\prime }\right)$. Such a transformation is equivalent to adding to $\kappa$ an arbitrary function of the time. With this process, we can always make $a\left(t\right)$ in (35) vanish. Without loss of generality, we can thus set

(36)$\lambda +\kappa =0.$

Equation (33) can be integrated and gives

(37)$g_{00}\equiv e^{-\lambda }\equiv e^{\kappa }=1+\frac{\beta }{r},$

where $\beta$ is an integration constant. To determine this constant, we recall that far from the gravitating body, the gravitational field is weak and can be represented by its Newtonian limit, in which case we have the relation

(38)$g_{00}=1+2\frac{\varphi }{c^2}$

with $\varphi$ the Newtonian potential

(39)$\varphi =-\frac{GM}{r}.$

Comparing Equations (37) and (38), the constant $\beta$ is found to be

(40)$\beta =-r_M\equiv -\frac{2GM}{c^2},$

with $r_M$ the radius of the black hole event horizon. The metric (23) can then be recast in the form

(41)$d{s}^2=f\left(r\right)c^{2}d{t}^2-\frac{d{r}^2}{f\left(r\right)}-r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)$

with:

(42)$f\left(r\right)=1-\frac{r_M}{r}.$

This is the usual Schwarzschild solution. Since

$\underset{r\to \infty }{lim}f\left(r\right)=1,$

the Schwarzschild metric reduces to Minkowski at infinity.

4.3. The de Sitter-Invariant Schwarzschild Solution

In contrast to Poincaré-invariant general relativity, where all solutions to the gravitational field in Equation (27) are spacetimes that reduce locally to Minkowski, in de Sitter-invariant general relativity, all solutions to the gravitational field in Equation (16) are spacetimes that reduce locally to de Sitter. In this case, Equations (28)–(31) are written with:

(43)${\Pi }^{\mu }{}_{\nu }=T^{\mu }{}_{\nu }-\frac{K^{\mu }{}_{\nu }}{4l^2}$

replacing the energy-momentum current $T^{\mu }{}_{\nu }$. Consequently, besides depending on the coordinate r, the parameters $\kappa$ and $\lambda$ turn out to also depend on the pseudo-radius l of the local de Sitter spacetime. After replacing $\kappa \to \kappa \left(l\right)$ and $\lambda \to \lambda \left(l\right)$, and setting the source current $T^{\mu }{}_{\nu }$ equal to zero outside the masses producing the gravitational field, Equations (32)–(34) assume the form

(44)$e^{-\lambda \left(l\right)}\left(\frac{{\kappa }^{\prime }\left(l\right)}{r}+\frac{1}{r^2}\right)-\frac{1}{r^2}=-\frac{8\pi G}{c^4}\frac{K^1{}_1}{4l^2},$

(45)$e^{-\lambda \left(l\right)}\left(\frac{{\lambda }^{\prime }\left(l\right)}{r}-\frac{1}{r^2}\right)+\frac{1}{r^2}=+\frac{8\pi G}{c^4}\frac{K^0{}_0}{4l^2},$

(46)$\dot{\lambda }\left(l\right)=\frac{8\pi G}{c^4}\frac{K^1{}_0}{4l^2}.$

Please note that the vacuum is defined as the vanishing of the energy-momentum tensor, the source of gravitation. Since $K^{\mu }{}_{\nu }$ is the source of the background de Sitter spacetime, not of gravitation, its presence does not spoil the vacuum condition. Actually, the presence of the proper conformal current in the field Equations (44) and (45) is crucial for the local de Sitter invariance of the solution. If the proper conformal current vanishes, the background de Sitter spacetime reduces to Minkowski and we obtain the usual Poincaré-invariant black hole solution.

For symmetry reasons, the component $K^1{}_0$ of the proper conformal current vanishes, yielding the constraint $\lambda =-\kappa$. Consequently, one obtains:

(47)$\frac{8\pi G}{c^4}\frac{K^0{}_0}{4l^2}=\frac{8\pi G}{c^4}\frac{K^1{}_1}{4l^2}\equiv F(r,l).$

with $F(r,l)$ an arbitrary function to be determined by the boundary conditions. Equations (44) and (45) can then be rewritten as:

(48)$e^{-\lambda \left(l\right)}\left(\frac{{\kappa }^{\prime }\left(l\right)}{r}+\frac{1}{r^2}\right)-\frac{1}{r^2}=-F(r,l),$

(49)$e^{-\lambda \left(l\right)}\left(\frac{{\lambda }^{\prime }\left(l\right)}{r}-\frac{1}{r^2}\right)+\frac{1}{r^2}=+F(r,l).$

To solve these equations, we recall that, far from the gravitating body, the gravitational field is weak and can be represented by its Newtonian limit. In the case of the de Sitter-invariant general relativity, the de Sitter-modified Newtonian potential is given by [12]

(50)${\varphi }_{dS}=-GM\left(\frac{1}{r}+\frac{r}{2l^2}\right).$

Substituting in the Newtonian relation

(51)$g_{00}=1+2\frac{{\varphi }_{dS}}{c^2},$

we obtain an expression for the de Sitter-invariant Schwarzschild metric,

(52)$g_{00}\equiv e^{-\lambda \left(l\right)}\equiv e^{\kappa \left(l\right)}=1+\beta \left(\frac{1}{r}+\frac{r}{2l^2}\right),$

with $\beta$ an integration constant. Substituting this metric in Equation (49), the function $F(r,l)$ is found to be

(53)$F(r,l)=\frac{\beta }{l^{2}r}.$

The field Equation (49) with the right-hand side given by (53), can then be easily integrated, yielding the de Sitter-invariant Schwarzschild metric (52), with the integration constant given by:

(54)$\beta =-r_M\equiv -\frac{2GM}{c^2}.$

Using the above results, the de Sitter-invariant Schwarzschild solution can be written in the form:

(55)$d{s}^2=f\left(r\right)c^{2}d{t}^2-\frac{d{r}^2}{f\left(r\right)}-r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)$

where²

(56)$f\left(r\right)=1-\left(\frac{r_M}{r}+\frac{r}{r_{\Lambda }}\right),$

with:

(57)$r_M=2M\mathrm{and}{r}_{\Lambda }=\frac{l^2}{M}\equiv \frac{2l^2}{r_M}.$

These expressions show that the de Sitter-invariant black hole has two horizons: an event horizon with radius $r_M$ and a cosmic horizon with radius $r_{\Lambda }$. In this approach, the proper conformal current $K^{\mu \nu }$ of ordinary matter is the source of the background de Sitter spacetime. Since $K^{\mu \nu }$ depends on the energy-momentum current $T^{\mu \nu }$, as can be seen from Equation (18), the radius $r_{\Lambda }$ of the cosmic horizon turns out to depend on the radius $r_M$ of the event horizon.

4.4. Asymptotic Behavior

The Newtonian limit of standard general relativity is obtained when the gravitational field is weak, and the particle velocities are small. In the presence of a cosmological term $\Lambda$, some subtleties related to the process of group contraction show up. For example, in the non-relativistic contraction limit $c\to \infty$, the Poincaré group reduces to the Galilei group. However, the Newton–Hooke group does not follow straightforwardly from the de Sitter group under the same contraction limit. The reason for this is that, under such a limit, the boost transformations are lost. To obtain the physically relevant result, in which the group dimension is preserved, one has to simultaneously consider the limits $c\to \infty$ and $\Lambda \to 0$, but in such a way that:

(58)$lim{c}^2\Lambda =\frac{1}{{\tau }^2}$

with $\tau$ a finite time parameter. This means that, in the presence of $\Lambda$, the usual Newtonian limit must be supplemented with the small $\Lambda$ condition [13]:

(59)$\Lambda r^2\ll 1,$

which is equivalent to $r^2/l^2\ll 1$. Consequently,

(60)$\underset{r\to \infty }{lim}{r}^2/l^2=0\mathrm{and}\underset{r\to \infty }{lim}r/l^2=0.$

As we have seen, the metric of the usual Schwarzschild solution reduces to the Minkowski metric at infinity. Using the limits (60), one sees that the de Sitter-invariant Schwarzschild metric also reduces to Minkowski at infinity. Such asymptotic behavior comes from the cosmological term $\Lambda$ being not constant in the de Sitter-invariant approach to gravitation. Recall that, in this approach, gravitation and $\Lambda$ are both sourced by ordinary matter. Since all source fields vanish at infinity, gravitation and $\Lambda$ will also vanish. This property is a crucial difference concerning ordinary general relativity, where the constancy of $\Lambda$ plagues the theory with divergences that obscure the physically relevant results [14].

5. Thermodynamic Variables

The horizons $r_M$ and $r_{\Lambda }$, given by Equation (57), are not manifest in the Sitter-invariant Schwarzschild metric (55). In fact, it is not singular for $r=r_M$ or $r=r_{\Lambda }$. Therefore, this metric is not appropriate for studying black hole thermodynamics.

5.1. Reparameterizing the Horizons

To obtain a metric such that the horizons become manifest, one has to impose the condition $f\left(r\right)=0$, which yields the polynomial equation

(61)$r^2-r_{\Lambda }r+r_M{r}_{\Lambda }=0.$

This polynomial equation has two real roots provided

(62)$r_{\Lambda }>4r_M.$

In this case, the two solutions, denoted $R_M$ and $R_{\Lambda }$, are found to be

(63)$R_M=\frac{r_{\Lambda }}{2}\left[1-{\left(1-\frac{4r_M}{r_{\Lambda }}\right)}^{1/2}\right],$

and:

(64)$R_{\Lambda }=\frac{r_{\Lambda }}{2}\left[1+{\left(1-\frac{4r_M}{r_{\Lambda }}\right)}^{1/2}\right].$

It follows from these expressions that $R_{\Lambda }>R_M$. In terms of $R_M$ and $R_{\Lambda }$, the original horizons are written as

(65)$r_M=\frac{R_{\Lambda }{R}_M}{R_{\Lambda }+R_M}\mathrm{and}{r}_{\Lambda }=R_{\Lambda }+R_M.$

Substituting in Equation (56), it assumes the form

(66)$f\left(r\right)=\frac{(r-R_M)(R_{\Lambda }-r)}{(R_{\Lambda }+R_M)r},$

where the two horizons are now manifest. Furthermore, condition (62) assumes the form

(67)${(R_{\Lambda }+R_M)}^2>4R_M{R}_{\Lambda }.$

5.2. Entropy

The entropy associated with a given horizon is defined as $S_i=A_i/4$, with $A_i$ the area of the horizon. In the case of the event horizon, it is given by

(68)$S_M=\pi R_M^2.$

In the case of the cosmic horizon, it reads

(69)$S_{\Lambda }=\pi R_{\Lambda }^2.$

It should be noted that, since $R_{\Lambda }>R_M$, then $S_{\Lambda }>S_M$.

5.3. Temperature

The horizon temperature is defined as $T_i={\kappa }_i/2\pi$, where

(70)${\kappa }_i=\frac{1}{2}{\left|\frac{df}{dr}\right|}_{r=R_i}$

is the surface gravity, with $f\left(r\right)$ given by Equation (66). For the event horizon, the temperature is

(71)$T_M=\frac{1}{4\pi }\left[\frac{R_{\Lambda }-R_M}{R_M(R_{\Lambda }+R_M)}\right].$

On the other hand, for the cosmic horizon, it has the form

(72)$T_{\Lambda }=\frac{1}{4\pi }\left[\frac{R_{\Lambda }-R_M}{R_{\Lambda }(R_{\Lambda }+R_M)}\right].$

Considering that $R_{\Lambda }>R_M$, then $T_{\Lambda }<T_M$.

6. A New Black Hole Solution

We now discuss some properties of the de Sitter-invariant black holes.

6.1. First Law of Black Hole Thermodynamics

In standard general relativity, the first law of black hole thermodynamics, which holds in the event horizon, is expressed by

(73)$d{E}_M=T_{M}d{S}_M.$

When there is a cosmological term $\Lambda$ in the universe, an independent similar law is as follows:

(74)$d{E}_{\Lambda }=T_{\Lambda }d{S}_{\Lambda }$

holds in the de Sitter cosmic horizon [15].

In the case of de Sitter-invariant general relativity, the Schwarzschild solution has both an event and a cosmic horizon. The conserved energy density of the solution, given by the $\left(00\right)$ component of the Noether current (17), is

(75)${\epsilon }_{\Pi }={\epsilon }_M-{\epsilon }_{\Lambda },$

where ${\epsilon }_M=T^{00}$ is the translational notion of energy density and ${\epsilon }_{\Lambda }=K^{00}/4l^2$ is the proper conformal notion of energy density [3]. In terms of energy, it reads

(76)$E_{\Pi }=E_M-E_{\Lambda }.$

Consequently, the corresponding first law of the black hole thermodynamics is written as

(77)$d(E_M-E_{\Lambda })=T_{M}d{S}_M-T_{\Lambda }d{S}_{\Lambda }.$

In this theory, therefore, the event and cosmic horizons are not independent but make up a unique entangled system. Since the cosmological term $\Lambda$ is not constant, the dark energy $E_{\Lambda }$ is not constant either, and the thermodynamic variables can evolve along with cosmic time while preserving the thermodynamic Equation (77). In the contraction limit $\Lambda \to 0$, corresponding to $l\to \infty$, it reduces to the thermodynamic Equation (73) of standard general relativity.

6.2. Black Holes as Reservoirs of Dark Energy

As already discussed, in the de Sitter-invariant approach to gravitation, any source field gives rise to an energy momentum and proper conformal currents. Whereas the energy-momentum current appears as the source of the dynamic curvature of gravitation, the proper conformal current is the source of the kinematic curvature of the background de Sitter spacetime. Since the cosmological term $\Lambda$, defined as the sectional curvature of the background de Sitter spacetime, is an inherent part of the theory, all solutions to the de Sitter-invariant Einstein’s Equation (16) comprise a gravitational field and a local cosmological term Λ.

However, although present in all solutions, the cosmological term will be relevant only for physical systems with high-enough energy densities. Such is the case of black holes, in which the dark energy associated with $\Lambda$ can represent a relevant part of the black hole energy. This means de the Sitter-invariant black hole carries both matter ($E_M$) and dark ($E_{\Lambda }$) energies. The amount of each kind of energy it carries is governed by the thermodynamic Equation (77) and can change along with cosmic time.

6.3. Connecting Black Holes with the Universe’s Evolution

The de Sitter-invariant black hole comprises an event and a cosmic horizon, which make up a unique entangled system. These horizons show a fundamental difference: where the event horizon has a singularity at the center of the horizon, the cosmic horizon does not have it (see Equation (56)). Consequently, where the event horizon is always tied to a definite position in space, the cosmic horizon is not tied to any definite position in space. This position-indefiniteness allows us to assume that all black holes contribute to an effective cosmological term for the universe.

Thus, if the dark energy stored in all black holes can provide the necessary amount of energy to account for the dark energy we measure today, one can assume that the dark energy stored in the black holes represents the universe’s dark energy. Such an assumption establishes a link between the black hole’s dynamics and the universe’s evolution.

Of course, if there exists a cosmological coupling of black holes, it should be at work at any cosmic time. In particular, it should hold at the primordial universe. In this case, to drive inflation and the subsequent universe expansion, it would be necessary to presuppose the existence of primordial black holes with $E_{\Lambda }\gg E_M$. This assumption represents an essential move because, today, black holes are mainly associated with the gravitational collapse of old giant stars.

7. Observational Evidence for Black Holes Cosmological Coupling

Recent cosmological observations [16] found evidence of a cosmological coupling of black holes. To establish this coupling, one has to assume that the gravitating mass of black holes increases with the universe’s expansion, independently of accretions or mergers. Such a mass increase produces a concomitant increase in the dark energy carried by the black holes, which in turn contributes a dark energy density to Friedmann’s equations. Furthermore, by considering the black hole production inferred from the cosmic star formation history, it is concluded that black holes could account for the universe’s total amount of dark energy, as measured today. As the remnants of stellar collapse, black holes are considered the origin of the universe’s dark energy. Such a mechanism could explain, for example, the late time acceleration of the universe’s expansion.

Despite the conceptual differences, the de Sitter-invariant approach to gravitation can be said to corroborate the experimental evidence of a cosmological coupling of black holes. There are some crucial differences, though. For example, in the de Sitter-invariant approach to gravitation, the source of dark energy is not the black hole itself but the proper conformal current of ordinary matter. It is also unclear how the cosmological coupling of black holes would work in other cosmic periods, such as the primordial universe. Notwithstanding the conceptual differences, we believe this topic deserves further studies as establishing a cosmological coupling of black holes could provide answers to crucial problems of modern cosmology.

8. Discussion

Proper (or local) conformal symmetry is a broken symmetry of nature, which is expected to play a significant role in the Planck scale’s physics [17]. There is a widespread belief that, at the Planck scale, where the masses can be neglected, it must be an exact symmetry of nature. However, since Poincaré-invariant Einstein’s special relativity does not include local conformal transformations in the spacetime kinematics, it is unclear how it could become relevant at the Planck scale. For this reason, proper conformal transformation is sometimes considered the missing component of spacetime physics [18].

Unlike Poincaré-invariant special relativity, de Sitter-invariant special relativity does include the proper conformal transformations in the spacetime kinematics. Such inclusion occurs because the de Sitter group is obtained from Poincaré’s by replacing translations with a combination of translations and proper conformal transformations. Please note that such inclusion does not change the dimension of the spacetime’s local kinematics as Poincaré and de Sitter are ten-dimensional groups. The only effect of the inclusion is to change the notion of spacetime’s local transitivity.

When Poincaré-invariant special relativity is replaced with the de Sitter-invariant special relativity, general relativity changes in the de Sitter-invariant general relativity, with the corresponding field equation given by de Sitter-invariant Einstein’s equation. The presence of proper conformal transformations has profound consequences for the theory. For example, all solutions to this equation are given by a gravitational field, whose source is the energy-momentum current of ordinary matter, and a local cosmological term, sourced by the proper conformal current of the same ordinary matter.

In particular, the Schwarzschild solution represents a black hole with an event and a cosmic horizon. Consequently, besides lodging matter energy $E_M$, black holes also lodge dark energy $E_{\Lambda }$. If a black hole absorbs an object, its mass increases, producing a concomitant increase in the dark energy lodged in the black hole. Conversely, if a black hole emits Hawking radiation, its mass decreases, producing a concomitant decrease in its dark energy content. Supposing the existence of a black hole cosmological coupling, such a mechanism, which allows $E_M$ and $E_{\Lambda }$ to change along with cosmic time, could contribute to elucidating some key problems of modern cosmology. For example, it could explain the current universe’s accelerated expansion and why, in the past, the universe was expanding slower than today.

As is well-known, the standard Schwarzschild black hole is specified by its mass M only. On the other hand, the de Sitter-invariant black hole is specified by its mass M and the local cosmological term $\Lambda$. Accordingly, $\Lambda$ can be interpreted as a hair of the black hole. Furthermore, since $\Lambda$ is encoded in the spacetime’s local kinematics, which has a quantum nature by construction, it can be considered a quantum hair. This additional structure, which comes from the proper conformal sector of the theory, gives rise to an entirely new black hole. The next step of this research is to study the thermodynamics of this black hole, including possible consequences for the Hawking information paradox [19].

Author Contributions

Conceptualization: J.G.P.; Methodology: D.F.L., S.A.N. and J.G.P.; Investigation: D.F.L., S.A.N. and J.G.P.; Resources: D.F.L., S.A.N. and J.G.P.; Writing—original draft preparation: J.G.P.; Writing—review and editing: D.F.L., S.A.N. and J.G.P.; Visualization: D.F.L. and S.A.N.; Supervision: J.G.P.; Project administration: J.G.P. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

No new data were created or analyzed in this study.

Acknowledgments

The authors thank A. Coley and R. van den Hoogen for useful discussions. DFL thanks Dalhousie University, Canada, for a Ph.D. scholarship. SAN thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil, for a post-doctor fellowship (Contract No. 166099/2020-1). JGP thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil, for a research grant (Contract No. 312094/2021-3).

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Notes

1. The generators ${\Pi }_{\nu }$ are not really translations but rotations in the planes $(4,\nu )$. Hence the quotation marks

2. From now on, we use natural unities, in which $c=G=\hslash =1$

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

References

1. Amelino-Camelia, G. Doubly-Special Relativity: Facts, Myths and Some Key Open Issues. Symmetry; 2010; 2, 230. [DOI: https://dx.doi.org/10.3390/sym2010230]

2. Aldrovandi, R.; Beltrán Almeida, J.P.; Pereira, J.G. De Sitter special relativity. Class. Quantum Grav.; 2007; 24, 1385. [DOI: https://dx.doi.org/10.1088/0264-9381/24/6/002]

3. Araujo, A.V.; López, D.F.; Pereira, J.G.; Salazar, J.R. De Sitter-Invariant Approach to Cosmology. arXiv; 2022; arXiv: 2203.04824

4. Pereira, J.G.; López, D.F. An Improved Framework for Quantum Gravity. Universe; 2020; 6, 243. [DOI: https://dx.doi.org/10.3390/universe6120243]

5. Aldrovandi, R.; Pereira, J.G. An Introduction to Geometrical Physics; 2nd ed. World Scientific: Singapore, 2016.

6. Hawking, S.W.; Ellis, G.F.R. The Large Scale Structure of Space-Time; Cambridge University Press: Cambridge, UK, 1973.

7. Gürsey, F. Introduction to the de Sitter group. Group Theoretical Concepts and Methods in Elementary Particle Physics; Gürsey, F.F. Gordon and Breach: New York, NY, USA, 1962.

8. Coleman, S. Aspects of Symmetry; Cambridge University Press: Cambridge, UK, 1985; Chapter 3

9. Araujo, A.; Jennen, H.; Pereira, J.G.; Sampson, A.C.; Savi, L.L. On the spacetime connecting two aeons in conformal cyclic cosmology. Gen. Rel. Grav.; 2015; 47, 151. [DOI: https://dx.doi.org/10.1007/s10714-015-1991-4]

10. Pereira, J.G.; Sampson, A.C. de Sitter geodesics: Reappraising the notion of motion. Gen. Rel. Grav.; 2012; 44, 1299. [DOI: https://dx.doi.org/10.1007/s10714-012-1338-3]

11. Landau, L.D.; Lifshitz, E.M. The Classical Theory of Fields; Butterworth-Heinemann: Oxford, UK, 1980; Chapter 12

12. Araujo, A.; López, D.F.; Pereira, J.G. de Sitter invariant special relativity and the dark energy problem. Class. Quantum Grav.; 2017; 34, 115014. [DOI: https://dx.doi.org/10.1088/1361-6382/aa6bbb]

13. Gibbons, G.W.; Patricot, C.E. Newton–Hooke spacetimes, Hpp-waves and the cosmological constant. Class. Quantum Grav.; 2003; 20, 5225. [DOI: https://dx.doi.org/10.1088/0264-9381/20/23/016]

14. Susskind, L. De Sitter Holography: Fluctuations, Anomalous Symmetry, and Wormholes. Universe; 2021; 7, 464. [DOI: https://dx.doi.org/10.3390/universe7120464]

15. Gibbons, G.W.; Hawking, S.W. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D; 1977; 15, 2738. [DOI: https://dx.doi.org/10.1103/PhysRevD.15.2738]

16. Farrah, D.; Croker, K.S.; Zevin, M.; Tarlé, G.; Faraoni, V.; Petty, S.; Afonso, J.; Fernandez, N.; Nishimura, K.A.; Pearson, C. Observational Evidence for Cosmological Coupling of Black Holes and its Implications for an Astrophysical Source of Dark Energy. Astrophys. J. Lett.; 2023; 944, L31. [DOI: https://dx.doi.org/10.3847/2041-8213/acb704]

17. Callan, C.G.; Coleman, S.; Jackiw, R. A New Improved Energy-Momentum Tensor. Ann. Phys.; 1970; 59, 42. [DOI: https://dx.doi.org/10.1016/0003-4916(70)90394-5]

18. ’t Hooft, G. Local conformal symmetry: The missing symmetry component for space and time. Int. J. Mod. Phys. D; 2015; 24, 1543001. [DOI: https://dx.doi.org/10.1142/S0218271815430014]

19. Hawking, S.W. Particle creation by black holes. Commun. Math. Phys.; 1975; 43, 199.Erratum in Commun. Math. Phys. 1976, 46, 206 [DOI: https://dx.doi.org/10.1007/BF01608497]