# Bootstrapped Newtonian stars and black holes

## Abstract

We study equilibrium configurations of a homogenous ball of matter in a bootstrapped description of gravity which includes a gravitational self-interaction term beyond the Newtonian coupling. Both matter density and pressure are accounted for as sources of the gravitational potential for test particles. Unlike the general relativistic case, no Buchdahl limit is found and the pressure can in principle support a star of arbitrarily large compactness. By defining the horizon as the location where the escape velocity of test particles equals the speed of light, like in Newtonian gravity, we find a minimum value of the compactness for which this occurs. The solutions for the gravitational potential here found could effectively describe the interior of macroscopic black holes in the quantum theory, as well as predict consequent deviations from general relativity in the strong field regime of very compact objects.

## 1 Introduction and motivation

The true nature of black holes is already problematic in the classical description given by general relativity and notoriously more so once one tries to incorporate the unavoidable quantum physics. Once a trapping surface appears, singularity theorems of general relativity require an object to collapse all the way into a region of vanishing volume and infinite density [1]. At the same time, a point-like source is well known to be classically unacceptable [2, 3, 4]. One would therefore hope that quantum physics cures this problem, the same way it makes the hydrogen atom stable, by affecting the gravitational dynamics, at least in the strong field regime (where the description of matter likely requires physics beyond the standard model as well [5, 6]).

*M*is the total ADM mass [11] of the system and

In Ref. [7], we showed that the qualitative behaviour of the complete solutions to that non-linear equation resembles rather closely the Newtonian counterpart. This result, which essentially stems from including a gravitational self-interaction in the Poisson equation, is what we call “bootstrapping” the Newtonian gravity. In fact, including those specific non-linear terms could be viewed as the first step in the perturbative reconstruction of classical general relativity (see, e.g. Refs. [12, 13, 14]). However, we could also entertain the idea that these terms are meaningful to determine the (mean field) gravitational potential of extremely compact objects if the *quantum* break-down of classical general relativity occurs at *macroscopic* scales and general relativity therefore fails at describing (the interior of) black holes [15, 16, 17, 18, 20, 21]. In this case, although its origin lies in the quantum nature of gravity (and matter), if this effective potential applies for macroscopic sources, it does not need to contain explicitly a dependence on the Planck constant \(\hbar \) such that general relativistic configurations are (formally) recovered for vanishing \(\hbar \).^{1} On the other hand, Newtonian physics is recovered (by construction) for sources with small compactness \(G_\mathrm{N}\,M/R\sim R_\mathrm{H}/R\ll 1\), and one can therefore consider that the compactness \(G_\mathrm{N}\,M/R\) is the parameter measuring deviations from general relativity in the bootstrapped potential.^{2} To be more specific, we expect that the bootstrapped potential admits a description in terms of a quantum state of bound gravitons, like the coherent state that can be used to reproduce the Newtonian potential [8, 9]. The quantum features of the system would hence become apparent only after such a quantum state is constructed explicitly (which we leave for future investigations). Moreover, by studying static sources of uniform density \(\rho \) in Ref. [7], we found a finite matter pressure *p* could support sources of arbitrarily large compactenss \(G_\mathrm{N}\,M/R\sim R_\mathrm{H}/R\gg 1\), so that there is no equivalent of the Buchdahl limit [23] of general relativity in the bootstrapped Newtonian gravity. Of course, the pressure becomes the dominant source of energy when \(G_\mathrm{N}\,M/R\gg 1\) and, although the strong energy condition \(\rho +p>0\) still holds, the dominant energy condition \(\rho \ge |p|\) is violated in this regime (see, e.g. Ref. [24]). This suggests that the source of highly compact configurations, such as black holes, must be matter in a quantum state with no purely classical analogue (like Bose–Einstein condensates [15, 16, 17, 18, 19, 20, 21] or degenerate neutron stars). This result is again consistent with the fact that classical general relativistic configurations are expected to become physically relevant only for astrophysical objects with small compactness \(R_\mathrm{H}/R\ll 1\).

Since we are mainly interested in investigating static sources which we found can be very compact, a pressure term which prevents the gravitational collapse needs to be included from the onset. For this reason, we here modify the effective theory used in Ref. [7] in order to consistently supplement the matter density with the pressure as sources of the gravitational potential. We then study systems with generic compactness \(G_\mathrm{N}\,M/R\sim R_\mathrm{H}/R\), from the regime \(R\gg R_\mathrm{H}\), in which we recover the standard post-Newtonian picture, to \(R\ll R_\mathrm{H}\) where we find the source is enclosed within a horizon. The latter is defined according to the Newtonian view as the location at which the escape velocity of test particles equals the speed of light. Of course, it should be possible to treat the single microscopic constituents of the source in this test particle approximation and the presence of an horizon therefore refers to their inability to escape the gravitational pull.

Like in Refs. [7, 9], we shall just consider (static) spherically symmetric systems, so that all quantities depend only on the radial coordinate *r*, and the matter density \(\rho =\rho (r)\) will also be assumed homogenous inside the source (\(r\le R\)) for the sake of simplicity. The pressure will instead be determined consistently from the condition of staticity. The paper is organised as follows: in Sect. 2, we briefly review the derivation of the equation for the potential with the inclusion of a pressure term; in Sect. 3, we solve for the outer and inner potential generated by the homogenous source using appropriate analytical methods for the diverse regimes. In particular, we study intermediate and large compact sources with \(R\lesssim R_\mathrm{H}\) as possible candidates for effectively describing collapsed objects which should act as black holes according to general relativity; their horizon structure is then analysed in Sect. 4; we finally comment about our results and possible outlooks in Sect. 5.

## 2 Bootstrapped theory for the gravitational potential

^{3}

*p*which prevents the system from collapsing becomes very large for compact sources with a size \(R\lesssim R_\mathrm{H}\), where \(R_\mathrm{H}\) is the gravitational radius of Eq. (1.1). We must therefore add a corresponding potential energy \(U_\mathrm{B}\), associated with the work done by the force responsible for the pressure

*p*, such that

^{4}for the three different currents \(J_V\), \(J_\mathrm{B}\) and \(J_\rho \) respectively. They also allow us to control the origin of non-linearities, as we recover the Newtonian Lagrangian (2.1) by setting all of them equal to zero.

*V*reads

*p*and the derivative self-interaction term in the right hand side.

## 3 Homogeneous ball in vacuum

^{5}

### 3.1 Outer vacuum solution

*M*for large

*r*. In fact, the large

*r*expansion now reads

### 3.2 The inner pressure

### 3.3 The inner potential

^{6}), that is

#### 3.3.1 Small and intermediate compactness

*R*of the source much larger or of the order of \(G_\mathrm{N}\,M\), an analytic approximation \(V_\mathrm{s}\) for the solution \(V_\mathrm{in}\) can be found by simply expanding around \(r=0\), and turns out to be

*r*in the Taylor expansion about \(r=0\) must vanish.

*M*. Using the first equation in (3.21), one then finds

*M*only as

*C*determined according to the theorem in Appendix C (with \(C>1\) for small compactness and \(C<1\) for intermediate compactness). This means that the approximate solution (3.18) overestimates the expected true potential \(V_\mathrm{in}\) for low compactness, whereas it underestimates \(V_\mathrm{in}\) when the compactness grows beyond \(G_\mathrm{N}\,M / R \simeq 1/20\). We also note that the gap between the above \(V_{-}\) and \(V_{+}\) increases for increasing compactness, which signals the need for a better estimate of \(M_0=M_0(M)\) in order to narrow this gap and gain more precision for describing the intermediate compactness. The latter regime is particularly useful for understanding objects that have collapsed to a size of the order of their gravitational radius.

^{7}We should remark that, in this analysis, we actually employed the comparison method in the whole range \(0\le r< \infty \) by defining \(V_\pm =C_\pm \,V_\mathrm{out}\), for \(r>R\), where \(V_\mathrm{out}\) is the exact solution in Eq. (3.7) (see Figs. 4 and 5). This means that we did not require that the lower function \(V_-\) (for \(G_\mathrm{N}\,M/R \lesssim 1/20\)) and the upper function \(V_+\) (for \(G_\mathrm{N}\,M/R > rsim 1/20\)) satisfy the boundary conditions (3.4) and (3.5) at \(r=R\). However, since we have the analytical form for \(V_\mathrm{out}\) in its entire range of applicability, all that is needed to ensure that \(V_\pm \) are the upper and lower bounding functions is for the constants \(C_\pm \) which multiply the expression for \(V_\mathrm{out}\) to be smaller, respectively larger than one.

*M*as

#### 3.3.2 Large compactness

*A*,

*B*and \(M_0\) can be fixed (for any value of

*R*) by imposing the boundary conditions (3.3), (3.4) and (3.5). Regularity at \(r=0\) in particular yields

*B*in terms of

*M*and

*R*. Putting everything together, we obtain

*A*,

*B*and \(M_0\) should again be computed from the three boundary conditions, so that \(V_\mathrm{in}\) eventually depends only on the parameters

*M*and

*R*. Since solving for \(f=f(r)\) is not any simpler than the original task, we shall instead just find lower and upper bounds, that is constants \(C_\pm \) such that

*C*. Equation (3.3) yields the same expression (3.31), whereas the l.h.s. of Eq. (3.32) is just rescaled by the factor

*C*and continuity of the derivative therefore gives the approximate solution

*R*and

*M*, one can then numerically determine a constant \(C_+\) such that \(E_+<0\) and a constant \(C_-<C_+\) such that \(E_->0\).

For example, for the compactness \(G_\mathrm{N}\,M/R=10^3\), we can use \(C_-\simeq 1\) and \(C_+\simeq 1.6\), and the plots of \(E_-\) and \(E_+\) are shown in Fig. 8. In particular, the minimum value of \(|E_+|\simeq 14\). The corresponding potentials \(V_\pm \) along with \(\tilde{V}=\tilde{C}\,\psi \), where \(\tilde{C}=(C_++C_-)/2\), are displayed in Fig. 9. It is easy to see that the three approximate solutions essentially coincide almost everywhere, except near \(r=0\) where they start to fan out, albeit still very slightly (the right panel of Fig. 9 shows a close-up of this effect). A similar behaviour is obtained for larger values of \(G_\mathrm{N}\,M/R\). For smaller values of the compactness up to \(G_\mathrm{N}\,M/R\simeq 50\), the approximation (3.40) is still quite accurate (see Fig. 10), even if the smaller the compactness the bigger the difference between \(V_\pm \). Actually, the error in the derivative of the potential at \(r=R\) is of the order of \(0.01\,\%\) and \(0.6\,\%\) for \(G_\mathrm{N}\,M/R=10^2\) and \(G_\mathrm{N}\,M/R=50\), respectively. In order to obtain a comparable precision for lower compactness, the approximate expression (3.40) should be improved, but we do not need to do that given how accurate is the perturbative expansion employed in Sect. 3.3.1.

*M*can then be obtained from the simple linear approximation

## 4 Horizon and gravitational energy

The approach we used so far completely neglects any geometrical aspect of gravity. In particular, it is well known that collapsing matter is responsible for the emergence of black hole geometries, providing us with the associated Schwarzschild radius (1.1). In general relativity, this marks the boundary between sources which we consider as stars (\(R\gg R_\mathrm{H}\)) and black holes (\(R\lesssim R_\mathrm{H}\)). Moreover, if the pressure is isotropic, stars must have a radius \(R>(9/8)\,R_\mathrm{H}\), known as the Buchdahl limit [23], otherwise the necessary pressure diverges.

*R*of the matter source, which occurs when

*M*. We can summarise the situation as follows

One can also calculate the three components of the gravitational potential energy in the regime of intermediate compactness \(G_\mathrm{N}\,M/R\sim 1\), but the explicit expressions would be too cumbersome to display. Instead, the left panel of Fig. 17 shows a comparison in the regime of low compactness between the above expression for \(U_\mathrm{G}\) and the one obtained starting from the analytic approximations from Eqs. (3.23) and (3.24), which are valid both for sources of low and intermediate compactness. It can be seen that the two approximations lead to similar results for objects that have low compactness. The center panel also shows the behaviour of \(U_\mathrm{G}\) for objects of intermediate compactness. As expected, the gravitational potential energy becomes more and more negative as the density of the source increases.

Of course, the total energy of the system should still be given by the ADM-like mass *M*, which must therefore equal the sum of the matter proper mass \(M_0\) and the energy associated with the pressure (see Appendix D for more details about the energy balance).

## 5 Conclusions and (quantum) outlook

In this work we have fully developed a bootstrapped model of isotropic and homogeneous stars, in which the pressure and density both contribute to the potential describing the gravitational pull on test particles. No equivalent of the Buchdahl limit was found, and the matter source can therefore be kept in equilibrium by a sufficiently large (and finite) pressure for any (finite) value of the compactness \(G_\mathrm{N}\,M/R\). When the compactness of the source exceeds a value of order 0.46, a horizon appears inside the source and its radius \(r_\mathrm{H}\simeq R\) for \(G_\mathrm{N}\,M/R\simeq 0.69\). For larger values of the compactness, the source is entirely inside \(r_\mathrm{H}\) and we can consider cases with \(r_\mathrm{H} > rsim R\) as representing bootstrapped Newtonian black holes.

*p*is required in order to support the matter core. In fact, if we assume that black holes have regular inner cores of finite proper mass \(M_0\) and thickness

*R*, from Eq. (3.43) we obtain

Correspondingly, the regular potential we obtained in the present work should be viewed as the mean field description of the quantum state of the (off-shell) gravitons in a (regular^{8}) black hole when \(R\lesssim r_\mathrm{H}\). It will be therefore a natural development to investigate the quantum features of this potential, as it affects both the quantum state of matter inside the black hole (or falling into it) and the dynamics of the gravitons themselves. Eventually, one would also like to identify the fully quantum state that generates this potential, like it was done for the Newtonian potential in Refs. [9, 33, 34], or in Refs. [40, 41, 42]. Finally, we would like to remark that, although we found that \(M\gg M_0\) for very large compactness, and one could thus infer that matter become almost irrelevant inside a black hole [15, 16, 17, 18, 19, 20, 21], the above picture inherently requires the presence of matter, whose role in black hole physics we believe needs more investigations [35, 36, 37, 38, 39].

## Footnotes

- 1.
Terms explicitly proportional to \(\hbar \) are usually obtained as perturbative corrections to classical solutions and they can therefore be trusted only as long as they remain smaller than the quantities they perturb (see, e.g. Ref. [22]).

- 2.
A detailed study of orbits in the region outside the source is underway.

- 3.
- 4.
Different values of \(q_V\), \(q_\mathrm{B}\) and \(q_{\rho }\) can be implemented in order to obtain the approximate potentials for different motions of test particles in general relativity and describe different interiors.

- 5.
More realistic energy densities with physically motivated equations of state will be considered in future developments.

- 6.
We just remark here that the comparison theorems do not require that the approximate solutions \(V_\pm \) have the same functional forms of the exact solution \(V_\mathrm{in}\).

- 7.
- 8.
For a review, see Ref. [43]

## Notes

### Acknowledgements

R.C. and M.L. are partially supported by the INFN Grant FLAG. The work of R.C. has also been carried out in the framework of activities of the National Group of Mathematical Physics (GNFM, INdAM) and COST action *Cantata*. O.M. is supported by the Grant Laplas VI of the Romanian National Authority for Scientific Research.

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