1 Introduction↩︎

The aim of this paper is to relate two ideas that have appeared in the recent literature. The first idea, reviewed in [1], is the suggestion that horizonless black hole mimickers termed “dynamical gravastars” form when matter under extremely high pressure undergoes a phase transition to a state with negative energy density, as is permitted when quantum effects are taken into account. These compact objects obey the “strong” and “null” energy conditions and since they are globally unitary, they thus eliminate the “information paradox” problem associated with black hole horizons. The second idea is the suggestion [2]–[3] that the “little red dots” observed by the James Webb Space Telescope (JWST) are a new type of compact relativistic object, taking the form of an ultramassive black hole surrounded by an opaque atmosphere, which absorbs energy arising from black hole accretion and emits it as very low energy (hence red) radiation. Our purpose here is to give an alternative mechanism for the formation of the “dots”, based on the large gravitational red shift associated with generation of a dynamical gravastar.

In Section 2 we briefly review the model for formation of black hole mimickers that we have developed in a series of papers [1]. The model is based on solving the Tolman-Oppenheimer-Volkoff (TOV) relativistic pressure balance equations, with an equation of state jump from a relativistic matter equation of state in the exterior, to an equation of state with positive pressure and negative matter energy density in the interior. In particular, we include a figure showing that at radii well above the equation of state jump, the metric component \(g_{00}\) is already exponentially small, showing that there will be a large red shift when latent energy emitted below the jump radius exits to infinity.

In Section 3 we briefly review the observations by the JWST leading to the discovery of what are now colloquially called “little red dots”, as well as some of the recent literature on the astrophysical modeling of what the dots signify.

In Section 4 we give a simplified model of a dynamical gravastar accreting matter from its environment, at a slow enough rate so that it grows by passing through the equilibrium states described in Section 2. Following this growth using the Mathematica notebook for dynamical gravastars shows that as the mimicker mass grows, the depth of the gravitational potential well in which latent energy is released correspondingly grows, and so does the red shift of exiting radiation. This calculation also shows that as the mimicker grows, it continues to obey the mathematical black hole radius–mass relation \(R=2GM\) (which in geometrized units with \(G=1\) as used in the Mathematica notebook is simply \(R=2M\)).

In Section 5 we pose questions that may be addressable in future astrophysical observations and modeling. Finally, in Section 6 we summarize the multiple features that all follow from the single assumption of a high pressure phase transition of matter to a state with negative energy density.

2 The dynamical gravastar model↩︎

The dynamical gravastar model reviewed in [1] consists of solving the TOV equations [4]–[5] (which are the relativistic generalization of the hydrostatic equilibrium equations [6] used in nonrelativistic calculations of stellar structure) assuming an equation of state undergoing a phase transition at very high fluid pressure. For a spherically symmetric fluid with pressure \(p(r)\) and energy density \(\rho(r)\), the TOV equations take the general form \[\begin{align} \label{newTOV} \frac{dm(r)}{d r}=&4\pi r^2\rho(r)~~~,\cr \frac{dp(r)}{d r}=&-\frac{\rho(r)+p(r)}{2} \frac{d\nu(r)}{d r} ~~~,\cr \frac{d\nu(r)}{d r}=&\frac{N(r)}{1-2m(r)/r}~~~,\cr N(r)=&(2/r^2)\big(m(r)+4\pi r^3 p(r)\big)~~~.\cr \end{align}\tag{1}\] Here \(m(r)\) is the volume integrated energy density within radius \(r\), and \(\nu(r)=\log\big(g_{00}(r)\big)\). The general form TOV equations become a closed system when supplemented by an equation of state \(\rho(p)\) giving the energy density \(\rho\) in terms of the pressure \(p\). In the model reviewed in [1] the equation of state used is a relativistic matter equation of state \(\rho(p)=3p\) for \(p\leq {\rm pjump}\), and \(p + \rho(p) =\beta\), for \(p > {\rm pjump}\), with \(\beta\) a small positive constant that is a parameter of the model. The constant \(\beta\) is a stand-in for more complex equation of state physics, and its positive value maintains decrease of pressure from the center of the black hole mimicker to the exterior. The model exhibits a metric structure very similar to that of a Schwarzschild black hole in an exterior region dictated by the equations, whereas \(g_{00}\) becomes exponentially small, but remains always positive, in the interior region. This is illustrated in Fig. 1, which shows that the exponential decrease in \(g_{00}\) commences in the exterior region, at pressures well below \({\rm pjump}\).

3 JWST, “little red dots”, and current astrophysical interpretation↩︎

Early JWST observations found many small, circular red objects present in the first billion years of the evolution of the universe. Initial interpretations of these as reddish mature galaxies or as galaxies shrouded in dust did not survive more detailed observations and analysis [3]. The dots are too compact to be galaxies, and dust shrouded objects should re-emit at longer millimeter wavelengths where nothing is seen. A currently favored hypothesis [7], [8] is that the dots are a new class of objects termed “black hole stars” [9], in which an accreting supermassive black hole is surrounded by an opaque layer of gas, which is heated by energy released by the black hole but then re-radiates this energy in the red or infrared much like a red giant star. This picture can explain the compactness and energetics of the dots, and their observed absence of short wavelength ultraviolet and x-ray emission.

4 An alternative interpretation based on the structure of dynamical gravastar black hole mimickers↩︎

In this section we propose an alternative mechanism for the formation of red dots. Like the “black hole star” proposal, it is a variant on conventional black hole ideas. But the redness in our proposal results not from absorption of energetic radiation by an opaque surround with subsequent re-emission at red wavelengths, but rather from the gravitational redshift of initially energetic radiation as it climbs out of the deep gravitational potential well associated with the gravastar. Moreover, in our proposal the source of the initial energetic radiation is not the violent accretion of matter by a central black hole, but rather the latent energy released in the first order phase transition to negative matter energy density at pressures above pjump, as postulated in the dynamical gravastar model. A brief sketch of this mechanism for “dot” formation was given in a post-publication Appendix C added to arXiv:2504.18690v4. We elaborate on our sketch in this section by studying a simple model for dynamical gravastar growth.

Consider an initially small dynamical gravastar embedded in a matter medium (likely hydrogen), from which it grows in mass by accretion. When infalling matter equilibrates with the interior of the dynamical gravastar, our postulate implies that the ultra high pressure causes the matter to undergo a phase transition to a state of negative energy density, incrementally adding to the pre-existing gravastar interior negative energy density. This is accompanied by emission of latent energy as radiation that diffuses off to infinity, emerging highly red shifted, giving the observed “red dot” signature. We assume the accretion is slow enough that the gravastar growth is quasi-equilibrium, in the sense that it can be approximated by a sequence of equilibrium structures described by the TOV equation Mathematica notebook of [1]. The specific model for which we present results has \(\beta=0.01\), and units of length chosen to make pjump=1. The sequence of structures is then generated by considering a sequence of central pressures \(p(0)\) starting from just above pjump, and trending to the largest value for which we could get reliable computational results from the Mathematica notebook. We will see that these central pressure values correspond to an increasing series of gravastar mass values \(M\).

Sample results of this study are given in Table I, and in Figs. 2 and 3. The columns of Table I, reading across, give the central pressure \(p(0)\), the mass \(M\) as seen at effective infinity, the radius \({\rm rjump}=r({\rm pjump})\) where the energy density jumps, the radius \(r_{\rm MIN}\) at the kink minima seen in Figs. 2 and 3 which plot \(1-2m(r)/r\), the central value of \(\log g_{00}(0)=\nu(0)\), and finally the jump value of \(\log g_{00}({\rm rjump})=\nu({\rm rjump})\). Scanning the table, one sees that (i) the mass \(M\) increases monotonically, and very nonlinearly, with \(p(0)\), (ii) the gravastar nominal horizon at the kink minimum obeys \(r_{\rm MIN}\simeq 2 M\), and (iii) the central and jump radius values of \(\log g_{00}(r)=\nu(r)\) decrease monotonically with increasing \(M\), becoming very large negative for large \(M\). From Figs. 2 and 3 we see that \({\rm denom}=1-2m(r)/r\) always has a clear minimum defining \(r_{\rm MIN}\). Also, when \(p(0)\) is not too close to \({\rm pjump}=1\), there is a sharp transition region from the exterior to the interior structure, with most of the transition taking place at radii above the jump radius, as already seen in the stacked plots of Fig. 1. This means that as the mass \(M\) increases, the latent energy released inside the jump radius has an increasingly large redshift to exit to infinity, giving the observational presentation of an energetic “red dot”.

5 Observational questions↩︎

We note two questions that may be addressable in future observations. First, is there an observed correlation between the dot radii and masses, and if so, does this approximate the geometrized black hole radius–mass relation \(R=2M\)? Second, is there a way of distinguishing observationally between redness arising from absorption of radiation by an opaque layer with subsequent thermalized re-radiation, and redness arising as a gravitational redshift effect?

6 Features following from the underlying assumption of a phase transition to a state with negative energy density↩︎

An important criterion for judging the merit of a hypothesis in physics is to look for multiple consequences stemming from a single assumption. As reviewed in [1] and as further elaborated in this paper, our starting hypothesis of a high pressure first order phase transition of matter to a state with negative energy density has the following consequences:

• No one-way horizon, so no associated “unitarity” and “information” paradoxes In the dynamical gravastar model, the metric component \(g_{00}\) is always positive down to the center of the compact object. So physics remains globally unitary, and there is no associated “black hole information paradox”. Globally the universe is causal, without the trillions of causally disconnected regions implied by the existence of horizons.

• Mechanism for forming “red dots” As elaborated in this paper, our starting hypothesis leads to a natural mechanism for the formation of the observed “little red dots” seen in the very early universe.

• An active role for astrophysical black holes in galaxy formation. If astrophysical black holes are dynamical gravastars, since there is no horizon they are “leaky” and can play an active role in galaxy formation, as proposed in [10], [11] and [12].

• Interesting mathematics of autonomous differential equations As briefly discussed in [1], the scale invariant rewriting of the exterior region TOV equations for our model takes the form of a two dimensional pair of autonomous differential equations, as previously noted in [13] and as discussed in more detail in [14]. This gives a powerful tool for analyzing the mathematical structure of the dynamical gravastar proposal.

7 Mathematica code for inversion of \(p(r)\)↩︎

Figs. 2 and 3 can be plotted with the Mathematica code given in Appendix A of [1]. Table I uses the following additional lines of code to invert the interpolating function \(p(r)\) and then calculate rjump:

References↩︎