How Black Holes Remain Neutron Stars

1. Introduction

Usually, the formation of a “black hole” (BH) begins with a neutron star (NS), which is the remnant core of a supernova explosion. The NS is an extremely compact, superdense star composed almost entirely of neutrons packed together. If the size of the NS exceeds a certain “critical mass”, its gravitational field is so strong (i.e. GR spacetime curvature is so great) that light itself cannot escape. The NS then behaves as a BH surrounded by an “event horizon”. This event horizon marks the boundary, within which the star is completely invisible to observation at any wavelength (from gamma rays to X-rays to ultraviolet to visible to infrared and radio). Therefore, no information about the BH can be accessed from within/behind the event horizon.

It is widely accepted that general relativity (GR) field equations, based on a distant external observer, break down at this event horizon, and they cannot be used to model the likely internal structure and dynamics within a BH. It has been assumed that within the BH event horizon, this neutron star core would entirely collapse to form a singularity. A singularity has finite mass but zero size, producing infinite density and infinite GR spacetime curvature.

2. Neutrons

Simplistically, a neutron is composed of three “valence” quarks – an “up” quark with a charge of +²/₃, and two “down” quarks with a charge of −¹/₃. The combination of these three charges produces no net charge (neutral). These quarks are held together by “gluons” which bind the quarks within the neutron. Protons are similar, except they are composed of two up quarks, with their charge of +²/₃, and a down quark, with its charge of −¹/₃, combining to form a net charge of +1 (equal and opposite charge to an electron).

However, the composition of a neutron or proton is far more complicated than this picture presents. In addition to the three valence quarks, there are possibly fourteen quark-antiquark pairs, also held together by gluons. The external outcome of all these interactions is the short-range “strong force” (or “strong interaction”) carried by the gluons, responsible for binding the neutrons and protons within an atomic nucleus (powerful enough to overcome the charge repulsion between protons).

Within a NS, the dense cluster of neutrons are prevented from collapsing under their own gravity as a result of the “Pauli Exclusion Principle”, which states that no two fermions (such as neutrons) can occupy the same quantum state simultaneously. This results in a “neutron degeneracy pressure” resisting the collapse and creating a hydrostatic equilibrium. This degeneracy pressure may be understood as a force derived from the strong nuclear interaction, opposing the breakdown of a neutron into its component quarks and gluons.

It has been widely assumed that when a NS has sufficient mass to form a BH, the gravitational field becomes so strong that this degeneracy pressure is overcome, and the star collapses into a singularity.

However, no causal connection has been reported, between the moment of formation of the event horizon, and the moment of alleged collapse of the neutron core. One event does not necessarily cause the other.

A further complication is to note that, within the nucleus of an atom, the nucleons (neutrons and protons) are not necessarily “touching spheres”, as might be imagined. Each nucleon may be separated by some short equilibrium distance (measured centre-to-centre), greater than the known free-nucleon diameter of ~1.68 fm. A striking example is that of the beryllium (⁹Be) nucleus, with its 9 nucleons structured as two helium nuclei and one neutron, orbiting each-other as three distinct entities [1].

Brodsky et al. [2] have experimentally determined the form and magnitude of the strong force, which rises to a maximum at ~3 fm (~1.8 diameters) separation, in the case of isolated (free) nucleons.

Brodsky et al. [3] explain that: “So far as we know, gluons and quarks are point-like,” and nucleons in isolation, behave differently from how they do in a cluster. “Overlapping nucleons would be very complex objects, possibly with no relation to the individual nucleons.” To date, the magnitude of the neutron degeneracy pressure cannot be accurately quantified, theoretically or empirically.

However, it is accepted that a bound cluster of neutrons is compressible under gravity (thereby increasing its average mass density in proportion to volumetric compression). As a neutron cluster grows and gains mass, the increasing compression of the cluster under its own increasing gravity, may initially involve a reduction in the nucleon separation, until this centre separation reaches their observed free size of ~1.68 fm diameter, prior to any compression of each individual nucleon volume.

3. Neutron stars

So, what is the possible NS size range, and what could be the value of the critical mass described above? Consider NS theoretical evidence (in publishing order):

• Haensel et al. [4] say that a NS has a physical radius of 11 to 13 km containing a mass up to 2 solar masses (2 M_ʘ i.e. 3.98E+30 kg), with a crust density of 0.3 times the normal nuclear saturation density (ρ_ns = 2.7E+17 kg/m³), rising to >2 ρ_ns at the core.

• Katayama and Saito [5] calculate the maximum theoretical NS mass at 2.08 M_ʘ (based on GR).

• Vidaña [6] says that NS’s are likely the densest objects in the universe, with a central density in the range 4 to 8 ρ_ns and a surface radius of 10 to 12 km.

• Giddings [7] says that: “perhaps some unknown laws of physics prevent larger stars from forming BHs and instead lead them to become a kind of massive remnant – more like a NS than a BH – (but) we cannot explain what would prevent their continued collapse under gravity.”

One might anticipate that such a cataclysmic and powerful event as the sudden collapse of a NS to form a BH singularity, as theorised in GR, should generate an observable energy pulse or shock wave, at least from an accretion disk (outside the event horizon). No such evidence is reported.

What may become of neutrons under extremely high compression by gravity has been a matter of conjecture. It was believed that these conditions could not be created in any Earth-bound laboratory, and that the equation of state for neutron-degenerate matter had not been determined [8].

But instead of possible “continued collapse” [7], it seems at least equally likely that neutrons (despite possibly transforming into neutron degenerate matter at some stage), can withstand any level of compression without collapsing into a singularity – being sustained by the neutron degeneracy pressure operating amongst and beyond the quarks of which neutrons are composed – a force understood to become increasingly repulsive at reducing short range and which asymptotes toward infinite repulsion at zero separation. Therefore, collapse should not be possible, and this condition is substantiated by the most recent high-energy physics laboratory evidence:

• Schmidt et al. [9] explain that strong interactions are described by the equations of quantum chromodynamics (QCD) which are not well constrained at short distances, such as in the cores of neutron stars. These interactions have been studied with the CEBAF Large Accelerator Spectrometer (CLAS Collaboration), at densities several times higher than the central density of the nucleus. “As the relative momentum between two nucleons increases and their separation thereby decreases, we observe a transition from a spin-dependent tensor force to a predominantly spin-independent scalar force … We find even at the highest densities; the nucleons seem to keep their identities and don’t turn into this bag of quarks … The strong nuclear force creates a repulsive force between neutrons that, at a neutron star’s core, helps keep the star from collapsing in on itself.”

This is physical/empirical evidence that large neutron stars (beyond the critical mass), do not necessarily collapse into a singularity. It is also noted that in accordance with Haug and Gianfranco [10], the only limit to the compressibility of matter (such as the nucleons in a neutron star), is likely linked to the Planck density itself (which is not “infinite” density).

4. Analysis

Timlin [11] has produced an extensive derivation of the implied neutron degeneracy pressure in equilibrium with gravity:

Where ħ = 1.054E-34 is the reduced Planck constant, M is the stellar mass, m = 1.674E-27 kg is the neutron mass and R is the surface radius. Substituting for the constants we obtain:

In view of Brodsky et al. [2, 3]., this equation can be regarded as a ‘best guess’ guide to quantifying the form and magnitude of the pressure within a neutron cluster, but this steep (5th power) inverse function illustrates how the degeneracy pressure approaches infinity at zero separation. In view of Schmidt et al. [9], this pressure is understood to resist degeneracy, without actual degeneracy (into quarks and gluons) necessarily ever being achieved.

Brodski et al. [3] explain that repulsion between free nucleons begins to dominate at centre separations of 0.25 fm, where quark-quark interactions remain strong.

Given that GR field equations cannot be relied upon to model the structure within a BH, a return to basic principles, with a reduction in complexity and the use of informative Figures is considered necessary to obtain a more intuitive, verifiable NS model.

Kahana [12] affirms that for a spherically symmetric object under a GR regime, in consequence of Birkhoff’s theorem, the metric inside of a shell is flat Minkowski space. So, the internal structure of a massive object can be investigated with the Shell theorem of Newton [13] or Gauss. Bautista [14] cautions that Birkhoff’s theorem may not hold for a body undergoing rotational or radial motions, which introduce velocity time dilation. Zhang [15] notes there is time dilation due to massive distant bodies. But for an isolated, non-rotating object, the Shell theorem is applicable to GR (and classical) physics, whereby the mass beyond a given shell radius has no gravitational effect.

The non-relativistic (Newtonian) gravitational potential energy U existing between two massive objects is:

where the gravitational constant G = 6.673*10⁻¹¹ m³/kg-sec², M and m are masses separated by displacement r (measured from their centres of mass). By the Shell theorem, the gravitational potential energy at the centre of any isolated massive object is zero – a robust small mass placed there would hover, weightless, because all the gravitational forces in all directions cancel and nullify.

Based on Eq. (1) we first obtain the Newtonian velocity (or more-correctly, the speed) of escape from an object of mass M:

where G is defined above, and r is the radial displacement from the object’s centre of mass. Augousti [16] points out that Eq. (4) coincides with the GR equation for the escape speed in the Schwarzchild metric.

Obtaining a reliable relativistic escape speed has proven to be more controversial. Rybczyk [17] provides a derivation of the escape speed: “based on the Lorentz invariant transformation while employing the principles of GR”. He equates the relativistic kinetic energy to the relativistic potential energy and in due course obtains:

where c is the speed of causality, or light (2.998E+08 m/sec), and we set Φ ≤ c². Eq. (3) reduces to the Newtonian escape speed when Φ/c is small. Eq. (5) is fully in agreement with an extensive derivation by Haug [18].

Alternatively, Camp [19] derives the following equation, which we then present in a form similar to Eq. (5):

While Leinenweber [20] derives a different equation which we have re-written in a form more consistent with Eqs. (5) and (6) to assist comparison:

All equations should produce results that match Newtonian escape speeds for non-relativistic objects like the Earth or Sun. However, for relativistic objects such as a 10 M_ʘ star, surprisingly, Eqs. (6) or (7) do not produce peak escape speeds that reach c. Whereas it will be shown that Eq. (4) can produce escape speeds that far exceed c. Overall, Eq. (5) has been found to produce the most consistent and credible data when compared with observations, and this will be relied upon for relativistic calculations here.

Commencing with a non-relativistic object such as the Earth, we apply the Shell theorem together with Eq. (5), summing small incremental steps, to compute the hypothetical “shell escape speed” at increasing shell radii (concentric with the centre of gravity). This method relies on the fact that matter beyond a shell does not exist for calculation purposes (in practice, escaping one shell would then involve the next shell, with a revised escape speed). After applying the Earth density profile data of Dziewonski et al. [21] to each shell, we obtain Figure 1.

Extending this analysis to a much more massive object, Figure 2 represents the shell escape speed for the Sun, based on solar density profile data reproduced by Hathaway [22]. Here we note that the peak shell escape speed is reached far below the surface of this fluid object – this is because of the relatively high density of the core (by ~53% of the surface radius, the density has already fallen to that of drinking water [22]).

What follows is a suggested methodology, open for discussion, that could in future be adjusted according to any new findings on the observed density profile of neutron stars.

The all-important surface radius of a neutron star (to be compared with the Schwarzchild radius) depends on the uncertain density profile. In Figure 3, a range of plausible density profiles has been developed for the 2.08 M_ʘ neutron star of Katayama and Saito [5].

Assuming a crust density of 0.3 ρ_ns as proposed by Haensel et al. [4], Figure 3 anticipates that all possible profiles would lie between the bounds of:

1. a maximum density of 8 ρ_ns [Vidaña, 6], which is achieved close to the surface – suggesting a crystalline-like structure, and

2. unlimited compressibility because the neutron degeneracy pressure asymptotes to infinity – suggesting an elastic structure.

Note that a wide range of possible NS density profiles (linear and variously curved) have been estimated and applied in this analysis (Figures not shown). Application of the various profiles was verified by summation of all the incremental shell masses, to ensure they match the specified star mass as a whole. In each case, the surface radii were determined from the density profile and the specified star mass.

It will be shown, however, that the closest match to observation data is achieved by specifying unlimited compressibility. This was also considered the most likely phenomenon, in light of Schmidt et al. [9] and Haug [10], and also because it is expected that the neutron degeneracy pressure asymptotes toward infinity at zero range.

Now, using relativistic speed Eq. (5), we calculate the shell escape speed (as a fraction of c), versus radial displacement for the predicted NS of 2.08 M_ʘ, according to the full span of possible density profiles in Figure 3, as presented in Figure 4. These extremes of density profile embrace a 6.0 to 7.96 km range of possible surface radii.

In Figure 4, in the case of a density profile that is compressible up to 8 ρ_ns, we note that a peak hypothetical escape speed of 0.792 c (2.37E+08 m/sec) occurs at the surface (7.96 km radius). Alternatively, in the case of a linearly compressible density profile, the peak hypothetical escape speed of 0.886 c (2.65E+08 m/sec) occurs at 5.5 km radius which is 0.5 km below the surface, whereas at the surface (6.0 km), the escape speed is 0.876 c (2.62E+08 m/sec).

Either result for surface-escape speed is significantly below the photon speed (1.000 c), so this object should be visible.

Due to high surface temperatures, neutron stars emit strongly in X-ray (consistent with blackbody radiation), which may also contain the Fe Kα spectral line [23, 24, 25]. This ~6.4 keV line would be subject to gravitational redshift.

The line-of-sight redshift, Z, associated with the escape speed at the surface, is obtained from Eq. (8):

In Figure 4 we found that the plausible surface escape speed ranges from 79.2% c at a possible 7.96 km radius, to 87.6% c at a possible 6.0 km radius. These determine a redshift of 1.9 to 2.9 respectively. This is a significant difference, so if the gravitational redshift of such stars can be detected and measured, this should inform the true density profile (discussed later).

Based on Katayama and Saito [4], we would expect a NS of 2.08 M_ʘ to be nearly massive enough to form a BH, but according to Figure 4, the formation of a BH should require substantially greater mass, or higher density (smaller surface radius). We are drawn to review the observational evidence for the size of BH’s as follows (in publishing order):

• Gelino [26] initially estimated GRS 1009–45 at 5.2 M_ʘ, but this has since been revised to 8.5 M_ʘ [StarDate Black Hole Encyclopedia 2012].

• Gelino and Harrison [27] report that the smallest stellar BH yet found, in GRO J0422 + 32, with a size range of 3.66 to 4.97 M_ʘ. Further analysis in 2012 revised this mass to 2.1 M_ʘ, which suggests this object is in fact a NS.

• Casares [28] lists 20 “confirmed cases” of stellar BH’s – binaries with mass functions exceeding 5–6 M_ʘ, and X-ray transients with mass estimated in the range 4–14 M_ʘ.

• Remillard and McClintock [29] report that H 1705–250 has a confirmed dynamical measurement for the mass of its BH in the range of 5.6 to 8.3 M_ʘ.

• Kreidberg et al. [30] report on several analyses, that indicate a significant “mass gap” between the anticipated maximum neutron star mass (3 M_ʘ) and the anticipated low end of the BH mass distribution (5 M_ʘ).

• Ozel et al. [31] concur with the existence of such a gap in the observed masses of these stars, occurring between 2 M_ʘ and 6 M_ʘ.

• Antoniadis et al. [32] find that one of the largest neutron stars yet detected is 2.01 ± 0.04 M_ʘ.

• Cromartie et al. [33] of Green Bank report discovery of a neutron star J0740 + 6620 with a mass of 2.14 M_ʘ.

• Sathraprakash [34] estimates neutron star masses of up to 2.2 M_ʘ and 2.6 M_ʘ involved in mergers with BH’s, based on LIGO-Virgo-KAGRA data.

• Bodensteiner et al. [35] compile a list of some 200 NS’s and BH’s detected using gravitational wave or electromagnetic detectors, revealing a paucity within the 2 to 5 M_ʘ range (with very wide error bars on gravitational detections).

The mass gap identified above could be due to an observational bias, or obscurity due to dimming and redshift at the higher masses, but according to the above observations, we should expect BH formation to occur at a minimum mass of ~5 M_ʘ.

A slightly lower mass of 4.75 M_ʘ is initially chosen for examination. Extending the criteria of Figure 4, we investigate the range of plausible density profiles for a neutron star of 4.75 M_ʘ, which obtains surface radii ranging from 7.56 km to 10.38 km.

Again using relativistic speed Eq. (5), Figure 5 explores a range of hypothetical shell escape speeds expressed as a fraction of the photon speed, c, for a neutron star of mass 4.75 M_ʘ according to the full span of possible density profiles. In the case of a density profile that is compressible up to 8 ρ_ns we find the peak escape speed of 0.950 c occurs at the surface (10.6 km radius). Whereas in the case of the linearly compressible density profile, the peak escape speed just reaches the photon speed at a radius of 6.8 km, being 0.8 km below the surface.

We shall note that this is the first occurrence of an event horizon for a neutron star. This horizon is contained wholly within the star, a scenario that calls into question whether the GR assumed collapse of all matter within the horizon, while leaving outer layers unaffected, is even conceivable.

In this linearly compressible case, the shell escape speed falls to 0.997 c at the surface (7.60 km). In accordance with Eq. (8), this highest escape speed of 0.997 c at the surface, produces an extremely high redshift of 26.6. Even in the case of the lowest density profile, the surface redshift is 5.0, which is still quite substantial. This range of possible redshift would result in significant reddening and dimming of the star, but the star may remain electromagnetically visible. This 4.75 M_ʘ neutron star having such a high gravitational redshift, is consistent with a minimum observed BH mass of ~5 M_ʘ.

Included for comparison in Figure 5 are the Schwarzchild GR results according to Eq. (4) [15], for the two extremes of density profile used (dashed curves). These results clearly exceed 1.00 c, which cannot be valid, confirming our decision to rely upon Eq. (5).

Consider now a slightly larger neutron star of 5 M_ʘ for which the range of plausible density profiles, produces a range of radii from 7.65 to 10.6 km. The shell escape speeds (again using relativistic speed Eq. (5)), for the range of plausible density profiles are shown in Figure 6.

In the case of a density profile that is compressible up to 8 ρ_ns we find a peak escape speed of 0.950 c occurring at the surface, at a radius of 10.6 km, producing a large redshift of 4.5. In the case of a linearly compressible density profile, the escape speed reaches c for all radii between 6.4 and 7.4 km, being just slightly below the surface of 7.65 km. The surface escape speed is 0.9994 c corresponding to an extreme redshift of 55.8, so this star may be nearly invisible, and may approximately represent the critical size at the cusp of invisibility.

Once again referring to Figure 6, when approaching from the direction of the core, a relativistic shell escape speed equal to the photon speed is reached at an “inner horizon”, formed at a radius of 6.4 km (not to be confused with the once proposed Cauchy horizon).

When approaching from an external direction, the photon speed is reached at an “outer horizon” at a radius of 7.4 km. Apparently, a pair of event horizons are thereby defined. The region between the inner horizon and the outer horizon forms a “photon trap” – a broad “event zone” from which photons cannot escape.

Any internal photons released anywhere inside the event zone would (subject to scattering or absorption) circulate within the zone indefinitely, as would any external photons entering the event zone from beyond either horizon.

As noted, only the linearly compressible density profile scenario shown in Figure 3, has allowed invisibility in neutron stars to begin as small as 5 M_ʘ.

If in fact there is a phase change in stellar composition occurring between 3 M_ʘ and 5 M_ʘ as suggested by Kreidberg et al. [30], then the stellar density would increase and the surface radius would correspondingly reduce, without affecting the outer event horizon radius.

Consider now the shell escape speeds for the same range of plausible density profiles in a neutron star of higher mass such as 6.5 M_ʘ as shown in Figure 7. In the case of a density profile that is compressible up to 8 ρ_ns we find a peak escape speed of 0.987 c at a surface radius of 11.5 km (redshift of 11.3).

In the case of linear compressibility, we find an inner horizon is developed at 5.5 km and an outer horizon is developed at 9.6 km. We note that at an event horizon, the redshift is infinite [34] (invisible). Since the outer horizon is well beyond the surface radius of 8.2 km, this star is certainly invisible. Such an invisible object is preferably termed a “black star” [34] (a star which is electro-magnetically invisible), which removes the connotation of the object necessarily containing or forming a wormhole.

If we investigate a significantly larger mass star of 13.2 M_ʘ for example, we find that the photon trap region is expanded. The inner horizon is at 7.0 km, and the outer horizon is at 19.5 km, being well outside the surface radius of 12.7 km (Figure not shown).

As we extend this analysis to consider possible black stars of larger mass such as 35 M_ʘ we obtain a nominal surface radius of 16.2 km and an expanded photon trap zone ranging from 3.5 to 52.5 km as shown in Figure 8.

Consider the implications of this trap region (or event zone) using Figure 7 as an example. Can matter or radiation continue to reside within this zone, including the compacted neutrons (or neutron degenerate matter) comprising the outer layers of this star?

Under GR it is widely anticipated that, in order to remain stable and avoid collapsing to the stellar centre, matter and radiation within the event horizon (the outer horizon) would need to move outward faster than the speed of light, which is accepted as impossible. It has been claimed that for this reason, collapse would be inevitable and unlimited, resulting in a singularity. This argument fails, because of the previously unknown inner horizon, which presents a barrier to any possible collapse (the shell escape speeds at lesser radii are subluminal).

If we challenge the model further, by taking a more extreme jump to 1000 M_ʘ we find a surface radius of 37.5 km, with the photon trap zone extending from an inner horizon of 2.5 km radius to an outer horizon of 1470 km radius (Figure not shown).

The linearly compressible density profile data proposed herein are summarised in Figure 9 (log scale). Here we see that the surface radius and the central density, as functions of stellar mass, each conform to a one-quarter power exponential curve, fitting data with high precision (R² = 1).

The critical mass value of 5 M_ʘ was selected, to be consistent with the observational evidence cited earlier [26, 27, 28, 29, 30, 31, 32, 33, 34, 35], but it should be regarded as an estimate which is tested using the calculations and Figures shown here. Future, more-comprehensive observational evidence may be used to refine this estimate.

If our methodology used herein holds, then based on this critical mass, if the gravitational redshift for a neutron star could be detected (after accounting for any velocity redshift), Figure 10 could inform or confirm its likely mass and radius (e.g., a redshift of 10 indicates a mass of 4.1 M_ʘ and a radius of 7.3 km). Figure 10 also demonstrates a smooth transition toward invisibility, rather than a sudden, cataclysmic event (implied by GR).

By collating all the foregoing data, we can chart the photon trap dimensions (the inner and outer horizons), together with the surface radii, for a range of stellar masses, in Figure 11.

The point of convergence of the inner and outer horizon radii (lower left of Figure 11, below the surface) represents the minimum mass for an event horizon to exist, being approximately 4.75 M_ʘ. We see that the outer horizon exceeds the surface radius at masses beyond 5 M_ʘ, whereby the star becomes invisible.

Importantly, when an outer horizon is developed, so too is an inner horizon. Given the existence of the inner horizon, and in light of Schmidt et al. [9], and also in light of Barcelo et al. [36] (who characterise the situation as: ‘quantum matter always seems to find new ways of delaying gravitational collapse’), there appears to be no compelling physical reason for a core collapse of an invisible (black) star that would result in the production of a singularity.

It seems unlikely that broadening the scope of this analysis to include charged, rotating and/or revolving stars, or the influence of other distant massive objects, could result in the formation of a real singularity.

In contrast to the inner horizon radius, the outer horizon radius increases linearly with mass, and is unrelated to the density profile assumptions made here.

On the assumption of a non-rotating, spherically symmetric body, the Schwarzchild radius equation [37] has been applied to Figure 11 (dashed line) as follows, with stellar mass M_S measured in M_ʘ:

By inspection, we find that the outer event horizon radius (based on relativistic speed Eq. (5)) calculates to exactly 50% of the Schwarzchild radius, consistently across all data obtained (>5 M_ʘ). Accordingly:

This is confirmed by Haug [18] who newly identified the escape radius (where the escape speed reaches the photon speed), is: “equal to a half of the Schwarzschild radius”.

5. Conclusions

The foregoing has used a methodology (based on a critical mass of ~5 M_ʘ), that is open for discussion. This methodology could in future be adjusted according to any new findings on the real density profile of neutron stars.

1. At least within the range of stellar masses up to 1000 M_ʘ modelled here, isolated neutron stars fit a continuum of increasingly massive neutron stars that happen to become electromagnetically invisible once their mass exceeds ~5 M_ʘ, described here as the “critical mass”.

2. Neutron stars may appear increasingly dim and red-shifted as masses increase from 3 to 5 M_ʘ, leading to the appearance of a mass gap in observations.

3. The closest match to observation data was achieved by specifying linear (elastic) neutron compressibility under gravity, resisted by neutron degenerate pressure. The magnitude of this pressure (based on the strong nuclear force) has not been reliably quantified.

4. A photon trap (or event zone) is developed between an inner event horizon and an outer event horizon. The inner horizon first appears below the surface at some 6.75 km radius, in a neutron star that has reached a mass of ~4.75 M_ʘ, reducing to 2.5 km radius at 1000 M_ʘ. The outer horizon first appears coincident with the inner horizon at ~4.75 M_ʘ, separating at higher masses, increasing to ~1475 km at 1000 M_ʘ.

5. The outer horizon radius corresponds mathematically to exactly 50% of the Schwarzchild radius.

6. The existence of an inner event horizon prevents the physical possibility of stellar collapse with the formation of a singularity. This is supported by cited high-energy physics experimental evidence that neutrons under compression may not transition into degenerate matter, nor collapse to a singularity.

7. It is apparent that the first occurrence of an event horizon, at a neutron star mass of ~4.75 M_ʘ, has a radius of ~6.8 km. This is below the surface radius of ~7.6 km and it seems inconceivable that the ‘core’ of the star (up to 6.8 km radius), could collapse to a singularity while the ‘crust’, from 6.8 km to 7.6 km, remains in place. Since the crust (with escape speeds less than c), cannot collapse, this suggests that the core does not collapse either.

8. There seems no necessary physical mechanism whereby neutron stars represent a special case within a relativistic universe, which can (at ~5 M_ʘ) abruptly transform into a singularity with zero surface radius and infinite density, producing infinite spacetime curvature. No observed evidence of such a cataclysmic and powerful event is reported, even from an accretion disk or jet (beyond the outer event horizon).

9. Accordingly, the singularity anticipated under GR field equations, apparent to an external observer, may be a mathematical illusion. Internally, no real singularity is formed, therefore a GR black hole per se does not physically exist. An invisible neutron star is more appropriately termed a “black star”.

10. Absent a real singularity existing within a black hole (or black star), no bridge can form between a pair of them – the wormhole has no physical reality.

11. It may be possible to inform the mass, surface radius and average density of a sufficiently luminous <5 M_ʘ neutron star based on gravitational redshift, observed in the Fe Kα spectral line emitted in X-ray.

Acknowledgments

This chapter represents a significant revision of the paper by Dr. M.T. Cole [38] (reproduced material is reprinted with permission). The encouragement, advice and guidance provided by Prof. M. Bailes (Swinburne University of Technology), Prof. E.G. Haug (Norwegian University of Life Sciences), Dr. K. Leschinski (University of Vienna), and Em Prof. S.J. Brodsky et al. [3]. has been very much appreciated.