Possible distribution of mass inside a black hole. Is there any upper limit on mass density?

Abstract

The maximum mass of a neutron star is about three solar masses. In this case the radius of such neutron star is approximately equal to the Schwarzschild radius. Adding a small amount of matter to this star, a black hole arises. Thus its interior could contain a star with neutron or quark density just below the event horizon instead of the proposed point singularity. We also show that the Hawking miniature black hole evaporation is improbable, since it would yield unrealistic mean mass densities.

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Fig. 1

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Acknowledgements

The author is indebted to Yurii V. Dumin, Attila Mészáros, Vladimír Novotný, Lawrence Somer, and Vladimír Wagner for fruitful discussions and to Filip and Pavel Křížek for drawing the figures. This paper was supported by RVO 67985840 of the Czech Republic.

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Appendix: Calculation of the mass of Sgr A∗

Appendix: Calculation of the mass of Sgr A

According to Schödel et al. (2002), the star S2 orbits the supermassive black hole Sgr A in the center of our Galaxy along elliptic trajectory. We shall assume that the Newtonian mechanics is a good approximation of reality in this case. By present measurements, S2 is about 26 500 ly from us and its orbital period is

$$ T=16.08~\mbox{yr} \approx 507\cdot 10^{6}~\mbox{s}. $$
(21)

First we show that the eccentricity \(e\) of the actual elliptical trajectory can surprisingly be derived from the observed trajectory.

Denote by \(F\)’ the point corresponding to the strong X-ray source Sgr A, which is the orthogonal projection of the focus \(F\) of the actual orbit. Consider the line \(S\)\(F\)’ and denote by \(A\)’ its intersection with the projected orbit. The semimajor axis \(a\) that contains the focus \(F\) is then projected to the line segment \(A\)\(S\)’. Therefore, we have (see Fig. 2)

$$ e=\frac{\varepsilon }{a}=\frac{|\mathit{FS}|}{|\mathit{AS}|}= \frac{|F'S'|}{|A'S'|}, $$
(22)

where the ratio on the right-hand side can be evaluated, \(|\cdot |\) denotes the length, and \(\varepsilon =|\mathit{FS}|=\sqrt{a^{2}-b^{2}}\) is the linear eccentricity. For the observed trajectory illustrated in Fig. 3 we get by (22) that the eccentricity of the actual trajectory is \(e=0.885\).

Fig. 2
figure2

Actual and observed trajectory of the star S2. The segments \(a\), \(b\), and \(c=\sqrt{a^{2}+b^{2}}\) are orthogonally projected on \(a\)’, \(b\)’, and \(c\)

Fig. 3
figure3

Projection of S2-orbit on the celestial sphere. Its semimajor axis \(\overline{a}=4.55\) ld (light days) is less than \(a\)

Further, we construct the point \(B\)’ on the observed orbit so that the line \(B\mbox{'}S\mbox{'}\) is parallel with the tangent at \(A\)’ and that the angle \(A\)\(S\)\(B\)’ is nonobtuse. Then the triangles \(\mathit{ABS}\), \(\mathit{AA}\)\(S\), and \(\mathit{BB}\)\(S\) are right and we have

$$ a^{2}+b^{2}=c^{2}={c\mbox{'}}^{2} + \bigl(\sqrt{a^{2}-{a\mbox{'}}^{2}} + \sqrt{b^{2}-{b\mbox{'}}^{2}} \bigr)^{2}, $$

where \(a\mbox{'}=|A\mbox{'}S\mbox{'}|\), \(b\mbox{'}=|B\mbox{'}S\mbox{'}|\), and \(c\mbox{'}=|A\mbox{'}B\mbox{'}|\). From this we obtain

$$ a{{\text{'}}}^{2}+b{{\text{'}}}^{2}-c{{\text{'}}}^{2} =2\sqrt{a^{2}-a{{ \text{'}}}^{2}} \sqrt{b^{2}-{b{\text{'}}}^{2}}. $$

Squaring this equation, the substitution \(b^{2}=(1-e^{2})a^{2}\) leads to quartic equation for one unknown \(a\),

$$\begin{aligned} &{\bigl(1-e^{2}\bigr) \bigl(a^{2}\bigr)^{2}- \bigl[\bigl(1-e^{2}\bigr){a{\text{'}}}^{2}+{b{\text{'}}}^{2} \bigr]a^{2}+ {a{\text{'}}}^{2}{b{\text{'}}}^{2}} \\ &{\quad {} -\frac{1}{4}\bigl({a{\text{'}}}^{2}+{b{ \text{'}}}^{2}-{c{\text{'}}}^{2} \bigr)^{2}=0.} \end{aligned}$$
(23)

Since this equation does not contain any cubic and linear term, it is, in fact, a quadratic equation for \(a^{2}\).

By angular measurements we know that \(a\mbox{'}=3.99\) ld, \(b\mbox{'}=2.49\) ld, and \(c\mbox{'}=4.01\) ld. Substituting these data into (23), we get

$$ a=5.61~\mbox{ld}=970~\mbox{au}=145\cdot 10^{12}~\mbox{m}. $$
(24)

The second positive solution of (23) is not physical, since it is smaller than \(a\)’. From (21), (24), and Kepler’s third law we obtain

$$ M_{\bullet }=\frac{4\pi ^{2} a^{3}}{G T^{2}}\approx 7\cdot 10^{36}~\mbox{kg}\approx 3.5\cdot 10^{6}\ M_{\odot }. $$

The resulting mass is, of course, very sensitive on precise measurements of \(a\) and \(T\). The corresponding Schwarzschild radius is

$$ R_{\bullet }=\frac{2GM_{\bullet }}{c^{2}}\approx 10^{10}~\mbox{m} \approx 0.07~\mbox{au} $$
(25)

and the mean mass density

$$ \rho _{\bullet }=\frac{M_{\bullet }}{V_{\mathrm{relativistic}}}< \frac{M_{\bullet }}{\frac{4}{3}\pi R^{3}_{\bullet }}\approx 1.67 \cdot 10^{6}~\mbox{kg}/\mbox{m}^{3}. $$
(26)

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Křížek, M. Possible distribution of mass inside a black hole. Is there any upper limit on mass density?. Astrophys Space Sci 364, 188 (2019). https://doi.org/10.1007/s10509-019-3679-9

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Keywords

  • Black hole
  • Neutron star
  • Relativistic volume
  • Chandrasekhar limit
  • TOV limit