Avoidance of singularities in spherically symmetric charged dust

In spherically symmetric charged dust, just like in neutral dust, two kinds of singularity may be present: the big bang/big crunch (BB/BC) singularity, and shell crossings. Quite unlike in neutral dust, the BB/BC singularity may be avoided. When the charge density ${\rho }_e$ and the mass-energy density $ϵ$ obey $|{\rho }_e|<D_e\stackrel{\mathrm{def}}{=}\sqrt{G}ϵ/c^2$, the conditions that allow the model to avoid the BB/BC singularity necessarily lead to shell crossings. However, when $|{\rho }_e|\to D_e$ at the center of symmetry while $|{\rho }_e|<D_e$ elsewhere, both kinds of singularity may be avoided for a sufficiently long period that a body of charged dust may go through the tunnel between the singularities in the maximally extended Reissner-Nordström spacetime, to emerge into another asymptotically flat region. An explicit example of such a configuration is presented and discussed. It does not contradict any astrophysical constraints.

Figure 1

Schematic graphs of the solutions (8.3), with different values of $M$ and $\Phi$. Only those parts of the curves that lie above the $s$ axis describe physical situations. From bottom to top (counted at the left edge), the curves correspond to: (1) $\{{\ell }_2=-1,M<0,\Phi >0\}$. This model has a finite time of existence. (2) $\{{\ell }_2=+1,M>0,\Phi <0\}$. (3) $\{{\ell }_2=+1,M>0,\Phi =0\}$. This is the uncharged (Lemaître-Tolman) model, included for comparison. (4) $\{{\ell }_2=+1,M>0,\Phi >0\}$. The collapsing branch (left) ends in a singularity at $R=0$, the expanding branch (right) starts at the singularity at $R=0$. Models 1, 3, and 4 are the only ones with singularities. (5) $\{{\ell }_2=+1,M<0,\Phi >0\}$. (6) $\{{\ell }_2=+1,M<0,\Phi =0\}$. (7) $\{{\ell }_2=+1,M<0,\Phi <0\}$. The instant of time-symmetry is in general different on different $M$-shells. It was made the same in the figure only for better readability.

Figure 2

Schematic graphs of the solutions (8.4), with $\ell =+1$ and different values of $M$. Curve I is the nonsingular solution with $M<0$, curve II is a solution with $M>0$ and a singularity at finite $s$. In solution I $R$ tends to zero asymptotically as $s\to -\infty$.

Figure 3

Schematic graphs of the solutions (8.5), with $\ell =+1$ and different values of $M$ and $\Phi$. Every solution has a singularity; only the slope of the curve at the singularity and the instant of the singularity depend on $M$ and $\Phi$.

Figure 4

Schematic graphs of the solutions (8.6), with $\ell =\pm 1$ for curve I and $\ell =+1$ for the other two curves, and with different values of $M$ and $Q{,}_N$. Curve I is the nonsingular solution with $M>0$ and $Q,_N{}^2<G/c^4$, curve II corresponds to $M<0$ and $Q,_N{}^2>G/c^4$, curve III corresponds to $M>0$ and $Q,_N{}^2>G/c^4$. Model II achieves maximal expansion at $R=-Q^2(Q,_N{}^2-G/c^4)/(2M)$ and then recollapses.

Figure 5

Schematic graphs of the solutions (8.8), with $\ell =+1$ and with different values of $M$ and $\Phi$. Only those parts of the curves that lie above the $s$-axis describe physical situations. The values of the parameters are as follows: Curve I—$M<0$ and $\Phi <0$; Curve II—$M>0$ and $\Phi >0$; Curve III—$M>0$ and $\Phi =0$ (this is the Lemaître-Tolman evolution, shown for comparison); Curve IV—$M>0$ and $\Phi <0$. Curve II is the nonsingular model, all the other models begin and end with a singularity, and the derivative $\mathrm{d}R/\mathrm{d}s$ is always infinite at the singularity. Different $M$-shells have different periods in $s$. The periods were made equal only to improve the readability of the picture.

Figure 6

A possible pathology of an oscillating model with $E<0$: the period of oscillations tends to zero as the center of symmetry is approached [the curves are graphs of the functions $R(t,\mathcal{M})$ for different values of $\mathcal{M}$]. The thicker curve goes through the minima of the evolution graphs. (The model shown in the figure is time symmetric with respect to the bounce in the middle of the axis.)

Figure 7

Graphs of the functions $p(x)$, $\mathrm{d}p/\mathrm{d}x$, $F_1(x)$, $F_2(x)$ and $F_3(x)$. Note that $-1<\mathrm{d}p/\mathrm{d}x<+1$ and $F_2(x)>0$, $F_3(x)>0$ for all $x>0$, thus conditions (5), (6), and (7) are fulfilled. All the functions are zero at $x=0$. As $x\to \infty$, $F_2$ and $F_3$ tend to $\infty$, $F_1$ tends asymptotically to 1, while $p$ and $\mathrm{d}p/\mathrm{d}x$ tend asymptotically to 0. The inset shows the functions $p(x)$, $F_2(x)$, and $\mu (x)$ in a vicinity of $x=0$. It shows that $\mu (x)>p(x)$ for $x>0$.

Figure 8

Graphs of the functions $F_4(x)$ and $F_5(x)$ with $b=0.75$. The graphs of $F_1(x)$, $F_2(x)$, and $F_3(x)$ are shown for comparison and for scale. Both $F_4(x)$ and $F_5(x)$ asymptotically tend to 1 as $x\to \infty$ and both have vertical tangents at $x=0$.

Figure 9

The maximal radius achieved in each cycle, $R_{-}$, and the radius of the outer R-N horizon, $R_{H+}$, as functions of mass. The numbers on the axes are multiples of $G{N}_0/c^4$. At this scale, the radius of the inner horizon and the minimal radius seem to coincide with the $x$-axis, but see Fig. 10. As required by the equations of motion, we have $R_{-}>R_{H+}>R_{H-}>R_{+}$ for all masses. Note that $R_{-}$ and $R_{H+}$ are increasing all the way.

Figure 10

The radius of the inner horizon, $R_{H-}$ (upper curve), and the minimal radius, $R_{+}$ (lower curve), as functions of mass. The numbers on the axes are multiples of $G{N}_0/c^4$. The figure shows that $R_{+}<R_{H-}$. The inset is a close-up view of the region where the two curves in the main figure seem to be tangent. They still seem to be tangent, but a numerical calculation at double precision confirmed that $R_{+}<R_{H-}$ everywhere. Both radii first increase with mass, but then decrease.

Figure 11

The period (in the time coordinate) as a function of $x=N/N_0$.

Figure 12

The curves $R(\mathcal{M},t)$ corresponding to several values of $\mathcal{M}$. The mass increases uniformly from $x=0.01$ on the lowest curve to $x=0.1$ on the highest curve. The range of masses was chosen so as to make the figure readable; the real range of periods and of amplitudes can be very large. See more explanation in the text. The bounce is always smooth and at a nonzero value of $R$. (The figure suggests otherwise, but this is only an illusion created by the scale.)

Figure 13

A schematic Penrose diagram for the configuration defined by Eqs. (11.1, 11.2, 11.5). See explanation in the text.