Semiclassical constant-density spheres in a regularized Polyakov approximation

We provide an exhaustive analysis of the complete set of solutions of the equations of stellar equilibrium under semiclassical effects. As classical matter we use a perfect fluid of constant density; as the semiclassical source we use the renormalized stress-energy tensor (RSET) of a minimally coupled massless scalar field in the Boulware vacuum (the only vacuum consistent with asymptotic flatness and staticity). For the RSET we use a regularized version of the Polyakov approximation. We present a complete catalogue of the semiclassical self-consistent solutions which incorporates regular as well as singular solutions, showing that the semiclassical corrections are highly relevant in scenarios of high compactness. Semiclassical corrections allow the existence of ultracompact equilibrium configurations which have bounded pressures and masses up to a central core of Planckian radius, precisely where the regularized Polyakov approximation is not accurate. Our analysis strongly suggests the absence of a Buchdahl limit in semiclasical gravity, while indicating that the regularized Polyakov approximation used here must be improved to describe equilibrium configurations of arbitrary compactness that remain regular at the center of spherical symmetry.

Figure 1

This table shows the complete set of classical and semiclassical stellar solutions of constant density. We distinguish whether the energy density $\rho$ takes values below, above, or at the critical value ${\rho }_c$; if the compactness is below or above the Buchdahl limit; and, for semiclassical stars, if their surface is located outside, inside, or at the neck itself. Each cell shows the behavior of pressure and compactness at the smallest value of $l$ in the domain of definition of the solution. Light-green (light grey) cells correspond to singular geometries. Yellow cells (grey) are strict stellar spacetimes. The orange color (black) percolating into the rightmost part of sub-critical cells and the leftmost part of supercritical cells represents $\epsilon$-strict spacetimes. This family of spacetimes includes the subset of subcritical solutions with small wormhole necks. See Fig. 2 for the respective numerical solutions for each cell of the table.

Figure 2

This table shows numerical integrations for each of the distinct regimes that we have found exploring classical and semiclassical stellar solutions of constant density. The criteria for classifying the solutions is the same followed in Fig. 1. The three regimes (subcritical, critical and supercritical) appear represented for sub-Buchdahl and super-Buchdahl stars in both classical and semiclassical theories. In the semiclassical case, distinction is made on whether the star surface is located outside, inside, or at the neck of the vacuum wormhole geometry. In the semiclassical regime we show $\epsilon$-strict solutions in the cases where no regular critical solutions exist. Each cell shows a numerical integration of a constant-density star. All but critical sub-Buchdahl stars, which are integrated from $l=0$ outwards, are integrated from the surface $l_{\mathrm{S}}=R=2$ inwards until the center of spherical symmetry or a singularity is reached. We stick to the following color criteria for the represented functions throughout the rest of the paper: the shape function $r(l)$ is represented in green, the pressure $p(l)$ in red, and the compactness function $C(l)$ in blue. The region where the classical fluid is present is filled in yellow for pictorical purposes. Spacetime singularities are depicted by a vertical zigzag line and correspond in every case with a divergent pressure. In super-Buchdahl supercritical plots the pressure grows inwards outside the plot window (with no divergences), to just come back inside the plot window when closer to the radial origin. Inside-the-neck sub-Buchdahl stars have $C(R)<8/9$ and their compactness function grows to 1 inside the structure, generating a local maximum in the shape function $r(l)$ just below the surface. The semiclassical critical profiles from the fourth row downwards are pictorial representations of how the respective exact solutions would look like, rather than complete numerical integrations. This is due to the numerical instability of our numerical algorithm at the critical density. We have attached a Mathematica notebook that generates all the plots appearing in this table [75]. The reader can access it in order to consult the values of $C(R)$, $\rho$ and $\alpha$ used for integrating the equations. See Fig. 1 for the asymptotic behaviors of the pressure and the compactness in each situation.

Figure 3

Plot of a positive pressure solution to the equations of structure. The above and below green lines represent the shape function $r(l)$ and the region in between both curves has been colored for pictorial purposes. The blue line denotes the compactness function of the geometry $C(l)$, which reaches 1 at the radial maximum and vanishes at the poles. The red curve is the pressure of the solution (in units of $\rho$) for a star with $\rho =0.03$ and $p(0)=2\rho$. Notice that the region of positive pressure corresponds to a relativistic star with $C(l_{\mathrm{S}})\simeq 0.82$. This solution has an enormous density compared to that of astrophysical objects, but the physics of classical critical solutions is scale-invariant.

Figure 4

Plot of a subcritical, sub-Buchdahl star with $R=2,C(R)=0.8$ and $\rho =0.84{\rho }_{c\text{−}\mathrm{clas}}$. Green lines represent the shape function $r(l)$, while red and blue lines denote the functions $p$ and $C$, respectively. The presence of a positive constant mass $M_0\simeq 0.07$ generates a (singular) event horizon at $l\simeq 7.25$.

Figure 5

Plot of the pressure profile for a supercritical, super-Buchdahl star with $C(R)=0.96$. The curves denote the pressure $p(r)$ (in units of $\rho$) for the values of the energy density (from right to left) $\rho /{\rho }_{c\text{−}\mathrm{clas}}=1$, 1.8, 2, 2.13, 2.26, 2.4 and 3. The dashed curve ($\rho /{\rho }_{c\text{−}\mathrm{clas}}\sim 2.13$) corresponds to a separatrix for which pressure diverges at $r=0$. Note how the divergence in pressure of super-Buchdahl stars moves inwards as the density increases. An increase of the negative mass $M_0$ finally regularizes the pressure, which tends to the value $p=-\rho$ at the origin.

Figure 6

Numerical representation of the Schwarzschild vacuum geometry. The right-hand side is the asymptotically flat region of the spacetime. In an inwards integration, a minimal surface or throat is encountered for the shape function $r(l)$ (green curve, in units of the neck radius), which connects to a null singularity at finite affine distance. This singularity has a negative infinite mass associated with it, which can be related to a runaway of vacuum polarization, as shown by the blue curve denoting the compactness $C(l)$ and the yellow curve representing the redshift function $e^{2\varphi (l)}$. We have chosen $M=0.05$ and $\alpha =1.01$ to better highlight the characteristics of the solution, but the qualitative behavior of the geometry does not change for greater values of the ADM mass.

Figure 7

Pictorial representation of an $R\gg l_{\mathrm{P}}$ slice of the phase space of semiclassical constant-density stars. The vertical and horizontal axes represent the energy density and the surface compactness of stars. The curve ${\rho }_c$ corresponds to a separatrix solution. The vertical dotted line here denotes the Buchdahl limit, in which the central black dot represents the most compact configuration strictly regular in both pressure and compactness, or Buchdahl solution. We distinguish four regions in the resulting figure: region I represents subcritical sub-Buchdahl stars, region II is for supercritical sub-Buchdahl, region III is for subcritical super-Buchdahl; and region IV represents supercritical super-Buchdahl stars. In the sub-Buchdahl semiplane (left-hand side of the vertical dotted line), the separatrix ${\rho }_c$ corresponds to strict stellar spacetimes. For super-Buchdahl configurations this separatrix correspond to nonregular solutions, but in its neighborhood we find quasi-regular configurations (i.e., $\epsilon$-strict stellar configurations).

Figure 8

Plot of the semiclassical counterpart of Fig. 3 with $\rho =0.03$, $p_c=2\rho$ and $\alpha -1={10}^{-3}$. We have plotted the functions $r(l)$, $C(l)$ and $p(l)$ (in units of $\rho$) and they appear in green, blue and red, respectively. The right pole of the geometry is shown in detail. Notice how the radial function has a minimal surface (vertical dashed line) at $l\sim 6.23$ and the geometry connects to a singular region (vertical zigzag line) located at $r\to \infty$ but at finite proper distance from $l=0$.

Figure 9

Plot of the RP-RSET components $-⟨{\widehat{T}}_t^t⟩$ (dark blue), $⟨{\widehat{T}}_l^l⟩$ (magenta) and $⟨{\widehat{T}}_{\theta }^{\theta }⟩$ (cyan) for the cosmological solution with $p_c=2\rho$, $\rho =0.03$ and $\alpha -1={10}^{-3}$.

Figure 10

Semiclassical stars integrated from the surface. The green and blue curves denote $r(l)$ and $C(l)$, and the red curve represents the function $p(l)$. All integrations correspond to stars with $R=1.8$ and $\alpha -1={10}^{-3}$. Their surface compactness and their $\rho /{\rho }_{c\text{−}\mathrm{clas}}$ quotients are, approximately and from top to bottom: (0.84, 0.71), (0.92, 1.16), (0.93, 1.34) and (0.96, 1.87). The second and third panels show a zoomed plot of the near-neck region, highlighting the neck (vertical dashed line) and the singularity (zigzag line). Increasing $\rho$ generates a well of negative mass. This negative mass slows down the increase in pressure, causing a shrinkage of the wormhole neck. Eventually, the neck disappears leaving a naked singularity at $r=0$. In between subcritical and supercritical configurations there is an infinite pressure separatrix solution. As the wormhole neck can be as small as desired by adjusting $\rho$, a mild deformation of the geometry at the core would suffice to make the whole construction regular. See Fig. 11 for the RP-RSET components from each solution.

Figure 11

RSET components $-⟨{\widehat{T}}_t^t⟩$ (dark blue), $⟨{\widehat{T}}_l^l⟩$ (magenta) and $⟨{\widehat{T}}_{\theta }^{\theta }⟩$ (cyan) for various stars integrated from the surface. All the integrations correspond to stars with $R=1.8$ and $\alpha -1={10}^{-3}$. They correspond to the solutions appearing in Fig. 10, whose surface compactness and $\rho /{\rho }_{c\text{−}\mathrm{clas}}$ quotients are, approximately: (0.84, 0.71) (top left), (0.92, 1.16) (top right), (0.93, 1.34) (bottom left), (0.96, 1.87) (bottom right). Note the abrupt change in the sign of the semiclassical energy density in the transition from the subcritical to the supercritical regime.

Figure 12

Numerical plot of the roots ${\mathcal{R}}_1$, ${\mathcal{R}}_2$ and ${\mathcal{R}}_3$ (green, blue and turquoise curves, respectively) together with the exact solutions ${\psi }_{\pm }$ (orange and red dashed curves) and an exact numerical solution in black (the neck radius $r_{\mathrm{B}}$ is represented by a vertical dashed line). The numerical solution corresponds to a subcritical super-Buchdahl star with $R=2$, $C(R)=0.95$ and $\rho /{\rho }_{c\text{−}\mathrm{clas}}\simeq 1.67$ with its neck at $l\simeq 0.45$ (vertical dashed line). These values have been chosen to aid visualization. The upper portion of the exact solution lives in the unconcealed branch, whereas the bottom portion lives in the concealed branch. The concealed part of the exact solution gets confined between ${\mathcal{R}}_3$ and ${\psi }_{+}$, converting toward the vacuum solution asymptotically.

Figure 13

Plot of ${\rho }_{\mathrm{reg}\text{−}p}$ in terms of the compactness for classical (orange) stars and of ${\rho }_c$ for semiclassical (green) stars with $R=2$ and $C(R)\in (0,1)$. The blue line corresponds to the classical critical density (17). The orange curve diverges in the $C(R)\to 1$ limit, whereas the green curve reaches a finite value, in this case ${\rho }_c(C(R)\to 1)\simeq 1.366$.

Figure 14

Plot of the Misner-Sharp mass at a central radius $r_{\mathrm{core}}=\mathcal{O}(l_{\mathrm{P}})$ in terms of the surface compactness for classical (orange) and semiclassical (green) stars with $R=2$. Notice how the orange curve diverges in the $C(R)\to 1$ limit, as infinite negative masses are required to regularize the pressure in that limit. In the semiclassical case, since the surface of $C(R)=1$ is a wormhole neck. As pressure at the neck is finite [see Eq. (71)] the required negative mass is finite, in this case $M_{\mathrm{core}}(C(R)\to 1)\simeq -41.20$.

Figure 15

Plot of ${\rho }_c$ for semiclassical stars of various radii and compactness $C(R)=1-{10}^{-10}$. As $R$ shrinks, the separatrix density diverges, whereas for larger radii (in Planck units) it decreases linearly.

Figure 16

Plot of the central negative mass $M_{\mathrm{core}}$ for semiclassical supercritical stars of various radii and surface compactness $C(R)=1-{10}^{-10}$. This estimated central mass is obtained by stopping the integration at a security radius far from the region where the Misner-Sharp mass diverges. Notice how the central mass has to be many orders of magnitude higher than the total mass of the star $M\simeq R/2$ for big stars. For small stars have their pressure regularized by negative central masses comparable to their total mass, indicating that the physics of Planckian stars may be different from that of astrophysical bodies.

Figure 17

Plot of a subcritical star located inside the neck. The blue green and blue curves represent the shape function $r(l)$ and the compactness $C(l)$, respectively. The red curve is the pressure $p(l)$. The dashed and dot-dashed vertical lines represents the radial minimum $r_{\mathrm{B}}$ and maximum $r_{\mathrm{M}}$, respectively. The singularity is represented by a zigzag line. The parameters chosen are $R=1.8$, $C(R)=0.96$, $\rho /{\rho }_{c\text{−}\mathrm{clas}}=43.4$ and $\alpha -1={10}^{-3}$.

Figure 18

RSET components $-⟨{\widehat{T}}_t^t⟩$ (dark blue), $⟨{\widehat{T}}_l^l⟩$ (magenta) and $⟨{\widehat{T}}_{\theta }^{\theta }⟩$ (cyan) for a subcritical star located beyond the neck. The parameters of the integration are $R=1.8$, $C(R)=0.96$, $\rho /{\rho }_{c\text{−}\mathrm{clas}}=43.4$ and $\alpha -1={10}^{-3}$.

Figure 19

Plot of ${\rho }_c$ in terms of the compactness for a star located inside the neck. The density required to regularize the structure grows linearly as compactness diminishes, reaching over Planckian densities very quickly.

Figure 20

Left panel: series of classical super-Buchdal stars approaching a finite pressure solution. The green lines represent the shape function $r$ and the red curves denote the pressures $p$, which diverge at $r(l_{\mathrm{div}})$ (zigzag lines). Lighter colors correspond to stars whose density is nearing ${\rho }_{\mathrm{reg}\text{−}p}$ (the critical solution ${\rho }_{c\text{−}\mathrm{clas}}$ has the darkest colors). Any attempt of regularizing the pressure makes the star highly supercritical, causing $p^{\prime }$ and $r^{\prime }$ to diverge at $l=0$. The shape function of a regular star is drawn in blue for comparison. Right panel: series of semiclassical subcritical super-Buchdahl stars approaching the critical solution. Again, the shape functions $r$ are shown in green, while the pressures $p$ are plotted in red. Lighter colors correspond to stars whose density is nearing ${\rho }_c$. The critical solution is approached together with the regularization in the pressure. The vertical dashed lines represent the necks of the solutions and singularities are represented by vertical zigzag lines. We have drawn in blue the shape function of a hypothetical regular star. Note that regularity requires $r^{\prime }(0)=1+\mathcal{O}(l^2)$. The classical supercritical configuration with finite pressure has $r^{\prime }(0)\to +\infty$, whereas semiclassical configurations with small wormhole necks are $\epsilon$-strict stellar spacetimes. Regularization of these profiles amounts to selecting a suitable regularization for the Polyakov RSET.