Loosely trapped surface and dynamically transversely trapping surface in Einstein-Maxwell system

Since a black hole creates a strong gravitational field, there exists unstable circular orbits for photons. For a spherically symmetric system, the collection of them makes a surface called a photon sphere. In the Schwarzschild spacetime, for example, the photon sphere exists at the surface $r=3m$, where $m$ is the Arnowitt-Deser-Misner (ADM) mass. Furthermore, a generalized concept of a photon sphere, which is called a photon surface, has been proposed Claudel:2000 . However, the definition of the photon surface requires a highly symmetric spacetime (e.g., Yoshino:2016 ). Moreover, the existence of photon surfaces does not necessarily mean that the gravitational field is strong there Claudel:2000 .

A photon sphere is directly related to observational phenomena. The quasinormal modes of black holes are basically determined by the properties of photon spheres cardoso2009 . The black hole shadow, whose direct picture has been taken by the recent radio observations of the Event Horizon Telescope Collaboration akiyama , is also determined by a photon sphere Virbhadra:1999 .

Motivated by the recent observations, the four of the present authors proposed concepts that characterizes strong gravity regions; a loosely trapped surface (LTS) shiromizu2017 , a transversely trapping surface (TTS) Yoshino:2017 , and a dynamically transversely trapping surface (DTTS) Yoshino:2020-1 ; Yoshino:2020-2 (see also Ref. galtsov for an extension of a TTS). For certain cases, we have proved inequalities analogous to the Penrose inequality penrose1973 , that is, their areas are equal to or less than $4\pi(3m)^{2}$, where $m$ is the ADM mass (see also Ref. hod for an earlier work for a photon sphere). The upper bound is realized for a photon sphere in the Schwarzschild black hole, where $3m$ comes from the areal radius of unstable circular photon orbits. For an LTS, the application is restricted to a spacelike hypersurfaces with a positive Ricci scalar. This restriction is natural because it is guaranteed by the positivity of the energy density for maximally sliced initial data. For a DTTS, on the other hand, one requires the non-positivity of the radial pressure on a DTTS in addition to the positivity of the Ricci scalar. This requirement of the non-positive pressure does not significantly restrict the situation because the vacuum cases do work. However, it remains a little mystery why the non-positivity of the radial pressure is required.

Therefore, in this paper, we shall discuss non-vacuum cases. As a first typical example, we will focus on Einstein-Maxwell systems. We adopt Jang’s work jang1979 to show the inequality, that is, we will employ the method of the inverse mean curvature flow geroch ; wald1977 ¹¹1The resulted inequality for the minimal surface in Ref. jang1979 has a lower bound. However, it is known that the lower bound is violated for multi-black-hole systems weinstein2004 . The proof for multi-black-hole systems in the Einstein-Maxwell theory is given in Refs. khuri2013 ; khuri2017 .. The upper bound is expected to be given by the area of the outermost photon sphere, i.e., the locus of the unstable circular orbits of photons in a spherically symmetric charged black hole spacetime, namely, a Reissner-Nordström spacetime with the same mass and charge. As a consequence, however, we see that the obtained inequalities depend on (the part of) the energy density and the pressure/tension of the electromagnetic fields which give corrections to the naively expected upper bound. This is impressive because the Penrose inequality for apparent horizons does not depend on such quantities.

The rest of this paper is organized as follows. In Sect. 2, we will briefly describe the Maxwell theory in a curved spacetime. Some of the notations will be explained together. In Sect. 3, we will present the definition of the LTS and prove the Penrose-like inequality in the Einstein-Maxwell system. Then, in Sect. 4, we will present the definition of the DTTS and prove the Penrose-like inequality in the Einstein-Maxwell system. In Sect. 5, we will revisit the problem of DTTSs in Majumdar-Papapetrou two-black-hole spacetimes, which was studied in our previous paper Yoshino:2020-2 , from the viewpoint of the current study. We will examine the properties of the correction term of the Penrose-like inequality through numerical calculations. The last section will give a summary and discussions. In Appendix, we will shortly discuss the case of a TTS defined for static/stationary spacetimes. Note that we use following units in which the speed of light $c=1$, the Newtonian constant of gravitation $G=1$ and the Coulomb constant $1/(4\pi\varepsilon_{0})=1$, where $\varepsilon_{0}$ is the permittivity of vacuum.

In this paper, we consider an asymptotically flat spacelike hypersurface $\Sigma$ in a four-dimensional spacetime with a metric $g_{ab}$. We suppose $n^{a}$ to be the future-directed unit normal to $\Sigma$ and the induced metric of $\Sigma$ is given by $\gamma_{ab}=g_{ab}+n_{a}n_{b}$. In $\Sigma$, we consider a two-dimensional closed surface, an LTS (denoted by $S_{0}$) or a DTTS (denoted by $\sigma_{0}$), with the induced metric $h_{ab}$. The outward unit normal to that surface is $r_{a}$ (and therefore, $\gamma_{ab}=h_{ab}+r_{a}r_{b}$).

We assume the presence of electromagnetic fields on $\Sigma$. The electromagnetic fields are specified by the anti-symmetric tensor, $F_{ab}$, and its Hodge dual, ${}^{*}F_{ab}=(1/2)\epsilon_{abcd}F^{cd}$, where $\epsilon_{abcd}$ is the Levi-Civita symbol in a four-dimensional spacetime. The tensors $F_{ab}$ and ${}^{*}F_{ab}$ follow Maxwell’s equations,

where $j^{b}$ is the four-current vector. The electric and magnetic fields, $E^{a}$ and $B^{a}$, are defined by

respectively. Obviously, these fields are tangent to $\Sigma$ because $E_{a}n^{a}=B_{a}n^{a}=0$ holds. The electric charge density $\rho_{\rm e}$ and the electric current $(J_{\rm e})^{a}$ are defined by

In our paper, we require the charge density to vanish (outside an LTS or a DTTS), i.e. $\rho_{\rm e}=0$, but do not necessarily require the electric current $(J_{\rm e})^{a}$ to be zero.

The electric and magnetic fields satisfy Gauss’ laws, $D_{a}E^{a}=4\pi\rho_{\rm e}$ and $D_{a}B^{a}=0$, where $D_{a}$ is the covariant derivative with respect to $\gamma_{ab}$. The total electric and magnetic charges are

where $S_{\infty}$ is a sphere at spacelike infinity and $r_{a}$ is the outward unit normal to $S_{\infty}$. It would be important to point out that $q_{\rm e}$ can have a nonzero value even if the electric charge density is zero throughout the spacetime, as one can understand by imagining a spacelike hypersurface with an Einstein-Rosen bridge and two asymptotically flat regions in the maximally extended Reissner-Nordström spacetime. Similarly, although we assume the absence of magnetic monopoles throughout the paper, the value of $q_{\rm m}$ can be nonzero. In a spherically symmetric spacetime, if both electric and magnetic fields are present, the total squared charge defined by

appears in the spacetime metric of the Reissner-Nordström solution. We handle the magnetic charge $q_{\rm m}$ in the following way. In Sects. 3 and 4, we will assume $q_{\rm m}=0$ (and therefore, $q^{2}=q_{\rm e}^{2}$), and derive the Penrose-like inequalities. In the final section, we will discuss the modifications to those inequalities when $q_{\rm m}$ is nonzero.

The energy-momentum tensor for the electromagnetic fields is given by

In addition to $T_{ab}^{\rm(em)}$, we consider the presence of ordinary matters whose energy-momentum tensor is $T_{ab}^{(m)}$. The total energy-momentum tensor is given by $T_{ab}=T_{ab}^{\rm(em)}+T_{ab}^{(m)}$. The relations of particular importance in this paper are the energy density,

where $\rho^{(m)}:=T^{(m)}_{ab}n^{a}n^{b}$, and the radial pressure,

where $P_{r}^{(m)}:=T^{(m)}_{ab}r^{a}r^{b}$.

In this section, we review the definition of an LTS following Ref. shiromizu2017 , and show the Penrose-like inequality for it in Einstein-Maxwell systems.

In this section, we first explain the observation that motivates the definition of a DTTS, and introduce the definition of a DTTS. Then we prove an inequality for its area in Einstein-Maxwell systems.

In Sect. 4.2, we have obtained the Penrose-like inequality for a DTTS. There, the quantity $\Phi^{-}_{0}$ appears, and this quantity depends on the configuration of electromagnetic fields. The purpose of this section is to examine the values of $\Phi^{-}_{0}$ in an explicit example. Specifically, in our previous paper Yoshino:2020-1 , we numerically solved for marginally DTTSs in systems of two equal-mass black holes adopting the Majumdar-Papapetrou solution. We revisit this problem from the viewpoint of our current work.

A Majumdar-Papapetrou spacetime is a static electrovacuum spacetime. The metric is

where the spatial structure is conformally flat, and we span the spherical-polar coordinates here. The electromagnetic four-potential is

Any solution to the Laplace equation $\bar{\nabla}^{2}U=0$ gives an exact solution, where $\bar{\nabla}^{2}$ is the flat space Laplacian. In this situation, $E^{a}$ and $B^{a}$ are calculated as

Setting $E_{\parallel}^{2}:=h_{ab}E^{a}E^{b}$, we have $8\pi\Phi^{-}_{0}=-\int_{\sigma_{0}}E_{\parallel}^{2}dA$. Therefore, the value of $\Phi^{-}_{0}$ is non-positive.

In our previous paper Yoshino:2020-1 , we chose the solution

that represents the system in which two extremal black holes with the same charge are located with the coordinate distance $2z_{0}$. Then, assuming the functional form $\tilde{r}=h(\tilde{\theta})$, we numerically solved for a marginally DTTS that surrounds both black holes for each value of $z_{0}$. The solution was found in the range $0\leq z_{0}/m\leq 0.79353$. We refer readers to our previous paper Yoshino:2020-1 for explicit shapes of the obtained solutions.

We examine the value of $\Phi^{-}_{0}$. The left panel of Fig. 1 presents the behavior of $E_{\parallel}^{2}$ as a function of $\tilde{\theta}$ on a marginally DTTS for $z_{0}/m=0.25$, $0.375$, $0.5$, $0.625$, $0.75$, and $0.79353$. The value of $E_{\parallel}^{2}$ is generally nonzero, but is less than $10^{-3}$. The right panel of Fig. 1 plots the value of $\Phi^{-}_{0}$ as a function of $z_{0}$. It is negative and its absolute value is less than $5\times 10^{-4}$.

Figure 2 shows the relation between the area $A$ of the marginally DTTS and $z_{0}$. We normalize the value of $A$ in two ways: One is $A/4\pi r_{\rm DTTS+}^{2}$ (a red solid curve), where $r_{\rm DTTS+}$ is defined in Eq. (39), and the other is $A/4\pi r_{p}^{2}$ (a green dotted curve), where $4\pi r_{p}^{2}$ is the area of a photon sphere with the same mass and charge [see Eq. (10)]. Because the value of $\Phi^{-}_{0}$ is small, the difference is scarcely visible. Both of these values are in agreement with the Penrose-like inequalities.

The reason why the value of $\Phi^{-}_{0}$ is so small is that the electric field is approximately perpendicular to the marginally DTTS. From the formula for $E^{a}$ given in Eq. (44), this means that the marginally DTTS approximately coincides with a contour surface of $U$. Figure 3 confirms this feature for $z_{0}/m=0.79353$. Here, the blue curve depicts the marginally DTTS, and the red curve shows the contour surface of $U=2.14$. They agree well.

The lesson from this numerical experiment is that if a spacetime is static, the quantity $\Phi^{-}_{0}$ is small and does not play an important role in the Penrose-like inequality for a DTTS. Of course, it is expected that the absolute value of $\Phi^{-}_{0}$ may become large if dynamical situations are considered. For example, if two black holes have opposite charges, the contribution of the electric pressure would become important, although such a situation is more difficult to study. Exploring such issues is left as a remaining problem.

In this paper, we have examined the properties of LTSs and DTTSs for Einstein-Maxwell systems, particularly focusing on the derivation of Penrose-like inequalities on their area. Similarly to the Riemannian Penrose inequality for charged cases, the electric charge comes into the inequalities, but there are additional contributions from the density or the pressure/tension of electromagnetic fields in general. This is a rather interesting result because one naively expects that the upper bound for the area of an LTS and a DTTS is that of the photon sphere $4\pi r_{p}^{2}$, where $r_{p}$ is the radius of an unstable circular orbit of a photon in the Reissner-Nordström spacetime given in Eq. (10). For an LTS, we have a tighter inequality than the naive one. For a DTTS, the obtained inequality can become both stronger and weaker depending on the configuration of electromagnetic fields. We have numerically examined the value of the correction term, represented by $\Phi^{-}_{0}$ in Eq. (32), for a Majumdar-Papapetrou two-black-hole spacetime. Although the correction term makes the inequality weaker, we have checked that the value of $\Phi^{-}_{0}$ is very small in that situation.

Up to now, we have assumed that the magnetic charge $q_{\rm m}$ vanishes. Here, we consider what happens to our inequalities when nonzero $q_{\rm m}$ is present. We first consider the case of an LTS. Since the total squared charge $q^{2}:=q_{\rm e}^{2}+q_{\rm m}^{2}$ appears in the metric of the Reissner-Nordström solution when both the electric and magnetic fields are present, we would like to present the Penrose-like inequality in terms of $q^{2}$. For this reason, we have to use the inequality

in the calculations of Eqs. (16) and (17). As a result, we must introduce the quantity

instead of $\Phi^{+}_{0}$ of Eq. (14). The resultant inequality is the one of Eq. (13) but $\Phi^{+}_{0}$ being replaced by $\Phi_{0}$. Next, we consider the case of a DTTS. Similarly to the case of an LTS, we must use the inequality of Eq. (46) (but $S_{0}$ being replaced by $\sigma_{0}$) in the calculations of Eqs. (34) and (35). As a result, instead of $\Phi_{0}^{-}$, we must introduce $-\Phi_{0}=-(1/8\pi)\int_{\sigma_{0}}\left[(E_{a}E_{b}+B_{a}B_{b})h^{ab}\right]dA$. The resultant inequality is the same as the one of Eq. (31), but $\Phi^{-}_{0}$ being replaced by $-\Phi_{0}$. These results are summarized as follows:

In the main article of this paper, we have not considered a TTS for the static and stationary spacetimes defined in our previous paper Yoshino:2017 . We note that the concepts of a TTS and a DTTS are related but independent of each other in the sense that no inclusion relationship can be found Yoshino:2020-1 . In Appendix A, we present a theorem on the Penrose-like inequality for a TTS in a static spacetime, which is very similar to Theorem 2.

Throughout this study, we have not used the property of Maxwell’s equations except for Gauss’ law. The information from Maxwell’s equation may further restrict the properties of LTSs, DTTSs, and TTSs, especially for static/stationary spacetimes with static/stationary electromagnetic fields.

T. S. and K. I. are supported by Grant-Aid for Scientific Research from Ministry of Education, Science, Sports and Culture of Japan (No. 17H01091). H.Y. is supported by the Grant-in-Aid for Scientific Research (C) (No. JP18K03654) from Japan Society for the Promotion of Science (JSPS). The work of H.Y. is partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).