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

We study the properties of the loosely trapped surface (LTS) and the dynamically transversely trapping surface (DTTS) in Einstein-Maxwell systems. These concepts of surfaces were proposed by the four of the present authors in order to characterize strong gravity regions. We prove the Penrose-like inequalities for the area of LTSs/DTTSs. Interestingly, although the naively expected upper bound for the area is that of the photon sphere of a Reissner-Nordstroem black hole with the same mass and charge, the obtained inequalities include corrections represented by the energy density or pressure/tension of electromagnetic fields. Due to this correction, the Penrose-like inequality for the area of LTSs is tighter than the naively expected one. We also evaluate the correction term numerically in the Majumdar-Papapetrou two-black-hole spacetimes.

We study the properties of the loosely trapped surface (LTS) and the dynamically transversely trapping surface (DTTS) in Einstein-Maxwell systems. These concepts of surfaces were proposed by the four of the present authors in order to characterize strong gravity regions. We prove the Penrose-like inequalities for the area of LTSs/DTTSs. Interestingly, although the naively expected upper bound for the area is that of the photon sphere of a Reissner-Nordström black hole with the same mass and charge, the obtained inequalities include corrections represented by the energy density or pressure/tension of electromagnetic fields. Due to this correction, the Penrose-like inequality for the area of LTSs is tighter than the naively expected one. We also evaluate the correction term numerically in the Majumdar-Papapetrou two-black-hole spacetimes.

Introduction
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 [1]. However, the definition of the photon surface requires a highly symmetric spacetime (e.g., [2]). Moreover, the existence of photon surfaces does not necessarily mean that the gravitational field is strong there [1]. A photon sphere is directly related to observational phenomena. The quasinormal modes of black holes are basically determined by the properties of photon spheres [3]. The black hole shadow, whose direct picture has been taken by the recent radio observations of the Event Horizon Telescope Collaboration [4], is also determined by a photon sphere [5].
Motivated by the recent observations, the four of the present authors proposed concepts that characterizes strong gravity regions; a loosely trapped surface (LTS) [6], a transversely trapping surface (TTS) [7], and a dynamically transversely trapping surface (DTTS) [8,9] (see also Ref. [10] for an extension of a TTS). For certain cases, we have proved inequalities analogous to the Penrose inequality [11], that is, their areas are equal to or less than 4π(3m) 2 , where m is the ADM mass (see also Ref. [12] 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 [13] to show the inequality, that is, we will employ the method of the inverse mean curvature flow [14,15] 1 . 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 [9], 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/(4πε 0 ) = 1, where ε 0 is the permittivity of vacuum.

Einstein-Maxwell theory and setup
In this paper, we consider an asymptotically flat spacelike hypersurface Σ in a fourdimensional spacetime with a metric g ab . We suppose n a to be the future-directed unit normal to Σ and the induced metric of Σ is given by γ ab = g ab + n a n b . In Σ, we consider a two-dimensional closed surface, an LTS (denoted by S 0 ) or a DTTS (denoted by σ 0 ), with the induced metric h ab . The outward unit normal to that surface is r a (and therefore, γ ab = h ab + r a r b ).
We assume the presence of electromagnetic fields on Σ. The electromagnetic fields are specified by the anti-symmetric tensor, F ab , and its Hodge dual, * F ab = (1/2) abcd F cd , where 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 Σ because E a n a = B a n a = 0 holds. The electric charge density ρ e and the electric current (J e ) a are defined by In our paper, we require the charge density to vanish (outside an LTS or a DTTS), i.e. ρ e = 0, but do not necessarily require the electric current (J e ) a to be zero. The electric and magnetic fields satisfy Gauss' laws, D a E a = 4πρ e and D a B a = 0, where D a is the covariant derivative with respect to γ ab . The total electric and magnetic charges are where S ∞ is a sphere at spacelike infinity and r a is the outward unit normal to S ∞ . It would be important to point out that q 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 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 m in the following way. In Sects. 3 and 4, we will assume q m = 0 (and therefore, q 2 = q 2 e ), and derive the Penrose-like inequalities. In the final section, we will discuss the modifications to those inequalities when q m is nonzero.
The energy-momentum tensor for the electromagnetic fields is given by In addition to T (em) ab , we consider the presence of ordinary matters whose energy-momentum tensor is T relations of particular importance in this paper are the energy density, 8πρ := 8πT ab n a n b = E a E a + B a B a + 8πρ (m) , where ρ (m) := T (m) ab n a n b , and the radial pressure, where P (m) r

Loosely trapped surface in Einstein-Maxwell system
In this section, we review the definition of an LTS following Ref. [6], and show the Penrose-like inequality for it in Einstein-Maxwell systems.

Definition of an LTS
The definition of an LTS is motivated by the following observation. As an example, we consider a Reissner-Nordström spacetime. The metric is where f RN (r) := 1 − 2m/r + q 2 /r 2 , and m and q are the ADM mass and total charge, respectively. dΩ 2 2 is the two-dimensional metric of the unit round sphere. From the behavior of a null geodesic, one can find unstable circular orbits of photons at where we suppose 9m 2 ≥ 8q 2 . Note that a photon sphere exists even if the spacetime possesses a naked singularity at the center for (9/8)m 2 ≥ q 2 > m 2 . Now we define a similar concept to a photon sphere for general setups in terms of geometry. Here, we recall the fact that an apparent horizon is the minimal surface on time-symmetric initial data. Therefore, one possibility to specify a strong gravity region is to employ the mean curvature, that is, the trace of the extrinsic curvature of two-dimensional surfaces. Therefore, we look at the mean curvature for the Reissner-Nordström spacetime. It is easy to see that the mean curvature of an r = constant surface on t = constant hypersurface is given by From the first derivative of k with respect to r, we find that the maximum value of k exists at r = r p . This is exactly the same location with that of unstable circular orbits of photons. In the region between the event horizon and the photon sphere at r = r p , the mean curvature satisfies k ≥ 0 and dk/dr ≥ 0. From the above argument, one may adopt the following definition of an LTS [6].
Definition 1. A loosely trapped surface (LTS), S 0 , is defined as a compact two-surface in a spacelike hypersurface Σ, and has the mean curvature k for the outward spacelike normal vector such that k| S0 > 0 and k | S0 ≥ 0, where is the derivative along the outward spacelike normal vector.

Penrose-like inequality for an LTS
In this section, we present the inequality for the area of an LTS in Einstein-Maxwell systems. Our theorem is as follows: Theorem 1. Let Σ be an asymptotically flat spacelike hypersurface with the Ricci scalar is the non-negative energy density for other matters. 2 We assume that Σ is foliated by the inverse mean curvature flow, and a slice of the foliation parameterized by y, S y , has topology S 2 . We also suppose the electric charge density ρ e to vanish outside the LTS, S 0 . Then, the areal radius r 0 = (A 0 /4π) 1/2 of the LTS, S 0 , in Σ satisfies the inequality where m is the ADM mass and q is the total charge. Φ + 0 is defined by where h ab is the induced metric of S 0 and r a is the outward-directed unit normal vector to S 0 in Σ.
Proof. On Σ, the derivative of the mean curvature k along r a is given by where D a is the covariant derivative of Σ, D a is the covariant derivative of S 0 , (2) R is the Ricci scalar of S 0 , (3) R is the Ricci scalar of Σ, k ab is the extrinsic curvature of S 0 and ϕ is the lapse function for y, that is, r a = ϕD a y. Then, the integration of Eq. (15) over S 0 gives us wherek ab := k ab − (1/2)h ab k. Note that Φ + 0 ≥ 0. Using the Gauss-Bonnet theorem and Cauchy-Schwarz inequality, we can derive the following inequality for the mean curvature where we used Gauss' law for the electric field, S0 E a r a dA = S∞ E a r a dA = 4πq. Here, S ∞ denotes the two-sphere at spacelike infinity.
Let us consider Geroch's quasilocal energy [13][14][15] E(y) := where A(y) is the area of S y . Here, we suppose that the surfaces y = 0 and y = ∞ correspond to the LTS and a sphere at spacelike infinity, respectively. Under the inverse mean curvature flow generated by the condition kϕ = 1, the first derivative of E(y) is computed as with the same procedure as the derivation of the inequality of Eq. (17), The integration of the inequality of Eq. (20) over y in the range 0 ≤ y < ∞ implies us In the above, we have used the well-known relation A(y) = A 0 exp(y) that holds in the inverse mean curvature flow at the first step, and the non-negativity of Φ + y and the inequality of Eq. (17) in the second step. Then, we find the inequality of Eq. (13).
There are four remarks. First, the minimum value of the right-hand side of the inequality of Eq. (13) implies Setting Φ + 0 = 0, this inequality is reduced to m ≥ 2 where This inequality must be interpreted carefully in the sense that the lower bound would not hold in general. In the case of the ordinary Riemannian Penrose inequality [16], it has been pointed out that the lower bound is expected to be incorrect for multi-black-hole systems. 6/15 We consider this also may be the case for an LTS with multiple components. This is not a contradiction to our proof since an LTS with multiple components is out of the application of our theorem. Therefore, in general, we would have only the upper bound for the area, Note that if we restrict out attention on an LTS with a single component, the lower bound must hold true. The physical reason is as follows. Let us consider a Reissner-Nordström spacetime in the parameter region (9/8)m 2 > q 2 > m 2 which possesses both a naked singularity and two photon spheres. The singularity of a Reissner-Nordström spacetime is known to be repulsive. This is because the energy of electromagnetic fields outside a small sphere near the singularity (e.g., in the sense of the Komar integral) exceeds the ADM mass m, and hence, the gravitational field is generated by negative energy in that region. The repulsive gravitational field is, of course, not strong. This is the reason why an LTS with a single component cannot exist in the vicinity of a naked singularity with an electric charge and a lower bound exists for its area in the Einstein-Maxwell theory.
The third remark is that there is no contribution from Φ + 0 in the Riemannian Penrose inequality for the Einstein-Maxwell system [13], whereas in our theorem it appears. This is because the Riemannian Penrose inequality discusses the minimal surface with k = 0, for which the inequality of Eq. (17) is unnecessary.
Finally, the presence of Φ + 0 makes the inequality tighter than the case of Φ + 0 = 0. From Eq. (15), one can see that the quantity Φ + 0 appears from (the part of) the energy density in the Hamiltonian constraint. The electromagnetic energy density increases if Φ + 0 is turned on, and through the relation given by Eq. (15), the positive energy density tends to make the formation of an LTS more difficult. This means that the area of S 0 will become smaller.

Dynamically transversely trapping surface in Einstein-Maxwell system
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.

Definition of a DTTS
The concept of a DTTS is inspired by the induced geometry of a photon surface in a spherically symmetric spacetime. The photon surface is defined as a timelike hypersurface S such that any photon emitted tangentially to S at an arbitrary point of S remains in S forever [1]. Let us consider spherically symmetric spacetimes with the metric, Solving the null geodesic equations, we can find that S satisfies dr/dt = ±f (1 − b 2 f /r 2 ) 1/2 , where b is the impact parameter. The induced metric of the photon surface S is obtained as where α(r) := bf (r)/r. The mean curvature of t = constant surface σ t in S is given bȳ 7/15 and the Lie derivative alongn a fork is computed as wheren a is the future-directed unit normal to σ t in S. For the Reissner-Nordström black hole, it becomes Thus, (3)£nk is negative and positive in the inside and outside regions of the photon sphere, respectively. Hence, the non-positivity of (3)£nk is expected to indicate the strong gravity. We now give the definition of a DTTS [8].
Definition 2. A closed orientable two-dimensional surface σ 0 in a smooth spacelike hypersurface Σ is a dynamically transversely trapping surface (DTTS) if and only if there exists a timelike hypersurface S that intersects Σ at σ 0 and satisfiesk = 0, max(K ab k a k b ) = 0, and (3)£nk ≤ 0 at every point in σ 0 , wherek is the mean curvature of σ 0 in S,K ab is the extrinsic curvature of S,n a is the unit normal vector of σ 0 in S, (3)£n is the Lie derivative associated with S and k a is arbitrary future-directed null vectors tangent to S.
Since the emitted photons do not form a photon sphere in general without spherical symmetry, here we emit photons in the transverse direction (to satisfy the condition k = 0), and adopt the location of the outermost photons as the surface S [the condition max(K ab k a k b ) = 0]. Then, we judge that the surface σ 0 exists in a strong gravity region if the condition (3)£nk ≤ 0 is satisfied. See our previous papers [8,9] for more details.

Penrose-like inequality for a DTTS
We present the following theorem on a Penrose-like inequality for a DTTS: ab . We suppose that an asymptotically flat spacelike hypersurface Σ is time-symmetric and foliated by the inverse mean curvature flow. We also assume that a slice of the foliation parameterized by y, where each of the y = constant surfaces, σ y , has topology S 2 , and σ 0 is a convex DTTS. We further assume the electric charge density ρ e to vanish outside σ 0 . Then, if ρ (m) := T (m) ab n a n b ≥ 0 in the outside region of σ 0 and P (m) r ab r a r b ≤ 0 on σ 0 , where n a is the future-directed timelike unit normal to Σ and r a is the outward spacelike unit normal to σ y in Σ, the areal radius r 0 = (A 0 /4π) 1/2 of the convex DTTS σ 0 satisfies the inequality where m is the ADM mass and q is the total charge. Here, Φ − 0 is defined by where h ab is the induced metric of σ 0 .

8/15
Proof. On σ 0 , the Lie derivative of the mean curvaturek along n a is given by [8] (3)£ where (2) R is the Ricci scalar of σ 0 , k ab is the extrinsic curvature of σ 0 in Σ, that is, k ab = (1/2) (3) £ r h ab , k is its trace, k L is the largest value of the eigenvalues of k ab and P r is the radial pressure defined in Eq. (8). Then, the condition (3)£ nk | σ0 ≤ 0 gives us on σ 0 , where we used Eq. (8) and the inequality for a convex DTTS [8], 2kk L + k 2 − k ab k ab ≥ (3/2)k 2 . With the condition P (m) r ≤ 0, the integration of the above over σ 0 gives us 3 where we used the Gauss-Bonnet theorem, the Cauchy-Schwarz inequality and Gauss' law as in the proof for Theorem 1.
We now consider the inverse mean curvature flow in which the foliation is given by y = constant surfaces. Similarly to the proof for Theorem 1, the surfaces y = 0 and y = ∞ are set to be the DTTS and a sphere at spacelike infinity, respectively. With the same procedure, one can derive the inequality of Eq. (20) again. Then, the integration of the inequality of Eq. (20) over the range 0 ≤ y < ∞ shows us where we used the inequality of Eq. (35) at the last step. Then, we arrive at the inequality given by Eq. (31).
There are four remarks. Similarly to Theorem 1, in general, the minimum value of the right-hand side of the inequality of Eq. (31) implies us the lower bound for m. However, unlike Φ + 0 in the case of an LTS, the quantity Φ − 0 does not have a definite signature. For 3 Note that the origin for Φ − 0 in the case of a DTTS is different from that for Φ + 0 in the case of an LTS. As we can see from the derivation, Φ ± 0 come from the energy density and the radial pressure of electromagnetic fields in the cases of the LTS and the DTTS, respectively. This is essential reason why we have different results for the LTS and the DTTS.
there is no such restriction for m, whereas, for Φ − 0 ≥ −1, m has a lower bound, Next, under the condition of the inequality of Eq. (37), a short rearrangement of the inequality of Eq. (36) gives where (39) However, from the same reason to the remark in Theorem 1, we expect that the lower bound is not correct for a DTTS with multiple components, and just the inequality would hold in a general context. For a DTTS with a single component, the lower bound must hold true with the same physical reason as the one given in Sect. 3.2.
As a third remark, in a similar way to the case of an LTS, the obtained inequality depends on the electromagnetic field. Interestingly, if Φ − 0 is negative, the contribution from Φ − 0 makes the inequality weaker than the cases of Φ − 0 = 0. Furthermore, for the case of Φ − 0 ≤ −1, the upper bound disappears. Let us discuss the effect of Φ − 0 physically. It is known that there are two kinds of pressure for magnetic fields. One is the negative pressure in the direction of magnetic field lines, called the magnetic tension. The other is repulsive interaction (i.e., positive pressure) between two neighboring magnetic field lines, called the magnetic pressure. A similar thing happens also to electric field lines (say, the electric tension and the electric pressure). We recall the formula for 8πP r , Eq. (8). In that formula, −(E a r a ) 2 − (B a r a ) 2 is the contribution of the electric/magnetic tension, while (E a E b + B a B b )h ab is the contribution of the electric/magnetic pressure. Then, Eq. (33) tells that the electric/magnetic tension makes the formation of a DTTS difficult, while the electric/magnetic pressure helps the formation of a DTTS. Therefore, in the presence of the electric/magnetic pressure, the area of a DTTS tends to be larger. This is the reason why the upper bound of the area of a DTTS becomes larger when Φ − 0 is negative. Nevertheless, the negativity of Φ − 0 would not change the situation so much in the following reason. If Φ − 0 is negative, the upper bound for the DTTS becomes weaker and the DTTS can exist at farther outside. However, Φ − 0 depends on the position of the DTTS and we naively expect that it is sharply decreasing according to the distance from the center, if the electromagnetic field is intrinsic to the compact object; namely, monopole or multi-pole fields. Therefore, when we take a farther surface, Φ − 0 becomes immediately negligible. Then, the area of the DTTS cannot be large. On the other hand, Φ − 0 could be large at some point by extrinsic effects, such as, external fields and/or dynamical generation of fields.
The final remark is on the relation between an LTS and a convex DTTS σ 0 on timesymmetric initial data. Recall Proposition 1 in Ref. [8], that is, a convex DTTS with k > 0 10/15 in time-symmetric initial data is an LTS as well if ρ + P r = 0 is satisfied on σ 0 . Since 8π ρ (em) + P (em) the presence of Φ ± 0 disturbs the equivalence between an LTS and a convex DTTS in general. This feature is reflected in the two inequalities obtained in this paper.

Numerical examination of the Majumdar-Papapetrou spacetime
In Sect. 4.2, we have obtained the Penrose-like inequality for a DTTS. There, the quantity Φ − 0 appears, and this quantity depends on the configuration of electromagnetic fields. The purpose of this section is to examine the values of Φ − 0 in an explicit example. Specifically, in our previous paper [8], we numerically solved for marginally DTTSs in systems of two equalmass 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∇ 2 U = 0 gives an exact solution, where∇ 2 is the flat space Laplacian. In this situation, E a and B a are calculated as Setting E 2 := h ab E a E b , we have 8πΦ − 0 = − σ0 E 2 dA. Therefore, the value of Φ − 0 is nonpositive.
In our previous paper [8], 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 formr = h(θ), 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 ≤ z 0 /m ≤ 0.79353. We refer readers to our previous paper [8] for explicit shapes of the obtained solutions. We examine the value of Φ − 0 . The left panel of Fig. 1 presents the behavior of E 2 as a function ofθ 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 2 is generally nonzero, but is less than 10 −3 . The right panel of Fig. 1 plots the value of Φ − 0 as a function of z 0 . It is negative and its absolute value is less than 5 × 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/4πr 2 DTTS+ (a red solid curve), where r DTTS+ is defined in Eq. (39), and the other is A/4πr 2 p (a green dotted curve), where 4πr 2 p is the area of a photon sphere with the same mass and charge [see Eq. The reason why the value of Φ − 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 Φ − 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 Φ − 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.  Fig. 3 The marginally DTTS for z 0 /m = 0.79353 (the blue curve) and the contour surfaces of U in the Majumdar-Papapetrou two-black-hole spacetime (gray and red curves). The red curve depicts the contour surface of U = 2.14.

Summary and discussion
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 4πr 2 p , 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 Φ − 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 Φ − 0 is very small in that situation. Up to now, we have assumed that the magnetic charge q m vanishes. Here, we consider what happens to our inequalities when nonzero q m is present. We first consider the case of an LTS. Since the total squared charge q 2 := q 2 e + q 2 m 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 S0 (E a r a ) 2 + (B a r a ) 2 dA ≥ A −1 0 (4πq e ) 2 + (4πq m ) 2 = (4πq) 2 A 0 (46) in the calculations of Eqs. (16) and (17). As a result, we must introduce the quantity instead of Φ + 0 of Eq. (14). The resultant inequality is the one of Eq. (13) but Φ + 0 being replaced by Φ 0 . Next, we consider the case of a DTTS. Similarly to the case of an 13/15 LTS, we must use the inequality of Eq. (46) (but S 0 being replaced by σ 0 ) in the calculations of Eqs. (34) and (35). As a result, instead of Φ − 0 , we must introduce −Φ 0 = −(1/8π) σ0 (E a E b + B a B b )h ab dA. The resultant inequality is the same as the one of Eq. (31), but Φ − 0 being replaced by −Φ 0 . These results are summarized as follows: Corollary 1. In the presence of nonzero q m , Theorems 1 and 2 hold by changing from Φ ± 0 to ±Φ 0 , where Φ 0 is defined by Eq. (47).
In the main article of this paper, we have not considered a TTS for the static and stationary spacetimes defined in our previous paper [7]. 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 [8]. 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.