Effects of a modified Reissner-Nordstr\"om spacetime

A modified Reissner-Nordstr\"om spacetime is considered here, where the central object (for example, a black hole or naked singularity) possesses a mass, with an ordinary, i.e., Standard Model (SM) electric charge, along with a dark electric charge associated with dark matter (DM). The inclusion of this dark charge modifies the gravitational properties of the spacetime, while not affecting ordinary electrodynamics. Geodesic motions of both charged and uncharged SM test particles are modified due to the presence of dark charge. The modifications may allow the detection and measurement of the dark charge. In particular, (1) the effective potential for orbiting particles is modified, and consequently, (2) the angular momenta and ISCOs of test masses in circular orbits are changed from those for the usual Reissner-Nordstr\"om case. (3) In the case of a naked singularity, it is possible that a"levitating atmosphere", due to short distance repulsive gravity, forms at the zero gravity radius, which depends upon both SM and DM charges. A levitating atmosphere may then cloak the naked singularity.


INTRODUCTION
The nature of dark matter remains somewhat mysterious, and its interaction with ordinary matter, that is, Standard Model (SM) matter, seems to be extremely weak or nonexistent, except via gravitational interactions.It is therefore reasonable to entertain the distinct possibility that dark matter (DM) may reside in the interiors of objects such as black holes or naked singularities.The types of DM particles and their mutual interactions, i.e., symmetries, is also unknown.One possibility is that the DM sector contains an unbroken U(1) abelian gauge symmetry associated with a "dark charge" and a massless "dark photon", which may be completely decoupled from the SM sector.
It therefore seems worthwhile to examine the possibility of a modified Reissner-Nordström (RN) spacetime.Specifically, we assume the coexistence of Standard Model (SM), or 'ordinary' matter along with 'dark' matter (DM) occupying the same region of space with metric g µν (x α ).The source (assumed to be some compact object such as a black hole or naked singularity) has a total mass M and a total "electric" charge Q composed of an unspecified proportion of SM and DM matter.The total mass is M = M S + M D and the total charge is Q T ot = Q S + Q D , where Q S(D) is the SM (DM) U(1) charge associated with an unbroken U(1) S × U(1) D group (i.e., the associated photons are massless).This modified Reissner-Nordström metric depends only on the total mass M and a (charge) 2 term proportional to (Q 2 S +Q 2 D ), and differs from the usual case by replacing It is assumed that the SM and DM sectors are completely decoupled from each other, i.e., noninteracting, other than gravitationally.But both SM and DM respond to gravitation in the same way, obeying the Einstein equations G µν = −κ 2 T T ot µν = −κ 2 (T SM µν +T DM µν ).(It is also assumed that both SM and DM sectors have the same gravitational coupling, i.e., κ S = κ D = κ = √ 8πG.)However, although a charge-neutral test mass m responds to the total mass M and total (charge) 2 ), an ordinary (SM) test charge q responds electromagnetically only to the SM charge Q S .In this case, SM charged particle geodesics differ from the normal case where Q D = 0, since the modified RN metric g µν (x) differs from the ordinary RN metric for which Q D = 0.This is due to the presence of the electromagnetic force qF S µν u ν appearing in the geodesic equations, where q responds only to the charge Q S , but a gravitational effect is due to both charges, Q 2 S + Q 2 D .We therefore anticipate an alteration in the expressions for the orbits of ordinary SM charged (and neutral) particles in comparison to the geodesics previously studied for the usual Reissner-Nordström case [1], [2], [3], [4] where Q = Q S and Q D = 0.In particular, we focus upon SM particles in circular orbits.
The idea has been entertained that on astrophysical scales, large masses, and possibly quite large (SM) charges Q S , may be encountered due to different effects.In fact, it has been speculated that enormous charges on the order of ∼ 10 20 C could be supported by highly charged compact stars, such as neutron stars [5], [6] or white dwarfs [7].Even if Q S is small, it has been speculated [8] that if a sufficiently heavy "dark electron" exists, the Schwinger effect [9] may be diminished for microscopic primordial black holes, allowing a large Q D .
A special case is that where the source has a mass M, and Q S = 0 but Q D = 0.In this case the source appears to all ordinary, SM particles to be a charge-neutral gravitating object with an extra energy density due to the dark electric field energy density.The modification of the usual single-charged Reissner-Nordström metric to the double-charged metric involves a simple substitution Gravitational effects on orbiting objects due to charge of the source are therefore modified if Q D = 0.This results in changes in angular momenta L(r), and therefore the radius of an innermost stable circular orbit (ISCO) for an object orbiting a black hole, and a change in the "zero gravity radius" of a spherical surface surrounding a naked singularity (see, for example, [10], [11]).A zero gravity sphere allows accreted matter to form a "levitating atmosphere", supported by gravitation and requiring no radiation pressure [10], [11].In this sense, the "levitating atmosphere" around the singularity differs from the radiation supported "levitating atmosphere" lying above the surface of a luminous neutron star [12], [13], with both having definite inner and outer radii.

A MODIFIED REISSNER-NORDSTR ÖM METRIC
We assume SM and DM electric charges Q S and Q D , respectively, to be generated by a gauge group U(1) S ×U(1) D .An ordinary (SM) electric charge q interacts only with the U(1) S photon A µ .(Here, natural units with = c = ǫ 0 = 1 are used along with Heaviside-Lorentz electromagnetic units, but G is allowed to remain visible.We use the (−, +, −) Misner-Thorne-Wheeler classification [14] for the metric, Riemann tensor, and Einstein equation.Conversion to usual gravitational-Gaussian units amounts to setting G → 1 and converting the Heaviside-Lorentz charge The usual Reissner-Nordström metric (using the above conventions) is given by ds 2 = f 0 (r)dt 2 − f 0 (r) −1 dr 2 − r 2 dΩ 2 , (dΩ 2 = dθ 2 + sin 2 θdϕ 2 ), where Q S denotes an ordinary (SM) charge, with The modified RN metric proposed here is of the same form, but with an additional (dark) where Q S and Q D are SM and DM charges, respectively 1 .The U(1) S and U(1) D gauge fields are noninteracting, except through gravitation.Since the U(1) gauge fields are assumed to be decoupled, the associated electromagnetic stress-energies are additive, which results in a replacement of the Q 2 S /r 2 term in the usual RN metric by a term (Q 2 S +Q 2 D )/r 2 in the function f (r) of the modified metric.Specifically, for the modified RN metric we have f 0 (r) → f (r): with µ and Z 2 given by (2b).
The action for the system consists of a field theoretic and a particle contribution for an "ordinary" (SM) test particle of mass m, S = S f + S p = d 4 x √ gL f + dτ L p , where we denote the SM electromagnetic field by The total charge of the object which appears in this modified RN metric is An ordinary (SM) charge q interacts only with the SM fields A µ and F µν , although both SM and DM charges contribute to gravitation through (Q The Lagrangians for the fields and test particles are given by with the overdot representing differentiation with respect to proper time τ for a massive particle (or an affine parameter for a photon).The equations of motion for the U(1) gauge fields following from L f are ∇ µ F µν = J ν , and ∇ µ F µν = J ν along with the associated Bianchi identities.The region of interest is outside of the source, where J ν = J ν = 0, and we will assume that the charged (SM + DM) source generates a static, spherically symmetric SM electric field E(r) with vanishing magnetic field, B = 0, and a DM electric field E(r) (not felt by SM particles), with a vanishing DM magnetic field B = 0.
The field equations outside of the source (with Bianchi identities) then reduce to where the dual tensors are Fµν = 1 2 ǫ µναβ F αβ and Fµν = 1 2 ǫ µναβ F αβ .In addition to the mass M of the source, both the SM and DM fields contribute to gravitation via a stressenergy tensor T = T µν + T µν due to the gauge fields.Assuming a static, spherical symmetry, the electric fields E r = F r0 and E r = F r0 follow from ( 5), e,g,, where The additivity of the stress-energy tensors for the abelian gauge fields outside of the mass M results in the RN metric with the replacement For this modified Reissner-Nordström metric with For |Z|/µ < 1 we have a black hole, and for |Z|/µ = 1 we have an extremal black hole.However, for |Z|/µ > 1 the metric represents a naked singularity.
To be seen from the geodesic equations or the effective potential, in the case of a naked singularity, a "zero gravity sphere" will exist for neutral test particles [1], [2], [10], [11] at a radius r 0 = Z2 /µ.At this radius the spatial acceleration a µ = u α ∇ α u µ of a neutral test particle placed at rest will vanish, according to a static observer.At this radius r 0 , accreting matter can collect to form a spherical "cloak" around the singularity [10], [11] as in the case of the usual Reissner-Nordström metric [1], [2], [10], [11].For r < r 0 there is a gravitational repulsion, and for r > r 0 there is a gravitational attraction.However, in the present case it is possible for the singularity to have a vanishing "ordinary" electric charge, Q S = 0, but a nonvanishing dark charge, Q D = 0, thereby appearing to be an electrically neutral object.For |Z|/µ > 1 this object may appear as an electrically neutral cloaked singularity.

GEODESICS
We consider the motion of an SM test particle of mass m which may, or may not, have SM charge q (q = 0 for a neutral test mass) in the presence of gravitational and electric fields, sourced by some compact object of total mass M = M SM + M DM and total charge Q T ot = Q S +Q D .The Lagrangian for a test particle of mass m and charge q in the spherically symmetric spacetime under consideration is taken to be where overdot indicates differentiation with respect to an affine parameter, such as proper time τ .We assume the particle motion to be restricted to the equatorial plane, with θ = π/2, with θ = θ = 0.The motion of a test particle is then obtained from L p using the Euler-Lagrange and constraint equations, with u α = ẋα and the condition u α u α = β constrains the trajectory to the particle worldline.We will focus only on the case of a test particle with nonzero mass (β = 1).
The resulting equation of motion for the test charge can be written in the equivalent forms where the terms on the right hand side are due to the Lorentz force acting on the particle.For our metric given by (3), the Lagrangian L p is cyclic in the variables t and ϕ, implying conservation laws for energy and angular momentum.Writing p µ = mu µ = mg µν u ν we have for µ = 0 and µ = ϕ, where E and L are constants.As a reminder, the charge q is assumed to be an ordinary (SM) charge which interacts only with the charge Q S which generates the SM electric field E. The (SM) potential A 0 is determined from the Maxwell field F r0 = ∂ r A 0 with A 0 (r) = F r0 dr = Q S /(4πr).

Circular orbits
For circular orbits, u r = 0, ur = 0, and V 2 = E 2 /m 2 , so that in this case (16) gives solutions (with For the case that Q D = 0 this reduces to the effective potential given in Refs.[1], [2] for a charge q in a circular orbit about a Reissner-Nordström black hole The charge product qQ S could be positive or negative, qQ S = ±|qQ S |.From this point forward we only consider the potential V + , which has an asymptotic value V + (∞) = 1 (i.e., E = m), and simply denote it by V : The angular momentum L of a charge q in a circular orbit of radius r is a constant on that particular orbit, but L varies with different orbital radii r, and is determined by the conditions For the special case qQ S = 0, i.e., either q = 0 (an SM charge neutral particle) and/or Q S = 0 (an SM charge neutral source), applying (19) to (18) yields the result where . When Q S = 0, this result holds for any SM particle -charged or neutral -in circular orbit of radius r.This agrees with, and generalizes, previous results for the usual Reissner-Nordström case [1] where The radius r ISCO of the innermost stable circular orbit (ISCO) is determined by the condition [17], [18] dL 2 dr r ISCO = 0 Differentiating (20), (21) gives the condition (for r = 0) This recovers the condition4 found in [1] for the special case Q 2 D = 0.In the limiting case of a Schwarzschild black hole (Z = 0) we obtain the Schwarzschild value, r ISCO = 6µ.Eq.( 22) can be solved for Z = 0, but the solutions are quite complicated (see, e.g., Eq.( 12) of Ref. [1] for the special case Q D = 0), and are not presented here.For qQ S = 0 the solutions to (19) and ( 21) are even more complicated, and are not considered here.
We can note the following points.
(i) Previous detailed analyses for neutral or charged particles in circular orbits in a Reissner-Nordström spacetime can be extended and generalized to the case of a modified Reissner-Nordström spacetime, where both SM and DM charges may comprise the source -e.g., a black hole or a naked singularity.
(ii) A source that appears to be SM charge-neutral (Q S = 0) and nonrotating, may in fact, contain an apparent mass function M(r) = M − Z 2 D /(2r) for the metric function r .The correction term −Z 2 D /r = −KQ 2 D /r (mimicing a negative mass distribution), disappears asymptotically, with M(∞) = M, but has an effect on all orbiting SM particles that differs from that expected for an ordinary Schwarzschild or Reissner-Nordström source.The angular momentum of a particle, described by (20), and hence the ISCO, is somewhat sensitive to the value of Z 2 , which generally depends upon both Q S and Q D .
(iii) The potential exists for a detection and determination of the set of parameters {µ, Q 2 S , Q 2 D } characterizing the (spinless) source.These parameters µ, Q 2 S , and Q 2 D can, in principle, be extracted through measurements by a distant observer of the orbital velocities of test particles in three different circular orbits of radii r i , i = 1, 2, 3, i.e., three independent equations containing {µ, Q 2 S , Q 2 D }.A distant observer in an asymptotic region of space at a coordinate distance R ≫ r i (and in the equatorial plane) could measure the rotational velocities v ϕ i (r i ) (say, by Doppler shifts) of three different test particles in circular (equatorial) orbits at radii r i from the center of the source.The physical rotational velocity v ϕ i (r i ) (velocity in the ϕ direction) according to a local static observer at a coordinate distance r i from the center of the source (origin) is where ω = dϕ/dt = 2π/T is simply the angular speed of the orbiting particle as observed by the distant (asymptotic) observer (for which g 00 = 1).If the distant observer can measure v ϕ i (r) (e.g., combined gravitational and velocity Doppler shifts) and ω(r) = 2π/T (r) (e.g., orbital period), then Solving these three equations simultaneously then yields (in principle) the set {µ, Q 2 S , Q 2 D }. (Detecting these parameters, in practice, is not expected to be easily done, requiring skills of experienced astronomers able to measure combined gravitational and velocity Doppler shifts, and deal with a realistic set of orbiting objects.)Other methods for obtaining estimates or bounds on charges, such as lensing effects, or parameters of noncircular stellar orbits (as for the S2 star around Sgr A*), and binary systems, may also exist (see, e.g., Ref. [19] and references therein), but are not considered here.

ZERO GRAVITY SPHERES AROUND NAKED SINGULARITIES
From the form of g 00 (r) = f (r) = 1 − 2µ/r + Z 2 /r 2 we find black hole solutions for Z 2 /µ 2 ≤ 1 with horizons given in (7), where g 00 = 0. On the other hand, for Z 2 /µ 2 > 1 the metric describes a naked singularity, with g 00 being nonzero at all spatial points.However, in this case an interesting situation arises wherein there are "zero gravity spheres" (see, e.g., [1], [2], [10], [11] and references within.)A "zero gravity sphere" is a spherical surface where a particle placed at rest remains at rest in a state of stable equilibrium, according to a distant static observer.For the metric considered here, a radius r 0 = Z 2 /µ locates the radius of a zero gravity sphere (see below) for a neutral particle (q = 0) and/or for a neutral singularity (Q S = 0).At radii r < r 0 a neutral particle experiences a gravitational repulsion, and for radii r > r 0 gravitation is attractive.By spherical symmetry, such particles can spherically accrete near the radius r 0 of the singularity, oscillate, and through dissipative processes, settle down, more or less, near the radius r 0 .After time, a spherical cloud, or a "levitating atmosphere", forms near this zero gravity sphere (see, for example, Refs.[10] and [11], and references within).A levitating atmosphere around a naked singularity is the analogue of the levitating atmosphere above the surface of a luminous neutron star, which is supported by radiation [12], [13].However, the levitating atmosphere around the singularity is supported solely by gravitation, requiring no radiative support [10], [11].The radius r 0 of the zero gravity sphere locates the minimum of the effective potential V (r), determined by the conditions ∂ r V | r 0 = 0 and ∂ 2 r V | r 0 > 0 at the equilibrium point r 0 .For Z 2 /µ 2 > 1 this minimum is radially stable.Similar statements can be made for charged particles (q = 0) and a charged singularity (Q S = 0), but the solutions for the corresponding radii r ± q are a little more complicated.Nevertheless, small oscillations about these radial positions are expected to become damped, and charged particles may combine to form neutral ones, allowing the formation of a levitating cloud, or "atmosphere" to form around the singularity.The objective here is to simply show the existence of these zero gravity spheres for the extended Reissner-Nordström metric, where Z 2 depends upon both Q 2 S and Q 2 D .What would normally be considered to be an "undercharged" Reissner-Nordström black hole (with Q 2 S < µ 2 ) may actually be an "overcharged" extended Reissner-Nordström singularity, with Q 2 S < µ 2 but Z 2 > µ 2 .To locate the radii of one of these equilibrium spheres, we assume a particle to be at rest at a radius r with u r = u θ = u ϕ = 0.Then, for a static observer, the condition ur = 0 for a particle locates the radius r 0 or r ± q .Equivalently, for the effective potential we require L = 0 and u r = 0 for the particle, and look for the minimum of V (r).The effective potential for motion at a constant radius, i.e., circular motion, is given by (18), so that for the case for which either or both the SM charge q of the test mass, or the charge Q S of the singularity, vanishes, the effective potential for a particle at rest, L = 0, is Since |Z|/µ > 1 for a singularity, where ), then g 00 and V 0 (r) are everywhere nonzero.An extremum of V 0 is determined by ∂ r V 0 | r 0 = 0 and stability at this radius r 0 requires that the extremum is indeed a minimum, , (zero gravity radius, qQ S = 0) ( and > 0, ensuring radial stability for µ 2 /Z 2 < 1.
5.2.L = 0, qQ S = 0 For this case the effective potential is more complicated.For a stationary particle with u ϕ = 0 (L = 0) and u r = u θ = 0, where V 0 = √ g 00 .The extrema are located by Now, for convenience, define the r -independent parameter C ≡ qQ S 4πm = q QS m which depends upon the charge Q S of the singularity and the charge-to-mass ratio q/m of the test particle.Then the potential V (r) can be written as (Note that in Gaussian units C = q QS /m.)The parameter C = ±|C| can be positive or negative.The potential for a charged particle, V , is that for a neutral particle, V 0 = √ g 00 , shifted by the term C/r.The local extrema r q of V (r) are given by Eq.( 28), which, for C 2 < µ 2 , has solutions 5 given by where r + is the solution for C > 0 (qQ S > 0, electric repulsion) and r − is the solution for C < 0 (qQ S < 0, electric attraction).For the case that |C| ≪ µ, we have, approximately, For C = 0 we recover the solution r 0 = Z 2 /µ for a neutral particle q = 0, and/or Q S = 0 for the singularity.The equilibrium radius r − q is a little smaller than r 0 for electrical attraction, while r + q is a little further away for electrical repulsion.

Levitating atmospheres
Although dark matter particles may also accumulate near some equilibrium sphere, little, if anything, is really known about their interactions among other particles.However, particles of ordinary matter will interact with each other, and some particles can eventually collect near the zero gravity radius r 0 .If a particle is assumed to have some angular momentum L, then L will minimize with L = 0 at an ISCO of radius [18] r ISCO ≈ r 0 .
The effective potential V 0 (r) = g 00 (r) for an "ordinary" neutral particle (Q S = 0) diverges at the singularity (V 0 (r) → +∞ as r → 0 for Z 2 > µ 2 ), so that no such particle of finite energy actually reaches it.The interactions of ordinary neutral particles near r 0 can dissipate energy, so that a particle with initial energy E/m V 0 (r 0 ) becoming trapped within the potential well will undergo radial oscillations about r 0 , and form a spherical "levitating atmosphere" near that radius.The characteristics of such atmospheres around naked singularities possessing zero gravity spheres has been studied previously by Vieira and Kluźniak (see, for example, [10], [11] and references therein).In this way the naked singularity becomes "cloaked" by the levitating atmosphere.It is proposed [11] that such an atmosphere can be formed in a very short time by the accretion of ambient matter.This "levitating atmosphere" around the singularity, is analogous to the radiation supported "levitating atmosphere" lying above the surface of a luminous neutron star [12], [13], both having definite inner and outer radii.However, the levitating atmosphere around the singularity is supported only by gravity, requiring no radiative support.
The situation for charged particles, however, is less clear, since V (r) = C/r + V 0 (r) → ±∞, depending on the sign and magnitude of C ∝ qQ S /m and V 0 (r).In the case that C < 0 (electric attraction) with |C| > |Z|, then V (r) decreases toward the origin and becomes negative with V (r) → −∞ as r → 0. An unperturbed electric field E r ∼ Q S /r 2 then grows in magnitude without bound at small r.If at some small r c this field reaches a critical strength e /e where the Schwinger effect [9] becomes important, then a cascade of e + e − pairs may be coaxed out of a destabilized vacuum at radii r r c , with the ensuing evolution becoming problematic.If enough charged particles of sign opposite to that of Q S fall onto the singularity, the result would be a reduction of |Q S | and |Z| and an increase in mass µ, possibly resulting in a state where Z 2 < µ 2 .One then expects the formation of horizons, resulting in the naked singularity evolving into a black hole.This process would support the claims of Cohen and Gautreau [20] that a Reissner-Nordström naked singularity can be destroyed but not created, while event horizons can be created but not destroyed.

CONCLUSIONS
We have considered the simple, but interesting, possibility that some compact objectsblack holes or naked singularities -can harbor an ordinary, Standard Model, charge Q S and/or a dark charge Q D due to an unbroken U(1) S × U(1) D symmetry of the source.The corresponding photons are massless, and the Reissner-Nordström solution to the Einstein equation is extended so that the metric components g 00 (r) = −g rr (r) = f (r) depend upon both charges Q S and Q D as in Eqs.( 2) and (3).The parameter Z 2 ∝ (Q 2 S + Q 2 D ) now appears in the metric component g 00 (r) = f (r).For µ 2 > Z 2 the object is a black hole, for µ 2 = Z 2 it is an extremal black hole, and for µ 2 < Z 2 the object is a naked singularity.The dark matter and dark charge are assumed to be completely decoupled from the SM sector, except via gravitation.The presence of a nonzero dark charge will alter the geodesics expected for an ordinary Reissner-Nordström source.Geodesic motions of both neutral and charged particles in circular orbits around ordinary Reissner-Nordström black holes and naked singularities have been well studied (see, for example, [1] and [2] and references therein), but the inclusion of a nonzero dark charge will modify these previous results with a replacement of Q ) in the metric.In principle, these modifications could allow the inference of the existence and inclusion of dark matter and dark charge hidden in the gravitational source.
For ordinary SM particles in circular orbits around a source with Q D = 0, the angular momentum per unit mass L(r)/m differs from that due to a standard Reissner-Nordström source.In fact, even in the extreme case Q S = 0, all SM particles, charged or neutral, have angular momenta L(r)/m determined only by the parameters M and Q D of the source (see (20)).In addition, the radius r ISCO of the ISCO will depend upon both types of charge (see (21) and ( 22)).This observation for an idealized situation concerning test particles may lead to insights for a more realistic case where a real accretion disc, with its complex internal interactions, may have properties dependent upon the dark charge Q D of the source.Since the metric components g 00 and g rr for the modified metric generally differ from those for the usual one, the presence of nonzero Q D changes the position and structure of a black hole horizon or naked singularity.
In the case of a naked singularity, a "zero gravity" spherical surface exists, where a particle having zero angular momentum L = 0, [1], [2] can remain at rest.Consequently, it is possible for a "levitating atmosphere" of accreted matter of the type previously described by Vieira and Kluźniak [10], [11] to form, so that the cloaked singularity may take the appearance of a more ordinary astrophysical object.This levitating atmosphere surrounding a naked singularity is analogous to the levitating atmosphere above a luminous neutron star [12], [13], both having a finite thickness.The radius of a zero gravity sphere for a neutral particle is given by r 0 = Z 2 /µ, which generalizes the results found [1], [10], [11] for the usual Reissner-Nordström spacetime with Q S = 0, Q D = 0. Since the radius of a zero gravity sphere depends upon both charges Q S and Q D , a possible consequence is that an inferred observational value for Q S may actually be an inferred value for The radius of a zero gravity sphere locates the stable minimum of the effective potential where u r = u θ = u ϕ = 0, and a neutral particle placed at rest there remains at rest.For r < r 0 there is a gravitational repulsion, which (for a naked singularity with Z 2 > µ 2 ) diverges to +∞ as r → 0, preventing any massive neutral particle with finite energy from reaching the singularity.As emphasized by Mishra and Vieira [21], no stable circular orbits with r < r 0 exist for a naked singularity.If a gravitating source is assumed to be a naked singularity, an observed lower bound for stable circular orbits can yield a bound on r 0 and |Z|/µ.As an interesting example, Mishra and Vieira [21] have considered the compact source Sgr A * at the center of the galaxy to possibly be a Reissner-Nordström singularity.In this case it is argued that, based upon the Event Horizon Telescope observations [22], the apparent size of Sgr A * allows an inference of its charge to mass ratio to lie within a restricted range of 1 < | QS |/µ < 2.32 (using geometrized Gaussian units).However, the replacement Q 2 S → Z 2 → ( Q2 S + Q2 D ) could be used to obtain bounds for a dark charge 1 < | QD |/µ < 2.32 for Sgr A * if it is assumed to have QS = 0.
In conclusion, we have studied the possibility that a gravitating source, such as a black hole or naked singularity, can possess an abelian U(1) symmetry associated with a dark charge, analogous to the corresponding symmetry of the Standard Model, and the possible consequences.If an abelian dark charge with an attendant electrodynamics does in fact exist, a new avenue of probing dark matter properties might be opened.Since the nature of dark matter remains so enigmatic, theoretical explorations of its possible properties and effects are essential for further understanding.