Light from Reissner-Nordstrom-de Sitter black holes
LLight from Reissner-Nordstrom-de Sitterblack holes
Ion I. Cot˘aescu
West University of Timi¸soara,V. Pˆarvan Ave. 4, RO-1900 Timi¸soara, Romania
January 19, 2021
Abstract
The light from the Reissner-Nordstrom black hole in de Sitterexpanding universe is studied deriving for the first time the formof the spiral null geodesics around the photon sphere and theradius of the sphere hosting the apparent sources near the blackhole shadow. We obtains thus the principal parameter givingthe redshift and the observed black hole shadow according to ourmethod we proposed recently giving its complete calculations andresults in an algebraic code on computer [1].PACS: 04.02.Cv and 04.02
Keywords: Reissner-Nordstrom black hole; de Sitter expanding uni-verse; black hole shadow; redshift; spiral null geodesics; conserved quan-tity. 1 a r X i v : . [ g r- q c ] J a n Introduction
The light emitted by black holes provides us with information about twoimportant astronomical observables, namely the black hole shadow andredshift. The shadow is related to the black hole geometry such that a geometric method was widely used for studying the Schwarzschild blackholes in the flat Minkowski space-time [2] or in expanding universes [3–8] and rotating black holes with Kerr [9, 10] or Kerr-de Sitter [11–14]metrics. A special attention was paid to the new object M87 [15, 16]which was discovered recently [17, 18].However, the geometric method is not suitable for studying the red-shift which depend on the energies of the emitted and observed photonswhose ratio gives information about the cosmic expansion and the pos-sible peculiar velocity of the observed black hole. For separating thesetwo contributions one combined so far the Lemaˆıtre rule [19, 20] of Hub-ble’s law [21], governing the cosmological effect, [22–24] with the usualtheory of the Doppler effect of special relativity [25] even though thereare evidences that our universe is expanding. Recently we proposed animprovement of this approach replacing the special relativity with ourde Sitter relativity [26, 27] which allowed us to focus on the transforma-tion rules of the conserved quantities finding a new method for deriv-ing simultaneously the black hole shadow and redshit observed from theasymptotic zone of the de Sitter expanding universe [28, 29].This method is mainly algebraic being based on the conserved quan-tities on the spiral null geodesics that can be measured by a remote ob-server when we know the relative motion of the black hole with respectto this observer. The spiral geodesics are rolled out on the photon sphereescaping outside near their specific singularities such that the photons onthis geodesics give the light around the black hole shadow defining thusits radius. A remote observer, which neglects the black hole gravity, mea-sures an apparent genuine de Sitter null geodesic of a photon of givenenergy and momentum, emitted by an apparent source near the blackhole shadow. Thus we may derive simultaneously the redshift related tothe measured photon energy and the shadow angular radius given by theobserved photon momentum [29]. In the case of the Schwarzschild-deSitter black hole we met spiral geodesics which are very similar to Dar-win’s ones [30, 31], derived for this black hole in the flat space-time [29].However, if we intend to extend this study to the Reissner-Nordstromblack hole we find that only its photon sphere was studied so far [32]but without deriving the form of the spiral geodesics and their conservedquantities. 2or this reason we would like to continue this study here focusing onthe light around the Reissner-Nordstrom black holes for deriving the spi-ral geodesics and the conserved quantities we need for writing down theclosed formula of the radius of the sphere hosting the apparent sourcesaround the black hole shadow and the observed energy giving the red-shift. For this purpose we consider the Reissner-Nordstrom-de Sitterblack holes in co-moving local charts (frames) with Painlev´e coordinates[33] that appear as genuine de Sitter co-moving frames [34] for the remoteobservers. Then the conserved quantities in the proper black hole framecan be related to the observed ones through suitable de Sitter isome-tries [26, 27] as in Ref. [29] and its attached code on computer [1]. Wederive thus for the first time the equation of the spiral geodesics of theReissner-Nordstrom-de Sitter black holes pointing out their propertiesand deriving the radius of the sphere of the apparent sources that maybe substituted in our general results [29, 1] for finding the shadow andredshift of the Reissner-Nordstrom black holes moving feely in the deSitter expanding universe.We start in the second section reviewing the proper co-moving frameswith Painlev´e coordinates [33] of the Reissner-Nordstrom black hole inthe de Sitter expanding universe observing that for a remote observerthese appear as de Sitter co-moving frames. The next section is devotedto the spiral geodesics and their conserved quantities solving analyticallythe equation of the null geodesics and deriving the radius of the apparentsources near the black hole shadow. In the third section we show how theredshift and black hole shadow can be studied with our general method[29, 1] by substituting the parameters derived here. Finally, we presentour concluding remarks.In what follows we use natural Planck units with c = (cid:126) = G = 1and the notations of Ref. [29] where ω H = (cid:113) Λ3 c is the de Sitter Hubbleconstant (frequency) while the Hubble time t H = ω H and the Hubblelength l H = cω H have the same form. Let us consider the (1 + 3)-dimensional isotropic pseudo-Riemannianmanifolds, (
M, g ), where we may introduce local frames { x } of coor-dinates x µ ( α, µ, ν, ... = 0 , , , r, θ, φ ), and differenttime coordinates. The traditional static frames, { t s , r, θ, φ } , depend on3he static time t s having line elements of the form ds = g µν dx µ dx ν = f ( r ) dt s − dr f ( r ) − r d Ω , (1)where d Ω = dθ + sin θ dφ . These line elements can be put at any timein Painlev´e’s forms [33], ds = f ( r ) dt + 2 (cid:112) − f ( r ) dtdr − dr − r d Ω , (2)substituting in Eq. (1) t s = t + (cid:90) dr (cid:112) − f ( r ) f ( r ) , (3)where t represents the cosmic time of the frames { t, r, θ, φ } which haveflat space sections.The static frame of a Reissner-Nordstrom black hole of mass M andcharge Q in de Sitter expanding universe, has the metric (1) with f ( r ) = 1 − Mr + Q r − ω H r , (4)where, as mentioned before, ω H is the de Sitter Hubble constant in ournotation. The corresponding frame with Painlev´e coordinates and met-ric (2), { t, r, θ, φ } BH , has the asymptotic behaviour of the de Sitter co-moving frame { t, r, θ, φ } with f ( r ) → f ( r ) = 1 − ω H r . (5)For this reason we say that { t, r, θ, φ } BH is the co-moving frame of theReissner-Nordstrom black hole in the de Sitter expanding universe. Inthe asymptotic zone the frame { t, r, θ, φ } BH can be related to the observerco-moving one, { t, r, θ, φ } , through de Sitter isometries [26]. We assumethat the observers stay at rest in the origins of their own co-movingframes evolving along the unique time-like Killing vector field of the deSitter geometry which is not time-like everywhere but has this propertyjust in the null cone where the observations are allowed [35].In the black hole proper frame { t, r, θ, φ } BH the equation f ( r ) = 0might give the horizons depending on the parameters M , Q and ω H but we do not have closed formulas giving the general solutions of thisequation. Therefore, we may resort to a short numerical analysis for4igure 1: Function f ( ρ ) near the exterior horizon ρ + = 1 . µ = 0 .
01 and q = 0 . q = 0 (2) and that of the extremal black hole obtained here for q = 1 .
005 (3).which it is convenient to introduce the dimensionless coordinate ρ andthe new parameters κ and µ defined as ρ = rM , q = QM , µ = ω H M , (6)allowing us to write f ( r ) → f ( ρ ) = 1 − ρ + q ρ − µ ρ . (7)When µ = 0 the genuine Reissner-Nordstrom black hole has two horizonsat r ± = M ρ ± , ρ ± = 1 ± (cid:112) − q , (8)such that the condition 0 ≤ | q | ≤ | q | = 1 weobtain the extremal black hole whose horizons are degenerated.When we add the de Sitter gravity we must take into account that theparameter µ is extremely small such that the black hole horizons remainvery close to the values r ± while the specific de Sitter horizon is very far,near r dS ∼ ω − H → ρ dS ∼ µ − . Now the black hole with q = 1 has twodistinguishing horizons atˆ ρ − = √ µ − µ , ˆ ρ + = 1 − √ − µ µ (9)5ut which remain very close each other when µ (cid:28) ρ + − ˆ ρ − = 2 µ + O ( µ ). This means that now the range of q is larger but our numericalexamples indicate that the upper limit is so close to 1 such that it isconvenient to keep the condition | q | ≤ µ remains small. Forexample, even for larger values as µ = 0 .
01 the extremal black hole isobtained for q = 1 .
005 as one can see in Fig.1.
The shape of the null geodesics in the equatorial plane (with θ = π ) ofthe black hole frame { t, r, θ, φ } BH are given by the functions r ( φ ) whichsatisfy the equation [29] (cid:18) dr ( φ ) dφ (cid:19) − r ( φ ) E ph L ph + r ( φ ) f [ r ( φ )] = 0 , (10)resulted from the conservation of the photon energy, E ph , and angularmomentum along the third axis, L ph . Note that this equation is thesame as in the static frame since this is static, giving only the shape oftrajectory in the same space coordinates. In fact, the time evolution ongeodesics is quite different in the static and co-moving frames [36].In what follows it is convenient to consider the function ρ ( φ ) and thenew parameters defined by Eq. (6) for bringing Eq. (10) in homogeneousform (cid:18) dρ ( φ ) dφ (cid:19) − ρ ( φ ) E ph M L ph + ρ ( φ ) f [ ρ ( φ )] = 0 , (11)where f ( ρ ) is given by Eq. (7). This equation has two types of solutions,namely circular geodesics on the photon sphere and the associated spiralones.We start with the circular geodesics which must satisfy simultaneouslythe conditions dρ ( φ ) dφ = d ρ ( φ ) dφ = 0 . (12)giving a system of two algebraic equations whose solutions are the radiusof the photon sphere ρ ph = κ = 32 + 12 (cid:112) − q → r ph = κM , (13)derived in Ref. [32] and the mandatory condition L ph = ± κM E ph (cid:112) λ − κ µ , λ = 1 − κ + q κ . (14)6igure 2: The function ρ ( φ ) defined by Eq. (18) with κ = 2 , φ = 0.The new parameters introduced above have restricted ranges2 ≤ κ ≤ , ≤ λ ≤ , (15)since q ∈ [0 , q = 0 → κ = 3 , λ = we recover the parameters ofthe Scwartzschild-de Sitter black hole while for the extremal black holewe may take q = 1 → κ = 2 , λ = with a reasonable accuracy.Furthermore, we substitute the condition (14) in Eq. (11) obtainingthe new equation (cid:18) dρ ( φ ) dφ (cid:19) = ( ρ ( φ ) − κ ) κ (cid:2) ( κ − ρ ( φ ) + κ ) − κ (cid:3) , (16)which is independent on the Hubble de Sitter constant ω H as in thecase of the Schwarzschild-de Sitter black holes [29]. This equation has7hree constant solutions giving circular geodesics. Apart from the doublesolution ρ ph = κ of the circular geodesics of the photon sphere there aremore two constant solutions ρ = − κ √ κ − − √ √ κ − , ρ = − κ √ κ − √ √ κ − ρ < ρ is negative. More surprising is that Eq. (16) can be integratedanalytically obtaining that the shapes of the spiral geodesics r ( φ ) = M ρ ( φ ) are given by the functions ρ ( φ ) = κ ( κ −
1) cosh ν ( φ − φ ) + ( κ − √ κ − κ − ν ( φ − φ ) − √ κ − , (18)depending on the arbitrary integration constant φ and the parameter1 √ ≤ ν = (cid:114) − κ ≤ . (19)Obviously, these functions are determined up to a rotation fixing theorigin of the angular coordinate.As the functions (18) are derived here for the first time it deserves toinspect briefly their properties in the simpler case of φ = 0. Then thefunction is symmetric, ρ ( φ ) = ρ ( − φ ), having a pair of symmetric verticalasymptotes at φ ± = ± (cid:114) κ κ − √ κ − . (20)Moreover, the limits lim φ →±∞ ρ ( φ ) = κ , (21)point our the horizontal asymptote corresponding to the radius of thephoton sphere. From Fig. 2 we see that this function makes sense onlyon the domain ( −∞ , φ − ) ∪ ( φ + , ∞ ) where this remains outside the photonsphere, ρ ( φ ) ≥ κ . The domain [ φ − , φ + ] is an opaque window where thefunction is negative having no physical meaning.Thus we get the image of the spiral geodesics rolled out around thephoton sphere escaping outside only when φ is approaching to the values(20) where the radial functions r ( φ ) = M ρ ( φ ) can take larger valuesnear singularities. It is worth noting that in the left part of the domainthe spirals are left-handed while on the right part we meet right-handed8igure 3: The spiral geodesic given by function (18) with κ = 2 . φ = 0 for φ ∈ [ − π, − π ] (left panel) and φ ∈ [ π , π ] (right panel).spirals as in Fig. 3. Therefore if we need to work simultaneously with apair of symmetric left-handed and right handed spiral geodesics we mustdefine a suitable pair of functions r ± ( φ ), with different initial conditionsas in Ref. [29].It is remarkable that all the results concerning the spiral geodesics areindependent on µ = ω H M which is the only parameter depending on thede Sitter gravity. This means that we find similar spiral geodesics aroundthe Reissner-Nordstrom black hole in Minkowski space-time where µ = 0.In fact the de Sitter gravity is encapsulated only in Eq. (14) giving thecrucial quantity in determining the redshift and the black hole shadow.Finally, we observe that for q = 0 → κ = 3 , ν = 1 , λ = werecover the results obtained for the Schwartzschild-de Sitter black hole[29], namely the radius of the photon sphere ˆ r ph = 3 M , the conditionˆ L ph = ± √ M E ph (cid:112) − µ , (22)derived in Ref. [3] while the functions (18) becomeˆ ρ ( φ ) = 3 cosh( φ − φ ) + 1cosh( φ − φ ) − . (23)Note that this function is the same as in the case of the Schwartzschildblack holes in Minkowski space-time, complying with Darwin’s formula,930] 1 r ( φ ) = 1ˆ ρ ( φ ) M = − M + 12 M tanh (cid:18) φ − φ (cid:19) , (24)since, as in the general case, the parameter ω H arises only in Eq. (22)which was used for deriving the results of Refs. [29, 1]. The spiral geodesics are the closest trajectories to the photon sphere ofthe first photons that can be observed at the limit of the black holeshadow. Therefore, for studying this shadow and the associated redshiftwe have to consider only these photons that, in general, may be emittedby a moving black hole and measured by a fixed observer.The photon is emitted in the black hole co-moving frame with themomentum k , energy E ph = k = | k | and angular momentum (14). Thisis observed as coming from an apparent source S situated on a sphere ofradius r S = | L ph | k = ρ S M , (25)depending on the new parameter ρ S = κ (cid:112) λ − κ µ = κλ + 13 κ λ µ + O ( µ ) . (26)The apparent trajectory of this photon is a de Sitter geodesic whose con-served quantities depend exclusively on k and r S [29]. Then if we knowthe relative motion of the black hole with respect to the fixed observer wemay derive the conserved quantities measured by this observer by usingsuitable isometries of the de Sitter relativity [26, 28].In Ref. [29] we assumed that the photon is emitted at the initialmoment t = 0 when the black hole is translated with d moving alongthe direction of d with the peculiar velocity V with respect to the fixedobserver. Then by using a translation followed by a Lorentzian isometrywe deduced the general formulas of the observed quantities from which weextracted the redshift and shadow of the Schwartzschild-de Sitter blackholes in terms of d , V and the specific parameter ˆ ξ = ˆ r S d which dependson the black hole geometry only trough ˆ r S = | ˆ L ph | k where ˆ L ph is given byEq. (22) [29].For deriving the similar results in the case of our Reissner-Nordstrom-de Sitter black hole we have to take over all the results presented in Refs.1029, 1] substituting the new parameterˆ ξ → ξ = r S d = ρ S Md , (27)given by the radius (26). For example, in the particular case when theblack hole does not have an initial relative velocity ( V = 0) we may writethe simple formulassin α = ξ = r S d , (28)11 + z = 1 − ω H d cos α = 1 − ω H d (cid:112) − ξ , (29)showing how the observations of the shadow angular radius sin α and theredshift z are related each other.Note that the results presented here cover three particular cases asthe Shwartzschild black hole ( q = 0) in Minkowski ( µ = 0) [2] and deSitter ( µ (cid:54) = 0) [3, 29] space-times or the Reissner-Nordstrom black hole( q (cid:54) = 0) in Minkowski space-time ( µ = 0). We derived in premiere the functions giving the shapes of the spiralgeodesics around the Reissner-Nordstrom black holes in the de Sitterexpanding universe and the parameter ξ giving the redshift and blackhole shadow according to our method we proposed recently [1, 29].This method is mainly algebraic since this is based on the trans-formation rules under isometries of the conserved quantities being thusindependent on the coordinates we use, static or Painlev´e ones. Nev-ertheless, we preferred the co-moving frames with Painlev´e coordinatesfor keeping under control the philosophy of the remote observers and theblack hole relative motion in the de Sitter space-time.In general, the study of the conserved quantities is not enough forunderstanding the entire information carried out by the light emitted bymoving black holes. 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