Obtaining mass parameters of compact objects from red-blue shifts emitted by geodesic particles around them.

# Obtaining mass parameters of compact objects from red-blue shifts emitted by geodesic particles around them.

Ricardo Becerril, Susana Valdez-Alvarado and Ulises Nucamendi, Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo. Edif. C-3, 58040 Morelia, Michoacán, México,
August 22, 2019
###### Abstract

The mass parameters of compact objects such as Boson Stars, Schwarzschild, Reissner Nordstrom and Kerr black holes are computed in terms of the measurable redshift-blueshift () of photons emitted by particles moving along circular geodesics around these objects and the radius of their orbits. We found bounds for the values of () that may be observed. For the case of Kerr black hole, recent observational estimates of SrgA mass and rotation parameter are employed to determine the corresponding values of these red-blue shifts.

preprint: version/03-09-16

## I Introduction

The increasing amount of evidence that many galaxies contain a supermassive black hole at their center evidence , motivated Herrera and Nucamendi (hereafter referred as H-N) to develop a theoretical approach to obtain the mass and rotation parameter of a Kerr black hole in terms of the redshift and blueshift of photons emitted by massive particles traveling around them along geodesics and the radius of their orbits ulises . They found an explicit expression of the rotation parameter as a function of , , the radius of circular orbits and the mass , whereas might be found by solving an eight order polynomial which can only be done numerically. These circular orbits should of course, be bounded and stable. If a set of observational data , that is, a set of red and blue shifts emitted by particles orbiting a Kerr black hole at different radii were given, what would be desirable to know is the mass and rotation parameter in terms of that data set. In this paper, we provide the details of how this can be accomplished. Particularly, the mass of the black hole for SgrA and its corresponding angular momentum that have been recently estimated estimates : and , are employed in our analysis. In addition, the mass parameter of axialsymmetric non-rotating compact objects such as Schwarzschild and Reissner-Nordstrom black holes as well as Boson-Stars is found in terms of the red-blue shift of light and the orbit radius of emitting particles. In order to have a self contained paper, we provide a brief summary of H-N theoretical scheme in the section II. In sections III and IV we deal with the non-rotating examples above mentioned and the rotating Kerr black hole respectively.

## Ii Theoretical Approach

H-N considered a rotating axialsymmetric space-time in spherical coordinates . The geodesic trayectory followed by a massive particle in this space-time can be obtained by solving the Euler-Lagrange equations

 ∂L∂xμ−ddτ(∂L∂˙xμ)=0, (1)

with the Lagrangian given by

 L=12[gtt˙t2+2gtϕ˙t˙ϕ+grr˙r2+gθθ˙θ2+gϕϕ˙ϕ2], (2)

being and the proper time. It is assumed that the metric depeds solely on and ; thus, the space time is endowed with two commuting Killing vectors which read: , . Since , there are two quantities that are conserved along the geodesics

 pt = ∂L∂˙t=gtt˙t+gtϕ˙ϕ=gttUt+gtϕUϕ=−E, pϕ = ∂L∂˙ϕ=gtϕ˙t+gϕϕ˙ϕ=gtϕUt+gϕϕUϕ=L, (3)

where is the 4-velocity which is normalized to unity rendering

 −1 = gtt(Ut)2+grr(Ur)2+gθθ(Uθ)2+gϕϕ(Uϕ)2 (4) +gtϕUtUϕ.

Two of these 4-velocity components can be found by inverting (3)

 Ut=gϕϕE+gtϕLg2tϕ−gttgϕϕ,Uϕ=−gtϕE+gttLg2tϕ−gttgϕϕ. (5)

Inserting (5) in (4) one obtains

 grr(Ur)2+Veff=0, (6)

where is an effective potential given by

 Veff=1+gθθ(Uθ)2−E2gϕϕ+L2gtt+2ELgtϕg2tϕ−gttgϕϕ. (7)

The goal is to write the parameters of an axialsymmetric space-time in terms of the observational red and blue shifts and of light emitted by massive particles moving around a compact object. These photons have 4-momentum that move along null geodesics . Using the same Lagrangian (2) one gets two conserved quantities

 −Eγ = gttkt+gtϕkϕ, Lγ = gϕtkt+gϕϕkϕ. (8)

The frequency shift associated to the emission and detection of photons is defined as

 1+z=ωeωd, (9)

where is the frequency emitted by an observer moving with the massive particle at the emission point and the frequecy detected by an observer far away from the source of emission. These frequencies are given by

 ωe=−kμUμ|e,ωd=−kμUμ|d. (10)

and are the 4-velocity of the emisor and detector respectively. If the detector is located very far away from the source () then since , whereas . The frequency is explicitly given by

,

with a similar expression for . As a result (9) becomes

 1+z=(EγUt−LγUϕ−grrUrkr−gθθUθkθ)|e(EγUt−LγUϕ−grrUrkr−gθθUθkθ)|d. (11)

This is an expression for the red and/or blue shifts of light emitted by massive particles that are orbiting around a compact object measured by a distant observer. The apparent impact parameter of photons, that is to say, the minimum distance to the origin was introduced for convenience. Due to the fact that and are preserved along null geodesics all the way from emission to detection one has that . On the other hand, a set of massive particles (that could be a set of stars) that may be orbiting around a compact object (that could be a black hole) is expanding as a whole and it has a redshift . Yet all those particles are individually moving having therefore, an individual redshift. Astronomers define a kinematic redshift as , and some report their data in terms of . corresponds to a frequecy shift of a photon emitted by a static particle located at thus

 1+zc=(EγUt)|e(EγUt)|d=UteUtd (12)

The kinematic redshift can be written as

 zkin=(Ut−bUϕ−1EγgrrUrkr−1EγgθθUθkθ)|e(Ut−bUϕ−1EγgrrUrkr−1EγgθθUθkθ)|d−UteUtd (13)

The analysis can be performed with either using (13) or using (11). We work with in this paper. The general expression (13) is simplified for circular orbits () in the equatorial plane ()

 zkin=UtUϕdbd−UtdUϕebeUtd(Utd−bdUϕd). (14)

In (14) what is still needed is to take into account light bending due to gravitational field, that is to say, to find . The criteria employed in ulises to construct this mapping is to choose the maximum value of at a fixed distance from the observed center of the source at a fixed . Inverting (8) to obtain and inserting this expression into with and one arrives at

 b±=−gtϕ±√g2tϕ−gttgϕϕgtt, (15)

can be evaluated at the emissor or detector position. Since in general there are two different values of , there will be two different values of of photons emitted by a receding () or an approaching object () with respect to a distant observer. These kinematic shifts of photons emitted either side of the central value read

 z1=UteUϕdbd−−UtdUϕebe−Utd(Utd−Uϕdbd−), (16)
 z2=UteUϕdbd+−UtdUϕebe+Utd(Utd−Uϕdbd+). (17)

In the next section we shall apply this formalism to non-rotating compact objects.

## Iii Non-rotating space-times

In order to apply H-N approach, it is necessary to have a Killing tensor of the space-time to be analyzed, this implies the existance of an additional constant of motion . is not needed in the case of non-rotating space-times, that is to say, for or when particles are orbiting just on the equatorial plane. In the present section, we study the relationship between the observed redshift (blueshift) of photons emitted by particles traveling along circular and equatorial paths around non-rotating compact objects and the mass parameter of these objects. Since vanishes, the apparent impact parameter becomes and the effective potential (7) acquires a rather simple form

 Veff=1+E2gtt+L2gϕϕ. (18)

For circular orbits and its derivative vanish. From these two conditions one finds two general expressions for the constants of motion and for any non-rotating axialsymmetric space-time

 E2=−g2ttg′ϕϕgttg′ϕϕ−g′ttgϕϕ, (19)
 L2=g2ϕϕg′ttgttg′ϕϕ−g′ttgϕϕ, (20)

where primes denote derivative with respect to . In order to guarantee stability of these circular orbits, must hold. The general expression for is

 V′′eff = −E2[g′′ttgtt−2(g′tt)2g3tt]−L2⎡⎣g′′ϕϕgϕϕ−2(g′ϕϕ)2g3ϕϕ⎤⎦ (21) = g′ϕϕg′′tt−g′ttg′′ϕϕgttg′ϕϕ−g′ttgϕϕ+2g′ttg′ϕϕgϕϕgtt,

where (19) and (20) were employed in the last step. Using the explicit form of and , (19) and (20), in (5) one obtains expression for the 4-velocities in terms of solely the metric components

 Uϕ= ⎷g′ttgttg′ϕϕ−g′ttgϕϕ,Ut=− ⎷−g′ϕϕgttg′ϕϕ−g′ttgϕϕ. (22)

from which the angular velocity of particles in these circular paths becomes

 Ω= ⎷−g′ttg′ϕϕ. (23)

Since , the redshift and blueshift are equal but with opposite sign: , the explicit expression is

 z1=−UteUϕdbd++UtdUϕebe+Utd(Utd+Uϕdbd+). (24)

Furthermore, if the detector is located far away from the compact object , and as we mentioned before, . Thus (24) becomes

 z1=Uϕebe+= ⎷−gϕϕg′ttgtt(gttg′ϕϕ−g′ttgϕϕ). (25)

### iii.1 Schwarzschild Black Hole

As our first working example of a non-rotating space-time, we consider the Schwarzschild black hole, for which the relevant metric components are and . Inserting these components in (25) with one finds

 z2=rcM(rc−2M)(rc−3M), (26)

which is a relationship between the measured red-shift , the mass parameter of a Schwarzschild black hole and the radius of a massive particle’s circular orbit that emitts light and of course, . The relationship (26) is equivalent to

 M=rcF(z)whereF±(z)=1+5z2±√1+10z2+z412z2. (27)

On the other hand, circular orbits are stable as long as that , from (21) reads

 V′′eff=2M(rc−6M)r2c(rc−2M)(rc−3M), (28)

which is positive provided that ; therefore, which is fulfilled if and only if and solely for the minus sign . Hence, a measurement of the redshift of light emitted by a particle that follows a circular orbit of radius in the equatorial plane around a Schwarzschild black hole will have a mass parameter determined by , and must be . The energy, angular momentum, velocities , and the angular velocity of the emitter, can be computed from (19), (20), (22) and (23) and written as function of the measurable redshift and radius of the circular photons source’s orbit by using (27)

 E2 = (rc−2M)2rc(rc−3M)=(1−2F−(z))2rc(1−3F−(z)), L2 = Mr2crc−3M=r2cF−(z)1−3F−(z), (29)
 Ut = √rcrc−3M=1√1−3F−(z), Uϕ = 1rc√Mrc−3M=1rc√F−(z)1−3F−(z), (30)
 Ω=√Mr3c=√F−(z)r2c. (31)

The function is in geometrized units (G=c=1). In order to plot it, we scale and by any multiple of the solar mass, this is to say, by , for Srg .. Figure 1 shows this scaled relation which is symmetric with respect to the shift (, ).

Given a set of pairs of observed redshifts (blueshifts) of emitters traveling around a Schwarzschild black hole along circular orbits of radii , a Bayesian statistical analysis might be carried out in order to estimate the black hole mass parameter.

### iii.2 The Reissner-Nordström Black Hole

Our next non-rotating working example is the Reissner-Nordström space-time which represents a electrically charged black hole, whose relevant metric components are where is the electric charge parameter and . For circular equatorial orbits of the photon source, the redshift reads

 z2=r2c(Mrc−Q2)(r2c−3Mrc+2Q2)(r2c−2Mrc+Q2). (32)

This relationship is equivalent to

 M=rcG±(rc,z2,Q2), (33)

where

 G± = 112z2[(5z2+1)+7Q2z2r2c ± (z4+10z2+1+z2Q2r2c[z2Q2r2c−2(z2+5)])1/2]

In this case, the conserved quantities and are

 E2=(Q2+rc(rc−2M))2r2c(2Q2+rc(rc−3M)), (35)
 L2=r2c(Mrc−Q2)2Q2+rc(rc−3M). (36)

and are real only if and . Therefore, is positive provided that . As it is known, in this metric, one distinguishes three regions: , and , where are the roots of , which are real and distinct only if stands. The surface is an event horizon similar to that for the Schwarzschild’s metric chandra . Since implies , our analysis is performed for , that is, outside the event horizon.

The stability of circular equatorial orbits requirement

 V′′eff=Mrc(18Q2+2r2c−12Mrc)−8Q4r2c(2Q2+rc(rc−3M))(Q2+rc(rc−2M))>0, (37)

tells us that . Inserting into this last condition would yield, in principle, an inequality that may bound the values of the redshift , as it was the case for Schwarzschild. This inequality turns out to be cumbersome to be analysed analytically; hence, the analysis was performed numerically in the following manner: given values of and , we vary and compute for each value of . With this value at hand, we check whether the four conditions are all satisfied: (i) , (ii) , (iii) and (iv) . The second and third inequalities guarantee that, one indeed, has circular and equatorial orbits, the fourth stems from . We look for the minimum and maximum value of for which these four condictions are fulfilled. This process is repeated for several values of and . For , the result for Schwarzschild () is recovered. Figure 2 shows the surfaces and . Only for frequency shifts such that , the corresponding values are acceptable.

The velocities and of photons emitters orbiting in circular and equatorial paths are

 Uϕ = 1rc√Mrc−Q2r2c(2Q2+rc(rc−3M)), Ut = √r2c2Q2+rc(rc−3M), (38)

and their angular velocity is given by

 Ω=√Mrc−Q2r4c. (39)

Since , these 4-velocity components and are actually functions of the redshift , the radius of the circular orbit and the parameter . Unlike the Schwarzshild black hole, there is not an analytic relationship of the mass parameter in terms only of the measurable variables and , it depends also on . At any rate, given a set of the observables , Bayesian statistical analysis would provide an estimate for both parameters and .

### iii.3 Boson Stars

Colpi et al performed a study of self-interacting Boson stars which were modeled by a complex scalar field endowed with a quartic potential . The stability analysis yielded equilibrium configurations along either an stable and unstable branch Colpi ; Ruffini . We will be concerned with stable equilibrium configurations of Boson stars for which the metric reads

 ds2=−α2(r)dt2+a2(r)dr2+r2(dθ2+sin2θdφ2). (40)

The components and are found by solving

where, for numerical purposes, we have introduced the following dimensionless variables: , , and , where is the mass of complex scalar field , its frequency and the dimensionless self-coupling of the scalar. Here represents the derivative with respect to .

For the complex scalar field, we consider a harmonic form and solve the Klein-Gordon equation, that in terms of the dimensionless variables, takes the form

 ^ϕ′′=(1−Ω2α2+Λ^ϕ2)a2^ϕ−(α′α−a′a+2x)^ϕ′. (42)

The boundary conditions for the metric functions and the scalar field, in order to guarantee regularity at the origin and asymptotic flatness at infinity, are: , , , , and .

The system is basically an eigenvalue problem for the frequency of the boson star as a function of a parameter, the so called, central value of the scalar field which determines the mass of a boson star. This system can be solved by using the shooting method NR . Figure 3 shows the metric component and for Boson stars with .

For circular orbits () with radius the effective potential and its derivative vanish. From these conditions and are obtained

 L2=x3cα′(xc)α(xc)−xcα′(xc),E2=α3(xc)α(xc)−xcα′(xc), (43)

here . Choosing and as in (43) guarantees circular orbits. Generally both, and are non-negative; therefore, given a numerical solution, we only need to determine the domain of the radial variable where and work exclusively in that domain. We then compute the values and with (43) and perform a survey in checking where the condition for stable circular orbits holds. Thereby one finds a set of parameters which give us circular orbits, .

According to the equation (25), the redshift of photons emitted by particles orbiting a boson star is calculated by

 z(x)=√xα′α2(α−xα′). (44)

Figures 4 show the behavior in function of for several boson stars with different masses and for two values of , and . The solid black curve represents the boson star correspoding to the critical mass . For , or equivalently, the boson star is stable, otherwise is unstable. . and . for and respectively. In figure 4 for , it is observed that the maximum value of the redshift increases as the central value of the scalar field increases. But for large values of , all the curves seem to get closer to the value of at large for a boson star with the critical mass . One also can observe that the curves corresponding to smaller masses than the critical, remain below the solid black curve .

The table below, shows the values of the masses corresponding to stable and unstable boson stars for both and .

Λ=0 Stable Unstable ϕ0 Λ=100 Stable Unstable ϕ0 MT ϕ0 MT 0.04 1.371 0.10 2.249 0.08 2.227 0.16 1.892

One can also note that for configurations with the same value of mass, but different self-interacting parameter, the maximun redshift increases as decreases. For large values of , the redshift for all configurations converge to the same values (see fig. 5).

## Iv Kerr Black Hole

Explicit expressions for the shifts and computed at either side of were found by H-N

 z1 = ±√M(2aM+rc√r2c−2Mrc+a2)r3/4c(rc−2M)√r3/2c−3Mr1/2c±2aM1/2, z2 = ±√M(2aM−rc√r2c−2Mrc+a2)r3/4c(rc−2M)√r3/2c−3Mr1/2c±2aM1/2. (45)

Upper signs corresponds to co-rotating orbits and lower signs to counter-rotating orbits. From (45) the rotating parameter as a function of the mass parameter , the radius of circular equatorial orbits of particles around the Kerr black hole emitting light and the corresponding and turns out to be

 a2(α,β,rc,M)=r3c(rc−2M)α4M2β−r2cα, (46)

where and . Nonetheless, there is not an explicit expression to find the mass parameter , instead, there is an eight order polynomial for it derived also from (45). In this section, we carry out a numerical analysis to study how varies with , and the shifts and detected by a far away observer. The metric components of the Kerr black hole in the Boyer-Lindquist coordinates are given by

 gtt = −(1−2MrΣ),gtϕ=−2Marsin2θΣ, gϕϕ = (r2+a2+2Ma2rsin2θΣ)sin2θ, grr = ΣΔ,gθθ=Σ, (47)

where

 Δ≡r2+a2−2Mr,Σ≡r2+a2cos2θ,

with the restriction . For circular and equatorial orbits, the two conserved quantities are Bardeen

 E = r3/2−2M√r±a√Mr3/4√r3/2−3M√r±2a√M, L = ±√M(r2∓2a√Mr+a2)r3/4√r3/2−3M√r±2a√M. (48)

Co-rotating orbits (upper signs) have whereas counter-rotatings (lower signs) orbits have . In order to have real values for and , and thereby circular orbits, it is necessary that

 r3/2−3M√r±2a√M≥0. (49)

Circular-equatorial orbits can be either bound or unbound. The later type are those for which, given a small outward perturbation, the particle will go to infinity, one has bound orbits otherwise. There are bound orbits provided that

 r>rmb=2M∓a+2√M√M∓a (50)

is satisfied. Not all bound orbits are stable, only those whose radius satisfies are stable Bardeen . This condition is akin to

 r≥rms=M[3+Z2∓√(3−Z1)(3+Z1+2Z2)], Z1≡1+(1−a<