Parametrized-Post-Newtonian Test of Black Hole Spacetime for Galactic Center Massive Black Hole Sgr A${}^{\ast\,}$: Formulation and $\chi^{2}$ Fitting

The dynamical variables in the Hamiltonian formalism of geodesic equations are the spacetime point on the geodesic $x^{\mu}(\tau)$ and the 1-form conjugate to the four velocity of the geodesic $u_{\mu}(\tau)$, where $\tau$ is the proper length along the geodesic. The 1-form $u_{\mu}(\tau)$ has no dimension, while the point $x^{\mu}(\tau)$ has the length dimension. For these dynamical variables, the Hamiltonian of geodesics is given by

The Hamilton equations are given by

The solution of these equations under the constraint (2.12b) is the timelike geodesic. The Hamilton equations (2.13) are the first order differential equations. We construct our PPN formulations of timelike geodesics from the Hamiltonian (2.12a). ²²2 Following the ordinary procedure of the analytical mechanics, the Lagrangian is related to the Hamiltonian through the Legendre transformation of the dynamical variables, $L(x,\dot{x})=u_{\mu}\dot{x}^{\mu}-H$, where $\dot{x}={\rm d}x/{\rm d}\tau$. The Euler-Lagrange equations of this Lagrangian are the second order differential equations of the same form with $u^{\nu}\nabla_{\nu}u^{\mu}=0$.

Due to the stationary axial symmetry of spacetime, there are two conserved quantities along timelike geodesics,

where ${\rm d}E/{\rm d}\tau=0$ and ${\rm d}L_{z}/{\rm d}\tau=0$ are shown from $\partial\mathcal{H}_{\rm u}/\partial t=0$, $\partial\mathcal{H}_{\rm u}/\partial\varphi=0$ and Eqs.(2.13). Physical meanings of $E$ and $L_{z}$ are respectively the energy and the angular momentum around z-axis of a test particle (the star S0-2) moving on the timelike geodesic, where $E$ has no dimension normalized by the mass energy of S0-2 and $L_{z}$ has the length dimension.

Note that, for Kerr spacetime, there exists the third constant of geodesic motions, Carter constant, due to the so-called hidden symmetry of spacetime described by Killing tensor. On the other hand, as shown by C.M.Will ref:will2009 in Newtonian gravity, a special case of stationary axisymmetric Newtonian gravitational potential allows the existence of a “Carter-like” constant for motions of test particles, which is different from the energy and angular momentum. Hence, in our PPN model (2.9) which possesses the stationary axisymmetry, there may exist a special set of values of PPN parameters ${\bf X}_{\rm ppn}$, other than the Kerr case (2.7), which generates a “Carter-like” constant for geodesic motions. However, even if such a special case exists in our PPN model, we do not fix the value of ${\bf X}_{\rm ppn}$ at the special case, because our aim in this paper is the search of the value of ${\bf X}_{\rm ppn}$ best-fitting with observational data. Hence, in this paper, we do not expect the existence of a third constant of geodesic motions. (The search for a “Carter-like” constant in our PPN model is an interesting issue, but not in the scope of this paper.)

The PPN expansion of $\mathcal{H}_{\rm u}(x,u)$ is obtained by substituting the metric (2.9) into Eq.(2.12a),

In Eq.(2.15), “$n$PN” means the terms of $O(\varepsilon^{n+1})$. The 0PN terms express the Newtonian gravity in the framework of the special relativity. The 1PN terms express the largest non-Newtonian effect, which depends on the mass $m$ but not on the spin $a$. The 1.5PN terms express the largest effect depending on the spin $a$. Note that Eq.(A.1) in Appendix A shows the PPN expansion of $\mathcal{H}_{\rm u}$ with retaining $\{N_{\rm t},N_{\rm s},C_{\perp}\}$ in the metric (2.6).

The PPN timelike geodesic equations are obtained by substituting Eqs.(2.15) into Eq.(2.13). This procedure, usually, gives the geodesic equations in which the dynamical variables are regarded as the functions of the proper time $\tau$. On the other hand, the observational data produce, for example, the position of S0-2 $x(t)$ as the function of the observer’s proper time which corresponds to the coordinate time $t$, because the observer is far from Sgr A${}^{\ast\,}$. Therefore we formulate our PPN geodesic equations with regarding the dynamical variables as the functions of $t$ through the following transformation,

where the right-hand sides are given by the Hamilton equations with regarding $\{x^{\mu}(\tau),u_{j}(\tau)\}$ as the functions of $\tau$, and $u_{0}$ is omitted because $u_{0}=-E$ is constant. The PPN timelike geodesic equations through the transformation (2.16) are as follows.

where $q_{j}=(u_{2},-u_{1},0)$. Concerning these geodesic equations, let us make two notes.

• •

  Newtonian gravity is recovered by focusing on 0PN terms in Eqs.(2.17), where 3D velocity in Newtonian mechanics is given by $v_{\rm Newton}^{j}:=u_{j}/(-u_{0})=u_{j}/E$.

• •

  Although six dynamical variables $\{x^{j}(t),u_{j}(t)\}$ appear in Eqs.(2.17), one of four variables $\{x(t),y(t),u_{1}(t),u_{2}(t)\}$ is dependent due to the conserved quantity $L_{z}$ in Eq.(2.14). In integrating Eqs.(2.17) numerically, five dynamical variables need to be solved, once the values of $E$ and $L_{z}$ are specified through the initial conditions of S0-2 (see Sect.2.5).

As for timelike geodesics, we adopt the Hamiltonian formalism for null geodesics. The dynamical variables are the spacetime point on the geodesic $x^{\mu}(\sigma)$ and the tangent 1-form of the geodesic $k_{\mu}(\sigma)$, where $\sigma$ is an affine parameter along the geodesic. Hereafter, let $\sigma$ have the length dimension, and the 1-form $k_{\mu}(\sigma)$ has no dimension, while the point $x^{\mu}(\sigma)$ has the length dimension. For these dynamical variables, the Hamiltonian and the null condition are, respectively, given by

where $x$ and $k$ denote symbolically the dynamical variables. The Hamilton equations are given by

The solution of these equations under the constraint (2.18b) is the null geodesic. As for the conserved quantities along timelike geodesics (2.14), there are two conserved quantities along null geodesics,

Physical meanings of $w$ and $l_{z}$ are respectively the energy and the angular momentum around z-axis of a photon propagating on the null geodesic, where $w$ has no dimension and $l_{z}$ has the length dimension. We do not expect the existence of a “Carter-like” constant, as discussed in Sect.2.3.1.

The parametrized post-Minkowskian (PPM) expansion of $\mathcal{H}_{\rm k}(x,k)$ is obtained by substituting the metric (2.9) into Eqs.(2.18),

In Eq.(2.21a), “$n$PM” means the terms of $O(\varepsilon^{n})$. The 0PM terms express the null geodesic in the framework of the special relativity. The 1PM terms express the largest gravitational effect, which depends on the mass $m$ but not on the spin $a$. Note that Eq.(A.3) in Appendix A shows the PPM expansion of $\mathcal{H}_{\rm k}$ with retaining $\{N_{\rm t},N_{\rm s},C_{\perp}\}$ in the metric (2.6).

The PPM null geodesic equations are obtained by substituting Eqs.(2.21) into Eq.(2.19). Here we summarize these equations with regarding the dynamical variables as the functions of the affine parameter $\sigma$, not of the coordinate time.

where $k_{0}=-w$ is used. In Sect.3.1, from these geodesic equations, we will obtain analytic perturbative solutions of the 0PM and 1PM null geodesics which connect the star S0-2 and a distant observer representing us. As expected by the nullity of 0PM term of the acceleration (2.22c), the 0PM solution is of a constant velocity and corresponds to null geodesics on Minkowski metric.

Let us introduce the observer representing us so as to match with the actual observation process. In the reduction of observational values from observational raw data, the following effects are removed; the effect of earth’s spin and revolution around the sun, and the effect of sun’s peculiar motion with respect to the Local Standard of Rest (LSR) reference frame. Therefore, we make our observer move with a velocity which is not removed in the above reduction process. The time scale of such observer’s motion is expected to be of a time scale determined by the size of our galaxy $L_{\rm gal}\sim 4\times 10^{4}$ pc, which gives $L_{\rm gal}/c\sim 1.3\times 10^{5}$ years. It is thus appropriate to assume that the observer’s relative velocity to Sgr A${}^{\ast\,}$is constant, because the time scale of S0-2 observations is of a few ten years which is very shorter than $1.3\times 10^{5}$ years.

From the known approximated values of some parameters (2.2), the distance from our sun to Sgr A${}^{\ast\,}$is $R_{\rm GC}\sim 2.4\times 10^{14}$ km and the Schwarzschild radius of Sgr A${}^{\ast\,}$is $r_{\rm sch}\sim 1.2\times 10^{7}$ km. Then, the difference of time lapse between the sun and Sgr A${}^{\ast\,}$due to Sgr A${}^{\ast\,}$’s gravity, which is estimated from the gravitational redshift, is $r_{\rm sch}/R_{\rm GC}\sim 5\times 10^{-8}$. This means that, during 20 years observation from 2000 to 2020, a temporal difference of $20\times r_{\rm sch}/R_{\rm GC}\sim 10^{-6}$ years arises between the sun and Sgr A${}^{\ast\,}$. Such small temporal uncertainty cannot be identified in the actual observation which needs about one day for obtaining one set of observational raw data. Further, when we estimate the magnitude of our observer’s velocity $V_{\rm obs}$ as an object bounded by Sgr A${}^{\ast\,}$’s gravity, it becomes $V_{\rm obs}/c\sim\varepsilon^{1/2}\bigr{|}_{r=R_{\rm GC}}\sim(r_{\rm sch}/R_{\rm GC})^{1/2}\sim 10^{-4}$. The difference of time lapse between the sun and the coordinate time $t$ due to the velocity $V_{\rm obs}$, which is estimate from the Lorentz factor, is $(V_{\rm obs}/c)^{2}\sim 10^{-8}$. This difference is also not identifiable in the present observational data. Thus we regard the coordinate time $t$ as the proper time of our observer.

The spatial coordinate system $(X,Y,Z)$ appropriate to the observation is introduced as shown in Fig.1. We set $(X,Y,Z)$ be related with the Cartesian-like coordinates $(x,y,z)$ by a spatial rotation which will be explained in Sect.2.5.3. The coordinate axes of $(X,Y,Z)$ are fixed by making use of the apocenter (the farthest point from Sgr A${}^{\ast\,}$) of the S0-2’s orbit.

• •

  The $Z$-axis points from Sgr A${}^{\ast\,}$to the spatial position of the observer where the photon emitted by S0-2 at the apocenter is received by the observer.

• •

  The $Y$-axis points the same direction as the right ascension (R.A.), from the west to the east seen from the observer.

• •

  The $X$-axis points the same direction as the declination (Dec.), from the south to the north seen from the observer.

In this observational coordinate system, we define the distance from the sun to Sgr A${}^{\ast\,}$, $R_{\rm GC}$, as the $Z$ coordinate of the crossing event of the observer’s orbit and $Z$-axis, at which the observer receives the photon emitted by S0-2 at the apocenter. Further, let $t_{\rm obs.apo}$ denote the observation time of the photon emitted by S0-2 at the apocenter, which occurred already in 2010. Then, in the observational coordinates $(X,Y,Z)$, the spatial position of our observer $\vec{r}_{\rm obs}(t)$ at a given observation time $t$ is given by

where $\vec{V}_{\rm obs}=(V^{X}_{\rm obs},V^{Y}_{\rm obs},V^{Z}_{\rm obs})$ is the constant velocity of the observer and $\vec{A}_{\rm obs}=(0,0,R_{\rm GC})$ in the observational coordinates $(X,Y,Z)$.

The visible 2D position of S0-2 on the sky plane (astrometric data) have been observed by European group (with VLT telescope) and American group (with mainly Keck telescope and partially Gemini telescope), while the redshift of photons coming from S0-2 (spectroscopic data) have been observed by those two groups and our Japanese group (with Subaru telescope). In Fig.1, two sky planes for VLT and Keck groups are depicted. As explained below, the use of the astrometric data of S0-2 raises some additional parameters to be evaluated by fitting observational data and theoretical predictions.

The observations of S0-2 have to be performed by infrared astronomical observations, because stars at the center of our galaxy can be observable by infrared photons. Further, although Sgr A${}^{\ast\,}$itself is visible by radio waves radiated by very dilute plasma gases surrounding Sgr A${}^{\ast\,}$, the infrared photons from the gases are so faint that Sgr A${}^{\ast\,}$is not visible for infrared telescopes. This means that the origin of the 2D sky plane can not be set exactly at Sgr A${}^{\ast\,}$in infrared observations. In the actual astrometric observations, the origin of the sky plane is set at a position of, for example, an infrared flare event observed in the past in the vicinity of Sgr A${}^{\ast\,}$. The position of such past flare event is not exactly at Sgr A${}^{\ast\,}$and may be moving relative to Sgr A${}^{\ast\,}$. Therefore, we assume that the astrometric origin is moving relative to Sgr A${}^{\ast\,}$with a constant velocity. Further, because the setup of astrometric origins by VLT and Keck groups are not the same, the relative motion of the origin to Sgr A${}^{\ast\,}$should be introduced individually to the two astrometric data sets of VLT and Keck groups. Hence, the 2D displacement vector $\vec{O}_{i}(t)$ ($i=$ VLT, Keck) from Sgr A${}^{\ast\,}$to the astrometric origin on the sky plane at a given observation time $t$ is expressed as

where $\vec{V}_{i}=(V^{X}_{i},V^{Y}_{i})$ is the 2D constant velocity of the astrometric origin relative to Sgr A${}^{\ast\,}$, and $\vec{A}_{i}=(A^{X}_{i},A^{Y}_{i})$ is the 2D displacement of the astrometric origin from Sgr A${}^{\ast\,}$at $t_{\rm obs.apo}$.

We consider that every astrometric data is the offset of the visible 2D position of S0-2 from the origin $\vec{O}_{i}(t)$ at observation time $t$. In fitting theoretical prediction with astrometric observational data, the theoretically calculated 2D position on the sky plane need to be corrected by $\vec{O}_{\rm VLT}(t)$ for VLT’s astrometric data and by $\vec{O}_{\rm Keck}(t)$ for Keck’s astrometric data. Then, the best-fitting values of the eight parameters of the two astrometric origins, $\{V_{\rm VLT}^{X},V_{\rm VLT}^{Y},A_{\rm VLT}^{X},A_{\rm VLT}^{Y},V_{\rm Keck}^{X},V_{\rm Keck}^{Y},A_{\rm Keck}^{X},A_{\rm Keck}^{Y}\}$, should be obtained at the same time with all the other parameters in our PPN modelling (see Sect.2.5.5).

Finally in this Sect.2.5.2, let us make a comment on our treatment of the astrometric origin. When using only one astrometric data set of, for example, Keck group, we can require reasonably that the astrometric origin is moving with the observer, $\vec{V}_{\rm Keck}=(V^{X}_{\rm obs},V^{Y}_{\rm obs})$. However, when using two astrometric data sets of both groups, we do not know how to fix the two velocities $\vec{V}_{i}$ ($i=$ VLT, Keck) in relation with $\vec{V}_{\rm obs}$. In this paper, we leave the two velocities $\vec{V}_{i}$ as free parameters to be evaluated by fitting observational data and theoretical predictions.

As shown in Fig.2, we determine the spatial rotation relating the observational spatial coordinates $(X,Y,Z)$ and the Cartesian-like spatial coordinates of the black hole $(x,y,z)$ by the following two conditions.

• •

  Let us express the direction of the spin axis of Sgr A${}^{\ast\,}$in the observational coordinates $(X,Y,Z)$ by the zenith and azimuth angles $(\Theta_{\rm BH},\Phi_{\rm BH})$ as shown in Fig.2. This spin axis is the $z$-axis of the Cartesian-like coordinates which describe the metric (2.9).

• •

  Let us fix the $y$-axis of the Cartesian-like coordinates $(x,y,z)$ in the observational coordinates $(X,Y,Z)$ so as to be parallel to the outer product $\vec{e}_{Z}\times\vec{e}_{z}$, where $\vec{e}_{i}$ denotes the unit spatial vector along $i$-axis and $i=z,Z$. Then, $x$-axis is automatically fixed as a right-handed system.

Under these conditions, the coordinate transformation is given by

It should be noted that the black hole spin effects of Sgr A${}^{\ast\,}$is not measurable by the present telescopes, and the spin magnitude $a$ and angles $(\Theta_{\rm BH},\Phi_{\rm BH})$ cannot be evaluated. Therefore, in comparing observational data with theoretical predictions in this paper, we assume $(\Theta_{\rm BH},\Phi_{\rm BH})=(0,0)$. Under this assumption, the coincidence of spatial coordinate systems $(X,Y,Z)\equiv(x,y,z)$ holds. When next generation telescopes, such as the Thirty-Meter-Telescope, starts scientific operations, the detection of the spin effect will be realized.

Given a initial time for calculating a stellar motion, the number of parameters for the initial condition of the stellar motion is six for the initial spatial position and the initial spatial velocity. We determine these six parameters for S0-2 as follows.

Let us note that the pericenter distance $r_{\rm p}$ of S0-2 is about a thousand times the Schwarzschild radius $r_{\rm sch}$ of Sgr A${}^{\ast\,}$, $r_{\rm p}\sim 10^{3}r_{\rm sch}$, as indicated by Eq.(2.3). Therefore the orbit of S0-2 is almost elliptic. Then, we set the initial condition at the apocenter observed in 2010. Because the radial component of S0-2’s velocity at the apocenter vanishes, the number of parameters for the initial condition at the apocenter is reduced from six to five.

Given the initial spatial position and velocity at the apocenter, one can imagine a Keplerian elliptic motion which is determined by the given initial condition with assuming Newtonian gravity of Sgr A${}^{\ast\,}$. Further the difference between the imaginary Keplerian motion and the geodesic motion by Eqs.(2.17) is minimized, because the gravity of Sgr A${}^{\ast\,}$on the orbit of S0-2 becomes weakest at the apocenter. As shown in Fig.3, with referring to the imaginary Keplerian motion, we introduce a spatial coordinate system $(x_{\rm ic},y_{\rm ic},z_{\rm ic})$ relating with $(X,Y,Z)$ by a spatial rotation which will be given in Eqs.(2.27).

• •

  The $z_{\rm ic}$-axis points the same direction as the spatial angular momentum of S0-2 at the apocenter observed in 2010.

• •

  The $x_{\rm ic}$-axis points from the apocenter to the pericenter of the imaginary Keplerian elliptic orbit,

• •

  The $y_{\rm ic}$-axis is automatically fixed as a right-handed system.

In this coordinate system, the imaginary Keplerian orbit is on the $x_{\rm ic}$-$y_{\rm ic}$ plane.

The five parameters for the initial condition at the apocenter can be expressed by the five orbital parameters of the imaginary Keplerian motion; the orbital period $T_{\rm star}$, the orbital eccentricity $e_{\rm star}$, the inclination angle $I_{\rm star}$, the ascending node angle $\Omega_{\rm star}$, and the pericenter angle from the ascending node $\omega_{\rm star}$. The definition of the angles $\{I_{\rm star},\Omega_{\rm star},\omega_{\rm star}\}$ are shown in Fig.3, and the other two parameters $\{T_{\rm star},e_{\rm star}\}$ are transformed to the apocenter distance $r_{\rm apo}$ and speed $v_{\rm apo}$ by the Keplerian formulas,

where $m$ is the mass of black hole, $r_{\rm apo}$ has the dimension of length, and $v_{\rm apo}$ has no dimension. Note that, as will be shown in Sect.4, the difference between the Keplerian orbital period $T_{\rm star}$ and the time interval between neighboring apocenters (or pericenters) in the framework of our PPN model is of a few days, while the duration of observational operation for obtaining one observational data is about one day. The Keplerian period $T_{\rm star}$ is a good approximation as the observational orbital period.

From these five parameters $\{r_{\rm apo},v_{\rm apo},I_{\rm star},\Omega_{\rm star},\omega_{\rm star}\}$, the initial spatial position $(X_{\rm apo},Y_{\rm apo},Z_{\rm apo})$ and velocity $(V^{X}_{\rm apo},V^{Y}_{\rm apo},V^{Z}_{\rm apo})$ in the observational spatial coordinate system are calculated by