Post-Linear Metric of a Compact Source of Matter

Post-linear metric of a compact source of matter

Sven Zschocke Institute of Planetary Geodesy - Lohrmann Observatory, Dresden Technical University, Helmholtzstrasse 10, D-01069 Dresden, Germany

The Multipolar Post-Minkowskian (MPM) formalism represents an approach for determining the metric density in the exterior of a compact source of matter. In the MPM formalism the metric density is given in harmonic coordinates and in terms of symmetric tracefree (STF) multipoles. In this investigation, the post-linear metric density of this formalism is used in order to determine the post-linear metric tensor in the exterior of a compact source of matter. The metric tensor is given in harmonic coordinates and in terms of STF multipoles. The post-linear metric coefficients are associated with an integration procedure. The integration of these post-linear metric coefficients is performed explicitly for the case of a stationary source, where the first multipoles (monopole and quadrupole) of the source are taken into account. These studies are a requirement for further investigations in the theory of light propagation aiming at highly precise astrometric measurements in the solar system, where the post-linear coefficients of the metric tensor of solar system bodies become relevant.

I. INTRODUCTION Our motivation to consider the post-linear term in the metric perturbation (2) is triggered by the rapid progress The field equations of gravity [1, 2] represent a set in astrometric science, which has recently made the im- of ten coupled non-linear partial differential equations pressive advancement from the milli-arcsecond level [14– 16] to the micro-arcsecond level [17–21] in angular mea- for the ten components of the metric tensor gαβ which governs the geometry of space-time. Despite of their surements of celestial objects like stars and quasars. A complicated mathematical structure, exact solutions of prerequisite of astrometric measurements is the precise the field equations have been obtained for gravitational modeling of the trajectory of a light signal which propa- systems which have a symmetry, for instance [3]: the gates from the celestial object through the curved space- Schwarzschild solution for a spherically symmetric body time of the solar system towards the observer. And be- [4], the Kerr solution for a spherically symmetric body cause the trajectory of a light signal depends on the ge- in uniform rotational motion [5], the Weyl-Levi-Civita- ometry of space-time, it becomes obvious why the metric Erez-Rosen solution for an axially symmetric body [6– perturbation (2) is of fundamental importance in the the- 10], the Reissner-Nordströmsolution for an electrically ory of light propagation and astrometry. Already at the charged spherically symmetric body [11, 12], and the micro-arcsecond level in positional measurements of ce- Kerr-Newman solution for an electrically charged spher- lestial objects, the linear term of the metric perturbation ically symmetric body in uniform rotational motion [13]. (2) is not sufficient for modeling the positional observa- However, for a body of arbitrary shape and inner struc- tions performed within the solar system [22–34]. Mean- ture and which can also be in arbitrary rotational mo- while, there are several mission proposals aiming at the tions and oscillations, the field equations of gravity can sub-micro-arcsecond and even the nano-arcsecond scale only be solved within some approximation scheme. of accuracy [35–39]. That is why post-linear effects of the For an asymptotically flat space-time it is convenient metric perturbation (2) are coming more and more into to decompose the metric tensor into the flat Minkowski focus of astronomers and in the theory of light propagation [40–46]. metric ηαβ and a metric perturbation hαβ, The Multipolar Post-Minkowskian formalism is based gαβ (t, x) = ηαβ + hαβ (t, x) . (1) on the Landau-Lifschitz formulation of Einstein’s theory. In this approach, instead of determining directly the met- The post-Minkowskian scheme is certainly one of the ric tensor,√ one operates with the gothic metric density, most important approximations in the theory of gravity, gαβ = −g gαβ where g is the determinant of the metric which states that for weak gravitational fields, |hαβ| 1, tensor. Like in case of the metric tensor, for an asymptot- the metric perturbation can be series expanded in powers ically flat space-time it is appropriate to decompose the of the gravitational constant, gothic metric density into the flat Minkowskian metric αβ αβ 1 (1PM) 2 (2PM) 3 η and a gothic metric perturbation h (t, x), hαβ (t, x) = G hαβ (t, x) + G hαβ (t, x) + O G ,

(2) αβ gαβ (t, x) = ηαβ − h (t, x) . (3) (1PM) (2PM) where hαβ is the linear term and hαβ is the post- linear term of the metric tensor, and G2|h(2PM)| αβ αβ For weak gravitational fields, |h | 1, the correspond- 1 (1PM) G |hαβ | 1. ing post-Minkowskian series expansion of the perturba- 2 tion of the gothic metric density reads as follows, metric including the quadrupole structure of a compact source, which can be considered as some massive solar αβ 1 αβ 2 αβ 3 h (t, x) = G h(1PM)(t, x) + G h(2PM)(t, x) + O G , system body. The manuscript is organized as follows: In Section II (4) the exact field equations of gravity in harmonic gauge are αβ αβ given. The residual harmonic gauge freedom is consid- where h(1PM) is the linear term and h(2PM) is the post- ered in Section III. The post-Minkowskian expansion and 2 αβ linear term of the gothic metric, and G |h(2PM)| some fundamental results of the MPM formalism which αβ are relevant for our considerations are summarized in Sec- G1|h | 1. The knowledge of the contravariant (1PM) tion IV and Section V. The gothic metric density in the components of the gothic metric perturbation (4) allows linear and post-linear approximation for time-dependent to determine the covariant components of the metric per- sources is given in Section VI. The metric tensor in the turbation (2); relations between the gothic metric and the linear and post-linear approximation for time-dependent metric tensor are given in Appendix D. sources is given in Section VII. Finally, in Section VIII The Multipolar Post-Minkowskian (MPM) formalism the metric tensor in the linear and post-linear approxima- has been developed within a series of articles [47–52] and tion is given explicitly for the case of a source with time- provides a robust framework in order to determine the independent monopole and spin and quadrupole struc- gothic metric perturbation (4) of compact sources of mat- ture. A summary can be found in Section IX. The no- ter. In the MPM formalism, the gothic metric density tations as well as details of the calculations are relegated is expressed in terms of so-called symmetric and trace- to several Appendices. free (STF) multipoles, allowing for arbitrary shape, inner structure, oscillations and rotational motions of the source. The MPM approach was mainly intended for the- II. THE EXACT FIELD EQUATIONS OF oretical understanding of the generation of gravitational GRAVITY waves by some isolated source of matter, like inspiralling binary stars which consist of compact objects like black The field equations of gravity [1, 2] relate the metric holes or neutron stars. The compact source of matter tensor gαβ of the physical space-time M to the stress- can of course also be interpreted as some massive solar energy tensor of matter Tαβ, which can be written in the system body, being of arbitrary shape and inner struc- following form (§17.1 in [54]), ture, and which can be in arbitrarily rotational motions 1 8 π G and oscillations. R − g R = T , (6) Within the MPM approach the linear term and the αβ 2 αβ c4 αβ post-linear term of the gothic metric perturbation (4) ρ ρ ρ σ ρ σ where Rαβ = Γαβ,ρ − Γαρ,β + Γσρ Γαβ − Γσβ Γαρ is the have been determined long time ago for the case of a Ricci tensor (cf. Eq. (8.47) in [54]), compact source of matter. Accordingly, the aim of this 1 investigation is to give the linear and the post-linear term Γα = gαβ (g + g − g ) , (7) of the metric perturbation (2) for a compact source of µν 2 βµ , ν βν , µ µν , β matter. µ are the Christoffel symbols, and R = Rµ is the Ricci The determination of post-linear metric coefficients scalar. involves quite ambitious computations and the results The field equations (6) represent a set of ten coup- of the MPM approach become rather cumbersome al- led non-linear partial differential equations for the ten ready for the very first few multipoles beyond the simple components of the metric tensor. Because of the con- monopole term [53]. However, for many applications, for tracted Bianchi identities (cf. Eq. (13.52) in [54]) there instance in the theory of light propagation, it is sufficient are only six field equations which are independent of to consider the stationary case, where the gravitational each other [55]. These six field equations determine the fields generated by the body become time-independent, ten components of the metric tensor up to a coordinate hence the post-Minkowskian expansion (2) simplifies as transformation which involves four arbitrary functions follows, x0 µ = x0 µ (xν ). This freedom in choosing the coordinate system is called general covariance of the field equations 1 (1PM) 2 (2PM) 3 hαβ (x) = G hαβ (x) + G hαβ (x) + O G . (5) of gravity. For practical calculations in celestial mechanics and In the stationary case the computations of the MPM in the theory of light propagation it is very convenient formalism are considerably simpler than in the case of to chose concrete reference systems instead of keeping time-dependent gravitational fields. In the theory of light the covariance of the field equations. A powerful tool propagation in the solar system, the impact of post-linear is to use harmonic coordinates xµ = (ct, x), which are terms of the metric tensor on the light propagation is only introduced by the harmonic gauge condition [54, 56–61] known for the monopole term, but not for higher multi- (cf. Eq. (67.02) in [57], Eq. (5.2a) in [56]) poles. It is, therefore, a further aim of this investigation √ αβ to determine, in a transparent manner, the post-linear −g g , β = 0 , (8) 3 where Eq. (5.2b) in [56], Eq. (1.6.1) in [60], Eqs. (2.4) - (2.6) in √ [61]) gαβ = −g gαβ , (9) αβ 16 π G αβ αβ h (x) = − τ (x) + t (x) , (13) is the gothic metric density [54, 56–61], with g being the c4 determinant of the covariant components of the metric where x = (ct, x) are curvilinear harmonic coordinates on µν tensor. It is very useful to operate with the gothic met- the flat background space-time and = η ∂µ∂ν is the αβ ric density g rather than the metric tensor gαβ, because flat d’Alembert operator given in terms of these curvilin- the field equations in harmonic coordinates become con- ear harmonic coordinates [66]. The field equations (13) siderably simpler in terms of the gothic metric density. are called Landau-Lifschitz formulation of Einstein’s the- It should not be surprising that (8) is not a general- ory of gravity in harmonic coordinates. The exact field covariant relation, because this condition just selects a equations of gravity (6) are general-covariant, while the specific type of reference system, namely the (class of) exact field equations in harmonic coordinate systems (13) harmonic reference systems. Although the harmonic are only Lorentz-covariant. The terms on the r.h.s. in coordinate condition (8) is not general-covariant, it is (13) are given by Lorentz-covariant in the slightly generalized meaning αβ αβ of linear orthogonal transformations in curvilinear har- τ (x) = (−g (x)) T (x) , (14) monic coordinates [57]. The choice of harmonic reference systems is in line with the philosophy of general relativ- αβ αβ t (x) = (−g (x)) tLL (x) ity that one may adopt concrete reference systems, while 4 c αµ βν αβ µν observables (coordinate-independent scalars) are deter- + h , ν (x) h , µ (x) − h , µν (x) h (x) , mined as the final step in the calculations. The har- 16 π G monic gauge condition (8) is called de Donder gauge in (15) honor of its inventor for the exact field equations [62]. where T αβ is the stress-energy tensor of matter, while The harmonic reference system for the exact field equa- tαβ is the stress-energy pseudotensor of the gravitational tions has also been introduced independently by Lanczos αβ field, and tLL is the Landau-Lifschitz pseudotensor of [63], while the harmonic gauge condition to first order gravitational field, in explicit form given by Eq. (20.22) (linearized gravity) was originally introduced by Einstein in [54] and by Eq. (101.7) in [69]. [64, 65] (cf. Eq. (4) in [64], Eq. (5) in [65]). It has already been emphasized that the usage of the An alternative form for the definition of harmonic co- harmonic gauge condition, either in the form (8) or in ordinates via the gauge condition (8) is given by the con- the form (10), implies the loss of the general covariance. dition (cf. Eq. (93.03) in [57], Eq. (3.270) in [58]) That is why the expressions τ αβ and tαβ are not general- covariant tensors, but they are Lorentz-covariant tensors. xµ = 0 , (10) g The vanishing of the covariant derivative of stress-energy αβ where tensor of matter, T ; β = 0, implies [54, 56, 58, 60, 61, 69] (cf. Eq. (5.4) in [56], Eq. (2.8) in [61]) 1 √ αβ g = √ ∂α −g g ∂β (11) αβ αβ h αβ αβi −g τ + t = 0 =⇒ (−g) T + tLL = 0 (16) , β , β = gαβ ∂ ∂ (12) α β which represents a local conservation law and admits is the covariant d’Alembert operator, in (11) given in ar- the formulation of a global conservation law for the bitrary curvilinear four-coordinates, while in (12) given four-momentum of the entire gravitational system; cf. in terms of harmonic curvilinear four-coordinates. It Eqs. (20.23a) - (20.23c) in [54] or Eqs. (1.1.7) and (1.2.1) is crucial to realize that the four functions xµ in (10) in [60]. are just functions, not components of a vector. A func- The gravitational system is assumed to be spatially tion which obeys the homogeneous d’Alembert equation, compact, meaning that there exists a three-dimensional sphere of finite radius R which completely contains the gf = 0, is called harmonic function. That evident sim- ilarity is the reason of why coordinates xµ are called source of matter, so that the stress-energy tensor of mat- αβ harmonic coordinates. The harmonic four-coordinates ter T (t, x) = 0 when |x| > R. Furthermore, the grav- (ct, x) provide the closest approximation to rectilinear itational system is assumed to be isolated, that means four-coordinates that one can have in curved space-time flatness of the metric at spatial infinity and the con- and that is why they are often called Cartesian-like co- straint of no-incoming gravitational radiation are im- ordinates. posed [23, 57, 58, 70–72], µν Besides of the harmonic gauge (8) also the decompo- lim h (t, x) = 0 , (17) r→∞ sition (3) is used, which implies that the gothic metric t+ r =const αβ c perturbation h (t, x) propagates as dynamical field on ∂ µν ∂ µν the flat background space-time M0. Then, the exact lim r h (t, x) + r h (t, x) = 0 , (18) r→∞ t+ r =const ∂r ∂ct field equations of gravity (6) read [54, 56, 58, 60, 61] (cf. c 4 where r = |x|. These conditions are called Fock- 퓜 -1 퓜 Sommerfeld boundary conditions. The formal solution * 0 ᶲ ᶲ g α β of the exact field equations (13) for an isolated system is given by [54, 58, 60, 61] (e.g. Eq. (36.38) in [54]), 퓟 q αβ 16 π G −1 αβ αβ g h (t, x) = − R τ + t (t, x) , (19) ᶲ α β c4 ᶯ α β , where the inverse d’Alembert operator reads [47–49, 51, x x y 53, 73–75] Z 흋 −1 1 3 0 1 0 f (t, x) = − d x f (u , x ) . (20) 휇 , 휇 휇 R 4 π |x − x0| x x y 휒 The time of retardation between the source point x0, for ℝ⁴ ℝ⁴ ℝ⁴ instance located inside the source of matter, and the field , , point x, for instance located outside of matter, is *g = + (x) *g = + (x ) g (y) ᶲ αβ ᶯαβ h αβ ᶲ αβ ᶯαβ h αβ αβ |x − x0| u = t − , (21) c Figure 1: A geometrical representation of the Lorentz- where the natural constant c is the speed of gravitational covariant gauge transformation (22) and its inverse (25). The action which equals the speed of light in vacuum [54, 57]. physical manifold M is covered by coordinates {y} and is The spatial integral in (19) runs over the entire three- endowed with the metric tensor gαβ (y) which is a solution dimensional space, that means it gets support inside and of the exact field equations (6). The flat background mani- outside of the matter source, because the integrand de- fold M0 is covered by Minkowskian coordinates {x} and is pends on the metric perturbation which extends to the endowed with the metric tensor ηαβ . The diffeomorphism entire three-dimensional spatial space. It should be em- Φ: M0 → M maps the flat background manifold to the phasized that (13) are the exact field equations of grav- physical manifold, e.g. a point q ∈ M0 to a point P ∈ M. −1 ity and (19) represents an exact solution of the field Its inverse diffeomorphism Φ : M → M0 maps the physical equations, because the only requirements to get these manifold to the flat background manifold, e.g. a point P ∈ M to a point q ∈ M . The metric tensor g (y) of the physical equations have been the harmonic gauge and the Fock- 0 αβ manifold M is pulled back on the flat background manifold Sommerfeld boundary conditions. However, the exact M0 (active coordinate transformation as given by Eq. (A.9) solution (19) is an implicit integro-differential equation, ∗ in [59]). The pulled back metric is denoted by Φ gαβ and is because the metric perturbation appears on both sides of defined by gαβ (x) = ηαβ + hαβ (x). The pulled back metric Eq. (19). gαβ (x) on M0 is physically equivalent to the metric gαβ (y) on M, that means: if the metric gαβ (y) is a solution of the exact field equations (6) on the physical manifold M, then ∗ III. THE RESIDUAL GAUGE FREEDOM hαβ = Φ gαβ − ηαβ (for relations between metric and metric density see Appendix D) will be a solution of the exact field equations (13) in the flat background manifold M (cf. In order to solve the field equations of gravity (6) so- 0 text below Eq. (7.51) in [95]). The background manifold can called harmonic coordinates have been imposed by (10) also be covered by another harmonic coordinate system {x0}, which have simplified the field equations in the form given which is related to the Minkowskian coordinate system {x} by (13). This coordinate condition (10) does not uniquely by (22) with its inverse (25). The pulled back metric in coor- 0 0 0 0 0 determine the coordinate system but selects a class of in- dinate system {x } is defined by gαβ (x ) = ηαβ + hαβ (x ). finitely many harmonic reference systems, and permits a The relation between these pulled back metric tensors, gαβ (x) 0 0 coordinate transformation from the old harmonic system and gαβ (x ), in M0 is given by (29) where a series expansion α 0 α 0 0 {x } to a new harmonic system {x } (cf. Box 18.2 in of the argument of gαβ (x ) around x has been performed. [54] or Eq. (11.5) in [56] or Eq. (3.521) in [58]) [76],

x0 α = xα + ϕα (x) , (22) the functions ϕα with respect to space and time are of α α the same order as the metric perturbation, ϕ , µ = O hµ where ϕα (x) is a vector field; see Figure 1. α hence |ϕ , µ| 1. The four-coordinates in both systems refer to one and For later purposes we note the Jacobian matrix of the the same point P of the physical manifold M, that means coordinate transformation (22), x0 = x0 (P) and x = x (P) denote the four-coordinates in both systems but of one and the same point P of the ∂x0 α Aα (x) = = δα + ϕα (x) . (23) physical manifold, which is arbitrary: ∀ P ∈ M. It µ ∂xµ µ , µ is implicitly assumed that the coordinate transformation (22) is infinitesimal in the sense that the derivatives of One may conclude from (10) that the coordinate transfor- 5 mation (22) preserves the harmonic coordinate condition A. The residual gauge transformation of the metric (10) if the functions ϕα obey the homogeneous Laplace- tensor Beltrami equation in the old coordinate system {xα} (cf. Eq. (3.522) in [58]), The covariant components of the metric tensor transform as follows [54, 56–58, 60] (e.g. Eq. (11.10) in [56]) µν α g (x) ϕ , µν (x) = 0 , (24) ∂x0 µ ∂x0 ν g (x) = g 0 (x0) . (28) where gµν (x) is the old metric tensor in the old coor- αβ ∂xα ∂xβ µν dinate system. The exact field equations of gravity in The arguments on the l.h.s. and r.h.s. in Eq. (28) re- harmonic gauge (13) are invariant under a gauge transfer to one and the same point P of the physical mani- formation (22) if the functions obey the homogeneous fold M, that means x0 = x0 (P) and x = x (P) denote α Laplace-Beltrami equation (24). The functions ϕ in the four-coordinates in both systems but of one and the (22) are nothing more than a change of coordinates and, same point P of the physical manifold, which is arbitrary: therefore, they contain no physical information about the ∀ P ∈ M. By inserting (22) into (28) and performing a gravitational system. They are called gauge vector and series expansion (recall that the residual gauge transfor- the coordinate transformation (22) with (24) is called mation is infinitesimal) of the metric tensor on the r.h.s. residual gauge transformation. These gauge functions around the old coordinates {x} of the same point P of α ϕ are obtained by solving the differential equation (24). the physical manifold, one obtains (cf. Eqs. (11.11a) - The coordinate transformation (22) is a passively con- (11.11c) in [56]) structed diffeomorphism, that means there is a differen- 0 µ 0 ν 0 µ ν 0 tiable inverse transformation from a new harmonic sys- gαβ = gαβ + ϕ , α gµβ + ϕ , β gνα + ϕ , α ϕ , β gµν 0 α α tem {x } to the old harmonic system {x }, ∞ 1 µ µ ν ν X 0 µ1 µn + δα + ϕ , α δβ + ϕ , β gµν , µ ...µ ϕ . . . ϕ , α 0 α α 0 n! 1 n x = x + χ (x ) , (25) n=1 (29) where χα (x0) is a vector field; see Figure 1. The four- coordinates in both systems refer to one and the same where all expressions are functions of one and the same point P of the physical manifold M, that means x = argument x = (ct, x). It should be noticed that this x (P) and x0 = x0 (P) denote the four-coordinates in both relation is not general-covariant but Lorentz-covariant, systems but of one and the same point P of the physical in line with the fact that the general-covariance of the manifold, which is arbitrary: ∀ P ∈ M. field equations (6) is lost when they are expressed in har- The Jacobian matrix of the inverse coordinate trans- monic reference systems: the exact field equations (13) formation (25) is given by are only Lorentz-covariant. For some reflections about the general-covariant gauge transformation of the metric α α 0 ∂x α α 0 tensor see Section III C. B (x ) = = δ + χ (x ) . (26) 0 α µ ∂x0 µ µ , µ The harmonic coordinates x on the l.h.s. in (22) are curvilinear harmonic coordinates in the flat back- The gauge functions χα obey the homogeneous Laplace- ground space-time, while the harmonic coordinates xα Beltrami equation in the new harmonic coordinate sys- on the r.h.s. in (22) are chosen as Minkowskian coordi- tem {x0 α} nates in the flat background space-time, hence the partial derivatives in (29) are just flat-space partial derivatives 0 µν 0 α 0 g (x ) χ , µν (x ) = 0 , (27) of Minkowskian coordinates. The partial derivatives in (29) would have to be replaced by flat-space covariant where g0 µν (x0) is the new metric tensor in the new har- derivatives if one would use curvilinear coordinates in monic coordinate system. The inverse coordinate trans- the flat background space-time; cf. text above Eq. (1.1) formation (25) is frequently used in the literature. Here in [56], text above Eq. (1.13a) in [56] as well as text be- it is emphasized that the gauge functions ϕα (x) in the low Eqs. (11.11a) - (11.11c) in [56] and see also Box 18.2 old harmonic system {xα} have to be distinguished from D in [54]. Further mathematical insights can be found in the gauge functions χα (x0) in the new harmonic system Section 7.1 in [59]. {x0 α}. However, the gauge-independent terms of the The gauge dependent degrees of freedom in (29), i.e. metric tensor remain unaffected by a coordinate trans- all those terms which depend on the gauge functions, formation, that means one is free in choosing either (22) are redundant in the sense that they have no impact on or (25), albeit one has to state clearly which of them is physical observables. That means, the two different met- 0 used. Here, throughout this investigation, the coordinate ric tensors gαβ (x) and gαβ (x) in (29) describe one and transformation (22) is used and the inverse coordinate the same gravitational system. Accordingly, the resid- transformation (25) will not be applied. ual gauge freedom (22) permits to identify and to isolate Let us now consider how the metric tensor and the non-physical degrees of freedom hidden in the old met- gothic metric density transform under an infinitesimal ric tensor gαβ (x) which allows to arrive at considerably 0 gauge transformation (22). simpler form for the new metric tensor gαβ (x). 6

B. The residual gauge transformation of the metric C. Some comments on the general-covariant gauge density transformation

The gothic metric (9) is a tensor density of the weight The gauge transformation considered above in Sections w = −1 and its contravariant components transform as III A and III B is Lorentz-covariant and can therefore follows [54, 56–58, 60, 77, 78] (e.g. Eq. (4.4.4) in [77]), be expressed in terms of partial derivatives. A general- covariant gauge transformation must necessarily be given 1 ∂x0 α ∂x0 β g0 αβ (x0) = gµν (x) , (30) in terms of Lie derivatives £ξ acting on the metric tensor |J (x)| ∂xµ ∂xν along a vector ﬁeld ξµ which is a general-covariant differential operation [80]. Such a general-covariant gauge where J (x) is the determinant of the Jacobi matrix (23), transformation has been developed during the last two

α decades [81–94]. It might be constructive to make some α Tr(ln Aµ ) J = det Aµ = e , (31) comments about the general-covariant gauge transformation and its relation to the Lorentz-covariant resid- where the second relation in (31) is a theorem which al- ual gauge transformation considered in Sections III A lows to compute the determinant [79] and which can be and III B. In the investigations [81–94] the metric ten- proven by Schur’s matrix decomposition. One obtains 0 sor is separated in the form gαβ = gαβ + hαβ, which generalizes (1) because the background metric g0 of 1 σ 1 σ ω 1 σ ω 3 αβ = 1 − ϕ + ϕ ϕ + ϕ ϕ + O ϕ , the curved background manifold M is not simply the |J| , σ 2 , ω , σ 2 , σ , ω 0 flat Minkowskian metric, but can be the Schwarzschild (32) metric or the Kerr metric or the Friedmann-Lemaˆıtre- Robertson-Walker metric or some other curved space- which is sufficient four our investigations in the post- time. The dynamical degrees of freedom, h , are gov- linear approximation. The arguments on the l.h.s. and αβ erned by field equations which are obtained by inserting r.h.s. in Eq. (30) refer to one and the same point P of the decomposition g = g0 + h into Einstein’s equa- the physical manifold M, that means x0 = x0 (P) and αβ αβ αβ tions (6) and describe a tensorial field h which propa- x = x (P) denote the four-coordinates in both systems αβ gates in the curved background space-time M endowed but of one and the same point P of the physical mani- 0 with the background metric g0 . fold, which is arbitrary: ∀ P ∈ M. By substituting (22) αβ The general-covariant formalism distinguishes between into (30) and performing a series expansion (recall that α the residual gauge transformation is infinitesimal) of the the physical manifold M covered by four-coordinates y and endowed with metric gαβ, the background manifold gothic metric on the l.h.s. around the old coordinates α M0 covered by four-coordinates x and endowed with {x}, one obtains 0 background metric gαβ, and a diffeomorphism and in- 0 αβ 1 αβ α µβ β να α β µν verse diffeomorphism between these manifolds, namely g = g + ϕ g + ϕ g + ϕ ϕ g −1 |J| , µ , ν , µ , ν φ : M0 → M and φ : M → M0, respectively; see Fig- ∞ ure 2. The diffeomorphism φ maps each point p ∈ M0 X 1 − g0 αβ ϕµ1 . . . ϕµn , (33) to another point u ∈ M and, vice versa, the inverse dif- n! , µ1...µn −1 n=1 feomorphism φ maps each point u ∈ M to another point p ∈ M0 (cf. Figure 7.1 in [59]). The diffeomor- where all expressions are functions of one and the same phism allows to pull back the metric tensor gαβ from M argument x = (ct, x). It should be noticed that this re- to M0 which is given by an active coordinate transfor- lation is not general-covariant but Lorentz-covariant, in ∂yµ ∂yν mation: φ∗ g (x) = g (y) (cf. Eq. (A.9) in line with the fact that the exact field equations in har- αβ ∂xα ∂xβ µν monic coordinates (13) are only Lorentz-covariant; for [59]). The metric gαβ in M and the pulled back metric ∗ some comments about the general-covariant gauge trans- φ gαβ in M0 are physically equivalent (cf. Section 7.1. formation see Section III C. The reason of why there are in [59] and Section 7.1 in [95]) and the metric pertur- flat-space partial derivatives of Minkowskian coordinates bation is defined in the background manifold as follows: ∗ 0 in (33) is the same as described in the text below Eq. (29). hαβ(x) = φ gαβ(x) − gαβ (x) (cf. Eq. (7.10) in [59]). Like in case of the metric tensors, the old gothic met- Furthermore, the general-covariant approach of gauge ric gαβ (x) and the new gothic metric g0 αβ (x) in (33) transformations considers a family of actively con- describe one and the same gravitational system; cf. text structed diffeomorphisms acting on the background man- below Eq. (7.14) in [59] and Theorem 4.5 in [47]. The ifold, ψ : M0 → M0, which maps each point p ∈ M0 to gauge dependent degrees of freedom are redundant in another point q ∈ M0 (cf. Figure 7.2 in [59]). These dif- the sense that the gauge terms in (33) have no impact on feomorphisms are distinguished from each other by some physical observables. Nevertheless, the gauge-dependent parameter and they are generated by a vector field ξµ(x) terms have to be treated carefully because they allow to acting on the background manifold; for explicit expres- transform the old gothic metric density into a consider- sions cf. Eq. (2.18) in [88] or Eq. (2.56) in [93]. The ably simpler form. composition of the family of diffeomorphisms ψ with the 7

fold to the background manifold which implies a family -1 ϕ 퓜 퓜 0 of metric perturbations defined on the background man- () ∗ 0 p ifold, hαβ(x) = Φ gαβ(x) − gαβ (x) (cf. Eq. (7.11) in -1 u O ψ ξ ( ϕ ε ) [59]). The dependence of the metric perturbation on the parameter , that reflects the dependence of the metric qε ψ 0 µ ε perturbation on the vector field ξ (x), is called gauge g α β g α β freedom: each member of the family of metric perturba- x y () tions hαβ is physically isometric (physically equivalent) to each other and any of them describes the same physical x휇 y휇 system (i.e. all observables are unchanged).

ℝ⁴ ℝ⁴

* g 0 (x) + (x) g (y) ϕ gαβ = αβ hαβ α β

0 ( ε ) This geometrical approach leads in a natural way to (ϕ O ψ )* gαβ = g (x) + h (x) ε αβ αβ the gauge transformation of the metric tensor in terms of multiple Lie derivatives acting on the metric tensor along the gauge vector [81–94]. The active coordinate transfor- Figure 2: A geometrical representation of the general- mation can be rewritten in terms of a passive coordinate covariant gauge transformation. The physical manifold M transformation which relates the four-coordinates of one is covered by coordinates {y} and endowed with the metric and the same point q ∈ M0 of the background manifold, µ 0µ gαβ (y) which is a solution of the exact field equations (6). x (q) and x (q), in two different charts of the back- The curved background manifold M0 is covered by coordi- ground manifold M0; explicit calculations and expres- 0 nates {x} and endowed with the metric gαβ (x). The diffeo- sions up to the third-order of the perturbation theory morphism φ : M0 → M (not shown in the diagram) maps are given, for instance, in [93]. In this way one arrives the curved background manifold to the physical manifold, e.g. at a general-covariant gauge transformation of the metric a point p ∈ M0 to a point u ∈ M. The inverse diffeo- −1 tensor in terms of Lie derivatives by means of a passive morphism φ : M → M0 maps the physical manifold to the curved background manifold, e.g. a point u ∈ M to a coordinate transformation. point p ∈ M0. The metric gαβ (y) of the physical manifold M is pulled back on the curved background manifold M0 (active coordinate transformation as given by Eq. (A.9) in ∗ [59]). The pulled back metric is denoted by φ gαβ and de- 0 fined by gαβ (x) = g (x) + hαβ (x). The pulled back metric αβ In order to make a bridge between the general- gαβ (x) on M0 is physically equivalent to the metric gαβ (y) on M, that means: if the metric gαβ (y) is a solution of the covariant approach in [81–94] and the Lorentz-covariant exact field equations (6) on the physical manifold M, then approach described in Sections III A and III B, one would ∗ 0 0 hαβ = φ gαβ − gαβ will be a solution of the exact field equa- have to assume a flat background metric, gαβ = ηαβ, tions in the curved background manifold M0. The set of dif- and one would have to use harmonic reference systems −1 −1 feomorphisms Φ ≡ (φ ◦ ψ) maps the same point u ∈ M as well as to impose the Laplace-Beltrami condition (24) of the physical space-time M to a set of points q ∈ M0 of for the gauge vector. In this way one would finally arrive the curved background space-time M0, where ψ represents at a Lorentz-covariant residual gauge transformation in a family of diffeomorphisms which are distinguished by the terms of Lie derivatives and based on passive coordinate parameter and which are acting on the curved background manifold and generated by a gauge vector field ξ. The com- transformations. But it should be emphasized that the results of such an approach would not differ from the position of the diffeomorphisms ψ with φ implies a family of ∗ ∗ Lorentz-covariant residual gauge transformation consid- pulled back metric tensors Φ gαβ ≡ (φ ◦ ψ) gαβ which reads () 0 () ered above, because the physical content of the gravita- gαβ (x) = gαβ (x)+hαβ (x) in the same chart {x}. The pulled () tional system is comprised in the gauge-independent met- back metric tensors gαβ (x) and gαβ (x) in M0 are related by a gauge transformation which can be expressed in terms ric perturbation and, therefore, is independent of whether the Lorentz-covariant gauge transformation in terms of of multiple Lie derivatives of gαβ (x) in the direction of the ∞ n partial derivatives or in terms of Lie derivatives is ap- () X n vector field ξ, that means g (x) = Lξ gαβ (x). plied. Here, the Lorentz-covariant residual gauge trans- αβ n! n=0 formation in terms of Lie derivatives will not further be exposed, because the Multipolar Post-Minkowskian (MPM) formalism makes use of relations (29) and (33) diffeomorphism φ, that is Φ = φ ◦ ψ, leads to a fam- and uses the series expansion (35) which, subject to the ily of diffeomorphisms Φ : M0 → M and its inverse Laplace-Beltrami equation (24), results in the sequence −1 Φ : M → M0. This family of diffeomorphisms allows of differential equations (54) - (56) for the gauge func- to pull back the metric tensor from the physical mani- tions which will be used in what follows. 8

IV. THE POST-MINKOWSKIAN EXPANSION which are called linear gauge terms since they are lin- AND GAUGE TRANSFORMATION (nPM) ear in the gauge functions. The gauge terms Ωαβ are called non-linear gauge terms since they contain either A. The Post-Minkowskian expansion of the metric products of gauge functions or products of gauge func- tensor tions and metric perturbations. One may obtain a closed (nPM) expression for Ωαβ from Eq. (29) and using Eqs. (34) In the weak-field regime the old metric tensor gαβ in and (35). Here it is sufficient to consider only the first α the old harmonic system {x } can be expanded in powers two orders, given by of the gravitational constant, (1PM) ∞ Ωαβ = 0 , (39) X n (nPM) gαβ (x) = ηαβ + G hαβ (x) , (34) n=1 (2PM) 0 (1PM) µ (1PM) 0 (1PM) µ (1PM) Ωαβ = hµβ ϕ , α + hµα ϕ , β which is called post-Minkowskian (PM) expansion. Each (nPM) 0 (1PM) ν (1PM) µ (1PM) ν (1PM) individual term hαβ is invariant under Lorentz trans- +hαβ , ν ϕ + ϕ , α ϕ , β ηµν , (40) formations; cf. text below Eq. (3.527) in [58]. The residual gauge transformation (22) from the old harmonic while the higher orders n ≥ 3 are not relevant for our α (1PM) coordinate system {x } to a new harmonic coordinate investigations. The linear 1PM term ∂ϕαβ is in agree- system {x0 α} is assumed to admit a series expansion in ment with Eq. (21) in [82], while the linear 2PM term powers of the gravitational constant (cf. Eq. (4.23) in (2PM) (2PM) ∂ϕαβ and the non-linear 2PM term Ωαβ are in [47]), agreement with Eq. (22) in [82] (to verify that agreement ∞ one has to adopt a flat background metric in [82]). X x0 α = xα + Gn ϕα (nPM) (x) , (35) n=1 B. The Post-Minkowskian expansion of the gothic α (nPM) α (nPM) metric density where ϕ , β = O hβ and each individual term ϕα (nPM) is a Lorentz four-vector. In what follows the total sum ϕα is called gauge vector, while the individual The weak-field regime admits a series expansion of the terms ϕα (nPM) are called gauge functions. These gauge gothic metric in powers of the gravitational constant [47, functions ϕα (nPM) to any order of the perturbation the- 51, 56, 58, 60] (cf. Eq. (1.1) in [47], Eq. (9.5) in [56]), ory are governed by a sequence of equations which are ∞ αβ αβ X n αβ given below by Eqs. (54) - (56). g (x) = η − G h(nPM) (x) , (41) The coordinate transformation (35) transforms the old n=1 metric tensor (34) in the old harmonic system {xα} to the new (primed) metric tensor in the new harmonic system which is called the post-Minkowskian expansion of the {x0 α}, and its post-Minkowskian expansion reads gothic metric. The post-Minkowskian expansion (41) implies a corresponding post-Minkowskian expansion of the ∞ expressions (14) and (15), 0 0 X n 0 (nPM) 0 gαβ (x ) = ηαβ + G hαβ (x ) . (36) ∞ n=1 αβ αβ X n αβ τ = T + G τ(nPM) , (42) By inserting the post-Minkowskian expansions (34) - n=1 (36) into (28) and performing a series expansion of (36) ∞ α αβ X n αβ around the four-coordinates x , one arrives at the post- t = G t(nPM) . (43) Minkowskian expansion of the gauge transformation of n=1 the metric perturbation, Taking account of (3), inserting of (41) - (43) into (13) ∞ ∞ yields a hierarchy of field equations, X n (nPM) X n 0 (nPM) (nPM) (nPM) G hαβ = G hαβ + ∂ϕαβ + Ωαβ , n=1 n=1 αβ 16 π αβ h(1PM) = − T , (44) (37) c4

αβ 16 π where all terms are given in the harmonic system {x}. h = − τ αβ + tαβ , (45) The equation (37) is nothing else than equation (29) ex- (2PM) c4 (1PM) (1PM) pressed in terms of a series expansion in powers of the . gravitational constant. . (nPM) The gauge terms ∂ϕαβ have the following structure, αβ 16 π (nPM) µ (nPM) µ (nPM) h = − τ αβ + tαβ . (46) ∂ϕαβ = ϕ , α ηµβ + ϕ , β ηµα , (38) (nPM) c4 ((n−1)PM) ((n−1)PM) 9

The sequence of field equations (44) - (46) is invariant C. The equations for the gauge functions under Lorentz-transformations. The post-Minkowskian expansion (41) of the gothic metric inherits that the har- The gauge functions are governed by Eq. (24) which monic gauge (8) must be satisfied order by order for the can also be written in the form (cf. Eq. (4.25) in [47]) metric perturbation (cf. Eq. (4.6b) in [47]), µν α αβ g (x) ϕ , µν (x) = 0 . (53) h(nPM) (x) = 0 for n = 1, 2, 3,.... (47) , β By inserting the post-Minkowskian expansion of the As discussed above, the harmonic gauge condition (47) gothic metric (41) and of the gauge function ϕα (x) = still allows for a residual gauge transformation (35). The ∞ P n α (nPM) post-Minkowskian expansion of the new gothic metric in G ϕ (x) into the Laplace-Beltrami equation 0 α n=1 the new harmonic coordinate system {x } reads (53) one obtains a sequence of equations for the gauge ∞ functions ϕα (nPM) given by 0 αβ 0 αβ X n 0 αβ 0 g (x ) = η − G h(nPM) (x ) . (48) α (1PM) n=1 ϕ = 0 , (54) By inserting the post-Minkowskian expansions (41) and (48) as well as (35) into (30) and performing a series α (2PM) µν α (1PM) ϕ = h(1PM) ϕ , µν , (55) expansion of (48) around the four-coordinates xα, one arrives at the post-Minkowskian expansion of the gauge . transformation of the gothic metric perturbation, . ∞ ∞ X n αβ X n 0 αβ αβ αβ G h(nPM) = G h(nPM) + ∂ϕ(nPM) + Ω(nPM) , n−1 µν n=1 n=1 α (nPM) X α (mPM) ϕ = h((n−m)PM) ϕ , µν , (56) (49) m=1 where all terms are given in the harmonic system {x} on µν where h (x) are the terms of the post-Minkowskian the flat background space-time by Minkowskian coordi- (nPM) αβ nates x = (ct, x). The equation (49) is nothing else than expansion (41) of the old gothic metric g (x) in the α equation (33) expressed in terms of a series expansion in old harmonic system {x }. This sequence of equations powers of the gravitational constant. allows to determine the gauge functions to any order in αβ the post-Minkowskian expansion. The gauge terms ∂ϕ(nPM) read αβ α (nPM) µβ β (nPM) µα µ (nPM) αβ ∂ϕ(nPM) = ϕ , µ η + ϕ , µ η − ϕ , µ η , V. THE MULTIPOLAR POST-MINKOWSKIAN (50) (MPM) FORMALISM which are called gothic linear gauge terms since they are linear in the gauge functions. The gauge functions The Multipolar Post-Minkowskian (MPM) formalism ϕα (nPM) are governed by a sequence of equations, which represents a powerful approach in order to determine the will be considered below; cf. Eqs. (54) - (56). The gauge gothic metric gαβ external to the compact source of mat- αβ ter in harmonic coordinates. The MPM formalism is a terms Ω(nPM) are called gothic non-linear gauge terms since they contain either products of gauge functions or considerable extension of previous investigations in [96– products of gauge functions and gothic metric perturba- 98] and of the pioneering work [56]. The formalism has αβ been developed within a series of articles [47–52], where tions. One may obtain a closed expression for Ω (nPM) the approach has thoroughly been described in detail; from Eq. (33) and using Eqs. (41) and (35). Here it is see also the descriptions of the MPM formalism in sub- sufficient to consider the first and second order, given by sequent developments [99–101]. αβ Ω = 0 , (51) The fundamental concept of the MPM approach is to (1PM) solve iteratively the hierarchy of field equations (44) ... (46) for the gothic metric density in a sequence of three αβ α (1PM) β (1PM) µν ν (1PM) 0 αβ Ω(2PM) = +ϕ , µ ϕ , ν η + ϕ h(1PM) steps: , ν

1 µ (1PM) ν (1PM) µ (1PM) ν (1PM) αβ (i) solving the field equations in the internal near- + ϕ , ν ϕ , µ − ϕ , µ ϕ , ν η 2 zone Di of the source in the post-Newtonian (weak- 0 µβ − ϕα (1PM) h + ∂ϕµβ field slow-motion) scheme; see Eq. (1) in [101] for , µ (1PM) (1PM) a concrete definition of what a post-Newtonian β (1PM) 0 µα µα source is. The internal near-zone is defined by −ϕ , µ h(1PM) + ∂ϕ(1PM) , (52) Di = {(t, x) with |x| < ri} where R < ri λ, while the higher orders n ≥ 3 are not relevant for our where R is the radius of a sphere which encloses the investigations. source and λ is the wavelength of the gravitational 10

radiation emitted by the source. So the internal gothic metric perturbation, near-zone is a spatial region which contains the in- αβ gen αβ αβ gen terior of the source and a region in the exterior of g (xgen) = η − h (xgen) . (57) the source but much smaller than the wavelength of the gravitational radiation emitted by the source. An important result of the MPM approach consists in a theorem (Theorem 4.2 in [47]) which states that outside (ii) solving the field equations in the external zone the matter source the most general solution of the post- Minkowskian hierarchy (44) ... (46) depends on a set of De of the source in the post-Minkowskian (weak field) scheme. The external zone is defined by altogether six STF multipoles [47, 100–102] (cf. Eq. (62) in [100], Eq. (50) in [101], Eq. (4.1) in [102]) De = {(t, x) with |x| > re} where R < re < ri. So the external zone contains the entire spatial region ∞ in the exterior of the source. αβ gen X n αβ gen h (xgen) = G h(nPM) [IL,JL,WL,XL,YL,ZL] , n=1 (iii) performing a matching procedure of both these (58) solutions for the metric tensor in the intermediate near-zone Di ∩ De of the source, where where the square brackets denote a functional depen- both the post-Newtonian expansion and the post- dence on these six STF multipoles. The MPM solution Minkowskian expansion are simultaneously valid. (58) is the most general solution of Einsteins vacuum The intermediate near-zone is defined by Di ∩De = equations outside an isolated source of matter. The STF {(t, x) with re < |x| < ri}. The definitions of the multipoles in (58) depend on the retarded time sgen de- internal near-zone Di and external zone De are ad- fined by justed such that the intermediate near-zone Di ∩De is not empty. The intermediate near-zone is a spa- |xgen| sgen = tgen − , (59) tial region in the exterior of the source but much c smaller than the wavelength λ of the gravitational which is the time of retardation between some field point radiation emitted by the source. xgen and the origin of the spatial axes of the general harmonic coordinate system {xgen} [103]. As stated above In what follows only those fundamental results of the by Eq. (47), the harmonic gauge (8) is satisfied order by elaborated MPM formalism are considered which are of order for the metric perturbation, which in terms of the relevance for our analysis. In particular, we will not MPM solution is given by (cf. Eq. (4.6b) in [47]), consider the specific issue related to the far-wave zone, where so-called radiative coordinates and radiative mo- ∂ αβ gen h [IL,JL,WL,XL,YL,ZL] = 0 . (60) ments VL and UL are introduced, which are uniquely re- β (nPM) ∂xgen lated to the mass-multipoles ML and spin-multipoles SL via non-linear equations; cf. Eqs. (6.4a) - (6.4b) in [99]. The MPM formalism is augmented by a matching proce- In the far-wave zone only the transverse traceless projec- dure described in detail in [99, 101] which allows to de- TT tion of the metric perturbation, hαβ , is relevant because termine these six multipoles as integrals over the stress- it contains the physical degrees of freedom of the gravi- energy tensor of the source of matter. For that reason tational radiation field; cf. Eq. (64) in [71]. That trans- these multipoles IL,JL,WL,XL,YL,ZL are collectively verse traceless projection of the metric perturbation has named as the source multipole moments [99]. In fact, such been given in several investigations in the 1PM approx- an explicit closed-form expression for the set of these six imation [71, 101, 102] (e.g. Eq. (64) in [71], Eq. (66) in STF multipoles has been derived by Eqs. (5.15) - (5.20) TT TT in [99]; see also Eqs. (85) - (90) in [100], Eqs. (123a) - [101], Eq. (2.1) in [102]); note that hαβ(1PM) = hαβ(1PM) (cf. Eq.(7.119) in [61]). Here, we will not consider the (125d) in [101]. transverse-traceless gauge but emphasize that all the subsequent statements about the gothic metric perturbation B. Residual gauge transformation of the general and about the metric perturbation are valid in the entire gothic metric region in the exterior of the source of matter. A further result of utmost importance of the MPM formalism (Theorem 4.5 in [47]) is that there exists a A. The general solution of the gothic metric residual gauge transformation (cf. Eq. (35)),

∞ X In the MPM formalism the most general solution of the xα = xα + Gn ϕα (nPM) (x ) , (61) gothic metric is called general gothic metric and denoted can gen gen αβ gen n=1 by g (xgen) given in the general harmonic reference system {xgen} = (ctgen, xgen). According to Eq. (3) it is which preserves the harmonic gauge (47) and which al- decomposed in the flat Minkowskian metric and a general lows to write the general metric perturbation in (58) in 11 the following form, governed by Eqs. (54) - (56), which in terms of the STF multipoles of the MPM formalism read (cf. Eqs. (4.26) - ∞ X n αβ gen (4.27) in [47]) G h(nPM) [IL,JL,WL,XL,YL,ZL] n=1 α (1PM) ∞ ϕ = 0 , (70) X n αβ can αβ αβ = G h(nPM) [ML,SL] + ∂ϕ(nPM) + Ω(nPM) (62) n=1 α (2PM) µν gen α (1PM) ϕ = h(1PM) ϕ , µν , (71) α where all terms depend on the four-coordinates xgen and the STF multipoles depend on the retarded time s in . gen . (59). The relation (62) is nothing else than relation (49) expressed in terms of STF multiples of the MPM formalism [104]. The relation (62) states that the general gothic n−1 α (nPM) X µν gen α (mPM) metric perturbation (58) in terms of six source multipoles ϕ = h((n−m)PM) ϕ , µν , (72) is physically isometric to the canonical gothic metric per- m=1 turbation (63) in terms of two canonical multipoles. That means that the general gothic metric perturbation (58) which are given in the harmonic system {xgen} and where contains the same physical information as the canoni- the general solution of the metric perturbations is given cal gothic metric perturbation (63); see also text below by Eq. (58). The sequence of differential equations for the Eq. (45) in [100], text below Eq. (52) in [101], text above gauge functions in (70) - (72) is nothing but the sequence below Eq. (4.26) in [102]. of differential equations for the gauge functions in (54) - The term (56) expressed in terms of STF source multipoles.

∞ αβ can X n αβ can h (xgen) = G h(nPM) [ML,SL] (63) n=1 C. The general solution of the metric tensor on the r.h.s. in Eq. (62) is called canonical gothic metric The most general solution of the metric tensor in the perturbation and exterior of a compact source of matter is uniquely deter-

αβ can αβ αβ can mined by the relation (cf. Eq. (D10) in Appendix D) g (xgen) = η − h (xgen) (64) g = p−det (gµν gen) g . (73) is the canonical gothic metric. The multipoles ML and αβ gen αβ gen SL are called canonical multipoles and they are related to the source multipoles via two non-linear equations (cf. The terms on the r.h.s. of (73) are given by (57) - (58) Eqs. (6.1a) - (6.1b) in [99] and text below Eq. (45) in and by the isometry relation of the gothic metric [54, 57, [99]), 58] (cf. Eq. (D4) in Appendix D)

ασ gen α ML = ML [IL,JL,WL,XL,YL,ZL] , (65) g gσβ gen = δβ . (74) SL = SL [IL,JL,WL,XL,YL,ZL] , (66) According to Eq. (1), the general metric tensor in (73) which are of complicated structure; cf. Eqs. (97) and is separated into the ﬂat Minkowskian metric and the (98) in [101] for the case of L = i1i2 and L = i1i2i3. In general metric perturbation, view of the highly involved structure of the relations (65) - (66) it seems impossible to achieve an explicit closed- gαβ gen (xgen) = ηαβ + hαβ gen (xgen) . (75) form expression for the canonical multipoles ML,SL to any order of the post-Minkowskian series expansion [99– From (73) follows that the general metric perturbation 101]. The gauge terms on the r.h.s. in (62) depend, in formally reads the general case, on the full set of all six STF source multipoles (cf. text below Eq. (4.23) in [47]), ∞ X (nPM) h (x ) = Gnh [I ,J ,W ,X ,Y ,Z ] . αβ αβ αβ gen gen αβ gen L L L L L L ∂ϕ(nPM)(xgen) = ∂ϕ(nPM) [IL,JL,WL,XL,YL,ZL] ,(67) n=1 αβ αβ (76) Ω(nPM)(xgen) = Ω(nPM) [IL,JL,WL,XL,YL,ZL] . (68) The explicit structure of these gauge terms will be consid- The square brackets denote a functional dependence on ered below in the linear and post-linear approximation. the six STF source multipoles which depend on the re- These gauge terms are functions of the gauge functions tarded time sgen in Eq. (59). The Eqs. (75) and (76) represent the most general solution of Einsteins vacuum α (nPM) α (nPM) ϕ (xgen) = ϕ [IL,JL,WL,XL,YL,ZL] , (69) equations outside an isolated source of matter. 12

D. Residual gauge transformation of the general is given by (cf. Eq. (4)) metric tensor αβ αβ 1 αβ 2 αβ g (t, x) = η − G h(1PM) (t, x) − G h(2PM) (t, x) The residual gauge transformation (61) allows to trans- + O G3 . (82) form the general metric perturbation in the following form, αβ ∞ In this Section the linear term h(1PM) and the post-linear X n (nPM) αβ G hαβ gen [IL,JL,WL,XL,YL,ZL] term h(2PM) are considered. n=1 ∞ X n (nPM) (nPM) (nPM) = G hαβ can [ML,SL] + ∂ϕαβ + Ωαβ (77) A. The linear term of the gothic metric density n=1 α where all terms depend on the four-coordinates xgen and The solution of the field equations in the first iteration the STF multipoles depend on the retarded time sgen in (44) reads (59). The relation (77) is nothing else than relation (37) expressed in terms of STF multiples of the MPM for- αβ 16 π h (t, x) = − −1 T αβ (t, x) , (83) malism [105]. The relation (77) states that if the source (1PM) c4 R multipoles and the canonical multipoles are related to each other via Eqs. (65) - (66), then the general metric where T αβ is the stress-energy tensor of matter and −1 perturbation on the l.h.s. of (77) and the canonical metric R is the inverse d’Alembert operator defined by perturbation on the r.h.s. of (77) are related by the resid- Eq. (20). The integration runs only over the finite three- ual coordinate transformation (61). They are physically dimensional volume of the compact source of matter isometric to each other and either of them contains the [47, 48]. The integral (83) is finite and has been deter- entire physical information in the exterior of the gravita- mined in [48] and has later been reconsidered in specific tional source of matter. The term detail in [72]. ∞ According to the fundamental theorem (58) of the X n (nPM) MPM formalism, the most general solution for the 1PM hαβ can (xgen) = G hαβ can [ML,SL] (78) n=1 term of the gothic metric perturbation (83) in the exterior of a compact source of matter depends on six STF on the r.h.s. in Eq. (77) is called canonical metric per- αβ gen turbation and source multipoles and is denoted by h(1PM) . The residual gauge transformation (61) in 1PM approximation trans- gαβ can (xgen) = ηαβ + hαβ can (xgen) (79) αβ gen forms the linear gothic metric perturbation h(1PM) in the is the canonical metric. The canonical multipoles ML following form [47, 52, 56, 99, 101, 102], and SL are related to the source multipoles via Eqs. (65) and (66). The gauge terms on the r.h.s. in (77) depend, αβ gen h(1PM) [IL,JL,WL,XL,YL,ZL] in the general case, on the full set of all six STF source αβ can αβ multipoles, = h(1PM) [ML,SL] + ∂ϕ(1PM) (t, x) . (84) αβ αβ ∂ϕ(nPM) (xgen) = ∂ϕ(nPM) [IL,JL,WL,XL,YL,ZL] ,(80) The canonical gothic metric in 1PM approximation for αβ αβ one body at rest with full multipole structure is given by Ω(nPM) (xgen) = Ω(nPM) [IL,JL,WL,XL,YL,ZL] . (81) ∞ l The explicit structure of these gauge terms will be consid- 00 can 4 X (−1) ML (s) h (t, x) = + ∂ , (85) ered below in the linear and post-linear approximation. (1PM) c2 l! L r They are functionals of the gauge functions (69) which l=0 are determined by means of Eqs. (70)- (72). To simplify the notations, in all of the subsequent Sec- ∞ l ˙ 0i can 4 X (−1) MiL−1 (s) tions, the four-coordinates of the general harmonic sys- h(1PM) (t, x) = − ∂L−1 α α c3 l! r tem xgen = (ctgen, xgen) will be denoted by x = (ct, x). l=1 This implies that the retarded time sgen in (59) is now ∞ l 4 X (−1) l SbL−1 (s) denoted by s = t − |x| /c. − ∂ , (86) c3 (l + 1)! iab aL−1 r l=1 ∞ l ¨ ij can 4 X (−1) MijL−2 (s) VI. THE GOTHIC METRIC DENSITY IN h (t, x) = + ∂ POST-LINEAR APPROXIMATION (1PM) c4 l! L−2 r l=2 ∞ l ˙ 8 X (−1) l ab(iSj)bL−2 (s) The post-Minkowskian expansion of the gothic metric + ∂ . (87) c4 (l + 1)! aL−2 r density in the second post-Minkowskian approximation l=2 13

αβ αβ The non-linear relations (65) and (66) simplify in the where τ1 and t1 denote the ﬁrst iteration of (14) and −1 1PM approximation as follows (cf. Eqs. (6.2a) - (6.2b) in (15), respectively, and FPB=0 R is the Hadamard reg- [99], Eqs. (4.25a) - (4.25b) in [102]), ularized inverse d’Alembert operator deﬁned by Eq. (F2); details of the Hadamard regularization are given in Ap- αβ ML = IL + O (G) , (88) pendix F. The expression of τ1 follows from (14) by SL = JL + O (G) . (89) series expansion of the determinant. The expression of αβ t1 follows from (15) by using the 1PM approximation The explicit expressions for the canonical multipoles ML of the gothic metric perturbation; cf. Eq. (3.3) in [102]. and SL are given by Eqs. (5.33) and (5.35) in [52] as integrals over the stress-energy tensor of the matter source, and they are represented by Eqs. (C1) and (C2) in Ap- According to the fundamental theorem (58) of the pendix C. The linear gauge term in (84) is given by (cf. MPM formalism, the most general solution for the 2PM Eq. (50)) term of the gothic metric perturbation (92) in the exte-

αβ α (1PM) µβ β (1PM) µα rior of a compact source of matter depends on six STF αβ gen ∂ϕ(1PM) (t, x) = ϕ , µ (t, x) η + ϕ , µ (t, x) η source multipoles and is denoted by h(2PM) . The residual µ (1PM) αβ − ϕ , µ (t, x) η . (90) gauge transformation (61) in 2PM approximation trans- αβ gen forms the post-linear gothic metric perturbation h α (1PM) (2PM) The gauge function ϕ is determined by Eq. (70). in the following form [47, 52, 99, 102] (cf. Eq. (4.26) in The gauge function depends on four source moments, [102]) α (1PM) α (1PM) ϕ (t, x) = ϕ [WL,XL,YL,ZL] , (91) αβ gen h [IL,JL,WL,XL,YL,ZL] and is given by Eqs. (5.31b) in [52]; see also Eqs. (4.13a) (2PM) αβ can αβ αβ - (4.13b) in [99] or Eqs. (3.560) - (3.561) in [58]. = h(2PM) [ML,SL] + ∂ϕ(2PM)(t, x) + Ω(2PM)(t, x) .(93)

B. The post-linear term of the gothic metric density

The solution of the ﬁeld equations in the second itera- The canonical gothic metric for a source of matter tion (45) reads with full multipole structure has not rigorously been determined in the second post-Minkowskian (2PM) αβ 16 π h (t, x) = − FP −1 τ αβ + tαβ (t, x) , scheme thus far, but in the following approximation (2PM) c4 B=0 R 1 1 (cf. Eqs. (2.28a) - (2.28c) and Eq. (2.29) together with (92) Eqs. (2.18a) and (2.5) in [51])

∞ l !2 00 can 7 X (−1) ML (s) h (t, x) = ∂ + O c−6 , (94) (2PM) c4 l! L r l=0

0i can −5 h(2PM) (t, x) = O c , (95)

∞ l ! ∞ l ! ij can 4 X (−1) ML (s) X (−1) ML (s) h (t, x) = − FP −1 ∂ ∂ ∂ ∂ (2PM) c4 B=0 R i l! L r j l! L r l=0 l=0 ∞ l !2 1 X (−1) ML (s) + δ ∂ + O c−6 . (96) c4 ij l! L r l=0

These expressions are also in agreement with Eqs. (3.5a) and the Will-Wiseman approach has been explained in - (3.5c) in [61]; the agreement of the MPM formalism Section 4.3 in [101]. In the second line of (96) we have 14 used the following relation [106]. Its solution reads formally (cf. Eq. (4.28) in [47])

2 ∞ l ! α (2PM) −1 µν gen α (1PM) X (−1) ML (s) ϕ (t, x) = FPB=0 h ϕ (t, x) FP −1 ∂ ∂ R (1PM) , µν B=0 R k l! L r l=0 (101) ∞ l !2 1 X (−1) ML (s) = ∂ + O c−2 . (97) where FP −1 is the Hadamard regularized inverse 2 l! L r B=0 R l=0 d’Alembertian (F2). The formal solution (101) leads to

On the other side, the integral in the ﬁrst line of (96) α (2PM) α (2PM) ϕ (t, x) = ϕ [IL,JL,WL,XL,YL,ZL] , (102) is complicated because of the retarded time argument. Thus, while the time-time components (94) are already that means the gauge function ϕα (2PM) depends on the given in terms of multipoles, the spatial components (96) full set of the STF source moments. The explicit expres- of the gothic metric are associated with a complicated sion for the gauge function in (102) is complicated, but integration procedure, FP −1, consisting of the in- B=0 R we will not pursue it here because one may show that verse d’Alembert operator and Hadamard’s regularization, which is explained in more detail in Appendix F. ∂ϕαβ (t, x) = O c−6, c−5, c−6 , (103) The spin-multipoles SL do not occur in (94) - (96) be- (2PM) cause they are terms of the order O c−6. A further comment should be in order. In the solution of Eqs. (94) which is of the same order of the neglected terms in the - (96) terms of the order O c−6, c−5, c−6 are neglected canonical gothic metric perturbation in (94) - (96). The [107], while the perturbations are presented in terms of non-linear gauge term of the coordinate transformation the retarded time argument, which is not further ex- reads (cf. Eq. (52)) panded in powers of the inverse of the speed of gravity. So the solution in (94) - (96) is a hybrid representation αβ Ω(1PM) (t, x) = 0 , (104) in the sense that it is mixing the post-Minkowskian expansion (series in powers of G) and the post-Newtonian αβ α (1PM) β (1PM) µν expansion (series in inverse powers of c). A good rea- Ω(2PM) (t, x) = ϕ , µ (t, x) ϕ , ν (t, x) η son of such a representation is that the expressions (94) ν (1PM) αβ can + ϕ (t, x) h(1PM) (t, x) - (96) adopt their most simple form. But the main rea- , ν son for the hybrid representation is that it permits to µβ can − ϕα (1PM) (t, x) h (t, x) + ∂ϕµβ (t, x) avoid problems regarding the convergence of the post- , µ (1PM) (1PM) Newtonian expansion of the metric for non-compact sup- µα can µα − ϕβ (1PM) (t, x) h (t, x) + ∂ϕ (t, x) port; cf. text below Eq. (2.5) in [51]. In this respect we , µ (1PM) (1PM) recall that the source of matter is assumed to be com- 1 + ϕµ (1PM) (t, x) ϕν (1PM) (t, x) ηαβ pact, but one has to keep in mind that the integral (92) 2 , ν , µ gets support inside and outside the matter source, that 1 − ϕµ (1PM) (t, x) ϕν (1PM) (t, x) ηαβ . (105) means it acquires a non-compact support; cf. text below 2 , µ , ν Eq. (21). The non-linear relations (65) and (66) simplify in the corresponding approximation as follows (cf. The gauge function ϕα (1PM) on the r.h.s. in (105) de- Eq. (6.3) in [99], Eqs. (99a) - (99b) in [101], Eqs. (5.11a) pends on four source multipoles (cf. Eq. (91)) and is ex- and (5.11b) in [102]), plicitly given by Eqs. (5.31b) in [52]; see also Eqs. (4.13a) - (4.13b) in [99] or Eqs. (3.560) - (3.561) in [58]. The −5 ML = IL + O c , (98) αβ gauge term ∂ϕ(1PM) is given by Eq. (90), while the 1PM −5 SL = JL + O c . (99) αβ can canonical gothic metric perturbation h(1PM) has been given by Eqs. (85) - (87). It has been checked that one The explicit expressions for the canonical multipoles ML αβ and SL are given by Eqs. (5.33) and (5.35) in [52] as inte- would obtain the same non-linear gauge term Ω(2PM) as grals over the stress-energy tensor of the matter source, given by Eq. (4.7a) in [102] if one would use the resid- and they are represented by Eqs. (C1) and (C2) in Ap- ual gauge transformation (25) instead of (22) and if one pendix C. The linear gauge term in (93) is given by (cf. would series-expand the gothic metric in the system {x0} Eq. (50)) which then would have to be endowed by Minkowskian coordinates; note the diﬀerent sign-convention for the αβ α (2PM) µβ β (2PM) µα gothic metric perturbation. Here it should be emphasized ∂ϕ(2PM)(t, x) = ϕ , µ (t, x) η + ϕ , µ (t, x) η again that the canonical piece of the gothic metric density µ (2PM) αβ −ϕ , µ (t, x) η . (100) (and of the metric tensor) is gauge-independent, hence is independent of whether one uses the residual gauge The gauge function ϕα (2PM) is determined by Eq. (71). transformation (22) or (25) (cf. text below Eq. (27)). 15

VII. THE METRIC TENSOR IN POST-LINEAR In order to get (109) - (111) we made use of the property APPROXIMATION that M¨ ii = 0 since the multipoles are trace-free, as well as ˙ of the identity ab(i Si)bL−2 = 0 due to antisymmetry of The post-Minkowskian expansion of the metric tensor the Levi-Civita symbol and the symmetry of multipoles. up to terms of the order O G3 is given by (cf. Eqs. (1) One may verify that Eqs. (109) - (111) agree with Eq. (2) and (2)) in [108]; just use the decomposition of the metric tensor as given by Eq. (A.1) in [108] and apply the orthogonality 1 (1PM) 2 (2PM) relation of the metric tensor in 1PM approximation. The gαβ(t, x) = ηαβ + G hαβ (t, x) + G hαβ (t, x) gauge term in (108) reads (cf. Eq. (38)) +O G3 . (106)

(1PM) (1PM) µ (1PM) µ (1PM) ∂ϕ (t, x) = ϕ , α (t, x) ηµβ + ϕ (t, x) ηµα , In this Section the linear term hαβ and the post-linear αβ , β (2PM) (112) term hαβ of the metric tensor are considered.

where the gauge function ϕα (1PM) on the r.h.s. in (112) A. The linear term of the metric tensor is governed by Eq. (70). The gauge function depends on four source multipoles (cf. Eq. (91)) and its explicit (1PM) In order to determine the linear term hαβ of the form is given by Eqs. (5.31b) in [52]; see also Eqs. (4.13a) metric, the following relation between - (4.13b) in [99] or Eqs. (3.560) - (3.561) in [58].

µν 1 h(1PM) = h η η − h η , (107) αβ (1PM) αµ βν 2 (1PM) αβ µν where h = η h . The relation (107) allows to (1PM) µν (1PM) B. The post-linear term of the metric tensor determine the covariant components of the metric tensor from the contravariant components of the gothic met- (2PM) ric in 1PM approximation. By inserting Eq. (84) into In order to determine the post-linear term hαβ of Eq. (107) with the expressions in (85) - (87) and (90), the metric, the following relation between the metric and one obtains the general solution for the metric perturba- gothic metric is used, which is shown in Appendix E (cf. tion in the 1PM approximation, Eq. (1.6.3) in [60]),

(1PM) hαβ gen [IL,JL,WL,XL,YL,ZL] (2PM) µν 1 1 2 h = h ηαµ ηβν − h(2PM) ηαβ + h ηαβ (1PM) (1PM) αβ (2PM) 2 8 (1PM) = hαβ can [ML,SL] + ∂ϕαβ (t, x) . (108) 1 µν ρν µσ − h(1PM) h ηαµ ηβν + h h ηµν ηαρ ηβσ The linear term of the canonical metric perturbation for 2 (1PM) (1PM) (1PM) 1 µν ρσ one body at rest with full multipole structure is given by − h h η η η , (113) 4 (1PM) (1PM) µρ νσ αβ ∞ l (1PM) 2 X (−1) ML (s) h (t, x) = + ∂ , (109) 00 can c2 l! L r µν l=0 where h(2PM) = ηµν h(2PM). The relation (113) allows to determine the covariant components of the metric tensor ∞ l ˙ (1PM) 4 X (−1) MiL−1 (s) from the contravariant components of the gothic metric h (t, x) = + ∂ 0i can c3 l! L−1 r in 2PM approximation. By inserting Eqs. (84) and (93) l=1 into (113) with the expressions in (85) - (87) and (90) as ∞ l 4 X (−1) l SbL−1 (s) + ∂ , (110) well as (94) - (96) and (100) and (105), one obtains the c3 (l + 1)! iab aL−1 r general solution for the metric perturbation in the 2PM l=1 approximation, ∞ l (1PM) 2 X (−1) ML (s) h (t, x) = + δ ∂ ij can c2 ij l! L r (2PM) l=0 hαβ gen [IL,JL,WL,XL,YL,ZL] ∞ l ¨ (2PM) (2PM) (2PM) 4 X (−1) MijL−2 (s) =h [M ,S ] + ∂ϕ (t, x) + Ω (t, x) .(114) + ∂ αβ can L L αβ αβ c4 l! L−2 r l=2 ∞ l ˙ The post-linear term of the canonical metric perturbation 8 X (−1) l ab(iSj)bL−2 (s) + ∂ . (111) for one body at rest with full multipole structure reads c4 (l + 1)! aL−2 r l=2 16

∞ l !2 (2PM) 2 X (−1) ML (s) h (t, x) = − ∂ + O c−6 , (115) 00 can c4 l! L r l=0

(2PM) −5 h0i can (t, x) = O c , (116)

∞ l ! ∞ l ! (2PM) 4 X (−1) ML (s) X (−1) ML (s) h (t, x) = − FP −1 ∂ ∂ ∂ ∂ ij can c4 B=0 R i l! L r j l! L r l=0 l=0

∞ l !2 2 X (−1) ML (s) + δ ∂ + O c−6 . (117) c4 ij l! L r l=0

The spatial components (117) of the metric tensor are VIII. STATIONARY SOURCES −1 associated with an integration procedure, FPB=0 R , consisting of the inverse d’Alembert operator and In many applications of general theory of relativity it Hadamard’s regularization, which is explained in Ap- is possible to neglect the time-dependence of the matter pendix F. The canonical mass-multipoles ML are given source and to consider a stationary source, deﬁned by by Eq. (5.33) in [52] as integrals over the stress-energy µν tensor of the matter source; note that there are no spin- T , 0 = 0 , (121) multipoles SL in (115) - (117) because they are terms of the order O c−6. The linear gauge term in (114) is that means an approximation where the stress-energy given by (cf. Eq. (38)) tensor is only a function of the spatial coordinates in the harmonic reference system. The condition (121) does ∂ϕ(2PM)(t, x) = ϕµ (2PM)(t, x) η + ϕµ (2PM)(t, x) η , not necessarily imply that the source of matter is static. αβ , α µβ , β µα Namely, a static source implies that there is no motion at (118) all inside the source of matter, while a stationary source

α (2PM) only requires that motions of matter (e.g. inner circu- where the gauge function ϕ on the r.h.s. in (118) lations) have to be time-independent. Stated diﬀerently, is governed by Eq. (71). It’s explicit form is formally for static sources not only Eq. (121) holds but in addition given by Eq. (101) and depends on all six STF source T 0i = 0, while for stationary sources T 0i = const 6= 0 multipoles; cf. Eq. (102). That explicit expression for is possible. The metric of a stationary source is time- the gauge function in (118) is complicated and we will independent, not pursue it here because one may show that gαβ , 0 = 0 , (122) (2PM) −6 −5 −6 ∂ϕαβ (t, x) = O c , c , c , (119) that means the metric tensor depends only on spatial which is of the same order of the neglected terms in the coordinates. Let us notice that for a stationary metric canonical metric perturbation in (115) - (117). The non- g0i = const 6= 0 is possible, while for a static metric linear gauge term in (114) reads (cf. Eq. (40)) g0i = 0 (cf. Eq. (56.02) in [57]). In the stationary case the post-Minkowskian expansion of the metric tensor up (2PM) (1PM) µ (1PM) to terms of the order O G3 reads (cf. Eq. (5)) Ωαβ (t, x) = hµβ can (t, x) ϕ , α (t, x) (1PM) µ (1PM) (1PM) (2PM) +hµα can (t, x) ϕ (t, x) 1 2 3 , β gαβ (x) = ηαβ + G hαβ (x) + G hαβ (x) + O G . (1PM) ν (1PM) +hαβ can , ν (t, x) ϕ (t, x) (123)

µ (1PM) ν (1PM) (1PM) +ϕ , α (t, x) ϕ (t, x) ηµν . (120) , β In this Section the linear term hαβ and the post-linear term h(2PM) are considered. The gauge function ϕα (1PM) on the r.h.s. in (120) αβ is formally given by Eq. (91) and explicitly given by Eqs. (5.31b) in [52]; see also Eqs. (4.13a) - (4.13b) in [99] or Eqs. (3.560) - (3.561) in [58]. The linear gauge A. The linear term of the metric tensor for (1PM) stationary sources term ∂ϕαβ is given by Eq. (112), while the canonical (1PM) linear metric perturbation hαβ can is given by Eqs. (109) According to Eq. (108) the residual gauge transforma- - (111). tion of the 1PM terms of the metric perturbation for 17 stationary sources reads which depends on the gauge function ϕα (1PM) determined by (1PM) hαβ gen [IL,JL,WL,XL,YL,ZL] α (1PM) (1PM) (1PM) ∆ ϕ (x) = 0 , (131) = hαβ can [ML,SL] + ∂ϕαβ (x) , (124) where the source multipoles IL,JL,WL,XL,YL,ZL and which follow from (70) in the case of time-independence the canonical multipoles ML,SL are time-independent of the gauge functions; ∆ = ∂k ∂k is the ﬂat Laplace now. For the time-independent canonical multipoles one operator and the gauge function is time-independent: obtains from Eqs. (C1) and (C2) in Appendix C (cf. ϕα (1PM) = 0. This gauge function is formally given by Eqs. (5.33) and (5.35) in [52]), , 0

α (1PM) Z T 00 + T kk ϕ (x) = [WL,XL,YL,ZL] , (132) M = d3x xˆ , (125) L L c2 that means it depends on four STF source multipoles Z 0k 3 j T which are time-independent now. An explicit expression SL = STFL d x xˆL−1 i jk x . (126) l c of (132) can be deduced from Eqs. (5.31b) in [52] by taking the limit of vanishing time argument. For stationary sources the canonical metric perturbation in (124) simpliﬁes considerably. From Eqs. (109) - (111) one obtains

∞ l B. The post-linear term of the metric tensor for (1PM) 2 X (−1) ML h (x) = ∂ , (127) stationary sources 00 can c2 l! L r l=0 ∞ l According to Eq. (114) the residual gauge transforma- (1PM) 4 X (−1) l SbL−1 h (x) = ∂ , (128) tion of the 2PM terms of the metric perturbation for 0i can c3 (l + 1)! iab aL−1 r l=1 stationary sources reads ∞ l (1PM) 2 X (−1) ML h (x) = δ ∂ . (129) h(2PM) [I ,J ,W ,X ,Y ,Z ] ij can c2 ij l! L r αβ gen L L L L L L l=0 (2PM) (2PM) (2PM) = hαβ can [ML,SL] + ∂ϕαβ (x) + Ωαβ (x) , (133) This is the linear term of the canonical metric in case of a stationary source. The gauge term ∂ϕ(1PM) in (124) is αβ where the source multipoles I ,J ,W ,X ,Y ,Z and given by L L L L L L the canonical multipoles ML,SL are time-independent (1PM) µ (1PM) µ (1PM) now. For stationary sources the canonical metric pertur- ∂ϕαβ (x) = ϕ , α (x) ηµβ + ϕ , β (x) ηµα , bation in (133) simpliﬁes considerably. From Eqs. (115) (130) - (117) one obtains

∞ l !2 (2PM) 2 X (−1) ML h (x) = − ∂ + O c−6 , (134) 00 can c4 l! L r l=0

(2PM) −5 h0i can (x) = O c , (135)

∞ l ! ∞ l ! (2PM) 4 −1 ∂ X (−1) ML ∂ X (−1) ML hij can (x) = − 4 FPB=0 ∆ ∂L ∂L c ∂xi l! r ∂xj l! r l=0 l=0 ∞ l !2 2 X (−1) ML + δ ∂ + O c−6 . (136) c2 ij l! L r l=0

This is the post-linear term of the canonical metric in the canonical post-linear metric in (136) are associated case of a stationary source. The spatial components of with an integration procedure via the Hadamard regu- 18

−1 larized inverse Laplace operator, FPB=0 ∆ , deﬁned by C. Monopole and spin and quadrupole terms of (2PM) 2PM metric for stationary sources Eq. (G2). The gauge term ∂ϕαβ reads

(2PM) µ (2PM) µ (2PM) ∂ϕαβ (x) = ϕ , α (x) ηµβ + ϕ , β (x) ηµα , The metric in 2PM approximation of an arbitrarily (137) shaped body is considered, where the monopole and spin and quadrupole terms of the metric tensor are taken into α (2PM) account. The expressions for the monopole and spin and which is time-independent, that means ϕ , 0 = 0. (2PM) quadrupole follow from (125) and (126), viz. The gauge term ∂ϕαβ depends on the gauge function α (2PM) ϕ which is determined by the equation Z 00 kk 3 T + T M = d x 2 , (142) α (2PM) ij gen α (1PM) c ∆ ϕ (x) = h(1PM) (x) ϕ , ij (x) , (138) Z T 0k S = d3x xj , (143) as it follows from (71) in the limit of time-independence i ijk c of the gauge functions; ∆ = ∂k ∂k is the ﬂat Laplace Z 00 kk 3 T + T operator. A formal solution is provided by Mab = d x xˆab , (144) c2

α (2PM) −1 ij gen α (1PM) ϕ (x) = FPB=0 ∆ h(1PM) ϕ , ij (x) , where the integrals run over the three-dimensional vol- (139) 1 2 ume of the body, andx âb = xaxb − 3 r δab. The mass- dipole terms are not considered here, because they can where FP ∆−1 is the Hadamard regularized inverse B=0 be eliminated, M = 0, by an appropriate choice of the Laplacian defined by Eq. (G2). This gauge function is i coordinate system (origin of the spatial axes are tied to formally given by the center of mass of the source; cf. comment below α (2PM) Eq. (C7). ϕ (x) = [IL,JL,WL,XL,YL,ZL] , (140) that means it depends on all the six STF source multipoles which are time-independent now. The explicit expression for the gauge function in (140) is complicated 1. The 1PM terms of canonical metric perturbation but not relevant because of (119). The non-linear gauge term Ω(2PM) in (133) reads αβ From Eqs. (127) - (129) one immediately obtains the 1PM terms of the canonical metric perturbation: (2PM) (1PM) µ (1PM) Ωαβ (x) = hµβ can (x) ϕ , α (x) (1PM) µ (1PM) (1PM) ν (1PM) 2 M 3 nˆ M +hµα can (x) ϕ , β (x) + hαβ can , ν (x) ϕ (x) (1PM) ab ab h00 can (x) = 2 + 2 3 , (145) µ (1PM) ν (1PM) c r c r +ϕ , α (x) ϕ , β (x) ηµν , (141) 2 S h(1PM) (x) = n b , (146) and depends on the gauge function ϕα (1PM) (x) which is 0i can c3 iab a r2 formally given by Eq. (132); for an explicit expression see text below that equation. Let us recall that the metric (1PM) 2 M 3 nâb Mab h (x) = δij + δij . (147) perturbations and the gauge functions in (141) are time- ij can c2 r c2 r3 (1PM) α (1PM) independent: hαβ can , 0 = 0 and ϕ , 0 = 0. In summary of this Section, Eq. (124) with (127) - These expressions are in agreement with Eqs. (1) - (2) (129) and Eq. (133) with (134) - (136) represent the in [109]. It should be noticed thatn âb Mab = nab Mab metric perturbation in the second post-Minkowskian ap- because of the STF structure of the multipoles. proximation for one body at rest as function of the time-independent multipoles ML and SL. The mass- multipoles ML in (125) allow to describe an arbitrary shape and inner structure of the body, while the spin- 2. The 2PM terms of canonical metric perturbation multipoles SL in (125) allow to account for stationary currents of matter, like circulations of matter inside the body or stationary rotational motions of the body as a From Eqs. (134) - (136) one obtains the 2PM terms of whole. the canonical metric perturbation: 19

2 M 2 MM M M 3 18 9 h(2PM) (x) = − − 6 ab nˆ − ab cd δ δ + δ nˆ + nˆ + O c−6 , (148) 00 can c4 r2 c4 r4 ab c4 r6 5 ac bd 7 ac bd 2 abcd

(2PM) −5 h0i can (x) = O c , (149)

1 M 2 4 MM 15 32 12 h(2PM) (x) = δ +n ˆ + ab nˆ + δ nˆ − δ nˆ ij can c4 r2 3 ij ij c4 r4 2 ijab 7 ij ab 7 a (i j )b M M 75 90 27 25 + ab cd nˆ − nˆ δ + nˆ δ − nˆ δ δ c4 r6 4 ijabcd 11 ijac bd 11 abcd ij 84 ij ac bd 83 16 18 5 + nˆ δ δ + δ δ δ + nˆ δ − nˆ δ δ 42 ad bc ij 35 ac bd ij 11 acd(i j)b 21 a(i j)c bd 10 23 6 + δ δ nˆ − δ δ nˆ − δ δ δ + O c−6 , (150) 21 ci dj ab 42 b(i j)c ad 35 ad b(i j)c

where the details of the calculations are relegated to Ap- are given by Eqs. (118) and (120). The canonical met- pendices H and I; for various careful checks see text below ric perturbations depend on the canonical mass and spin Eq. (I10) in Appendix I. multipoles, ML and SL, which are functions of the retarded time s = t − |x| /c. These multipoles are given by Eqs. (C1) and (C2), allowing to account for arbitrary IX. SUMMARY shape, inner structure, oscillations, and rotational motions of the source of matter. The metric tensor gαβ (t, x) In this article four issues have been considered: in (151) represents the most general solution for the spa- 1. A coherent exposition of the Multipolar Post- tial region in the exterior of a compact source of matter. Minkowskian (MPM) formalism has been presented, 3. Furthermore, the metric of a stationary source of where we have focused on those results of the MPM for- matter has been considered, malism which are relevant for our investigations. Special 1 (1PM) 1 (1PM) care has been taken about the gauge transformation of gαβ (x) = ηαβ + G hαβ can (x) + G ∂ϕαβ (x) the metric tensor and metric density. It has been em- + G2h(2PM) (x) + G2∂ϕ(2PM)(x) + G2Ω(2PM)(x) phasized that the canonical piece of the metric tensor αβ can αβ αβ 3 and metric density is gauge-independent, hence is inde- +O G . (152) pendent of whether one uses the residual gauge transfor- (1PM) mation (22) or (25). The Lorentz covariant gauge trans- The canonical linear metric perturbation hαβ can is given formation and the general-covariant gauge transforma- by Eqs. (127) - (129) and the canonical post-linear met- (2PM) tion and how they are related to each other has been ric perturbation hαβ can is given by Eqs. (134) - (136) expounded in some detail. up to terms of the order O c−6, c−5, c−6. The gauge 2. The MPM formalism has been used in order to ob- (1PM) term ∂ϕ is given by Eq. (130), while the gauge tain the metric coeﬃcients in harmonic coordinates in αβ (2PM) (2PM) the post-linear approximation in the exterior of a com- terms ∂ϕαβ and Ωαβ are given by Eqs. (137) and pact source of matter, (141). The canonical metric perturbation depends on the canonical mass and spin multipoles, ML and SL, 1 (1PM) 1 (1PM) gαβ (x) = ηαβ + G hαβ can (x) + G ∂ϕαβ (x) which are time-independent. These multipoles are given by Eqs. (125) and (126), allowing to account for arbitrary + G2h(2PM) (x) + G2∂ϕ(2PM)(x) + G2Ω(2PM)(x) αβ can αβ αβ shape and inner structure as well as inner stationary cur- 3 +O G , (151) rents of the source of matter. The metric tensor gαβ (x) in (152) represents the most general solution for the spa- where x = (ct, x). The linear canonical metric per- tial region in the exterior of a stationary compact source (1PM) turbation hαβ can is given by Eqs. (109) - (111) and of matter. (2PM) 4. The spatial components of the canonical post-linear the post-linear canonical metric perturbation hαβ can is given by Eqs. (115) - (117) up to terms of the order metric perturbation are associated with an integration procedure: in (151) by the inverse d’Alembertian (F1) O c−6, c−5, c−6. The gauge term ∂ϕ(1PM) is given by αβ and in (152) by the inverse Laplacian (G1). That inte- (2PM) (2PM) Eq. (112), while the gauge terms ∂ϕαβ and Ωαβ gration procedure has been performed explicitly in (152), 20

ij where the first multipoles (monopole and quadrupole) are • δij = δ = diag (+1, +1, +1) is the Kronecker taken into account. The linear and post-linear metric co- delta. efficients are given by Eqs. (145) - (147) and (148) - (150), ijk respectively. • εijk = ε with ε123 = +1 is the three-dimensional The investigations are motivated by the rapid progress Levi-Civita symbol. in astrometric science, which has recently succeeded in αβµν • εαβµν = ε with ε0123 = +1 is the four- making the giant step from the milli-arcsecond level [14– dimensional Levi-Civita symbol. 16] to the micro-arcsecond level [17–21] in angular mea- αβ surements of celestial objects, like stars and quasars. • metric of Minkowskian space-time is ηαβ = η = A fundamental issue in relativistic astrometry concerns diag (−1, +1, +1, +1). the precise modeling of the trajectories of light signals emitted by some celestial light source and propagating • covariant components of metric of Riemann space- through the curved space-time of the solar system. The time are gαβ. light trajectories are governed by the geodesic equation, • contravariant components of metric of Riemann which implies the knowledge of the metric coefficients for αβ solar system bodies. Accordingly, interpreting the com- space-time are g . pact source of matter just as some massive body of ar- • the metric signature is (−, +, +, +). bitrary shape and inner structure, the post-linear metric coefficients allow to determine the light trajectory in the • g = det (gαβ) is the determinant of the covariant gravitational field of such a massive solar system body components of the metric tensor in the post-linear approximation. Thus far, the impact 1 αβγδ µνρσ of higher multipoles on the light trajectories in the post- that means: g = ε ε gαµ gβν gγρ gδσ. linear approximation is unknown. So the results of this 4! investigation are a fundamental requirement in order to • covariant and contravariant components of three- i 1 2 3 determine the impact of higher multipoles on the light vectors: ai = a = a , a , a . trajectory in the post-linear approximation. • n! = n (n − 1) (n − 2) ··· 2 · 1 is the factorial for positive integer (0! = 1). X. ACKNOWLEDGMENT • n!! = n (n − 2) (n − 4) ··· (2 or 1) is the double factorial for positive integer (0!! = 1). This work was funded by the German Research Foun- dation (Deutsche Forschungsgemeinschaft DFG) under • L = i1i2...il and M = i1i2...im are Cartesian multi- grant number 263799048. Sincere gratitude is expressed indices of a given tensor T , that means TL ≡ to Prof. Sergei A. Klioner and Prof. Michael H. Soffel for Ti1i2 . . . il and TM ≡ Ti1i2 . . . im , respectively. kind encouragement and enduring support. The author also wish to thank Dr. Alexey G. Butkevich, Prof. Las- • two identical multi-indices imply summation: A B ≡ P A B . zlo P. Csernai, Prof. Burkhard Kämpfer,PD Dr. Günter L L i1 . . . il i1 . . . il i . . . i Plunien, and Prof. Ralf Schützholdfor inspiring discus- 1 l sions and conversations about the general theory of rel- • triplet of spatial coordinates (three-vectors) are in ativity and astrometric science during recent years. Dr. boldface: e.g. a, b. Sebastian Bablok and Dipl.-Inf. Robin Geyer are kindly acknowledged for computer assistance. • the absolute value of three-vector is determined by p i j |a| = δija a .

• covariant components of four-vectors: aµ = Appendix A: Notation (a0, a1, a2, a3).

The notation of the standard literature [52, 54, 56] is • contravariant components of four-vectors: aµ = 0 1 2 3 used: a , a , a , a . ∂ • G is the Newtonian constant of gravitation. • ∂ = is partial derivative w.r.t. xi. i ∂xi • c is the speed of light in ﬂat space-time which equals i • f , i is partial derivative of f w.r.t. x . the speed of gravitational waves in a ﬂat back-

ground manifold. • ∂L = ∂a1 ... al denotes l partial derivatives w.r.t. xa1 . . . xal . • lower case Latin indices take values 1, 2, 3.

• f , a1 ... al denotes l partial derivatives of f w.r.t. • lower case Greek indices take values 0,1,2,3. xa1 . . . xal . 21

∂ α in [47, 56, 110–113]. As instructive examples of (B2) let • ∂α = is partial derivative w.r.t. x . ∂xα us consider the cases l = 2, l = 3, and l = 4: • f is partial derivative of f w.r.t. xα. , α 1 T = T(ij) − δij T(aa) , (B5) • f , µ1 ... µn denotes n partial derivatives of f w.r.t. 3 xµ1 . . . xµn . 1 df T = T(ijk) − δij T(kaa) + δik T(jaa) + δjk T(iaa) , • f˙ = is total time-derivative of f. 5 dt (B6) d2f 1 • f¨ = is double total time-derivative of f. T = T − δij T + δik T dt2 (ijkl) 7 (klaa) (jlaa) α α α ν 1 • A ; µ = A , µ + Γµν A is covariant derivative of ﬁrst − δil T(jkaa) + δjk T(ilaa) + δjl T(ikaa) + δkl T(ijaa) rank tensor. 7 1 αβ αβ α νβ β αν + δijδkl T(aabb) + δikδjl T(aabb) + δilδjk T(aabb) , • B ; µ = B , µ + Γµν B + Γµν B is covariant 35 derivative of second rank tensor. (B7)

• repeated indices are implicitly summed over (Ein- where the expressions in (B5) and (B6) were also given stein’s sum convention). by Eqs. (2.3) and (2.4) in [52]. Especially, the following Cartesian tensor is of primary importance, which just consists of products of unit three-vectors, Appendix B: Some useful relations of Cartesian tensors x x x n = i1 i2 ... il where r = |x| . (B8) L r r r The irreducible Cartesian tensor technique has been developed in [110–112] and is a very useful tool of the The symmetric trace-free part of the Cartesian tensor MPM formalism. Here we summarize some relevant re- (B8) reads lations of the Cartesian tensor technique. x x x nˆ = . (B9) The symmetric part of a Cartesian tensor TL is, cf. L r r r Eq. (2.1) in [56]: Using Eq. (A 20a) in [47], we present the explicit struc- 1 X T = T = A , (B1) ture of the following STF Cartesian tensors, (L) (i1...il) l! iσ(1)...iσ(l) σ 1 nˆ = n − δ , (B10) where σ is running over all permutations of (1, 2, ..., l). ab ab 3 ab The symmetric trace-free part of a Cartesian tensor TL ˆ 1 is denoted as TL and given by (cf. Eq. (2.2) in [56]) nˆ = n − (δ n + δ n + δ n ) , (B11) abc abc 5 ab c ac b bc a ˆ T = T = TL [l/2] nˆabcd = nabcd X 1 = alk δ(i i ...δi i Si , (B2) 1 2 2k−1 2k 2k+1...il) a1a1...akak − (δabncd + δacnbd + δadnbc + δbcnad + δbdnac + δcdnab) k=0 7 1 where [l/2] means the largest integer less than or equal + (δab δcd + δac δbd + δad δbc) , (B12) 35 to l/2, and SL ≡ T(L) abbreviates the symmetric part of tensor TL. The coeﬃcient in (B2) is given by which were also given by Eqs. (1.8.2) and (1.8.4) in [60]. Frequently, the following relations are needed, which con- l! (2l − 2k − 1)!! a = (−1)k . (B3) vert a non-STF tensor into a STF tensor, lk (l − 2k)! (2l − 1)!! (2k)!! l STF tensors vanish whenever two of their indices are n nˆ =n ˆ + δ nˆ , (B13) a L aL 2l + 1 a equal, l + 1 3 n nˆ = nˆ , (B14) X a aL 2 l + 1 L T = T = 0 , (B4) a=1 where (B13) has been given by Eq. (A 22a) in [47] (see because a summation of these indices is implied accord- also Eq. (2.7) in [52], Eq. (A7) in [113]), while (B14) is ing to Einstein’s sum convention; of course, the indi- given by Eq. (A 23) in [47] (see also Eq. (A.8) in [113]). vidual components of STF tensors do not vanish, e.g. Some explicit examples of (B13) are noticed which are of

T 6= 0. Further STF relations can be found relevance for our investigations, 22

1 2 n nˆ =n ˆ + (δ nˆ + δ nˆ ) − δ nˆ , (B15) a bc abc 5 ac b ab c 15 bc a 1 2 n nˆ =n ˆ + (δ nˆ + δ nˆ + δ nˆ ) − (δ nˆ + δ nˆ + δ nˆ ) , (B16) a bcd abcd 7 ab cd ac bd ad bc 35 bc ad cd ab bd ac 1 n nˆ =n ˆ + (δ nˆ + δ nˆ + δ nˆ + δ nˆ ) a bcde abcde 9 ae bcd ab cde ac bde ad bce 2 − (δ nˆ + δ nˆ + δ nˆ + δ nˆ + δ nˆ + δ nˆ ) , (B17) 63 be acd cd abe ce abd de abc bc ade bd ace 1 n nˆ =n ˆ + (δ nˆ + δ nˆ + δ nˆ + δ nˆ + δ nˆ ) a bcdef abcdef 11 af bcde ab cdef ac bdef ad bcef ae bcdf 2 − δ nˆ + δ nˆ + δ nˆ + δ nˆ + δ nˆ 99 bf acde cf abde df abce ef abcd bc adef + δbd nâcef + δbe nâcdf + δcd nâbef + δce nâbdf + δde nâbcf . (B18)

We recall thatn â = na but, nevertheless, we keep the Eqs. (5.33) and (5.35) in [52] and read notation in (B15) as is, in order to emphasize that, ac- +1 cording to the meaning of relation (B13), there are STF Z Z 3 tensors on the r.h.s. of each of these relations (B15) - ML = d x dz (B18). We also need the following relations, which are −1 specific cases of the general relation given by Eq. (2.13) i ij in [52]: × δl (z)x ˆL σ + al δl+1 (z)x îL σ˙ + bl δl+2 (z)x îjL σ¨ , (C1) 4 nab nîjcd =n îjabcd + (ˆnab +n ˆba) 11 +1 Z Z 12 a b 1 3 + nˆ δ δ + δ nˆ , (B19) SL = d x dz STFL 63 11 ab ijcd −1 2 × δ (z)x ˆ xj σk + c δ (z) xˆ σ˙ ks , nab nˆcd =n âbcd + (ˆnab +n ˆba) l L−1 iljk l l+1 iljk jsL 7 (C2) 2 a b 1 + δ + δab nˆcd . (B20) 15 7 where the integrals (C1) and (C2) run only over the finite Finally, we notice three-dimensional space of the compact source of matter, and where

1 l (2 l − 1)!! +1 ∂L = (−1) nˆL , (B21) Z r rl+1 (2 l + 1)!! l δ (z) = 1 − z2 with dzδ (z) = 1 ,(C3) l 2l+1l! l which agrees with Eq. (A 34) in [47]. −1 4 (2l + 1) al = − 2 , (C4) Appendix C: The STF mass-multipoles and c (l + 1) (2l + 3) spin-multipoles 2 (2l + 1) b = , (C5) l c4 (l + 1) (l + 2) (2l + 5) In the MPM formalism the solution of the gothic met- (2l + 1) c = − , (C6) ric is given in terms of irreducible symmetric and trace- l c2 (l + 2) (2l + 3) free (STF) Cartesian tensors. It is a fundamental result of the MPM formalism [47, 49, 52] that in the exterior of a and where source of matter the gothic metric (41) to any order in G depends on a set of only two kind of irreducible STF ten- T 00 + T kk T 0i σ = , σi = , σij = T ij , (C7) sors (Theorem 4.5 in [47], see also Eq. (3.1) and (3.2) in c2 c [49]), namely mass-type multipoles ML and current-type αβ multipoles SL [52, 56]. The explicit expressions for these with T being the energy-momentum tensor of the iso- multipoles up to terms of the order O G2 are given by lated system taken at the time-argument t − |x| /c + 23 z |x| /c, and a dot in (C1) and (C2) means partial deriva- By calculating the determinant of (D1) one finds that tive with respect to coordinate time. The mass-type mul- the determinant of the contravariant components of the tipoles ML and the spin-type multipoles SL are STF ten- gothic metric equals the determinant of the covariant sors, but we adopt the notation as frequently used in the components of the metric tensor [54, 57, 58, 60] (e.g. literature and do not write the multipoles with a hat, say Eqs. (D.67) and (D.68) in [57]) ML ≡ Mˆ L and SL ≡ SˆL. µν It should be noticed that the multipoles are functions det (g ) = det (gµν ) . (D7) of time, except the mass-monopole M, mass-dipole Mi, and spin-dipole Si, which are strictly conserved quan- The relations (D5) - (D7) imply that the determinant tities, that means M˙ = M˙ i = S˙i = 0. The system of the covariant components of the gothic metric equals may emit gravitational radiation which would change the the determinant of the contravariant components of the mass M and the mass-dipole Mi and the spin-dipole Si of metric tensor, the compact source of matter, but these effects occur at µν higher order beyond the 1PM and 2PM approximation. det gµν = det (g ) , (D8) Furthermore, if the origin of the spatial coordinate axes is located at the center-of-mass of the source, then the which can also be obtained by calculating the determi- mass-dipole vanishes, i.e. Mi = 0. nant of (D2) and by means of (D5). These relations (D7) A further note is in order about the mass-type multi- and (D8) allow to derive from (D1) and (D2) the follow- poles ML and current-type SL as given by Eq. (C1) and ing relations between the metric tensor and gothic metric Eq. (C2), respectively. Usually, for practical applications density, their explicit form as given by Eqs. (C1) and (C2) is not 1 needed. Instead, these multipoles can be related to ob- gαβ = gαβ , (D9) servables of the massive bodies of the solar system, and p−det (gµν ) can be determined by fitting astrometric observations. and

p µν Appendix D: Relations between metric tensor and gαβ = −det (g ) gαβ . (D10) gothic metric density

The contravariant and covariant components of the Appendix E: Derivation of Eqs. (107) and (113) gothic metric density are deﬁned by [54, 57, 58] (e.g. text below Eq. (3.506) in [58]) In what follows the relation (D10) between the covariant components of the metric tensor and the covariant αβ q αβ g = −det (gµν ) g , (D1) components of the gothic metric density is important, and p µν gαβ = −det (g ) gαβ . (E1) 1 gαβ = gαβ . (D2) Let us consider the evaluation of the determinant in the p−det (g ) µν r.h.s. of (E1). By taking the Minkowskian metric tensor The orthogonality relation of the metric tensor as factor in front, we rewrite (3) as follows, ασ α gµν = ηµσ Cν (E2) g gσβ = δβ , (D3) σ implies, subject to (D1) and (D2), the orthogonality where relation of the gothic metric density (e.g. text below ν ν ρν Eq. (3.506) in [58]), Cσ = δσ − h ηρσ . (E3)

ασ α g gσβ = δβ , (D4) The determinant in (E2) is calculated by means of the product law of determinants and the theorem in (31), which is sometimes called isomorphism identity. From ν (D3) one gets µν µσ ν Tr(ln Cσ ) det (g ) = det (η ) det (Cσ ) = − e , (E4) 1 µσ det (gµν ) = , (D5) where det (η ) = −1 has been taken into account. Using det (gµν ) (E4) one obtains while from (D4) one gets 1 µν ρσ 1 2 det (gµν ) = −1 + h + h η h η − h + O G3 , 1 2 µρ σν 2 det g = . (D6) µν det (gµν ) (E5) 24

µν where h = h ηµν . By inserting (E5) into (E1) one ob- which allow to determine the covariant components of tains by series expansion of the square root the gothic metric from the contravariant components of the gothic metric. Finally, by inserting (E11) into (E9) as 1 1 µν ρσ 1 2 g = 1 − h − h η h η + h g + O G3 . well as inserting (E11) and (E12) into (E10), one conﬁrms αβ 2 4 µρ σν 8 αβ (107) and (113), respectively. (E6)

Appendix F: Hadamard regularization of the inverse For the covariant components of the metric tensor gαβ we have (cf. Eq. (2)) d’Alembertian

1 (1PM) 2 (2PM) 3 The spatial components (96) of the post-linear terms gαβ = ηαβ + G h + G h + O G . (E7) αβ αβ of the gothic metric density as well as the spatial compo- The post-Minkowskian series expansion of the contravari- nents (117) of the post-linear terms of the metric tensor ant components of the gothic metric density, gαβ, is given are associated with an integration procedure, which is ab- −1 by Eqs. (3) and (41). What we also need is the post- breviated by the symbol FPB=0 R . In this Appendix Minkowskian series expansion of the covariant compo- some details of that integration procedure will be given. The symbol −1 denotes the inverse d’Alembertian nents of the gothic metric density, gαβ, which is defined R by acting on some function f, which is defined by (cf. Eq. (20)) 1 (1PM) 2 (2PM) 3 gαβ = ηαβ + G hαβ + G hαβ + O G . (E8) Z 3 0 0 −1 1 d x |x − x | 0 R f (t, x) = − 0 f t − , x , Here we emphasize that (E8) is a definition of the co- 4 π |x − x | c variant components of the perturbations of the gothic (F1) metric density. That means, the relations between the contravariant and covariant components of the perturba- for notational conventions see also text below Eq. (3.4) in tions of the gothic metric density follow from the isomor- [47]. The inverse d’Alembert operator (F1) is standard in phism identity (D4) of the gothic metric density. These the literature [47–49, 51, 53, 73–75]. As explained in de- relations are given by Eqs. (E11) - (E12) in 2PM approx- tail in the original work of the MPM formalism [47], the imation. Inserting (E7) and (E8) into (E6) and equating integral in (F1) is not well-defined in general because, de- pending of the behavior of function f, the integral might the powers of the gravitational constant yields for the 0 0 0 terms proportional to G1: become singular at r → 0, where r = |x |. As it has already been described in the original work of the MPM (1PM) 1 formalism [47], the reason for this difficulty is caused by h(1PM) = h − h η , (E9) αβ αβ 2 (1PM) αβ the fact that the gothic metric density (94) - (96) as well as the metric tensor (115) - (117) are only valid in the µν where h(1PM) = h(1PM) ηµν . Similarly, for the terms pro- exterior of the source of matter, while the integration of portional to G2 one obtains: the inverse d’Alembertian extends over the entire three- dimensional space, hence includes the inner region of the (2PM) 1 1 (1PM) matter source, where the multipole decomposition be- h(2PM) = h − h η − h h αβ αβ 2 (2PM) αβ 2 (1PM) αβ comes infinite at the origin. This issue is of course a 1 2 1 µν ρσ pure mathematical problem and not a physical one. + h η − h η h η η , 8 (1PM) αβ 4 (1PM) µρ (1PM) σν αβ A way out of this problem is found by the fact that 0 (E10) the limit r → 0 is impossible because each real body is of finite size while the gothic metric and the metric µν tensor are strictly valid only in the exterior of the body. where h(2PM) = h(2PM) ηµν . However, because the MPM formalism determines the contravariant components of Therefore, the inverse d’Alembert operator is replaced by the gothic metric, we have to express the relations (E9) the Hadamard regularized inverse d’Alembert operator, and (E10) solely in terms of the contravariant compo- FP −1f (t, x) nents of the gothic metric. From the above mentioned B=0 R Z 0 B 3 0 0 isomorphism identity of the gothic metric density (D4) 1 r d x |x − x | 0 = − lim 0 f t − , x , and from the series expansion (E8) as well as the series B→0 4 π r0 |x − x | c expansion (41) (with Eq. (3)), we obtain the following (F2) relations, 0 B (1PM) µν where a factor (r /r0) is imposed and where B ∈ C is hαβ = h(1PM) ηαµ ηβν , (E11) some complex number and r0 is an auxiliary real con- (2PM) µν µν ρσ stant with the dimension of a length. The abbreviation hαβ = h(2PM) ηαµ ηβν + h(1PM) h(1PM) ηµα ηρν ησβ , FP denotes Hadamard’s partie finie of the integral. If (E12) the real part of B, denoted by < (B), is large enough, 25

0 0 0 then all singularities at r = 0 are cancelled. So the pro- where r = |x | and r0 is an auxiliary real constant with cedure to determine the finite part (FP) consists of three the dimension of a length, and B ∈ C is some complex consecutive steps: number. Because there is no time-dependence, the integration procedure in (G2) is considerably simpler than (i) computation of the integral (F2) with suffi- (F2). ciently large real part of B, According to the 2PM metric perturbations in (134) and (136), we need to determine the following integral, (ii) inserting the limits of integration, Z 3 0 0 B 0 (iii) performing the limit B → 0. −1 nˆL 1 d x r nˆL FPB=0 ∆ k = − 0 k , r 4 π |x − x | r0 (r0) The final results of that procedure are equivalent to (G3) Hadamard’s technique of partie finie [114]; for mathematical rigor of Hadamard’s procedure we refer to Sec- 0 where the abbreviated notationn ˆL means tion 3 in [47] and the article [115], where the approach x0 x0 . . . x0 has been described in specific detail. 0 0 0 0 nˆL ≡ nˆL (ϕ , ϑ ) = . (G4) In many subsequent investigations of the MPM for- (r0)l malism that approach for determining the finite part has been applied [49, 51, 53, 99–101]. In particular, we refer The integral (G3) is for sufficiently large values of the to the important relations (4.24) in [49] or Eq. (A.11) in real part < (B) of the complex number B well-defined. In [53], which are very useful in order to perform that inte- order to determine that integral the following expansion gration procedure, where these effects occur in associa- of the denominator is used (cf. Eq. (8.188) in [120]) tion with gravitational radiation (tail effects, retardation  ∞ 0 m  effects, divergencies at spatial infinity, etc.) which have 1 X r 0  Pm(cos θ) if r > r  been elaborated in [47, 48, 51, 53, 56, 75].  r r  1  m=0  Hadamard regularization leads to consistent results in = , |x − x0| ∞ different approaches up to 2.5PN order. Later it has been  1 X r m   P (cos θ) if r0 > r  discovered that from the 3PN order on the Hadamard  r0 m r0  concept is not sufficient and the gauge invariant dimen- m=0 sional regularization approach is introduced. This ap- (G5) where Pm are the Legendre polynomials and θ is the an- proach and its implementation in the MPM formalism 0 was a serious work over a longer period of time [116– gle between x and x . By inserting (G5) into (G3) one 119]. In this investigation we are interested in the met- encounters the following angular integration −6 −5 −6 ric up to terms of the order O c , c , c and will 2π π not consider that specific issue of the MPM formalism. Z Z I = dϕ0 dϑ0 sin ϑ0 nˆ0 (ϕ0, ϑ0) P (cos θ) (G6) But it should be kept in mind that for higher orders of L m the post-Minkowskian or post-Newtonian expansion the 0 0 Hadamard concept has to be replaced by the gauge in- which deserves special attention. The addition theorem variant dimensional regularization. for Legendre polynomial states (cf. Eq. (8.189) in [120])

m 4π X P (cos θ) = Y (ϕ, ϑ) Y ∗ (ϕ0, ϑ0) , Appendix G: Hadamard regularization of the inverse m 2m + 1 mn mn Laplacian n=−m (G7) ∗ where Ymn and Ymn are the spherical harmonics and com- In this Section Hadamard’s concept for the case of plex conjugated spherical harmonics, respectively. The time-independent integrals will be considered in some spherical harmonics can be expanded in terms of the STF more detail. In case of stationary sources, the 2PM met- tensor in (B9), which reads (cf. Eq. (2.11) in [56]) ric perturbations in (134) and (136) are associated with an inverse Laplace operator, ˆ mn Ymn (ϕ, ϑ) = YM nˆM (ϕ, ϑ) , (G8) 1 Z d3x0 ˆ mn ∆−1f (x) = − f (x0) , (G1) where the coeﬃcients YM are independent of the an- 4 π |x − x0| gles ϕ and ϑ. Their explicit expressions are given by Eq. (2.12) in [56] or by Eq. (2.20) in [113], but they are where Hadamard’s regularization of the inverse Laplacian not needed here, because we use the following relation is given by (cf. Eq. (2.23) in [113]),

−1 m FPB=0 ∆ f (x) X (2m + 1)!! Yˆ mn Y ∗ (ϕ0, ϑ0) = nˆ0 (ϕ0, ϑ0) . Z 0 B 3 0 M mn 4 π m! M 1 r d x 0 n=−m = − lim 0 f (x ) , (G2) B→0 4 π r0 |x − x | (G9) 26

By inserting (G7) - (G9) into (G6) one obtains for the integration constant r0 = ∞ in the second line do not angular integration contribute and one arrives at

−1 nˆL (2m − 1)!! FPB=0 ∆ I = nˆM (ϕ, ϑ) rk m! B 2π π nˆ r 1 1 Z Z = L r2−k − . 0 0 0 0 0 0 0 0 0 2l + 1 r B − l − k + 2 B + l − k + 3 × dϕ dϑ sin ϑ nˆL (ϕ , ϑ )n ˆM (ϕ , ϑ ) .(G10) 0 0 0 (G16) The angular integration (G10) yields (cf. Eq. (2.5) in The limit B → 0 yields finally [56]) −1 nˆL 1 nˆL FPB=0 ∆ k = k−2 , (G17) 2π π r (k + l − 2) (k − l − 3) r Z Z 0 0 0 0 0 0 0 0 0 4 π m! which is meaningful for k ≥ 3 as well as k 6= l + 3; note dϕ dϑ sin ϑ nˆL (ϕ , ϑ )n ˆM (ϕ , ϑ ) = δlm . (2m + 1)!! that always l ≥ 0. The solution of the integral in (G17) 0 0 (G11) is a specific case of the integrals given by Eqs. (A.11) Inserting (G11) into (G10) yields finally for the angular and (A.16) in [53], respectively, and has been presented integration in (G6) the following result (cf. Eq. (B.3) in within several investigations, for instance by Eq. (3.9) in [121]), [47] and by Eq. (3.9) in [121] and by Eq. (3.42) in [122]. One might wonder about the global sign of the solu- 2π π Z Z tion (G17). For instance, if one considers the case l = 0 0 0 0 0 0 0 0 and k ≥ 4, then (G17) is a positive-valued expression, I = dϕ dϑ sin ϑ nˆL (ϕ , ϑ ) Pm(cos θ ) irrespective of the negative-valued integral in (G3). In 0 0 order to understand the global sign in (G17), one has 4 π = nˆL δlm , (G12) to realize that the partie finie procedure in (G3) implies 2m + 1 that the lower integration constant r0 = 0 in the first line where in (G15) does not contribute. Stated differently, in case of l = 0 and k ≥ 4 the partie finie procedure eliminates x nˆL ≡ nˆL (ϕ, ϑ) = l . (G13) a (infinitely) large negative term from the entire expres- (r) sion, so that the final result becomes positive-valued in It is important to realize that relation (G12) necessitates the case under consideration. the irreducible STF tensorn ˆL as integrand. If the integrand would not be of irreducible STF structure, then Appendix H: The proof of Eqs. (148) relation (G12) would not be valid. Accordingly, by means of (G5) and owing to relation (G12) one obtains for the integral (G3), In this Appendix some details of the computation of the matrix coefficients in Eqs. (148) are given. Ac- nˆL counting for monopole and quadrupole, one gets from FP ∆−1 B=0 rk Eqs. (134) r Z 0 B 0 l k 2 nˆL 1 2 r r 1 2 M ∂ M = − dr0 (r0) (2PM) ab ab −6 0 h00 can (x) = − + + O c . (H1) 2l + 1 r r0 r r c4 r 2! r 0 ∞ By means of (B21) one obtains Z 0 B l k nˆL 0 0 r r 1 − dr r . (G14) 2 0 0 (2PM) 2 M nâb 2l + 1 r0 r r h (x) = − + 3 MM r 00 can c4 r2 ab r4 The radial integration yields 2 9 nˆ nˆ − M M ab cd + O c−6 . (H2) c4 4 ab cd r6 −1 nˆL FPB=0 ∆ rk The last term is rewritten in the form Mab Mcd nâb nˆcd = " #r0=r Mab Mcd nabcd, then relations (B12) and (B10) are ap- nˆ 1 B 1 (r0)B+l−k+3 L plied; note Mab δab = Mcd δcd = 0. One arrives at = − l+1 2l + 1 r0 r B + l − k + 3 2 r0=0 (2PM) 2 M 6 nâb 0 h (x) = − − MM " #r =∞ 00 can 4 2 4 ab 4 nˆ 1 B (r0)B−l−k+2 c r c r − L rl . (G15) 2l + 1 r0 B − l − k + 2 3 1 δacδbd 18 1 δacnˆbd r0=r − M M − M M 5 c4 ab cd r6 7 c4 ab cd r6 For < (B) + l − k + 3 > 0 > < (B) − l − k + 2, the lower 9 1 nâbcd 0 − M M + O c−6 . (H3) integration constant r = 0 in the first line and the upper 2 c4 ab cd r6 27

Appendix I: The proof of Eqs. (150) By means of (B21) one obtains

In this Appendix some details of the computation of the matrix coeﬃcients in Eqs. (150) are given. Account- ing for the monopole and quadrupole term, one obtains from Eqs. (136)

2 M ∂ M 2 h(2PM)(x) = + δ + ab ab ij can c4 ij r 2! r 4 M ∂ M M ∂ M − FP ∆−1 ∂ + iab ab ∂ + jab ab c4 B=0 i r 2! r j r 2! r + O c−6. (I1)

2 M 2 nˆ 9 nˆ nˆ h(2PM)(x) = + δ + 3 MM ab + M M ab cd ij can c4 ij r2 ab r4 4 ab cd r6 4 n n nˆ 225 nˆ nˆ − FP ∆−1 M 2 ij + 15 MM (i j)ab + M M iab jcd + O c−6 . (I2) c4 B=0 r4 ab r6 4 ab cd r8

Before relation (G17) can be applied, one has to express expression in terms of irreducible STF tensors, the enumerator in the second line of (I2) in terms of irreducible STF tensors. The ﬁrst term of the second line of Mab Mcd nˆiab nˆjcd (I2) is rewritten in terms of irreducible STF tensors by means of relation (B10), 8 12 = M M nˆ + nˆ δ + nˆ δa δb ab cd ijabcd 11 ab 63 1 4 2 + M M δ nˆ + nˆ δ + δ a δb 1 7 ab cd ij abcd 7 ab 15 n =n ˆ + δ , (I3) ij ij 3 ij 2 4 2 + M M δ nˆ + nˆ δ + δ a δb 7 ab cd ic abjd 7 ab 15 4 4 2 − M M δ nˆ + nˆ δ + δ a δb 35 ab cd jc abid 7 ab 15 while for the second term of the second line of (I2) one 2 1 2 4 obtains − M M δ nˆ + nˆ δ + nˆ δ − nˆ δ . 5 ab cd ib jacd 7 cd aj 7 jc ad 35 ac jd (I5)

1 6 Taking account for the STF structure of the quadrupoles, Mab n(i nˆj)ab = Mab nîjab + δij nâb + δa (inˆj )b , 7 35 one may combine the second term and the fourth term (I4) of the last line, but here we keep these terms as given. The r.h.s. of Eq. (I5) has now been expressed in terms of irreducible STF tensors. But the structure of these terms is presented in a rather compact notification. A more explicit form is arrived with the aid of relations (B5) and (B7), by means of which one obtains where relation (B16) and the STF structure of the multipole Mab has been used. The last term in the second line 1 M M nˆ δ = + M M nˆ δ of (I2) is expressed in terms of irreducible STF tensors ab cd ab 2 ab cd acij bd by means of relations (B11) and (B16) as well as (B19) 3 1 + M M nˆ δ − M M nˆ δ , (I6) and (B20). After some steps one obtains the following 14 ab cd acd (i j ) b 14 ab cd abcd ij 28

1 M M nˆ δaδb = + M M nˆ δ δ ab cd 6 ab cd ij ac bd 10 43 + Mab Mcd nˆc (i δj ) a δbd + Mab Mcd nˆab δic δjd a b 21 210 Mab Mcd δ = +Mab Mcd δad δbi . (I9) 2 4 − M M δ δ nˆ − M M δ δ nˆ , (I7) 21 ab cd ij bd ac 21 ab cd ic bj ad

1 Mab Mcd nâb = + Mab Mcd nâi δdb By inserting (I3) - (I5) into (I2) by taking into account 2 1 1 the relations (I6) - (I9) as well as the solution for the + M M nˆ δ − M M nˆ δ , (I8) integrals (G17), one finally arrives at 2 ab cd ad bi 3 ab cd ab di

1 M 2 4 h(2PM)(x) = δ +n ˆ ij can c4 r2 3 ij ij 15 1 nˆ 32 δ nˆ 12 δ nˆ + MM ijab + ij MM ab − MM a (i j )b 2 c4 ab r4 7 c4 ab r4 7 ab r4 M M 75 90 27 25 + ab cd nˆ − nˆ δ + nˆ δ − nˆ δ δ c4 r6 4 ijabcd 11 ijac bd 11 abcd ij 84 ij ac bd 83 16 18 5 + nˆ δ δ + δ δ δ + nˆ δ − nˆ δ δ 42 ad bc ij 35 ac bd ij 11 acd(i j)b 21 a(i j)c bd 10 23 6 + δ δ nˆ − δ δ nˆ − δ δ δ + O c−6 . (I10) 21 ci dj ab 42 b(i j)c ad 35 ad b(i j)c

Several careful checks have been performed in order to be The gauge transformation (35) leads, up to terms of the certain about the correctness of these metric coefficients. order O G3, to For instance, one may see that the terms proportional to M 2 are in agreement with the same terms of Eq. (25) in αβ αβ 1 αβcan αβ g (x) = η − G h(1PM) (x) − ∂ϕ (x) [24]. Furthermore, it has been checked that inserting the (1PM) 2 αβcan αβ αβ gothic metric coefficients (J3) - (J8) in Appendix J into −G h(2PM) (x) − ∂ϕ(2PM) (x) − Ω(2PM) (x) , (J2) (113) yields the same metric coefficients as presented by Eqs. (148) - (150). In addition, each metric coefficient where the gauge terms are time-independent and for- has been determined in different ways and assisted by mally given by Eqs. (90) and Eqs. (100) and (105), re- the computer algebra system Maple [123]. spectively. Accounting for the monopole and spin and quadrupole terms, one arrives at

Appendix J: Monopole and spin and quadrupole 00 can M Mab h = 4 + 6 nˆ , (J3) terms of 2PM gothic metric for stationary sources (1PM) c2r c2 r3 ab

0i can 2 S In case of stationary source the post-linear gothic met- b h(1PM) = − 3 iab na 2 , (J4) ric (4) simpliﬁes as follows, c r ij can αβ αβ 1 αβ 2 αβ 3 h(1PM) = 0 , (J5) g (x)=η − G h(1PM) (x) − G h(2PM) (x) + O G . (J1) for the linear coeﬃcients, and 29

2 00 can M MM 63 M M h = 7 + 21 ab nˆ + ab cd n + O c−6 , (J6) (2PM) c4r2 c4 r4 ab 4 c4 r6 abcd

0i can −5 h(2PM) = O c , (J7)

2 ij can M 1 MM 15 1 h = n + ab n + δ n − 6 n δ + δ δ (2PM) c4 r2 ij c4 r4 2 ijab 2 ij ab a(i j)b ai bj M M 75 90 9 25 + ab cd nˆ − nˆ δ + nˆ δ − nˆ δ δ c4 r6 4 ijabcd 11 ijac bd 44 abcd ij 84 ij ac bd 29 11 18 5 + nˆ δ δ + δ δ δ + nˆ δ − nˆ δ δ 42 ad bc ij 70 ac bd ij 11 acd(i j)b 21 a(i j)c bd 10 23 6 + δ δ nˆ − δ δ nˆ − δ δ δ + O c−6 , (J8) 21 ci dj ab 42 b(i j)c ad 35 ad b(i j)c

for the post-linear coefficients. These gothic metric coef- tioned that these gothic metric coefficients (J3) - (J8) ficients in (J3) - (J8) have been calculated by the same have also been presented by Eq. (16) in [124]; the in- 6 approach as presented in the previous Appendix I; the correct coefficient z0 of Eq. (16) in [124] has later been last term in (J6) and the first line in (J8) are not ex- corrected by Eq. (21) in [125]. pressed in terms of irreducible STF multipoles, but it could be done by means of relations (B10) and (B12). The quadrupole-quadrupole gothic metric density for a time-dependent compact source of matter has been determined in [53] which allows to deduce the gothic metric References coefficients (J3) - (J8). Furthermore, it should be men-

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