Analytical coordinate time at first post-Newtonian order

In this letter, we exploit the Damour-Deruelle solution to derive the analytical expression of the coordinate time in terms of the polar angle. This formula has advantageous applications in both pulsar timing and gravitational-wave theory.

Introduction. -During the past few decades, General Relativity (GR) has been subjected to a number of experiments beyond the weak-field regime of the Solar System. A crucial astrophysical testbed of GR is represented by compact binary systems. However, due to the nonlinear geometric structure of GR, the dynamics of a binary system is contained in the gravitational field equations and it is governed by retarded-partial-integro differential equations [1, 2,3]. These issues complicate subsequent calculations, making it difficult to test GR against observations.
The aforementioned criticality has been solved via approximation strategies and by adopting special coordinate systems (e.g., harmonic coordinates), at the price of losing the general covariance of the GR theory. Furthermore, the gravitational source is assumed to be post-Newtonian (PN), i.e., slowly moving, weakly self-gravitating, and weakly stressed [1,2]. This allows us to apply in the near zone the PN approximation method [4], which produces instantaneous potentials with no retardation effects [1,2]. In addition, assuming that the bodies are well separated, we can deal with test particles and ordinary differential equations [2,3]. The bodies' motion occurs in the Newtonian absolute Euclidean space, in which we add the PN corrections. Moreover, these dynamical equations preserve their relativistic nature, since they remain invariant under (a) vittorio.defalco-ssm@unina.it (b) emmanuele.battista@univie.ac.at, emmanuelebattista@gmail.com (c) john@ia.forth.gr a global PN-expanded Lorentz transformation [2]. The PN method has been extensively employed to investigate several aspects of the relativistic two-body dynamics, including the back-reaction of gravitational radiation in binary pulsars [5,6,7,8] and the direct detection of gravitational waves (GWs) from coalescing compact binaries [2]. The PN formalism has also played a central role in precision tests of gravity theories [9] and neutron star mass measurements in binary pulsars [10]. These assessments extensively use the quasi-Keplerian analytical solution of Damour and Deruelle, describing the quasi-elliptic 1PN-accurate GR motion of a two-body system [11].
In this letter, we start from the latter reference to derive, for the very first time, an analytical expression of the coordinate time, t, in terms of the polar angle, ϕ, at the 1PN level, namely the function t = t(ϕ). The motivation for this work takes its origin from the need to speed up our numerical simulations for computing the gravitational waveforms and the fluxes from inspiralling binaries framed in GR (and, as recently proposed in the literature, also in Einstein-Cartan theory [12]). As the two-body dynamics can be readily analysed via the relative distance R = R(ϕ) [11], then all dynamical quantities can depend on ϕ and hence it is natural to invest efforts for determining t(ϕ).
This analytical formula can be used to replace numerical derivation schemes currently exploited in pulsar timing software such as TEMPO [5], TEMPO2 [13], and PINT [14]. Similarly, it may also be exploited in coherent pulsar search algorithms [15,16], in which a very large number p-1 arXiv:2301.02472v2 [gr-qc] 27 Mar 2023 of trial timing solutions must be generated and compared with the data, in a manner similar to template matching in ground-based GW astronomy.
Let us define the relative position, R := r 1 (t) − r 2 (t), and the relative velocity, Then, likewise the Newtonian case, the description of the motion can be simplified by choosing a fixed orthogonal reference frame (x, y) centered in the barycenter of the binary system, which, without loss of generality, is supposed to be static. We also introduce ϕ as the angle between R and the x-axis, which is measured counterclockwise (see Fig. 1). In this frame, one can write where R := |R| and V := |V |. In the GR framework, we consider the 1PN-accurate Damour-Deruelle treatment, where the total conserved energy E = E 0 + 1 c 2 E 1 +O c −4 and the angular momentum where It is important to note that the first integrals (2) can be easily calculated once the initial conditions are assigned. Therefore, at the initial time t in we have where 0 < β ≤ 1 with β = 1 corresponding to circular orbits. Having defined the parameters From the above equations, it is clear that the Damour and Deruelle solution links the relative radius R with the polar angle ϕ.

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Analytical expression of time. -We start from Eq. (2.16) in Ref. [11], which naturally expresses the time t in terms of the polar angle ϕ up to O(c −4 ) corrections, where Substituting the PN expansions (2), (4), and (5) into Eq. (6), we expand the ensuing expression in power series of 1/c → 0 and up to the first order. In this way, after a lengthy calculation, we obtain the PN-expanded differential equation for the coordinate time, which, up to O(c −4 ) terms, reads as This amounts to solve the integral 1 t = g(Kφ)dφ, whereφ := ϕ − ϕ in and g(Kφ) := g 1 (Kφ) + g 2 (Kφ) + g 3 (Kφ), with The coefficients A 1 , A 2 , A 3 occurring in Eq. (11) can be obtained by substituting the expression (5b) into Eq. (9). In this way, we end up with If we integrate Eq. (9) numerically, we get a monotonically increasing function t(ϕ) (see Fig. 2).
1 Since the two-body equations of motion in GR constitutes an autonomous dynamical system (i.e., independent of the time t) [11], we can assume t in = 0 without loss of generality. It is useful to write Eq. (5a) equivalently as where with B 1 ≥ B 2 ≥ 0 (cf. Eq. (4c)). Let us also define The solution t(ϕ) is obtained by adding the above three integrals (cf. Eqs. (9)-(11)), which we now solve separately.
For each integral, we use the same integration strategy: we first make the substitution x =Kφ and then convert the trigonometric functions into polynomials through Weierstrass substitution τ = tan(x/2), leading to the following transformations: For the function f 1 , we get where we have set (cf. Eq. (14)) For f 2 , we have the following result: Finally, the expression of f 3 is given by The plot of the function f (ϕ) := f 1 (ϕ) + f 2 (ϕ) + f 3 (ϕ) is shown in Fig. 3. It is clear that f (ϕ) is discontinuous, since its different periodic branches are not smoothly connected to each other; furthermore, it does not reproduce the behavior of Fig. 2. Therefore, we have to add to it an "accumulation function", which takes into account that time is a monotonically increasing function, while trigonometric functions and their related information are reset after each period. To this end, we define the following characteristic period and then the accumulation function reads as where n ∈ N. For a generic ϕ, the related value of n can be calculated considering q := [(φ − P ϕ )/P ϕ ], where [·] stands for the integer part of a number. Thus, if q is an even number, then n = (q + 2)/2; on the other hand, if q is an odd number, then n = (q + 1)/2. Therefore, we can conclude that the correct analytical form of t(ϕ) is At the 1PN level, this formula can be used to replace numerical schemes due to its simpler and faster implementation with no approximation costs. Importantly, our method can be generalized and extended to higher PN orders, where an analytical formula (in whatever form is presented) may improve speed and accuracy.
We stress that the variable ϕ naturally stems from the PN equations of motion for conservative binary systems. In fact, these are usually given as second-order differential equations of position with respect to the time t. Due to the conservation of energy and angular momentum, they can be divided into two coupled first-order ordinary differential equations of the radius R and the angle ϕ with respect to the time t. Then, it is possible to obtain an orbital equation, where we derive R with respect to ϕ. This normally allows us to determine R(ϕ), rather than R(t). This constitutes a very fundamental step that then permits to find out t(ϕ). We would like to underline again that the angle ϕ assumes a fundamental role in dealing with the two-bodies' equations of motion both in Newtonian physics and in GR. Indeed, it replaces the time t and in this way all dynamical quantities can be expressed in terms of ϕ, offering thus the possibility to achieve analytical results.
It is worth noting that pulsar timing recipes often make use of the inverse of Eq. (23), i.e., the function ϕ(t), whose calculation still requires a numerical inversion. Here, our simplified solution can be exploited as a mean of speeding up such computations. Our finding can be generally applied for timing the orbital period of compact binary systems during the inspiral phase [17]. In addition, this formula could be extremely useful for extracting fundamental information from the holy grail binary system, constituted by a pulsar and a black hole, as well as for providing significant tests of gravity [18]. * * * V.D.F. and E.B. are grateful to Gruppo Nazionale di Fisica Matematica of Istituto Nazionale di Alta Matematica for support, to Caterina Tiburzi for continuous help, and to Luigi Stella for illuminating discussions.