The first law of binary black hole scattering
Riccardo Gonzo Jack Lewis Adam Pound




rgonzo@ed.ac.uk
Higgs Centre for Theoretical Physics, School of Physics and Astronomy, 
University of Edinburgh, EH9 3FD, UK
J.E.Lewis@soton.ac.uk 
School of Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom
A.Pound@soton.ac.uk 
School of Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom




§ ABSTRACT 
In the last decade, the first law of binary black hole mechanics played an important unifying role in the gravitational two-body problem. More recently, binary black hole scattering and the application of high-energy physics methods have provided a new avenue into this classical problem. In this Letter, we connect these two themes by extending the first law to the case of scattering orbits. We present derivations based on classical S-matrix, Hamiltonian, and pseudo-Hamiltonian methods, the last of which allows us to include dissipative effects for the first time.  Finally, a “boundary to bound” map links this first law to the traditional bound-orbit version. Through this map a little-known observable for scatter orbits, the elapsed proper time, is mapped to the Detweiler redshift for bound orbits.





The first law of binary black hole scattering
    Adam Pound
    September 9, 2024
=============================================



Introduction.—The discovery of gravitational waves (GWs) from compact binary systems opened a new chapter in astronomy. Given the enhanced sensitivity and expanded frequency range of future GW detectors, we expect a dramatic increase in the number and variety of detectable compact binary sources <cit.>. Increasingly accurate waveform models will be needed to detect and analyse these sources <cit.>, calling for the development of new tools to study the classical two-body problem. 

Motivated by GW modeling, a host of techniques have been developed to solve the two-body problem in general relativity, including numerical relativity, which numerically solves the fully nonlinear Einstein equations <cit.>; gravitational self-force (GSF) theory, a perturbative method that applies when one body is much smaller than the other <cit.>; and Post-Newtonian (PN) and Post-Minkowskian (PM) theory, weak-field expansions that apply when the two bodies are widely separated <cit.>. Historically, focus has been on the bound, inspiralling systems that are the dominant sources for GW detectors. However, the case of hyperbolic, scattering encounters is now of great interest: it is now known that data for scattering orbits can inform bound-orbit models using the effective one-body framework <cit.> or through an analytic continuation from scattering to bound observables <cit.>, spurring the development of new particle physics tools <cit.> that have enabled analytical computations of the two-body scattering Hamiltonian and related observables at high PM order <cit.>. 

In the bound case, synergies between different methods have consistently helped drive progress <cit.>. An important tool in those synergies has been the first law of binary black hole (BH) mechanics <cit.>, which describes how a binary system responds to variations of its parameters (see also <cit.>).  This law has played an important role in the most accurate GSF waveform model <cit.> and in utilizing GSF results within PN, effective one body, and numerical relativity calculations <cit.>; see <cit.> for a review. For spinless particles, the binary's response to variations is determined by a basis of observables ℬ^< consisting of the periastron advance ΔΦ, the radial frequency Ω_r, and the averaged Detweiler redshift ⟨ z ⟩ <cit.>. 

To date, a first law for scattering scenarios has not been derived. In this Letter, we establish such a law and find the corresponding basis of scattering observables ℬ^>. Two of these observables are well studied: the deflection angle χ and the time delay Δ t. We complete the basis with a third observable: the elapsed proper time Δτ. Our approach is based on a pseudo-Hamiltonian formulation of GSF theory <cit.>. This allows us to include dissipative contributions in the first law, unlike all previous formulations for bound orbits. By comparing to bound-orbit formulations, we also establish a novel analytic continuation between the elements of the scattering ℬ^> and bound ℬ^< bases of observables.

Finally, we link our calculations to high-energy physics methods, proving that the exponential representation of the classical S-matrix <cit.> provides a generating functional for the basis of scattering observables and deriving a first law from a PM Hamiltonian.




  
Conventions We use geometric units with G=c=1 and the (-+++) metric signature.

First law in the probe limit.—In the GSF approach, the smaller body (of mass m_1) is treated as a point-particle perturbing the spacetime of the large body, which we take to be a Schwarzschild BH of mass m_2. We first consider the probe limit (0SF order), in which the particle moves on a geodesic of the Schwarzschild metric g_αβ^Schw. The particle's motion is governed by the geodesic Hamiltonian H_0 = (1/2) g^μν_Schw p_μ p_ν, where p_α = m_1 g_αβ^Schwd x^β/dτ is the particle's 4-momentum and τ is its proper time. Assuming, without loss of generality, that the motion lies on the equatorial plane θ = π/2, we label the position of the particle with x^α(τ)=(t(τ), r(τ), π/2, φ(τ)). Because of Schwarzschild's Killing symmetries, the particle's energy and angular momentum E = -p_t,0 and  L= p_ϕ,0 are conserved (here and below, a subscript 0 indicates the on-shell geodesic value). We now consider unbound geodesic orbits that begin and end at r=∞; such orbits have E>m_1 and L > L_crit(E), where L_crit(E) is a critical value of the angular momentum <cit.>.

Following Carter's application of Hamilton-Jacobi theory <cit.>, we use the constants of motion P_i=(m_1,E,L) as canonical momenta and transform to canonical coordinates (X^i,P_i) using the type-2 generating function

    W(t,r,φ;P_i)    = -E t + Lφ + I_r,0(r;P_i) , 
    
        I_r,0(r;P_i)    = ∫_r_m^r d r   p_r,0(r;P_i) ,

where r_m(P_i) is the geodesic's minimum radius (i.e., closest approach to the BH). g^μν_ Schwp_μ,0 p_ν,0 = -m_1^2 implies 

    p_r,0(r;P_i) = √(E^2 r^4 -r (r-2 m_2) (L^2+m_1^2 r^2))/r (r -2 m_2) .

In the coordinates (X^i,P_i), where X^i=∂ W/∂ P_i, the Hamiltonian is simply its on-shell value, H_0=-m_1^2/2, meaning that Hamilton's equations become <cit.>

    m_1 d X^i/dτ=∂ H_0/∂ P_i = -m_1δ^i_1 .

Therefore X^2 and X^3 are constants, while X^1 is linear in τ. Since X^i=∂ W/∂ P_i, this implies

    ∂ W/∂ E|_out   = ∂ W/∂ E|_in ,      ∂ W/∂ L|_out = ∂ W/∂ L|_in , 
    τ_out - τ_in   = - [∂ W/∂ m_1|_out - ∂ W/∂ m_1|_in] ,

where “in” and “out” denote the initial, incoming state and final, outgoing state. 

Equations (<ref>) and (<ref>) imply that the total changes in coordinate time, azimuthal angle, and proper time between initial and final states are

    Δ t_0 = t_out - t_in = ∂/∂ E[I_r,0(r_out;P_i) - I_r,0(r_in;P_i) ], 
       Δφ_0 = φ_out - φ_in = -∂/∂ L[I_r,0(r_out;P_i) - I_r,0(r_in;P_i) ], 
       Δτ_0 = τ_out - τ_in = -∂/∂ m_1[I_r,0(r_out;P_i) - I_r,0(r_in;P_i) ].


We are interested in the limit where the initial and final states are defined at past and future timelike infinity, with r_ in=∞ = r_ out, passing through the single radial turning point r_m. In this limit, Δφ_0 remains finite, but I_r,0(r_ in/out;P_i), Δ t_0, and Δτ_0 all diverge. However, we can define regularized versions. Using a convenient dimensionless regulator ϵ <cit.>, we first define the scattering radial action 


    I^>,ϵ_r,0   (P_i) = 2 ∫_r_m^+∞d r   r^ϵ  p_r,0(r;P_i)  .

This converges for Re(ϵ)<-1, but for physical observables we will be able to analytically continue to ϵ=0. In terms of I^>,ϵ_r,0 we can write the regularized r_ in/out→∞ limit of Eq. (<ref>) for the full scattering path as

    Δφ^ϵ_0 = π + χ^ϵ_0    = - ∂/∂ L I^>,ϵ_r,0(P_i)  ,

where χ_0=lim_ϵ→0χ^ϵ_0 is the physical scattering angle, and 

    Δ t^ϵ_0 = ∂/∂ E I^>,ϵ_r,0(P_i) ,     Δτ^ϵ_0 = - ∂/∂ m_1 I^>,ϵ_r,0(P_i) .

Unlike χ^ϵ_0, the elapsed times Δ t^ϵ_0 and Δτ^ϵ_0 diverge as ϵ→0. The associated physical observables, which are well defined when ϵ→0, are relative measurements: the difference between Δ t_0 (P_i) along the geodesic and Δ t_0 (P_i,ref) along some reference orbit. In the Supplemental Material, we provide exact expressions for the geodesic scattering observables as well as the first few terms in their PM expansions (corresponding to m_1 m_2 / L≪ 1). 

Finally, Eqs. (<ref>) and (<ref>) can be immediately combined into a single equation,

    δ I^>,ϵ_r,0 = -(π + χ_0) δ L+ Δ t^ϵ_0 δ E - Δτ^ϵ_0δ m_1 .

This is our first law for scattering geodesics. Here and below, we discard terms that vanish at ϵ=0, and equalities should be understood in this sense.

First law at all SF orders.—Beyond leading order in the mass ratio m_1/m_2, the particle generates a metric perturbation h_αβ on the Schwarzschild background. m_1 then moves on a geodesic of a certain effective metric g̃_αβ = g_αβ^Schw + h^ R_αβ <cit.>, where h^ R_αβ is a certain regular piece of h_αβ. At 1SF order, we can write h^ R_αβ in terms of the Detweiler-Whiting Green's function G_R^αβα^'β^' <cit.>: 

    h_R^αβ(x^μ;Γ)= 1/m_1∫_Γ G_R^αβα^'β^'(x^μ, x'^μ(τ̃')) p̃_α^'p̃_β^' dτ̃' .

Here p̃_α:=g̃_αβ dx^β/ dτ̃, τ̃ is proper time in g̃_αβ, and Γ denotes the particle's phase-space trajectory. Due to curvature-induced tail effects, G_R^αβα^'β^' is nonzero for all points x'^μ in the past of x^μ, implying h_R^αβ at a point on Γ depends on the entire prior history of Γ.  At higher SF orders, there is no known Green's-function form analogous to (<ref>), but an appropriate h^ R_αβ(x^μ;Γ) exists at all SF orders <cit.>.   

In this setting, we again consider scattering orbits with initial parameters P_i=(m_1,E,L). The particle's energy and angular momentum evolve due to dissipation, but the orbit remains planar (θ=π/2, p̃_θ=0). For L above a critical threshold, the orbit remains close to a Schwarzschild geodesic with the same initial P_i <cit.>.

Since the motion is geodesic in g̃_μν, it obeys Hamilton's equations with the test-mass pseudo-Hamiltonian H = (1/2) g̃^μνp̃_μp̃_ν <cit.>. However, we deviate from <cit.> by restricting  to the 4D phase space (x^A, p̃_A) satisfying the on-shell condition H = -m_1^2/2, with x^A=(r,φ). Solving the on-shell condition for p̃_t=-ℋ(t, x^A, p̃_A;Γ) gives the new pseudo-Hamiltonian

    ℋ
         = 1/g̃^tt[g̃^tAp̃_A-√((g̃^tAp̃_A)^2-g̃^tt(g̃^ABp̃_A p̃_B+m_1^2))]
       = -p_t(r,p̃_A) - h^μν_ R(t,x^A;Γ) p̃_μp̃_ν/2g^tt_ Schw p_t+ O(m_1^2/m_2^2),

with t now the parameter along the trajectory. ℋ is referred to as a pseudo-Hamiltonian because it depends on the trajectory Γ={(x^A(t), p̃_A(t)) | t ∈ℝ}. Hamilton's equations in this context read

    d x^A/ d t= [∂ℋ/∂p̃_A]    and    dp̃_A/ d t=-[∂ℋ/∂ x^A] ,

where [·] indicates specification of Γ as the self-consistent trajectory <cit.> passing through (x^A,p̃_A); prior to that specification, Γ is treated as independent, and derivatives do not act on it. We go further by replacing m_1 by m_1' in Eq. (<ref>), setting m'_1=m_1 only when [·] is applied. Importantly, Eq. (<ref>) captures the full dynamics, including dissipation, unlike an ordinary Hamiltonian description.

We derive the first law from H. Doing so will require its relationship to the redshift z: 

    z := dτ̃/d t = [∂ℋ/∂ m_1]  .

To establish this relationship, we consider the normalization condition

    -m_1=p̃_μd x^μ/dτ̃
         =(-ℋ+p̃_A [ ∂ℋ/∂p̃_A]  ) z^-1 ,

where we used p̃_t = -ℋ together with (<ref>). Next we note the first line of (<ref>) shows that, at fixed m'_1, ℋ is a homogeneous function of (m_1,p̃_A) of order 1. Euler's homogeneous function theorem hence implies 
ℋ= m_1 ∂ℋ/∂ m_1+p̃_A ∂ℋ/∂p̃_A.  Comparing this with (<ref>), we obtain (<ref>).

Now, to derive the first law, we loosely follow <cit.> by considering how H changes under variations δ P_i of the initial data, with δ x^A_ in=0. Given (<ref>) and (<ref>), we find

    [δℋ]     = [ ∂ℋ/∂ x^Aδ x^A ] + [ ∂ℋ/∂p̃_Aδp̃_A ] + [ ∂ℋ/∂ m_1δ m_1 ] 
       = - dp̃_A/d tδ x^A + d x^A/d tδp̃_A + dτ̃/d t δ m_1  .

Since h^ R_μν vanishes for an inertial particle in Minkowski <cit.>, its contribution to p̃_A and H vanishes in the initial state, such that p̃^ in_φ=p^ in_φ,0=L and H_ in=-p^ in_t,0=E.  We isolate δ L and δ E in (<ref>) by defining `interaction' quantities 𝓅̃_φ:=p̃_φ-L, 𝓅̃_r=p̃_r, and ℋ:= H-E, such that

    δ E + [δℋ] = - d𝓅̃_A/d tδ x^A + d x^A/d tδ𝓅̃_A + dφ/d tδ L + dτ̃/d tδ m_1  .

Next, we integrate (<ref>) along the physical scattering trajectory from t=-∞ to t=+∞, introducing the regularized integral ⟨ f⟩_Γ :=∫_Γdt  r^ϵ f as in the 0SF case. Integrating the term d𝓅̃_A/d tδ x^A by parts, we obtain

    Δ t^ϵδ E+⟨ [ δℋ ]  ⟩_Γ
    
         
    = δ⟨𝓅̃_A d x^A/d t⟩_Γ + Δφ^ϵδ L  + Δτ̃^ϵδ m_1   ,


where  Δ t^ϵ:=⟨ 1 ⟩_Γ, Δφ^ϵ := ⟨dφ/d t⟩_Γ, and Δτ̃^ϵ := ⟨dτ̃/d t⟩_Γ.
We have discarded boundary terms by choosing Re(ϵ) sufficiently negative and discarded terms that arise from derivatives acting on r^ϵ because they vanish when analytically continued to ϵ=0. Defining also the regularized `interaction' action

    I^>,ϵ := ∫_Γ dt  r^ϵ𝓅̃_A d x^A/d t=I^>,ϵ_r+∫_Γdφ 𝓅̃_φ ,

with I^>,ϵ_r:=∫_Γ dr  r^ϵp̃_r, we rewrite (<ref>) as

    Δ t^ϵδ E+⟨ [ δℋ]  ⟩_Γ   = δ I^>,ϵ +  Δφ^ϵδ L + Δτ̃^ϵδ m_1   .


Equation (<ref>) is valid at all SF orders. We can write it in two alternative forms by absorbing ⟨ [ δℋ ]  ⟩ into either `renormalized' observables B^>_ ren or variables {I^>,ϵ_ ren,E_ ren,L_ ren}. In the first approach, we write ⟨ [ δℋ ]  ⟩ = ⟨ [ ∂_P_iℋ ]  ⟩δ P_i and define

    χ^ϵ_ ren   := χ^ϵ -⟨ [ ∂_Lℋ ]  ⟩,  Δ t^ϵ_ ren := Δ t^ϵ +⟨ [ ∂_Eℋ ]⟩,
    Δτ̃^ϵ_ ren   := Δτ̃^ϵ -⟨ [ ∂_m_1ℋ ]⟩.

Equation (<ref>) then becomes

    δI^>,ϵ = -(π + χ^ϵ_ ren) δ L + Δ t^ϵ_ renδ E - Δτ̃^ϵ_ renδ m_1  .


In the second approach, following <cit.>'s treatment of the bound case, we define renormalized variables 

    E_ ren =  λ E ,    L_ ren =  λ L ,  I^>,ϵ_ ren =  λ I^>,ϵ .

Choosing λ(P_i) appropriately to eliminate ⟨ [ δℋ ]  ⟩ from Eq. (<ref>), we are left with 

    δI^>,ϵ_ ren = -(π + χ^ϵ) δ L_ ren + Δ t^ϵδ E_ ren - Δτ̃^ϵδ m_1   .

Equations (<ref>) and (<ref>) are two equivalent forms of the first law for scattering orbits, valid at all SF orders and including all dissipative effects.

From scattering to bound.—There is a well-known analytic continuation between the deflection angle χ for unbound orbits and the periastron advance ΔΦ for bound orbits, as well as between the scattering and bound radial actions <cit.>. Here, using the first laws for unbound and bound motion, we extend these analytic continuations to include all the observables in the scattering and bound bases, ℬ^> = (χ,Δ t^ϵ,Δτ^ϵ) and ℬ^< = (ΔΦ,Ω_r,⟨ z ⟩). We limit our analysis to 0SF order, as the analytic continuation for the radial action is not known to be valid beyond 0SF order due to nonlocal-in-time tail effects <cit.>.

We write the first law for bound geodesics in terms of the bound radial action,

    I^<_r,0   (P_i)  =  2 ∫_r_-(P_i)^r_+(P_i) d r   p_r,0(r;P_i)  ,

where r_∓ are the orbit's minimum and maximum radius (i.e., the radii at periapsis and apoapsis).
Following the same arguments as for unbound orbits, one can write the accumulated φ, t, and τ over a single radial period (T_r,0=2π/Ω_r,0) as derivatives of I^<_r,0, leading to the first law for bound orbits <cit.>:

    δ I^<_r,0   = -(2π + ΔΦ_0) δ L  + 2 π/Ω_r,0δ E- 2 π/Ω_r,0⟨ z ⟩_0 δ m_1 ,

where ⟨ z ⟩_0 := (1/ T_r,0) ∫_0^T_r,0dt (dτ /dt)_0. 

Knowing the scatter-to-bound map for the radial action <cit.>,

    I^<_r,0   (P_i)  =  lim_ϵ→0[I^>,ϵ_r,0(E,L,m_1)-I^>,ϵ_r,0(E,-L,m_1)],

and comparing Eq. (<ref>) to Eq. (<ref>), we immediately conclude that there is an analytic continuation between the full set of scattering and bound observables:

    ΔΦ_0    = χ_0 (E,L,m_1) + χ_0 (E,-L,m_1)  , 
    2 π/Ω_r,0   = lim_ϵ→0[Δ t^ϵ_0 (E,L,m_1) - Δ t^ϵ_0 (E,-L,m_1)] , 
    2 π⟨ z ⟩_0 /Ω_r,0   = lim_ϵ→0[Δτ^ϵ_0 (E,L,m_1) - Δτ^ϵ_0 (E,-L,m_1)] .

We note that the infrared  divergences in Δ t^ϵ_0 and Δτ^ϵ_0 cancel in these expressions because the divergences are independent of L; see the Supplemental Material.

First law from the S-matrix.—In this section, we put our first law in the broader context of the quantum S-matrix description of the classical two-body problem <cit.>. Given the two-body initial state of well-separated massive point particles Ψ_in = p_1 p_2 of mass m_1 and m_2, the action of the unitary S-matrix operator,


    Ŝ = 𝒯exp(-i/ħ∫d t   H_int(t)) ,


describes the time evolution of the state in terms of the interaction Hamiltonian H_int, with 𝒯 denoting time ordering. The classical two-body scattering dynamics in the asymptotic ħ→ 0 limit is equivalently obtained by evaluating the action, and therefore H_ int, on-shell.

Motivated by (<ref>), we define the exponential representation Ŝ = exp(i N̂ /ħ) <cit.>, where N̂ is a Hermitian operator. We then study the real-valued two-body matrix element 

    N(𝔼,q,m_1,m_2)    := ⟨ p_1' p_2'|N̂|p_1 p_2⟩ ,

where we defined Ψ_out = p'_1 p'_2, the initial total energy 𝔼 of both particles, and the exchanged momentum  q^μ = p_1'^μ-p_1^μ=p_2^μ-p_2'^μ. To make contact with the incoming total angular momentum 𝕃= (m_1 m_2 √(γ^2 - 1) b)/𝔼, where γ := p^μ_1+p_1'^μ/2p_2μ+p'_2μ/2 and b is the impact parameter in the center of mass (CM) frame,  we perform the Fourier transform

    N^>,ϵ(𝔼,𝕃,{m_a}) 
        = 1/4 m_1 m_2 √(γ^2-1)∫d^2+2 ϵ q/(2 π)^2+2 ϵ e^-i b(𝕃) · q/ħ  N(𝔼,q,{m_a})  ,

using dimensional regularization with d=4+2 ϵ. Infrared, 1/ϵ divergences arise due to the long-range nature of gravity, but their analytic structure is understood  <cit.>. 

In complete generality, the expectation value (<ref>) is a function of the kinematic data (𝔼,𝕃,{m_a}), and its variation in the phase space is 

    δ N^>,ϵ = c^ϵ_𝕃 δ𝕃+ c^ϵ_𝔼 δ𝔼 + ∑_a=1,2 c^ϵ_m_a δ m_a  ,

where c^ϵ_𝕃, c^ϵ_𝔼, and c^ϵ_m_a are gauge-invariant coefficients. 

Using insights from the PM Hamiltonian description <cit.> and the relation with the radial action <cit.>, we now identify the coefficients with the observables B^>. First, by matching the scattering angle χ in the CM frame, it was shown that the matrix element (<ref>) in the conservative case agrees with the radial action up to a constant proportional to 𝕃 <cit.>:

    N^>,ϵ|_cons = ∫_𝒞_r^>,ϵd r  p̃_r,c.m.(r;𝔼,𝕃,{m_a})_𝕀^>,ϵ_r + π𝕃 ,

where p̃_r,c.m. is the radial relative momentum in the CM frame and 𝒞_r^>,ϵ is the contour of integration for scattering orbits, which implicitly includes a regulator ϵ inherited from the dimensional regularization. In the Supplemental Material, using the PM conservative Hamiltonian and its symmetries in the CM frame <cit.>, we then provide a proof of the following conservative first law for the two-body scattering problem,

    δ N^>,ϵ =  -χδ𝕃+ Δ t^ϵδ𝔼 - ∑_a=1,2Δτ^ϵ_aδ m_a ,

where Δτ^ϵ_a is the elapsed proper time of particle a and Δ t^ϵ is the elapsed global time.

In general, the two-body incoming energy 𝔼 and angular momentum 𝕃 do not agree with the conserved one-body counterparts E and L of the GSF calculation performed earlier. Moreover, while N^>,ϵ is computed here in the CM frame, usually the GSF calculation is in the initial rest frame of the heavy BH <cit.>. However, at 0SF order, we can trivially identify 𝔼→ E + m_2 and 𝕃→ L and both frames, as the relative radial momentum p̃_r,c.m. coincides with the geodesic one p_r,0. Then (<ref>) identically matches  (<ref>), and we have χ^ϵ→χ^ϵ_0, Δ t^ϵ→Δ t^ϵ_0 and Δτ^ϵ_1→Δτ^ϵ_0. Notice that δ N^>,ϵ includes also a variation with respect to m_2, which we held fixed in our earlier variations. In the GSF context, we would expect the coefficient of such a variation to be related to the surface gravity of the primary BH <cit.>, while in the PM limit it should asymptote to the elapsed proper time of the BH <cit.>. At nSF order, the matching with our SF first law is more challenging as the dynamics of the heavy BH is integrated out compared to the S-matrix case; we leave this for future work. 





Incorporating dissipation in this framework is possible by combining the in-in expectation value <cit.> with the exponential representation of Ŝ <cit.>: for every observable 𝒪,[Notice that in the limit ħ→ 0, the commutators (-i/ħ)[ · , · ] are equivalent to suitable Dirac brackets {·,·} which generate the time evolution in the classical phase space <cit.>.]

    ⟨Δ𝒪⟩   = Ψ_inŜ^† 𝒪 ŜΨ_in - Ψ_in𝒪 Ψ_in
       = ∑_j=1^+∞(-i)^j/ħ^j j! [N̂,  [N̂, …,  [N̂_j  times, 𝒪] … ] ] ,

where now also the N̂ matrix elements with on-shell gravitons are relevant. This suggests a physical principle to connect the coefficients (c^ϵ_𝕃,c^ϵ_𝔼,c^ϵ_m_a) to observables at all orders, with dissipation included; see for example Eq. (3.48) of <cit.>.

Conclusion.—In recent years, the study of unbound orbits through the S-matrix formalism has transformed the gravitational two-body problem. In this Letter, we developed a powerful new tool for such studies: an extension of the first law of binary BH mechanics to the unbound (and dissipative) case. Our derivation at all SF orders utilizes a novel version of the pseudo-Hamiltonian formalism <cit.>. More generally, we showed how the first law can be derived from the variation of the classical S-matrix in the phase space of the kinematic data P_i =(ℰ,L,{m_a}); see Fig. <ref>. In that context, the S-matrix can be interpreted as a generating functional of classical observables. Among those observables, we have highlighted the elapsed proper time Δτ^ϵ as a new, core element of the (regularized) basis of scattering observables ℬ^> = (χ, Δ t^ϵ, Δτ^ϵ). 

Using the relation between the scattering and bound radial action, we also established a full correspondence (<ref>) between the bases of scattering observables ℬ^> and bound observables ℬ^< at 0SF order, as again summarized in Fig. <ref>. This extends the well-known map between the deflection angle and periastron advance.

Given the first law's varied applications for bound orbits <cit.>, we expect our work to open many new avenues for scattering calculations. We particularly encourage self-force scattering calculations <cit.> of the observables Δ t^ϵ and Δτ^ϵ. Natural extensions also present themselves. First, one might consider scattering orbits of spinning BHs <cit.>. Second, we considered only two-body matrix elements, but nothing prevents us from studying the variation of matrix elements involving on-shell graviton states, which should be related to the gravitational waveform. Finally, we hope that linking the first laws for scattering and bound orbits beyond 0SF can shed light on the tail effects that have limited the applicability of scatter-to-bound maps <cit.>.



Acknowledgments.
We thank Alex Le Tiec for comments on a draft of this Letter. RG would like to thank A. Ilderton, M. Porrati, K. Rajeev and C. Shi for useful discussions. JL and AP gratefully acknowledge helpful conversations with Soichiro Isoyama and Takahiro Tanaka about pseudo-Hamiltonian dynamics and with Leor Barack, Chris Kavanagh, Rafael Porto, Davide Usseglio, and Maarten van de Meent about scatter-to-bound maps. RG is grateful to the Mainz Institute of Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 390831469), for its hospitality and its partial support during the completion of this work. JL acknowledges the support of an STFC studentship. AP acknowledges the support of a Royal Society University Research Fellowship and a UKRI Frontier Research Grant (as selected by the ERC) under the Horizon Europe Guarantee scheme [grant number EP/Y008251/1].













§ SUPPLEMENTAL MATERIAL


In this Supplemental Material, we provide closed-form and PM-expanded formulas for the regularized scattering observables ℬ^>=(χ,Δ t^ϵ,Δτ^ϵ) at 0SF order. We then discuss how they relate, through analytic continuation, to the analogous observables for bound geodesics. Finally, we provide a derivation of the conservative first law for scattering orbits from the S-matrix formalism.



 §.§ Observables at 0SF order


All of the observables can be obtained from the scattering radial action (<ref>) via Eqs. (<ref>) and (<ref>). Changing variable to u=1/r, we write the radial action as

    I^>,ϵ_r,0(P_i)    = 2 ∫_0^u_ md u/u^2+ϵ √(B(u))/(1 -2 m_2 u) , 
    
         B(u)    = 2 L^2 m_2 (u-u_ m)(u-u_1)(u-u_2) ,

where the analytic form of the radial roots (u_ m,u_1,u_2) of B(u) = 0 follow from Cartan's formula <cit.>:

    u_ m   = 1/6 m_2(1 +  2 βcos(ζ-2 π/3)) , 
    
       u_1    = 1/6 m_2(1 +  2 βcos(ζ/3)) , 
    
       u_2    = 1/6 m_2(1 +  2 βcos(ζ+2 π/3)) ,

and we have defined the constants

    ζ   =arccos(L^2+(36 m_1^2-54 E^2) m_2^2/β ^3 L^2) , 
           β =√(1-12 m_1^2 m_2^2/L^2) .

 
Deflection angle.—The deflection angle χ=Δφ-π describes the total change of the azimuthal angle in the scattering process, relative to the value (π) for a straight line in Minkowski spacetime. This quantity does not require any regularization as it is infrared finite. To see how that finiteness follows from the radial action, note that the 1/ϵ divergence of the radial action is L-independent:

    I^>,ϵ_r,0   (P_i) |_1/ϵ = 2 m_2 (2 E^2 -m_1^2)/√(E^2-m_1^2)1/ϵ ,
 
in agreement with amplitude calculations <cit.>. Therefore, the deflection angle for a scattering geodesic, which is calculated from ∂ I^>,ϵ_r,0/∂ L, is always infrared finite.

Now, to evaluate the deflection angle from (<ref>), we notice that the action of the L-derivative operator on the integration boundary u_ m vanishes because B(u_ m) = 0, and therefore we only need to act at the integrand level. This allows us to express (see the strategy adopted in <cit.>) the geodesic deflection angle as

    χ_0 = 4 L u_ m/√(E^2-m_1^2) F_D^(2)(1,1/2,1/2,3/2; u_ m/u_1,u_ m/u_2)  ,

where F_D^(n) is Lauricella's hypergeometric function

    F_D^(n)(α,{β_j}_j=1^n, γ ;{x_j}_j=1^n)=Γ(γ)/Γ(α) Γ(γ-α)
         ×∫_0^1  d t  t^α-1(1-t)^γ-α-1∏_j=1^n(1-x_j t)^-β_j .


Coordinate time delay.—The total change in coordinate time along the geodesic is trivially infinite. Physically it only makes sense to measure the difference between this time and the time as measured along a reference path; for an appropriate reference path, that difference is the (Shapiro) time delay <cit.>, which represents the amount by which signals are slowed down in the presence of a gravitational source.  Inspired by <cit.>, we take as a reference the clock of an observer moving along a scattering geodesic at very large L_ref≫ L but fixed E_ref = E and define our physical infrared-finite time-delay observable as

    Δ t^rel_0,L_ref = lim_ϵ→ 0[Δ t^ϵ_0(P_i) - Δ t^ϵ_0 (P_i,ref) |_𝒪(m_1m_2/L_ref)],

where it is understood that we neglect subleading terms in m_1 m_2/L_ ref. Given (<ref>) and (<ref>), we notice that this definition is always infrared safe at geodesic order, as expected.  We also note that this definition is completely analogous to χ_0, which we can write as

    χ_0 = lim_ϵ→ 0[Δφ_0^ϵ(P_i) - Δφ_0^ϵ (P_i,ref) ]

because lim_ϵ→ 0Δφ_0(P_i, ref)=π.

Using the definition (<ref>) and the representation (<ref>) of the radial action, we can write the regularized elapsed coordinate time as

    Δ t^ϵ_0 = 2 √(π)m_1/√(E^2-m_1^2) u_ m^1-ϵΓ(-1+ϵ)/Γ(-1/2+ϵ)
       × F_D^(3)(-1+ϵ,1,1/2,1/2,-1/2+ϵ; 2 m_2 u_ m, u_ m/u_1,u_ m/u_2),

where we have used the definition (<ref>). Turning to the physical time delay (<ref>), at geodesic order the PM expansion gives the familiar structure <cit.>:

    Δ t^rel_0,L_ref   = 2 m_2 E (3 m_1^2-2 E^2)/(E^2-m_1^2)^3/2log(L/L_ref) 
            + 15 π E m_2^2/2 L + 𝒪(1/L^2)  .


Relative elapsed proper time.—The elapsed proper time along scattering orbits has received, to our knowledge, very little attention in the literature (except, possibly, for plunging orbits related to holography applications <cit.>). Very recently,  <cit.> and <cit.> investigated the proper time from the theoretical and experimental perspective, but without focusing on the two-body problem. In the body of this Letter, we showed that this is a crucial ingredient in our basis of scattering observables. 

Using the same strategy as before, we find that the definition in Eqs. (<ref>) and the radial action (<ref>) give

    Δ   τ^ϵ_0 = 2 m_1 √(π)/√(E^2-m_1^2) u_ m^1-ϵΓ(-1+ϵ)/Γ(-1/2+ϵ)
       × F_D^(3)(-1+ϵ,1/2,1/2,-1/2+ϵ; u_ m/u_1,u_ m/u_2)  ,

which is the regularized elapsed proper time along scattering orbits. The elapsed proper time relative to the large-L_ ref reference orbit is, in analogy with Eq. (<ref>),

    Δτ^rel_0,L_ref = lim_ϵ→ 0[Δτ^ϵ_0(P_i) - Δτ^ϵ_0 (P_i,ref) |_𝒪(m_1m_2/L_ref)],

which has the PM expansion

    Δτ^rel_0,L_ref   = 2 m_1^3 m_2 /(E^2-m_1^2)^3/2log(L/L_ref) 
            + 3 π m_1 m_2^2/2 L + 𝒪(1/L^2)  .

It is worth noticing that, unlike the time delay  Δ t^rel, the proper time is not defined for massless geodesics and therefore Δτ^rel→ 0 as m_1 → 0.



 §.§ Scatter-to-bound maps at 0SF order


In the body of the Letter, we derived the scatter-to-bound maps (<ref>) directly from the first laws for unbound and bound geodesics. Here we observe that these maps also follow from direct inspection of the integral representation of the observables. 

We begin with those representations for unbound geodesics. The observables are simply the (regularized) net change in φ, t, and τ over the orbit:

    π + χ_0 = Δφ_0    = 2 ∫_r_ m(P_i)^∞ d r  (dφ/d r)_0 ,
    Δ t^ϵ_0    = 2 ∫_r_ m(P_i)^∞ d r   r^ϵ(d t/d r)_0  , 
    Δτ^ϵ_0    =  2  ∫_r_ m(P_i)^∞ d r   r^ϵ(dτ/d r)_0 .

We next write the observables for bound orbits in analogous forms. The periastron advance ΔΦ_0 corresponds to the azimuthal precession over a period of the bound orbit, 

    2π + ΔΦ_0    = 2 ∫_r_-(P_i)^r_+(P_i) d r  (dφ/d r)_0 ,

where, recall, r_±(P_i) are the two radial turning points of the bound geodesic. Similarly, the radial coordinate-time frequency Ω_r,0 and the averaged redshift ⟨ z ⟩_0 can be written as <cit.>

    2 π/Ω_r,0   = 2 ∫_r_-(P_i)^r_+(P_i) d r  (d t/d r)_0  , 
    (2 π/Ω_r,0) ⟨ z ⟩_0    =  2  ∫_r_-(P_i)^r_+(P_i) d r  (dτ/d r)_0 .


Each of the bound-orbit integrals can be written in terms of unbound-orbit integrals by analytically continuing to E>m_1 and negative L. The key properties in this exercise are

    p_r,0(r;E,L,m_1) = p_r,0(r;E,-L,m_1),

which follows immediately from Eq. (<ref>), and the relationship between turning points for bound and unbound  geodesics <cit.>,

    r_±(E,L,m_1) = r_ m(E,∓ L,m_1)  .

Expressed in terms of p_t,0=-E, p_φ,0=L, and p_r,0, the observables read

    2π + ΔΦ_0    = 2 ∫_r_-(P_i)^r_+(P_i) d r  g^φφL/g^rrp_r,0(r;P_i) ,
    2 π/Ω_r,0   = 2 ∫_r_-(P_i)^r_+(P_i) d r  -g^ttE/g^rrp_r,0(r;P_i) ,
    (2 π/Ω_r,0) ⟨ z ⟩_0    = 2  ∫_r_-(P_i)^r_+(P_i) d r/g^rrp_r,0(r;P_i) .
 
In each case, we pull the constant (E or L) out of the integral and use Eqs. (<ref>) and (<ref>) to write the integral as

    ∫_r_-(P_i)^r_+(P_i)F(r)d r/p_r,0(r;P_i)
       = ∫_r_-(P_i)^∞F(r)d r/p_r,0(r;P_i)   - ∫_r_+(P_i)^∞F(r)d r/p_r,0(r;P_i)
       = ∫_r_m(P_i)^∞F(r)d r/p_r,0(r;P_i)  - ∫_r_ m(E,-L,m_1)^∞F(r)d r/p_r,0(r;E,-L,m_1).

Here F(r) is one of g^φφ/g^rr, g^tt/g^rr, or 1/g^rr, which in each case depends only on r. 
Now comparing the observables  (<ref>) to the observables (<ref>) for unbound orbits, and noting Eq. (<ref>), we recover Eq. (<ref>).



 §.§ Conservative first law from the S-matrix formalism


We prove here the validity of the first law in the conservative case for the S-matrix, combining the relation of the PM Hamiltonian <cit.> with amplitudes and with the radial action <cit.>, which is then linked directly to the exponential representation of the S-matrix <cit.>. Albeit the Hamiltonian extracted from the amplitude is gauge-dependent, the following derivation—and therefore our first law—is independent of the choice of gauge.

We first define the center of mass frame for the massive spinless particles 1 and 2 as the one where the incoming states have momenta (E_1, p⃗) and (E_2,-p⃗) while the outgoing states have momenta (E_1, p⃗^') and (E_2,-p⃗^'), with E_j = √(|p⃗|^2+m_j^2). Conservation of energy then implies |p⃗| = |p⃗^'|^2, and the exchanged momentum in the scattering process becomes q^μ = (0 ,q⃗) = (0 ,p⃗ - p⃗^').

We then introduce polar coordinates in the plane of the motion, x⃗ = (r cos(ϕ),r sin(ϕ),0), where r is the distance between the two bodies and ϕ is the azimuthal phase. We can then define an effective conservative Hamiltonian in the CM frame, 

    H^PM_c.m.(r, |p⃗|)=∑_a=1,2√(|p⃗|^2+m_a^2)+V(r, |p⃗|)  ,

where V(r, |p⃗|^2) is the interaction potential for spinless particles, which can be extracted systematically from conservative amplitudes order by order in the PM expansion <cit.>. Here x^i and p_i are canonically conjugate to each other <cit.>, and denoting the norm of the center of mass spatial momentum at infinity |p⃗|(r → +∞) as 

    p_∞ = 1/4 𝔼^2 (𝔼^2-(m_1-m_2)^2)(𝔼^2-(m_1+m_2)^2) ,

we obtain the conservation laws for the energy and angular momentum, 

    𝕃 = p_ϕ = b   p_∞ ,   𝔼 = H(r,|p⃗|^2= p_r,c.m.^2+L^2/r^2).

At this point, we can proceed as in <cit.> and consider the variation of our Hamiltonian H^PM_c.m.(r,p_r,c.m.,ϕ,𝕃,{m_a}) in the phase space:

    δ H^PM_c.m.   = ∂ H^PM_c.m./∂ rδ r+∂ H^PM_c.m./∂ p_r,c.m.δ p_r,c.m.
            +∂ H^PM_c.m./∂𝕃δ𝕃  + ∑_a=1,2∂ H^PM_c.m./∂ m_aδ m_a 
       = -d p_r,c.m./d tδ r+d r/d tδ p_r,c.m.
            +dϕ/d tδ𝕃  + ∑_a=1,2∂ H^PM_c.m./∂ m_aδ m_a ,

where we have used Hamilton's equations in the last line. We can now recognize the redshift variables z_1,z_2 <cit.>,

    z_a  = ∂ H^PM_c.m./∂ m_a = dτ_a/d t ,

where τ_a is the elapsed proper time of particle a in the CM frame. We can now proceed as in our pseudo-Hamiltonian derivation, using the regularized integration

    ⟨ F(·) ⟩  = ∫d t   r^ϵ F(·)

on both sides of the equality (<ref>). In this case, though, 

    ⟨ H(r,p_r,ϕ,𝕃,{m_a}) ⟩ = 𝔼 ⟨ 1⟩ = 𝔼 Δ t^ϵ ,

where Δ t^ϵ is the elapsed global time. Therefore, following similar steps to the ones in the main text and dropping total boundary terms (which are zero on-shell), we obtain

    δ𝕀_r^>,ϵ   =  -(π + χ)  δ𝕃+ Δ t^ϵδ𝔼
                    - ∑_a=1,2Δτ^ϵ_aδ m_a ,

where we have defined

    𝕀_r^>,ϵ  = ⟨ p_r,c.m.d r/d t⟩ , 
    χ   = ⟨dϕ/d t⟩ ,   Δτ^ϵ_a = ⟨dτ_a/d t⟩ .

Finally, we need to link the radial action defined here with the expectation value of the N̂ operator for the two-body conservative case. We now recall the proof in Sec. 3.1.3 of <cit.>, where it was shown (non-perturbatively) that  

    χ  = - ∂ N^>,ϵ(𝔼,𝕃,{m_a})/∂𝕃 ,

where all the other variables are kept fixed; showing therefore that, as expected, 

    N^>,ϵ  = 𝕀_r^>,ϵ + π𝕃 .

This concludes the proof of the conservative first law (<ref>) in the S-matrix formalism.


