# The 3PN radiative current quadrupole and the associated gravitational amplitude mode

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## Derivation of EoM and conserved quantities from a generalized Lagrangian

Let us consider a generalized Lagrangian L(~yA, ~vA,~aA, . . . ,~a (n) A ). We can derive [82] the EoM of such a Lagrangian using the generalized Euler-Lagrange formula where (yiA )(`) is the `th time derivative of the positions. Eq. (4.17) gives an expression of the accelerations with respect to the other dynamical variables, namely ai A = ai A(~yA, ~vA,~aA, . . . ,~a (n) A ). However, the acceleration entering in the right-hand side are of lower order (because they are multiplied by inverse powers of c). This implies that we can replace iteratively its value in order to find ai A(~yA, ~vA).

### The NNL reduced Fokker Lagrangian in a general frame

With the above methods in hand, we are now able to apply them to our problem. We first need to compute the tidal tensors using the unreduced values of the point-particle potentials. Then we can insert them in the definition of the Lagrangian in order to reduce it. Finally, we will be able to derive the EoM and conserved quantities.

#### Computation of the regularized tidal tensors

At the 2PN order, the metric of a general matter system in harmonic coordinates, given in Eq. (3.35), can be parametrized by the set of potentials {V, Vi, ˆWij , ˆRi, ˆX } whose definitions are given in Eq. (3.36). To perform a consistent Fokker reduction of the original action, the solutions of Eqs. (3.36) must be in principle constructed with the symmetric Green function, which kills all contributions of odd powers of 1/c at the current approximation level. As discussed above, thanks to the properties of the Fokker action, we only need the metric produced by point particles and can neglect tidal effects when inserting the metric (3.35) into the Fokker action. Therefore, we just require the potentials for point particles without including any internal structure effect. The relevant potentials have already been published in, e.g., [108], except that we compute here their off-shell values, without replacement of accelerations by means of the EoM (we then call them the “unreduced” potentials). However, it is known that the replacement of accelerations in the action is equivalent to performing an unphysical shift of the particles’ worldlines [109]. I have checked that, indeed, by inserting the reduced (on shell) versions of the potentials into the action, the final gauge invariant result for the conserved energy reduced to circular orbits, comes out the same. As already said, in practical calculations, I used the Hadamard regularisation, which is equivalent to DR up to the relatively low NNL/2PN order [100, 83]. The method to regularise quantities at point A using Hadamard partie finie is explained in Sec. 6.3.2. After performing a (3 + 1) splitting of the tidal tensors and injecting the 2PN point-particle metric, I found3.

The EoM and conserved quantities

The method for computing the EoM and conserved quantities are detailed in Sec. 4.2.4. The generalized Euler-Lagrange formula has been used to derive the NNL/2PN accelerations. Since the problem is perturbative, I replaced iteratively the on shell accelerations with their value in terms yiA and viA only. The result is displayed in Eq. (D.1). I have checked that their value is in agreement with the literature (up to NL/1PN, since the 2PN was not derived yet) and that they are invariant under a Lorentz boost as they should be.

I General introduction
1 General relativity and linearized theory of gravitational waves
1.1 General relativity in a nutshell
1.2 Linearized theory
1.3 Gravitational waves propagation
2 Detections
2.1 Possible sources
2.1.1 Binary systems of compact objects
2.1.2 Other types of sources
2.2 The detectors
2.2.1 Second generation detectors
2.2.2 Third generation detectors
2.3 Direct detections of GWs
2.3.1 The O1 run
2.3.2 The O2 run
2.3.3 The O3 run
2.4 Approximation methods for compact binary systems
2.4.1 Post-Newtonian
2.4.2 Effective field theory
2.4.3 Scattering amplitudes
2.4.4 Gravitational self force
2.4.5 Numerical relativity
2.4.6 Effective one body
3 The post-Newtonian formalism
3.1 The PN-MPM formalism
3.1.1 Linearized theory
3.1.2 The MPM algorithm
3.1.4 Matching the source moments to a Post-Newtonian source
3.1.5 Explicit expression of the source multipole moments for a PN source
3.1.6 The PN metric
3.2 Binary systems of compact objects
3.2.1 Choice of stress-energy tensor
3.2.2 The regularisation schemes
3.3 Practical computations
II Tidal effects
4 Conservative sector
4.1 The effective matter action for static tides
4.1.1 Generalities
4.1.2 The total 2PN action for static tides
4.2 General methods
4.2.1 The Fokker method
4.2.2 Properties of the Fokker action
4.2.3 The reduction method
4.2.4 Derivation of EoM and conserved quantities from a generalized Lagrangian
4.3 The NNL reduced Fokker Lagrangian in a general frame
4.3.1 Computation of the regularized tidal tensors
4.3.2 The Lagrangian
4.3.3 The EoM and conserved quantities
4.4 Lagrangian and conserved quantities in the CoM frame
4.5 The tidal Hamiltonian in isotropic coordinates
4.6 Tidal effects for quasi-circular orbits
5.1 The stress-energy tensor from the tidal effective action
5.1.1 Derivation of the covariant stress-energy tensor
5.2 The potentials
5.3 The source multipole moments
5.4 The GW flux
5.4.1 Computation of the tail terms
5.4.2 Computation of the flux
5.5 The GW phase
5.6 Conclusion of Part II
III Towards the 4PN phase
6 The 4PN source mass quadrupole
6.1 The general expression of the source mass quadrupole
6.2 The quadrupole moment as a function of potentials
6.2.1 The elementary potentials
6.2.2 The method of super-potentials
6.2.3 Integrations by part and surface terms
6.2.4 Expression in terms of potentials
6.3 The properly UV regularised quadrupole
6.3.1 Computation of C and distributional terms
6.3.2 Hadamard regularisation: computation of NC terms in 3d
6.3.3 Computation of the UV DDR for NC terms
6.3.4 Computation of surface terms in 3d
6.3.5 Final result for the UV regularised mass quadrupole
6.4 The properly IR regularised quadrupole
6.4.1 Computation of the DDR of potentials at point 1 and C terms
6.4.2 Computation of the d-dimensional potentials at spatial infinity
6.4.3 Computation of NC terms
6.4.4 Computation of the extra term and surface terms
6.4.5 Applying the IR shift
6.4.6 Final result for the source mass quadrupole
6.4.7 The IR DDR of the mass octupole
7 The 3PN radiative current quadrupole and the associated gravitational amplitude mode
7.1 The current multipole moments in d dimensions
7.2 The current quadrupole moment of compact binaries
7.2.1 Computation of the current quadrupole
7.2.2 Example of the super-potential method
7.2.3 Final result
7.2.4 The IR DDR for the current quadrupole
7.3 The gravitational-wave mode h21 at 3PN order
7.4 Test of the current quadrupole with a constant shift
7.5 Conclusion of Part III
Appendices
A Conventions and some technical aspects of General Relativity
A.1 General relativity
A.2 Notations and conventions in the PN approach
B The d-dimensional 4PN metric as function of potentials
C Proof that the Weyl tensor trace terms can be removed by a redefinition of the metric in the effective tidal action
D Lengthy expressions of the project on tidal effects
D.1 Conservative sector