Further examples of diagrams corresponding to known effects

In order to further elucidate the rationale behind the decomposition of gravitational perturbations in high frequency modes and a slowly varying background (2.38), it is instructive to present here some Feynman diagrams that correspond to physical effects pertaining the different regimes of conservative and radiative dynamics. An interesting case is the one related with spin phenomena. The inclusion of spin degrees of freedom in NRGR has been addressed in refs. [97, 98]: here, the conservative dynamics is studied at leading order and the corresponding spin-orbit and spin-spin potentials are reproduced.

Spin effects are incorporated by adding the world-line degrees of freedom Λ^J_a(λ), where λ is the affine parameter of the world-line and Λ^J_a is the boost that transforms the locally flat frame, labelled by a, to the co-rotating frame labelled by J. These frames are defined by vierbeins, which are given by e^µ_a and e^µ_I =e^µ_aΛ^a_I for the locally flat and co-rotating frame respectively and verify e^a_µe^b_νg^µν = η^ab. Then one can introduce the generalized angular velocity given by Ω^µν = e^µJ(De^ν_J/dλ). Finally, the spin is defined as the tensor S_µν that is the conjugate momentum to Ω_µν so that the form of the world-line action reads [98] action one can derive the Mathisson-Papapetrou equations of motion for spinning test particles [99, 100]. The spin-gravity coupling in (2.89) can be rewritten by introducing the Ricci rotation coefficients ω^ab_µ =e^b_νDµe^aν [97]

Sspin−grav =−1 2

Z

SLabω^ab_µu^µdλ , (2.90)

7. See the discussions about eq. (2.23) in Sec. 2.3.1 and eq. (1.52) in Sec. 1.4.

v¹ v⁰

(a)

v²

(b)

Figure 2.19: Panel (a): Feynman diagrams representing the spin-orbit interaction at leading order. Panel (b): Feynman diagram representing the spin-spin interaction at leading order. A grey blob represents an insertion of the spin operator in the point particle action; the factors of vⁿ correspond to the suppression with respect to the leading order coupling of a potential mode to a world-line (see eq. (2.50)). Figures taken from ref. [97].

with S_L^ab ≡ S^µνe^a_µe^b_ν, the spin in the locally flat frame. Expanding (2.90) in the weak gravity limit one obtains the Feynman rules for NRGR with spin [97, 98]; for example, at lowest order, the spin coupling of a potential graviton to a matter source reads

L⁽⁰⁾_{SP IN} = 1 2MPl

S_L^ik∂_kH_i0 . (2.91)

By making use of similar rules for higher order terms, one can derive the conservative dynamics for a binary system in the case of spinning bodies. At leading order the spin-orbit and spin-spin interactions are represented by the Feynman diagrams of Fig. 2.19:

as one can see from the figure, with respect to Newtonian potential, the spin-orbit interaction is O(v³) while the spin-spin is O(v⁴), i.e. these interactions constitute higher PN corrections than the EIH Lagrangian of Fig. 2.11: this explains why we did not take spin into account so far. From Sec. 1.3 we know that it has recently become possible to measure and constrain the general relativistic spin-orbit interaction by pulsar timing, even if with a slightly worse accuracy than other post-Keplerian parameters (cfr. Fig. 1.13).

As discussed at the end of the previous section, future GW detections at inter-ferometers will probe the non-linear dynamics of GR in both its conservative and its radiative regimes through monitoring the phase of the waveform. Concerning the dis-sipative dynamics, the presence of spin affects the radiative multipole moments, like the mass quadrupole Iij which equals Qij at leading order. At next-to-leading order, Iij receives contributions from the spin of both objects: these terms have been recently calculated in ref. [8] and the corresponding diagrams are represented in Fig. 2.20. The confrontation of Fig. 2.19 with Fig. 2.20 constitutes another manifestation of the dif-ferent roles that potential and radiation modes have in NRGR.

Another interesting example of the complementarity ofH and ¯his provided by phe-nomena which are still to be investigated in NRGR: the tail of the radiation, discussed in ref. [66], and the non-linear memory effect, predicted in refs. [101, 102]. These are

78 2.3 The effective theory for radiation

Figure 2.20: NRGR diagrams describing some of the spin contributions to the quad-rupole moment at next-to-leading order. The spin-graviton vertex of particle i is still represented with a grey blob in these diagrams it is labeled by S_i. Figure taken from ref. [8].

peculiar non-linearities of GR and their detection would then be an invaluable confirm-ation of the validity of the theory. The possibility of testing GR through detection of the tail effect has been considered in ref. [103] while the importance of the non-linear memory has been recently investigated in ref. [104].

The tails of GWs result from the non-linear interaction between the quadrupole radiation generated by an isolated system with total mass–energy M and the static monopole field associated with M. Their contributions to the field at large distances from the system include a particular effect of modulation of the phase in the Fourier do-main, which has M as a prefactor and depends on the frequency ωGW as ”ωGWlnωGW”.

On the other hand, the non-linear memory effect is a slowly-growing, oscilla-tory contribution to the GW amplitude. It originates from the fact that, in a non-linear theory like GR, GWs can themselves constitute the source of GWs. In an ideal interferometer a GW with memory causes a permanent displacement of the test masses that persists after the wave has passed. Remarkably, the non-linear memory affects the signal amplitude starting at leading order, i.e. the same as the Newtonian quadrupole;

however, due to thenon-oscillatory behavior, detecting the memory at interferometers is cumbersome.

In both the tail and memory phenomena, gravity non-linearities affect the propaga-tion of the emitted waves. Therefore, in NRGR both effects will be represented by diagrams where the orbital scale has been integrated out and the outermost field line is a radiation graviton ¯h resulting from a non-linear interaction with other gravitons.

The situation is described in Fig. 2.21 where the arrows on the double lines indicate the flow of time for the composite binary source. In the case of the non-linear memory, panel (b), the outgoing graviton results from the interaction between a wave emitted at time t₁ and a wave emitted at time t₂ > t₁, so the three-graviton vertex involves only gravitons of radiative type. For what concerns the tail instead, panel (a), the GW produced at a timet2 back-scatters off the pre-existent background gravitational field⁸: for this reason, one of the gravitons in the three-vertex is indicated with a dashed line.

8. This long-range static field is due to the two-body system so it is described by a potential mode that is not the same as the one integrated out to take care of the orbital dynamics.

(a) (b)

Figure 2.21: NRGR description of the non-linearities affecting the vacuum propagation of a GW: panel (a) reports the tail of the radiation, panel (b) the non-linear memory.

Figures stemming from a discussion with Ira Rothstein at Carnegie Mellon University.