Hot Topics in General Relativity and Gravitation EQUATIONS OF MOTION OF COMPACT BINARIES at THE FOURTH POST-NEWTONIAN ORDER Luc Blanchet Gravitation et Cosmologie ( G R ε C O ) Institut d’Astrophysique de Paris 31 juillet 2017 Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 1 / 20
Summary of known PN orders Method Equations of motion Energy flux Waveform 3.5PN non-spin 1 Multipolar-post-Minkowskian & post-Newtonian 4PN non-spin 3.5PN non-spin (MPM-PN) 3.5PN (NNL) SO 4PN (NNL) SO 1.5PN (L) SO 3PN (NL) SS 3PN (NL) SS 2PN (L) SS 3.5PN (NL) SSS 3.5PN (NL) SSS Canonical ADM Hamiltonian 4PN non-spin 3.5PN (NNL) SO 4PN (NNL) SS 3.5PN (NL) SSS Effective Field Theory (EFT) 3PN non-spin 2PN non-spin 2.5PN (NL) SO 4PN (NNL) SS 3PN (NL) SS Direct Integration of Relaxed Equations (DIRE) 2.5PN non-spin 2PN non-spin 2PN non-spin 1.5PN (L) SO 1.5PN (L) SO 1.5PN (L) SO 2PN (L) SS 2PN (L) SS 2PN (L) SS Surface Integral 3PN non-spin Many works devoted to spins: Spin effects (SO, SS, SSS) are known in EOM up to 4PN order SO effects are known in radiation field up to 4PN SS in radiation field known to 3PN 1 The 4.5PN coefficient is also known Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 2 / 20
Summary of known PN orders Method Equations of motion Energy flux Waveform 3.5PN non-spin 1 Multipolar-post-Minkowskian & post-Newtonian 4PN non-spin 3.5PN non-spin (MPM-PN) 3.5PN (NNL) SO 4PN (NNL) SO 1.5PN (L) SO 3PN (NL) SS 3PN (NL) SS 2PN (L) SS 3.5PN (NL) SSS 3.5PN (NL) SSS Canonical ADM Hamiltonian 4PN non-spin 3.5PN (NNL) SO 4PN (NNL) SS 3.5PN (NL) SSS Effective Field Theory (EFT) 3PN non-spin 2PN non-spin 2.5PN (NL) SO 4PN (NNL) SS 3PN (NL) SS Direct Integration of Relaxed Equations (DIRE) 2.5PN non-spin 2PN non-spin 2PN non-spin 1.5PN (L) SO 1.5PN (L) SO 1.5PN (L) SO 2PN (L) SS 2PN (L) SS 2PN (L) SS Surface Integral 3PN non-spin Many works devoted to spins: Spin effects (SO, SS, SSS) are known in EOM up to 4PN order SO effects are known in radiation field up to 4PN SS in radiation field known to 3PN 1 The 4.5PN coefficient is also known Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 2 / 20
The 4PN equations of motion THE 4PN EQUATIONS OF MOTION Based on collaborations with Laura Bernard , Alejandro Boh´ e , Guillaume Faye & Sylvain Marsat [PRD 93 , 084037 (2016); PRD 95 , 044026 (2017); PRD submitted (2017)] Tanguy Marchand , Laura Bernard & Guillaume Faye [PRL submitted (2017)] Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 3 / 20
The 4PN equations of motion The 1PN equations of motion [Lorentz & Droste 1917; Einstein, Infeld & Hoffmann 1938] � � � d 2 r A Gm B Gm C Gm D 1 − r AB · r BD � � � = − 1 − 4 − n AB r 2 r 2 d t 2 c 2 r AC c 2 r BD AB BD B � = A C � = A D � = B � �� + 1 B − 4 v A · v B − 3 v 2 A + 2 v 2 2( v B · n AB ) 2 c 2 G 2 m B m D Gm B v AB [ n AB · (3 v B − 4 v A )] − 7 � � � + n BD c 2 r 2 c 2 r AB r 3 2 AB BD B � = A B � = A D � = B Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 4 / 20
The 4PN equations of motion 4PN: state-of-the-art on equations of motion d v i d t = − Gm 2 1 n i 12 r 2 12 1PN Lorentz-Droste-Einstein-Infeld-Hoffmann term � �� � �� 5 G 2 m 1 m 2 � � + 4 G 2 m 2 1 2 n i + + · · · 12 + · · · c 2 r 3 r 3 12 12 � 1 � + 1 + 1 + 1 + 1 1 c 4 [ · · · ] c 5 [ · · · ] c 6 [ · · · ] c 7 [ · · · ] + c 8 [ · · · ] + O c 9 � �� � � �� � � �� � � �� � � �� � 2PN 2.5PN 3PN 3.5PN 4PN radiation reaction radiation reaction conservative & radiation tail [Otha, Okamura, Kimura & Hiida 1973, 1974; Damour & Sch¨ afer 1985] ADM Hamiltonian [Damour & Deruelle 1981; Damour 1983] Harmonic coordinates 2 PN [Kopeikin 1985; Grishchuk & Kopeikin 1986] Extended fluid balls [Blanchet, Faye & Ponsot 1998] Direct PN iteration [Itoh, Futamase & Asada 2001] Surface integral method Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 5 / 20
The 4PN equations of motion 4PN: state-of-the-art on equations of motion d v i d t = − Gm 2 1 n i 12 r 2 12 1PN Lorentz-Droste-Einstein-Infeld-Hoffmann term � �� � �� 5 G 2 m 1 m 2 � � + 4 G 2 m 2 1 2 n i + + · · · 12 + · · · c 2 r 3 r 3 12 12 � 1 � + 1 + 1 + 1 + 1 1 c 4 [ · · · ] c 5 [ · · · ] c 6 [ · · · ] c 7 [ · · · ] + c 8 [ · · · ] + O c 9 � �� � � �� � � �� � � �� � � �� � 2PN 2.5PN 3PN 3.5PN 4PN radiation reaction radiation reaction conservative & radiation tail [Jaranowski & Sch¨ afer 1999; Damour, Jaranowski & Sch¨ afer 2001ab] ADM Hamiltonian [Blanchet-Faye-de Andrade 2000, 2001; Blanchet & Iyer 2002] Harmonic EOM 3 PN [Itoh & Futamase 2003; Itoh 2004] Surface integral method [Foffa & Sturani 2011] Effective field theory [Jaranowski & Sch¨ afer 2013; Damour, Jaranowski & Sch¨ afer 2014] ADM Hamiltonian 4 PN [Bernard, Blanchet, Boh´ e, Faye, Marchand & Marsat 2015, 2016, 2017ab] Fokker Lagrangian [Foffa & Sturani 2012, 2013] (partial results) Effective field theory Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 5 / 20
The 4PN equations of motion Fokker action of N particles [Fokker 1929] Gauge-fixed Einstein-Hilbert action for N point particles 1 � c 3 d 4 x √− g � R − 1 � 2 g µν Γ µ Γ ν S g.f. = 16 πG � �� � Gauge-fixing term � � � m A c 2 − ( g µν ) A v µ A v ν A /c 2 − d t A � �� � N point particles Fokker action is obtained by inserting an explicit PN solution of the Einstein 2 field equations g µν ( x , t ) − → g µν ( x ; y B ( t ) , v B ( t ) , · · · ) The PN equations of motion of the N particles (self-gravitating system) are 3 � ∂L F � δS F ≡ ∂L F − d + · · · = 0 δ y A ∂ y A d t ∂ v A Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 6 / 20
The 4PN equations of motion Problem of point particles and UV divergences x + Gm 1 Gm 2 m 1 m 2 U ( x , t ) = | x − y 1 ( t ) | + | x − y 2 ( t ) | y 1 y (t) (t) 2 d 2 y 1 y 1 − y 2 ? d t 2 = ( ∇ U ) ( y 1 ( t ) , t ) = − Gm 2 | y 1 − y 2 | 3 For extended bodies the self-acceleration of the body cancels out by Newton’s action-reaction law For point particles one needs a self-field regularization to remove the infinite self-field of the particle Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 7 / 20
The 4PN equations of motion Dimensional regularization for UV divergences [t’Hooft & Veltman 1972; Bollini & Giambiagi 1972; Breitenlohner & Maison 1977] Einstein’s field equations are solved in d spatial dimensions (with d ∈ C ) with 1 distributional sources. In Newtonian approximation ∆ U = − 4 π 2( d − 2) d − 1 Gρ For two point-particles ρ = m 1 δ ( d ) ( x − y 1 ) + m 2 δ ( d ) ( x − y 2 ) we get 2 � d − 2 � � � k = Γ U ( x , t ) = 2( d − 2) k Gm 1 Gm 2 2 | x − y 1 | d − 2 + with d − 1 | x − y 2 | d − 2 d − 2 π 2 Computations are performed when ℜ ( d ) is a large negative number, and the 3 result is analytically continued for any d ∈ C except for isolated poles Dimensional regularization is then followed by a renormalization of the 4 worldline of the particles so as to absorb the poles ∝ ( d − 3) − 1 Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 8 / 20
The 4PN equations of motion Fokker action in the PN approximation We face the problem of the near-zone limitation of the PN expansion Lemma 1: The Fokker action can be split into a PN (near-zone) term plus a contribution involving the multipole (far-zone) expansion � r � r � � � B � B S g d 4 x d 4 x F = FP L g + FP M ( L g ) r 0 r 0 B =0 B =0 Lemma 2: The multipole contribution is zero for any “instantaneous” term thus only “hereditary” terms contribute to this term and they appear at least at 5.5PN order � r � � B S g d 4 x F = FP L g r 0 B =0 The constant r 0 will play the role of an IR cut-off scale IR divergences appear at the 4PN order Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 9 / 20
The 4PN equations of motion Gravitational wave tail effect at the 4PN order [Blanchet & Damour 1988; Blanchet 1993, 1996] field point At the 4PN order there is an imprint of gravitational wave tails in the local 1.5PN (near-zone) dynamics of the source This leads to a non-local-in-time contribution in the Fokker action 4PN This corresponds to a 1.5PN modification of the radiation field beyond the quadrupole approximation (already tested by LIGO) matter source �� = G 2 M d t d t ′ | t − t ′ | I (3) ij ( t ) I (3) S tail ij ( t ′ ) 5 c 8 Pf F s 0 where the Hadamard partie finie (Pf) is parametrized by an arbitrary constant s 0 Luc Blanchet ( G R ε C O ) 4PN equations of motion HTGRG 10 / 20
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