From quantum to classical scattering in post-Minkowskian gravity Early Stage Researcher: Andrea Cristofoli Niels Bohr Institute, Copenhagen November 08, 2019 This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764850 (SAGEX). 1 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Outline Post-Minkowskian (PM) physics ( | h µν | << 1 , v c ∼ 1) has been studied in General Relativity by more than 70 years 2 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Outline Post-Minkowskian (PM) physics ( | h µν | << 1 , v c ∼ 1) has been studied in General Relativity by more than 70 years The PM scattering angle can be used to construct improved gravitational wave templates 2 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Outline Post-Minkowskian (PM) physics ( | h µν | << 1 , v c ∼ 1) has been studied in General Relativity by more than 70 years The PM scattering angle can be used to construct improved gravitational wave templates Scattering amplitudes naturally provides this observable: can we improve the method to high PM order? 2 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Outline Post-Minkowskian (PM) physics ( | h µν | << 1 , v c ∼ 1) has been studied in General Relativity by more than 70 years The PM scattering angle can be used to construct improved gravitational wave templates Scattering amplitudes naturally provides this observable: can we improve the method to high PM order? Formula connecting M and θ PM to all orders (no potentials) Main result � ˜ � k 1 � ∞ ∞ M cl . ( r , p ∞ ) r 2 2 b � du ( ∂ b 2 ) k θ PM = p 2 r 2 k ! 0 ∞ k = 1 � u 2 + b 2 r = 2 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
The two-body problem in General Relativity is given by R µν − 1 2 g µν R = 8 π G N a = − Γ µ u µ αβ ( g µν ) u α a u β T µν ˙ , a c 4 3 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
The two-body problem in General Relativity is given by R µν − 1 2 g µν R = 8 π G N a = − Γ µ u µ αβ ( g µν ) u α a u β T µν ˙ , a c 4 .... no general solution is known! 3 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
The two-body problem in General Relativity is given by R µν − 1 2 g µν R = 8 π G N a = − Γ µ u µ αβ ( g µν ) u α a u β T µν ˙ , a c 4 .... no general solution is known! If we split the dynamics into several regimes of motion, we can use the EOB approach to provide an approximate solution 3 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
The two-body problem in General Relativity is given by R µν − 1 2 g µν R = 8 π G N a = − Γ µ u µ αβ ( g µν ) u α a u β T µν ˙ , a c 4 .... no general solution is known! If we split the dynamics into several regimes of motion, we can use the EOB approach to provide an approximate solution 3 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Improving the EOB (2016) Damour has proven that the fully relativistic scattering angle with arbitrary masses, θ PM , could be used in the EOB to improve gravitational waves templates (1609.00354) 4 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Improving the EOB (2016) Damour has proven that the fully relativistic scattering angle with arbitrary masses, θ PM , could be used in the EOB to improve gravitational waves templates (1609.00354) � + ∞ Classical physics ∆ p µ a p β a = − 1 −∞ d σ a ∂ µ g αβ ( x a ) p α a 2 4 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Improving the EOB (2016) Damour has proven that the fully relativistic scattering angle with arbitrary masses, θ PM , could be used in the EOB to improve gravitational waves templates (1609.00354) � + ∞ Classical physics ∆ p µ a p β a = − 1 −∞ d σ a ∂ µ g αβ ( x a ) p α a 2 P µ Covariant approaches ( Kosower et al. ): � ψ | ∆ˆ a | ψ � 4 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Improving the EOB (2016) Damour has proven that the fully relativistic scattering angle with arbitrary masses, θ PM , could be used in the EOB to improve gravitational waves templates (1609.00354) � + ∞ Classical physics ∆ p µ a p β a = − 1 −∞ d σ a ∂ µ g αβ ( x a ) p α a 2 P µ Covariant approaches ( Kosower et al. ): � ψ | ∆ˆ a | ψ � Potential based approaches: M PM V PM θ PM 4 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Improving the EOB (2016) Damour has proven that the fully relativistic scattering angle with arbitrary masses, θ PM , could be used in the EOB to improve gravitational waves templates (1609.00354) � + ∞ Classical physics ∆ p µ a p β a = − 1 −∞ d σ a ∂ µ g αβ ( x a ) p α a 2 P µ Covariant approaches ( Kosower et al. ): � ψ | ∆ˆ a | ψ � Potential based approaches: M PM V PM θ PM State of the art Bern et al. has computed θ PM ∼ G 3 N with a V PM based approach, but the method is hard to implement at higher PM orders 4 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
<latexit sha1_base64="9jZdKXrQUNruzT8aXYXAg30ZUcM=">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</latexit> 1 Compute the scattering amplitude of a 2 → 2 process between two scalar massive particles exchanging gravitons (C.M. frame) p 2 p 4 p ′ ) p ′ | = M ( � p , � , | � p | = | � p 1 p 3 5 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
<latexit sha1_base64="9jZdKXrQUNruzT8aXYXAg30ZUcM=">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</latexit> 1 Compute the scattering amplitude of a 2 → 2 process between two scalar massive particles exchanging gravitons (C.M. frame) p 2 p 4 p ′ ) p ′ | = M ( � p , � , | � p | = | � p 1 p 3 2 Calculate a post-Minkowskian potential V PM from M (e.g. Lippman-Schwinger equation / EFT approaches) 5 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
<latexit sha1_base64="9jZdKXrQUNruzT8aXYXAg30ZUcM=">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</latexit> 1 Compute the scattering amplitude of a 2 → 2 process between two scalar massive particles exchanging gravitons (C.M. frame) p 2 p 4 p ′ ) p ′ | = M ( � p , � , | � p | = | � p 1 p 3 2 Calculate a post-Minkowskian potential V PM from M (e.g. Lippman-Schwinger equation / EFT approaches) n ) ˜ � M ( � p , � V PM ( � n , � p ′ ) ˜ V PM ( � p ′ ) = M ( � p ′ ) − p , � p , � E p − E n + i ǫ n � 5 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
3 The fully relativistic scattering angle θ PM is given by � + ∞ � p 2 ( r ) − L 2 θ PM = − 2 ∂ r 2 − π dr ∂ L r min 6 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
3 The fully relativistic scattering angle θ PM is given by � + ∞ � p 2 ( r ) − L 2 θ PM = − 2 ∂ r 2 − π dr ∂ L r min p 2 ( r ) is the curve in the phase space ( p , r ) which solves H ( r , p ( r )) = E p r ( r min ) = 0 , 6 / 11 Andrea Cristofoli Scattering amplitudes and General Relativity
Recommend
More recommend
