Enhanced terms in the time delay and the direction of light propagation. Discussion for some solar system experiments

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1 Enhanced terms in the time delay and the direction of light propagation. Discussion for some solar system experiments P. Teyssandier Observatoire de Paris, Dépt SYRTE/CNRS-UMR 8630,UPMC Séminaire IHES, 6 Mars 2014 Séminaire IHES, 6 Mars /

2 Contents Introduction Interest of the time transfer functions (TTFs) for gravitational tests General considerations Expression of the static, spherically symmetric metric A natural idea which is useful, but reveals to be a false good idea... New methods for determining TTF for static, spherically symmetric metrics Expression of the TTF up to order G 3 Enhanced terms in TTF up to order G 3 Discussion for some solar system experiments Enhanced terms in the direction of light propagation Conclusion Séminaire IHES, 6 Mars /

3 Introduction General framework : metric theories of gravity post-newtonian parameters β, γ,... in GR: γ linked to the curvature of space and light propagation β linked to the non-linearity of the field The best current value for γ is due to the CASSINI Mission: γ 1 = (2.1 ± 2.3) 10 5 (Bertotti et al 2003) Recent improvements on γ (and β) from INPOP13a (Verma et al 2013). Scalar field in primordial cosmology expected violations of GR: γ 1 < 0 and, possibly, in the range (Damour & Lilley 2008) Many projects designed to measure γ at this level in the solar system: LATOR, ASTROD, ODYSSEY, SAGAS, GAME,... Séminaire IHES, 6 Mars /

4 Interest of the time transfer functions (TTFs) The basic ingredients: Space-time (V 4, g), g = physical metric (+ ). Geometric approximation : light is propagating along null geodesics of g. V 4 covered by a single quasi-cartesian coordinate system : x α = (x 0, x), x 0 = ct, g 00 > 0. Light rays emitted at x A at instant t A and received at x B at instant t B. In many tests of GR, the relevant tool is the expression of the light travel time t B t A between x A and x B as a function of (x A, t B, x B ) (resp. (t A, x A, x B )): t B t A = T r (x A, t B, x B ) = T e (t A, x A, x B ). T r (resp. T e ) = reception (resp. emission) time transfer function. Séminaire IHES, 6 Mars /

5 Interest of the time transfer functions (TTFs) Synchronization of distant clocks. Doppler tracking and time/frequency transfers ν B = (g g 0i β i + g ij β i β j ) 1/2 A ν A (g g 0i β i + g ij β i β j ) 1/2 B ( ). where β A/B = ( β i A/B) = 1 c dx i A/B dt 1 T r t B cβ B. xb T r, 1 + cβ A. xa T r Direction of light propagation at x A and x B (astrometry) ( ) ( ) li li la = = c xa T r, lb = = c x B T r l 0 l A 0 B 1 T, r t B where l α = g αβ dx β /dλ along the ray. Séminaire IHES, 6 Mars /

6 General considerations Determining the TTFs in a given space-time is inextricably complicated! Infinity of null geodesics passing through x A and x B (cf. Schwarzschild, e.g.). Even in Schwarschild space-time, no full expression of any TTF. Current assumptions : (V 4, g) asymptotically flat weak field: g µν (x, G) = η µν + n=1 g (n) µν (x, G), g (n) µν (x, G) = G n h (n) µν (x) light rays are quasi-minkowskian null geodesics: = T r (x A, t B, x B, G) = x B x A c + n=1 T r (n) (x A, t B, x B, G). Séminaire IHES, 6 Mars /

7 General considerations General methods at your disposal: Integration of the null geodesic equations (Shapiro 1964, Richter & Matzner 1983, Brumberg 1987, Kopeikin & Schäfer 1999, Klioner 2003) Iterative determination of Synge s world function (John 1975, Linet & Teyssandier 2002, Le Poncin-Lafitte et al 2004) Iterative solution of an eikonal equation (Teyssandier & Le Poncin-Lafitte 2008) A lot of results at order G contributions due to the non-sphericity or/and the motions of bodies. Séminaire IHES, 6 Mars /

8 General considerations However, higher order terms for the mass monopole contributions will be needed in future experiments (Minazzoli & Chauvineau 2011 and refs. therein). Several works devoted to the calculation of TTF in static, spherically symmetric (sss) metrics (John 1975, Richter & Matzner 1983, Brumberg 1987, Le Poncin-Lafitte et al 2004, Teyssandier & Le Poncin 2008, Klioner & Zschocke 2010). In sss space-time T r (x A, t B, x B ) T e (t A, x A, x B ) = T (x A, x B ). Until recently, T (x A, x B ) was known only up to order G 2. However, appearance of enhanced terms (Bertotti and Ashby, 2010) order G 3 is necessary. New methods developped to determine T at any order (Linet & Teyssandier 2013). Séminaire IHES, 6 Mars /

9 Expression of the static, spherically symmetric metric For a central body of mass M (m = GM/c 2 ), in isotropic coordinates: ds 2 = A(r)(dx 0 ) 2 B 1 (r) δ ij dx i dx j. Assumption : r h m r > r h A(r) > 0, B 1 (r) > 0 and A(r) = 1 2m r + 2β m2 r β m 3 3 r 3 + β m 4 4 r 4 + n=5 ( 1) n n 2 n 2 β m n n r n, B 1 (r) = 1 + 2γ m r ɛm2 r γ m 3 3 r γ m 4 4 r 4 + (γ n 1) mn r n. n=5 In GR: β = β 3 = β 4 = β 5 = = 1, γ = ɛ = γ 3 = γ 4 = = 1. Séminaire IHES, 6 Mars /

10 Expression of the static, spherically symmetric metric A null geodesic of g is also a null geodesic of the conformal metric where d s 2 = (dx 0 ) 2 U(r) δ ij dx i dx j, U(r) = 1 A(r)B(r) = 1 + 2(1 + γ)m r + 2κm2 r n=3 κ n m n r n. The parameters κ and κ 3 are given by κ = 2(1 + γ) β ɛ, κ 3 = 2κ 2β(1 + γ) β γ 3. In GR: κ = 15 4, κ 3 = 9 2. Séminaire IHES, 6 Mars /

11 A natural, useful idea... Assume that the ray has a unique pericentre P(r = r P ), has no apocentre and passes through P. The null geodesic equations lead to t B t A = 1 [ r 2 A r c 2 P + ] r 2 B r 2 P [ (1 + γ)m + ln (r A + r 2 A r 2 P )(r B + r 2 B r 2 P ) ra r P rb r P c r P r A + r P r B + r P { [ + m2 2κ arccos r P + arccos r ] [ P (1 + γ)2 ra r P rb r P cr P r A r B 4 r A + r P r B + r P ( ) 3/2 ( ) ]} 3/2 ( ) ra r P rb r P m 3 + O r A + r P r B + r P cr 2, P ] where r A = x A and r B = x B. = Determining γ at the level 10 8 t B t A at the level 0.7 ps. Séminaire IHES, 6 Mars /

12 which is a false good idea... So, is there a problem? YES! This expansion doesn t solve our problem : we don t know r = r P of x A and x B. as a function Fortunately, several other methods may be used (see above). Preference for procedures natively adapted to an emitter and a receiver both located at a finite distance of the origin of coordinates. Séminaire IHES, 6 Mars /

13 New methods for determining T for sss metrics First method: integration of an eikonal equation (Teyssandier & Le Poncin 2008, Linet & Teyssandier 2013) From ( ) li la = l 0 A = c xa T, lb = ( li l 0 ) B = c xb T. and g αβ l α l β = 0, it is inferred that T (x, x B ) obeys the eikonal equation c 2 x T (x, x B ) 2 = U(r) = 1 + 2(1 + γ) m r + 2 ( m κ n r n=2 ) n for any x (r = x ) and any x B such that r > r h and r B > r h. We are looking for light rays such that T (x A, x B ) = x B x A c + T (n) (x A, x B ). n=1 Séminaire IHES, 6 Mars /

14 New methods for determining T for sss metrics The T (n) (x A, x B ) s are given by T (n) (x A, x B ) = 1 c x B x A F (n) (x A, x B ), where the functions F (n) are determined by the recurrence law F (1) (x A, x B ) = (1 + γ)m F (n) (x A, x B ) = κ n m n 1 c n 1 1 p=1 dξ z(ξ) n 0 dξ z(ξ), [ ] x T (p) (x A, x). x T (n p) (x A, x) x=z(ξ) dξ, for n 2, with integrations on the straight segment joining x A and x B z(ξ) = x A + ξ(x B x A ). Séminaire IHES, 6 Mars /

15 New methods for determining T for sss metrics The recurrence law implies the uniqueness of T for quasi-minkowskian light rays a property of analyticity: The T (n) (x A, x B ) s are analytic functions if x A x B and n B n A. of crucial interest for the second method. The validity of the method is ensured only if x A and x B are such that x A + ξ(x B x A ) > r h ξ [0, 1]. Séminaire IHES, 6 Mars /

16 New methods for determining T for sss metrics Second method: integro-differential eq. inferred from the geodesic eqs. (Linet & Teyssandier 2013) Since T (x A, x B ) is analytic, we may assume that x A and x B are such that: The light ray joining x A and x B does not pass through a pericentre. At any point of the light ray r c < r A r r B, where r c is the Euclidean distance between the origin and the line passing through x A and x B, i.e. r c = r Ar B 1 µ 2, x B x A with µ = n A.n B, n A = x A r A, n B = x B r B. Séminaire IHES, 6 Mars /

17 New methods for determining T for sss metrics Second method: integro-differential eq. inferred from the geodesic eqs. (Linet & Teyssandier 2013) Then c(t B t A ) = b arccos µ + rb 1 r A r r 2 U(r) b 2 dr where b is the impact parameter (intrinsic quantity). Since b is linked to T by b = c 1 µ 2 T (r A, r B, µ) µ we have an integro-differential equation for T (r A, r B, µ)., Séminaire IHES, 6 Mars /

18 New methods for determining T for sss metrics Assuming that we find b = r c [1 + ( ) n m q n ], r c n=1 T (n) (r A, r B, µ) = 1 c ( ) n m rb r c r A 3n 4 k=0 U kn (q 1,..., q n 1 )r 3n k 2 c r k n+1 (r 2 rc 2 ) 2n 1 2 dr, where the U kn s are polynomials in q 1,..., q n 1. Using b = c 1 µ 2 T (r A, r B, µ) µ it may be seen that = q n = c ( rc ) n 1 µ 2 T (n) (r A, r B, µ), m r c µ T (1),..., T (n 1) q 1,..., q n 1 T (n). The analytic extension theorem = expressions of the T (n) s valid even if the initial assumptions are not met. Séminaire IHES, 6 Mars /

19 Expression of T up to order G 3 All the integrals can be calculated with any symbolic computer program. The two methods lead to the same results for n = 1, 2, 3: T (1) (x A, x B ) = (1 + γ)m c T (2) (x A, x B ) = m2 x B x A r A r B c ( 1 T (3) (x A, x B ) = m3 r A r B + r A ( ) ra + r B + x B x A ln r A + r B x B x A [ κ arccos µ (1 + γ)2 1 µ µ ) [ xb x A arccos µ κ 3 (1 + γ)κ 2 c(1 + µ) 1 µ r B (1 + γ)3 1 + µ ],, (Shapiro 1964) ], (Le Poncin et al 2004) where µ = n A.n B, n A = x A r A, n B = x B r B. Séminaire IHES, 6 Mars /

20 Enhanced terms up to order G 3 Case where x A and x B are in almost opposite directions (i.e. µ 1) o B N AB P AB A o n A o O n B where µ 1 = µ 2r Ar B r A r B (r A + r B ) 2 rc 2, r c = r Ar B 1 µ 2 x B x A. Séminaire IHES, 6 Mars /

21 Enhanced terms up to order G 3 Asymptotic expressions of the T (n) in a conjunction enhanced terms: ( ) T (1) enh (x (1 + γ)m 4rA r B A, x B ) ln, c (cf. Ashby & Bertotti 2010) T (2) enh (x A, x B ) 2 (1 + γ)2 m 2 c(r A + r B ) T (3) enh (x A, x B ) 4 (1 + γ)3 m 3 c(r A + r B ) 2 r 2 c r A r B, r 2 c ( ra r B r 2 c ) 2. Séminaire IHES, 6 Mars /

22 Enhanced terms up to order G 3 These expressions are reliable for n = 1, 2, 3 for configurations such that T (n) (x A, x B ) T (n 1) (x A, x B ), with T (0) (x A, x B ) = 1 c x B x A These conditions are satisfied if 2m r A + r B r A r B r 2 c 1 (C) Condition (C) is met in the solar system. For r B = 1 au and r A r B, one has 2m r A + r B r A r B r 2 c R2 r 2 c, R = solar radius. = Our results may be applied to the solar system experiments. Séminaire IHES, 6 Mars /

23 Discussion for solar system experiments Determination of γ in a SAGAS-like scenario: r A 50 au, r B 1 au Shapiro s formula = T 1 (1) γ T 2 R < r c < 5R = 158 µs > T (1) enh > 126 µs γ at the level 10 8 T at the level 0.7 ps Séminaire IHES, 6 Mars /

24 Discussion for solar system experiments In this configuration (times given in ps): r c /R T (1) S T (1) J 2 T (2) enh T κ (2) T (3) enh Conclusion: T (3) enh must be taken into account for rays almost grazing the Sun. Moreover, T (3) enh can be greater than the 1rst-order gravitomagnetic effect: with S kg m 2 s 1 ; T (1) S 2(1+γ)GS c 4 r c, the 1rst-order mass quadupole effect: T (1) J 2 with J (1+γ)m c J 2 R 2 r 2 c, Séminaire IHES, 6 Mars /

25 Discussion for solar system experiments This irruption of a G 3 -enhanced term is paradoxical. Remember that t B t A = 1 [ r 2 A r c 2 P + ] r 2 B r 2 P [ (1 + γ)m + ln (r A + r 2 A r 2 P )(r B + r 2 B r 2 P ) ra r P rb r P c r P r A + r P r B + r P { [ + m2 2κ arccos r P + arccos r ] [ P (1 + γ)2 ra r P rb r P cr P r A r B 4 r A + r P r B + r P ( ) 3/2 ( ) ]} 3/2 ( ) ra r P rb r P m 3 + O r A + r P r B + r P cr 2. P ] Numerically : m c = 4.9 µs, m 2 cr = 10.4 ps, m 3 cr 2 = 0.02 fs. Séminaire IHES, 6 Mars /

26 Discussion for solar system experiments Assume r A = r B = 1 au and r P = R 0th-order term : 499 s 1st-order term : 139 µas 2nd-order term : 162 ps = no enhanced effect predicted 3rd-order term < 1 fs = no enhanced effect predicted; negligible How can we surmount this apparent contradiction? Séminaire IHES, 6 Mars /

27 Discussion for solar system experiments r P and b are linked by b = r P U(rP ). Eliminating b between this relation and b = r c [ 1 + q 1 m r c + q 2 m 2 r 2 c + q 3 m 3 r 3 c + ], and then using q n = c ( rc ) n 1 µ 2 T (n) (r A, r B, µ), m r c µ r P may be expressed as a function of x A and x B : { r P = r c 1 (1 + γ q 1 ) m 1 ] m [2κ (1 + γ) 2 2 2q 2 r c 2 rc 2 1 ] } m [2κ 3 2κq 1 + (1 + γ) 2 3 q 1 2q r 3 c Séminaire IHES, 6 Mars /

28 Discussion for solar system experiments Substituting this expansion in the above expression of t B t A leads to t B t A = x B x A + T (1) (x A, x B ) + T (2) (x A, x B ) c +enhanced term of order G 3 + some regular terms of order G 3 ). When µ 1 q 1 = (1 + γ) r A + r B x B x A q 2 q 2 1, q 3 2q µ 1 + µ 2(1 + γ) ra r B r A + r B ra r B r c, r A = r B = 1 au, r c R = q = r P r c 630 km r A = 50 au, r B = 1 au, r c R = q = r P r c 1240 km So r P and r c are significantly different in a configuration of quasi-conjunction. Compare with b r P (1 + γ)m 3 km. Séminaire IHES, 6 Mars /

29 Enhanced terms in the direction of light propagation Substituting the T (n) (x A, x B ) into ( ) li la = = c xa T r, lb = l 0 A ( li l 0 ) B = c xb T r expansions of l A and l B in series in powers of m r c Séminaire IHES, 6 Mars /

30 Enhanced terms in the direction of light propagation l(0) B (x A, x B ) = N AB, l(1) (x (1 + γ)m B A, x B ) = r c l(2) B (x A, x B ) = m2 rc 2 [ + l(3) (x B A, x B ) = m3 r 2 A x B x A 2 r A [ 1 µ2 N AB (1 µ)p AB], x B x A {[ ] N AB + (1 + γ) 2 ( rb r 3 A (1 µ) rc 3 x B x A 3 ( [2(1 + γ) 3 ) (1 µ) 3/2 + 1 r A {{(1 + γ) 3 [ 2 + r B 1 + r B r A ) 2 1 µ 1 + µ µ ] P AB }, ] 1 µ (1 µ) r A ]P AB }, 1 + µ + } N AB where N AB = x B x A x B x A, P AB = N AB ( na n B n A n B ). Séminaire IHES, 6 Mars /

31 Enhanced terms in the direction of light propagation When µ 1 no enhanced term in l (1) B enhanced terms at higher orders: ( ) ( l(2) B l(3) B ) enh enh 4(1 + γ)2 m 2 r 2 c 16(1 + γ)3 m 3 r 3 c r B r c r 2 B rc 2 r 2 A (r A + r B ) 2 P AB, r 3 A (r A + r B ) 3 P AB. Séminaire IHES, 6 Mars /

32 Enhanced terms in the direction of light propagation 1. Deflection of light in a LATOR-like experiment: r A = r B = 1 au For a ray passing near the Sun ( ) ( χ (2) l(2) B enh l(2) A ) enh 4(1 + γ)2 m 2 r 2 c r A r B r c (r A + r B ) and For r c = R, χ (3) ( l(3) B ) enh ( l(3) A ) enh 16(1 + γ)3 m 3 r 3 c r A r B r A r B (r A + r B ) 2 rc 2. χ (2) 1.6 mas, χ (3) 3 µas (compare with Hees et al 2013). Séminaire IHES, 6 Mars /

33 Enhanced terms in the direction of light propagation 2. Deflection of light coming from infinity, with r B = 1 au (GAME, e.g.) χ (2) 4(1 + γ)2 m 2 r 2 c r B r c (Klioner & Zschocke 2010) and For r B = 1 au and r c = R, χ (3) 16(1 + γ)3 m 3 r 3 c ( rb r c ) 2. χ (2) 3.2 mas, χ (3) 12 µas. Séminaire IHES, 6 Mars /

34 Enhanced terms in the direction of light propagation The choice of r c is not truly relevant. Using the impact parameter b and inverting the expansion b = r c [1 + ( ) n m q n ], r c n=1 we get for the dominant terms in l B.P AB lb.p AB + m3 b 3 2(1 + γ)m r A b r A + r B r A r A + r B r A + κπm2 b 2 r A + r [ B 1 + { 4κ 3 + 2(1 + γ)κ r 2 A (r A + r B ) 2 = The enhanced contributions are absorbed by using b. ]} + O ( ) m 4. b 4 Séminaire IHES, 6 Mars /

35 Enhanced terms in the direction of light propagation This conclusion could be expected from the total bending of light 2(1 + γ)m χ = + κπm2 }{{ b }} b {{ 2 + [4κ m3 } b (1 + γ)κ 23 ] (1 + γ)3 +O }{{} ( ) m 4, b 4 with {}}{ 1.75 as {}}{ 11 µas for a light ray grazing the Sun {}}{ 0.1 nas is maintained using r P b = r P U(rP ) = r P { 1 + instead of b. One has (1 + γ)m r P [ 2κ (1 + γ) 2 ] m 2 r 2 P + O ( m 3 r 3 P ) }. Séminaire IHES, 6 Mars /

36 Conclusion For quasi-minkowskian light rays: Two new methods available for directly calculating the TTF and the direction of light propagation in s.s.s. symmetric space-times, at any given order in G. These calculations can be performed with any symbolic computer program. They can be extended to more general cases (stationary metrics, e.g.). It is explicitly checked that the two procedures lead to the same expressions of TTF up to order G 3. We confirm the occurrence of enhanced terms of order G 3 which must be taken into account for modelling time measurements reaching 1 ps light deflection at the level 1 µas They can be absorbed by using expansions in series in powers of m r P instead of m r c. or m b Séminaire IHES, 6 Mars /

37 References Ashby N & Bertotti B 2010 Class. Quantum Grav Bertotti B, Iess L & Tortora P 2003 Nature Brumberg V A 1987 Kinematics Phys. Celest. Bodies 3 6 Damour T & Lilley M 2008 String Theory, Gravity and Experiment, Les Houches Summer School in Theoretical Physics Session 87; arxiv: v1 Hees A, Bertone S & Le Poncin-Lafitte C 2013 Journées 2013 (poster) Hees A, Bertone S & Le Poncin-Lafitte C 2014 arxiv: John R W 1975 Exp. Tech. Phys Klioner S A 2003 Astron. J Klioner S A & Zschocke S 2010 Class. Quantum Grav Kopeikin S M & Schäfer G 1999 Phys. Rev. D Le Poncin-Lafitte C, Linet B & Teyssandier P 2004 Class. Quantum Grav Linet B & Teyssandier P 2002 Phys. Rev. D Linet B & Teyssandier P 2013 Class. Quantum Grav Minazzoli O & Chauvineau B 2011 Class. Quantum Grav Richter G W & Matzner R A 1983 Phys. Rev. D Shapiro I I 1964 Phys. Rev. Lett Teyssandier P& C. Le Poncin-Lafitte 2008 Class. Quantum Grav Teyssandier P 2012 Class. Quantum Grav Velma A K, Fienga A, Kaskar J, Manche H & Gastineau M 2013 arxiv: v2 Séminaire IHES, 6 Mars /