Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as...
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![Page 1: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/1.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
World function and as astrometry
Christophe Le Poncin-Lafitte and Pierre Teyssandier
Observatory of Paris, SYRTE CNRS/UMR8630
![Page 2: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/2.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Modeling light deflection
Shape of bodies (multipolar structure)
• We must take into account Motion of the
bodies
Several models based on integration of geodesic differential equations to obtain the path of the photon :- Post-Newtonian approach Klioner & Kopeikin (1992)
Klioner (2003)
- Post-Minkowskian approach Kopeikin & Schäfer (1999) Kopeikin & Mashhoom (2002)
We propose- Use of the world function spares the trouble of geodesic
determination
- Post-Post-Minkowskian approach for the Sun (spherically symmetric case)
- Post-Minkowskian formulation for other bodies of Solar System
![Page 3: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/3.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
The world function• 1. Definition
SAB= geodesic distance between xA and xB
for timelike, null and spacelike geodesics, respectively
• 2. Fundamental properties
- Given xA and xB, let be the unique geodesic path joining xA and xB , vectors tangent to at xA and xB
(xA,xB) satisfies equations of the Hamilton-Jacobi type at xA and xB :
AB is a light ray
Deduction of the time transfer function
![Page 4: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/4.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Post-Minkowskian expansion of (xA,xB)
• The post-post-Minkowskian metric may be written as
Field of self-gravitating, slowly moving sources :
• The world function can be written as
where and
![Page 5: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/5.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Using Hamilton-Jacobi equations, we find
• and the general form of (2)
where
and
the straight line connecting xA and xB
(Cf Synge)
![Page 6: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/6.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Relativistic astrometric measurement
• Consider an observer located at xB and moving with an unite 4-velocity u
• Let k be the vector tangent to the light ray observed at xB. The projection of k obtained from the world function on the associated 3-plane in xB orthogonal to u is
• => Direction of the light ray :
![Page 7: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/7.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Applications to the as accuracy
• For the light behaviour in solar system, we must determine :
– The effects of planets with a multipolar structure at 1PN
– The effect of post-post-Minkowskian terms for the Sun (spherically symmetric body)
• We treat the problem for 2 types of stationary field :– Axisymmetric rotating body in the Nordtvedt-Will PPN formalism
– Spherically symmetric body up to the order G²/c4 (2PP-Minkowskian approx.)
![Page 8: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/8.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Case of a stationary axisymmetric body within the Will-Nordtvedt PPN formalism
• From (1) , it has been shown (Linet & Teyssandier 2002) for a light ray
where F(x,xA,xB) is the Shapiro kernel function
•For a stationary space-time, we have for the tangent vector at xB
![Page 9: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/9.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
• As a consequence, the tangent vector at xB is
Where
With a general definition of the unite 4-velocity
=> Determination of the observed vector of light direction in the 3-plane in xB
![Page 10: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/10.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Post-Post-Minkowskian contribution of a static spherically symmetric body
• Consider the following metric (John 1975, Richter & Matzner 1983)
• We obtain for (xA,xB)
with
and
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Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Time transfer and vector tangent at xB up to the order G²/c4
where
We deduce the time transfer (for a different method in GR, see Brumberg 1987
Vector tangent at xB is obtained
![Page 12: Relativity and Reference Frame Working group – Nice, 27-28 November 2003 World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649f1c5503460f94c32034/html5/thumbnails/12.jpg)
Relativity and Reference Frame Working group – Nice, 27-28 November 2003
Conclusions
• Powerful method to describe the light between 2 points located at finite distance without integrating geodesic equations.
• Obtention of time transfer and tangent vector at the reception point with all multipolar contributions in stationary space-time at 1PN approx.
• Obtention of time transfer and tangent vector at the reception point in spherically symmetric space-time at 2PM.
• Possibility to extend the general determination of the world function at any N-post-Minkowskian order (in preparation).
• To consider the problem of parallax in stationary space-time.
• To take into account motion of bodies.