1. Parameterized Newtonian Theory
Tomoyuki Nakayama
December 11, 2008

2. Outline Introduction Why is it useful?/How is it used? Theory
General formulation of PPN formalism
Application to GR and Brans-Dicke theory
Experimental tests
Time delay/Light deflection
PPN enables us to compare and classify alternative metric theories.

3. What is PPN formalism?
Parameterized Post-Newtonian formalism express Einstein's’ equation in terms of deviation from Newtonian theory.
Well describes weak field. Bring us further comprehension on Solar-system.
Theoretical foundation for experimental tests for alternative (metric) gravitational theory.

4. Alternative Gravitational Theories
Metric theories
Have symmetric metric
Test bodies follow geodesics of the metric
In local Lorentz frames, non-gravitational laws are those of SR.
GR is promising so should be similar to GR.

5. Historical Background
A. S. Eddington (1922) – Analysis of vacuum gravitational field outside the Sun.
K. Nordtvedt (1968) developed the first full PPN formalism.
C. M. Will introduced hydrodynamical description.

6. PPN Formalism by Will
γ - How much space curvature gij is produced by unit rest mass ?
β - How much nonlinearity is there in the superposition law for gravity g00 ?
β1 - How much gravity is produced by unit kinetic energy  ?
β2 - How much gravity is produced by unit gravitational potential energy ρ0 / U ?
β3 - How much gravity is produced by unit internal energy ρ0Π ?
β4 - How much gravity is produced by unit pressure p ?
ζ - Difference between radial and transverse kinetic energy on gravity
η - Difference between radial and transverse stress on gravity
Δ1 - How much dragging of inertial frames g0j is produced by unit momentum ρ0v ?
Δ2 - Difference between radial and transverse momentum on dragging of inertial frames

7. Metric & Stress Tensors in PPN Formalism
in standard gauge
T== (e+p)uu-pg ABVW is density velocity displacement integral

8. Application of PPN formalism to GR & Brans-Dicke theory
Solution in PPN approximation
Brans-Dicke
metric correction

9. PPN parameters in GR and Bran-Dicke Theory

10. New Version of PPN parameterization

11. Deflection/Time Delay of Light
One of the PPN parameter (γ) is related to the deflection of light.
It also describes the time delay of light.

12. Experimental Results VLBI light deflection 0.02 % from unity (1995).
Cassini spacecraft agrees with GR to 10-3 %. γ-1 = (2.1±2.3)×10-5 (2003)
VLBI(very long baseline interferometry) harvard astronomical interferometry
thousands of kilometers in diameter
Doppler tracking of the cassini spacecraft

13. Current limits on PPN parameters

14. Summary
PPN formalism, with sophistication of experimental technique, enabled to test the validity of metric theories.
GR has survived through all the experimental testing so far, while myriads of alternative gravitational theories disappeared.

15. Thank you! Reference Misner, Thorne, Wheeler, (1973) Gravitation
Nordvedt (1976) Phys. Rev. 169,
Will. Living Rev. Relativity 9 (2006), 3
Chandrasekhar, Phys. Rev. Lett. vol 14, 1965
Nutku, Astro. Journal, vol 155, March 1969
Blanchet, Living Rev. Relativity 9 (2006), 4
Clifford Will