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Parameterized Newtonian Theory. Tomoyuki Nakayama December 11, 2008. 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. What is PPN formalism?.

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Parameterized newtonian theory

Parameterized Newtonian Theory

Tomoyuki Nakayama

December 11, 2008


Outline
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


What is ppn formalism
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.


Alternative gravitational theories
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.


Historical background
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.


Ppn formalism by will
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



Application of ppn formalism to gr brans dicke theory

GR

Solution in PPN approximation

Brans-Dicke

Application of PPN formalism to GR & Brans-Dicke theory




Deflection time delay of light
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.


Experimental results
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)



Summary
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.


Thank you
Thank you!

  • Reference

  • Misner, Thorne, Wheeler, (1973) Gravitation

  • Nordvedt (1976) Phys. Rev. 169, 1017-1025

  • 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


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