Higher Post Newtonian Gravity Calculations

Welcome to my page on using the techniques usually associated with perturbative (quantum) field theory -- see here and here -- to calculate general relativistic corrections to the Newtonian gravitational lagrangian for point masses.

Update Dec 2011: I have written the Mathematica package TensoriaCalc to compute geometric tensors - Riemann, Ricci, Christoffel symbols, etc. - within a Riemannian geometry framework. This is closely related to a parallel effort of mine to write a package that can manipulate abstract tensors in physicists' index notation. (I already have a beta version of TensoriaLite, which reproduces some of the tensor contraction capabilities of FeynCalc, including the computation of the N-graviton Feynman rules for the Einstein-Hilbert lagrangian.) Stay tuned!

Motivation    To measure gravitational waves produced by binary black holes and/or neutron stars as they spiral towards each other, gravitational wave observatories such as GEO, LIGO, TAMA, and VIRGO need the effective lagrangian of 2 point masses up to 3 PN and beyond: O[(v/c)6], and higher, relative to Newtonian gravity, where v is the typical speed of the binaries in its center of energy frame and c is the speed of light. The effective lagrangian is used to construct theoretical templates so that the raw data can be compared against them to see if there is significant enough correlation to claim detection; see Blanchet for a review. For now, I am focusing on the integer PN order, conservative portion of the dynamics.

On the other hand, experiments such as APOLLO, GTDM, LATOR, BEACON, etc. are beginning to probe the non-Euclidean nature of the solar system's geometry beyond the currently well tested O[(v/c)2], by measuring the timing and deflection of light propagation within the solar system to higher accuracy than before. Knowing the geometry of the solar system at a theoretical level amounts to knowing the effective lagrangian for n point masses, and vice versa.

In view of the gravitational wave detection efforts, because the calculation at such a high order is very arduous, it is imperative that the necessary software be developed to automate the computation as far as possible.

Code for post Newtonian diagrams, related tools

This section contains the code I used to compute the n-body effective lagrangian up to O[(v/c)4]. Please do let me know if you find errors / bugs.

Update 28 Jan 2009: I have added in the 0 thru 2 PN Mathematica notebooks some description for FeynD, the notation used to represent Feynman diagrams.
  • FeynCalc    I assume you have Mathematica. Install the high energy physics package 
  • Generator of fully connected diagrams    To generate the fully connected Feynman diagrams whose evaluation would yield the effective lagrangian at a given PN order, first load the following and do Kernel > Evaluation > Evaluate Initialization:
  • Sample files for Newtonian thru 2 PN    Next, look at the following files for Newtonian thru 2 PN to understand the format / notation of the "Feynman diagrams" generated. A portion of the diagrams can be calculated automatically.
  • N-graviton Feynman rules    The diagrams that need to be evaluated by a human being can be done so manually within each cell. To calculate diagrams involving graviton vertices, we need the N-graviton Feynman rules for N > 1. Here I have used the linearized de Donder gauge. Simply open the following file, do Kernel > Evaluation > Evaluate Initialization and the Feynman rule command GravitonNVert is ready for use.
  • Differentiation and integration    For manipulation and differentiation of Euclidean distances, vectors, velocities, accelerations, etc., as well as the computation of the Feynman integrals, it helps to have some tools handy. Again, do Kernel > Evaluation > Evaluate Initialization after opening the file.
Future developments

To push our knowledge of binary dynamics beyond the currently known 3.5 PN, O[(v/c)7], using field theory, the first steps to take would be to improve the efficiency of the codes in IntegerPN_FullyConnectedDiagrams.nb and EinsteinHilbertLagrangianFeynmanRules_LinearizeddeDonderGauge.nb.