Motions of Self-Gravitating bodies to the Second Post- Newtonian Order of General Relativity

Primary motivation: the detection of gravitational waves –Currently ground based interferometers, such as LIGO, are conducting science runs in search of gravitational waves –Plans for a space based interferometer, LISA –The systems that provide the best chance of producing detectable gravitational waves in the frequency range of these detectors are inspiralling compact binary systems –Because of the data analysis method that will be used very accurate theoretical waveforms and equations of motion are needed

It is a major theoretical challenge to derive equations of motion for these systems Two primary methods used –Numerical relativity Uses computers to solve the full Einstein equations Used for the final few orbits –Approximation methods We use the post-Newtonian approximation Valid for early orbits

The post-Newtonian Approximation –Assumes slow motion and weak gravity –First order recovers Newtonian gravity –The second order, called the first post-Newtonian order (1PN), provides the first relativistic corrections to Newtonian gravity –Equations of motion are known through the 3.5PN order Half orders (2.5PN and 3.5PN) are the radiation reaction terms

When calculating the post-Newtonian equations of motion three types of terms are encountered: –Terms that depend on positive powers of the size of the body, s Tidal terms and their relativistic generalizations –Terms that are of the order s^0 Give the point mass equations of motion –Terms that depend on negative powers of s Self-energy terms

Self-energy terms should not contribute to the final equation of motion –They depend on the internal structure of the bodies, this would violate the strong equivalence principle We want to solve the post-Newtonian potentials keeping self-energy terms and show that these terms either vanish, using various virial theorems, or can be renormalized into the definition of the mass –This has been done for the 1PN equations of motion –Previous 2PN work by Kopeikin assumed spherical bodies and internal pressure If it can be shown that the equations of motion are independent of internal structure then this provides confidence that they could be valid for neutron stars and black holes

Calculation to 1PN Order –The first term is of order s^0 –The second term is of order s^(-5/2) since –The third term is of order s^(-1) An example of a 1PN term: –where

Keeping the terms that are of the order of s^0 and lower, the 1PN equation of motion is

Virial Theorems –The assumption that the bodies are stationary allows us to set any time derivative of the moment of inertia tensor to zero –Using the Newtonian acceleration

Redefine the mass of bodies to be The Newtonian term becomes Remaining self-energy terms cancel Final equation of motion in terms of the redefined masses

Calculation to 2PN order Expand and integrate 2PN terms using similar methods as 1PN terms –A few highly non-linear terms (quadruple integrals over the density) are quite difficult Some 1PN terms contain accelerations - use 1PN equations of motion to produce 2PN terms 2PN terms are produced by substituting the new mass into 1PN terms Calculate the virial theorems to 1PN order Define new masses, M_1 and M_2, to 2PN order, including distortion of self-energy expression due to motion and the gravity of the companion

Results –Using the virial theorems and a consistent definition of the masses, we have shown that the 2PN equation of motion does not contain self-energy terms of the order s^(-1) and s^(-5/2) –Terms of order s^(0), s^(-1/2), s^(-2), and s^(-7/2) are still under study