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