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Post-Newtonian Approximations and Applications

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Post-Newtonian Approximations and Applications Monash University MTH3000 Research Project Coming out of the woodwork: Post-Newtonian approximations and applications Author: Supervisor: Justin Forlano Dr. Todd Oliynyk March 25, 2015 Contents 1 Introduction 2 2 The post-Newtonian Approximation 5 2.1 The Relaxed Einstein Field Equations . 5 2.2 Solution Method . 7 2.3 Zones of Integration . 13 2.4 Multi-pole Expansions . 15 2.5 The first post-Newtonian potentials . 17 2.6 Alternate Integration Methods . 24 3 Equations of Motion and the Precession of Mercury 28 3.1 Deriving equations of motion . 28 3.2 Application to precession of Mercury . 33 4 Gravitational Waves and the Hulse-Taylor Binary 38 4.1 Transverse-traceless potentials and polarisations . 38 4.2 Particular gravitational wave fields . 42 4.3 Effect of gravitational waves on space-time . 46 4.4 Quadrupole formula . 48 4.5 Application to Hulse-Taylor binary . 52 4.6 Beyond the Quadrupole formula . 56 5 Concluding Remarks 58 A Appendix 63 A.1 Solving the Wave Equation . 63 A.2 Angular STF Tensors and Spherical Averages . 64 A.3 Evaluation of a 1PN surface integral . 65 A.4 Details of Quadrupole formula derivation . 66 1 Chapter 1 Introduction Einstein's General theory of relativity [1] was a bold departure from the widely successful Newtonian theory. Unlike the Newtonian theory written in terms of fields, gravitation is a geometric phenomena, with space and time forming a space-time manifold that is deformed by the presence of matter and energy. The deformation of this differentiable manifold is characterised by a symmetric metric, and freely falling (not acted on by exter- nal forces) particles will move along geodesics of this manifold as determined by the metric. Although this notion of curved spaces is an elegant picture, when we seek to understand it at a mathematical level we naturally need some way of describing this manifold through a coordinate system. Crucially, gravity is associated with actual curved space-time, that is to say we should not be able to make a global coordinate transformation so that in another description the space is no longer curved. This is why the foundations of general relativity are built on a covariant formalism, where their form is invariant under coordinate trans- formations. Tensors are therefore unavoidable in the mathematics since tensor equations have this invariance property and when one wishes to impose a certain coordinate system, it is relatively straightforward to do so. This is in stark contrast to the Newtonian theory, where it is difficult to describe systems in anything other than Cartesian coordinates. Unfortunately the price we pay for such coordinate freedom is that the equations dealt with are far more challenging to solve exactly. The Einstein field equations which govern how the space-time is curved, take the form of 6 independent, non-linear, coupled partial differential equations. Apart from the most idealised geometries such as spherical symmetry where exact solutions exist (the Schwarzschild metric [2] for instance), we must instead resort to approximations. There are many different approaches ranging from the post-Newtonian approximation (PN), perturbation approaches which assume one mass is much larger than the other masses and the field of numerical relativity which seeks to obtain solutions to the field equations for complicated systems computationally. Here we consider the post-Newtonian method which proceeds iteratively generating ever higher order approximations. As the name suggests, the lowest order result is the Newtonian gravitational potentials. The next iteration obtains the first post-Newtonian order which is labelled as 1PN and represents a first correction past Newtonian gravity. After this one can then obtain higher post-Newtonian corrections such as 2PN, 3PN and so on. In this scheme there are two main assumptions: firstly that the gravitational fields are weak which is to say that they are dominated by the Newtonian order potentials and secondly that the velocities of the system are small enough to allow the weak gravity assumption to hold. This is known collectively as a weak field, slow motion approximation. The weak field approximation can be made by considering the metric as a sum of the flat- space, unperturbed Minkowski metric plus an additional term which describes deviations from this. By slow motion, we mean that the dimensionless parameter := vT =c 1 where vT is some characteristic velocity of the system and c is the speed of light in vacuum. Here 2 Chapter 1 can be thought of as the \slowness" of the system, a larger implies a more vigorously moving system. This allows us to then consider the metric to be expanded in powers of while the limit as ! 0 is to recover the Newtonian equations. An example is the case of a binary system where the post-Newtonian approximation seeks to determine the equations of motion (they give accelerations in terms of positions and velocities) of the bodies to a high order. The relative acceleration of the bodies can be schematically expanded as Gm a = −n + 2A + 3A + 4A + 5A + 6A + 7A + ··· ; r2 1PN 1:5PN 2PN 2:5PN 3PN 3:5PN (1.0.1) where m := m1 +m2, r := jx1 −x2j and n := (x1 −x2)=r. We can see that the Newtonian level is obtained when ! 0 and the higher order terms represent corrections past the Newtonian order given by approximations to General relativity. One can also include spin effects in this expansion, for instance as is done by Will [3] in the context of the direct integration of the relaxed field equation (DIRE) approach to post-Newtonian theory. This is allows the orbiting bodies to have some intrinsic angular momentum, much like how the Earth rotates about its own axis as it orbits the sun. The spin effects are first noticeable at the 1PN order where the spin can couple to the orbital angular momentum. The equations of motion to 1PN order for N- orbiting bodies was given by Eisntein, Infeld and Hoffman in 1938 [4]. At the other end of the scale are the highest completed calculations to 3.5PN order which have been tackled by Pati and Will [5], Futamese and Itoh [6,7], Blanchet [8] and 4PN Hamiltonians have been found [9]. The search for higher orders is complicated because the number of terms appearing in each PN term of (1.0.1) increases dramatically. The number of terms in the metric potentials at 1PN order is 10, at 2PN it is around 100 terms, and the 3PN is of the order of 103 ∼ 104 terms. Going to 4PN requires handling tens of thousands of terms, which is a hugely challenging task. A prediction of general relativity is that freely orbiting bodies are not stable forever. They in fact emit energy through gravitational radiation which gradually causes the bodies to spiral inwards towards closer orbits before plunging together in a coalescence phase. As they inspiral, the velocities increase so will in fact not be uniform forever, it will increase in this case. This implies that the post-Newtonian corrections can in fact become larger than the Newtonian terms which causes a breakdown of the Newtonian limit as the system moves to relativistic orbits. The effect of emitted gravitational waves on the system itself which is known as radiation-reaction is first seen at the 2.5PN order which is at O(5) in (1.0.1). The most likely sources for the detection of gravitational radiation are compact binary systems such as orbiting black holes or neutron stars because of the closeness of their orbits and large masses. The compact binary case is especially useful as tidal effects can be neglected for much of the motion right up until the close inspiral. The existence of gravitational waves is perhaps the greatest as yet undetected predic- tion of general relativity and is a large reason why the PN formulation exists at all. Just as matter curves space-time, it is possible that for accelerating, non-spherically symmetric systems, such as a compact binary or the violent collision between two black holes, pro- cesses such as these can form propagating waves in the space-time. This prediction was largely theoretical until a gradual shift in the technological capability around the 1980's opened up the possibility of detecting these waves. The experimental difficulty is that the amplitude of a gravitational wave is extremely small, even for a binary black hole system. The effects felt in a detector are on the order of one part in 1022. Modern gravitational wave detectors such as LIGO and VIRGO have to not only be built with sensitivities in this range but they must also contend with numerous sources of noise which obscures the true signal. The noise ranges from the small: thermal noise and quantum effects, to the large scale, such as seismic noise. The goal of the post-Newtonian approximations is to provide wave templates which are theoretically derived waveforms written in terms 3 Chapter 1 of some observable parameters such as masses of bodies, orbital frequency and distance to source. These waveforms can then be used to sift the true signal from the noisy data observed. These waveforms are computed in the far-field of the source and are expressed in terms of the orders as 4G n o hij = Qij + Qij + Qij + Qij + Qij + Qij + Qij + Qij + ··· : c2r 0:5PN 1PN 1:5PN 2PN 2:5PN 3PN 3:5PN (1.0.2) The lowest order term, Qij is in fact the quadrupole which leads to the quadrupole formula. It is this reason why the qaudrupole formula is sometimes considered as a Newtonian term even though it appears nowhere in Newtonian gravity.
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