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Does GR Really Predict Gravitational Waves in the Weak Field Limit?

The amazing confirmation of gravitational waves (GWs) by LIGO has been heralded as the greatest confirmation of General Relativity (GR) to date. But does GR really predict the existence of GWs in the weak field limit?


The conventional derivation of the wave equation for GWs consists in the linearization of the Ricci tensor, by setting the metric coefficients g equal to a flat background plus a small perturbation, h. The latter will satisfy the wave equation provided:


1. its divergence vanishes, and

2. its trace also vanishes.


The vanishing of its divergence is usually referred to as a Lorentz gauge in analogy with electrodynamics. But analogies can be deceiving!


A total of 10 constraints must be placed on the perturbed metric coefficients. Six of them come from the Einstein equations themselves while the other four come from the Lorentz gauge condition.


What is done is to first linearize the Ricci tensor and then throw out all pieces in which the divergence of h, or its trace-reversed perturbation, H, and second to introduce it into the Einstein equations, where the right-hand side of the equation is proportional to the energy-stress tensor. Far from the sources, the linear wave equation is obtained in the weak-field limit. It is then concluded that the vanishing of the d'Alembertian of the metric perturbation "tells us that a weak gravitational field can propagate as a wave with the same speed as light." [Moore, "A General Relativity Workbook"] But, is the analogy between the vanishing of the divergence of h and the Lorentz condition actually viable?


In GR, all motion must follow geodesics. Unlike electromagnetism (EM), this follows from the Einstein equations. Hence, the latter determines the motion of the particle whereas in EM, the motion is independent of Maxwell's field equations, and the introduction of a force--the Lorentz force--is necessary to discern the effects of the field on the motion of matter.


So why not start with the geodesic equations and then introduce the fact that we are working in the weak field limit? Certainly, the same linear wave equation must result, and as a bonus we can observe the nonlinear terms that have been neglected that would give the correct semi-linear wave equation [viz. N Koiso, "On a wave equation corresponding to geodesics" Osaka J Math 33 (1996) 93-98]. Unfortunately, the desire is not realized in GR.


Geodesic motion requires the acceleration perpendicular t the tangent plane at each point on the surface to vanish. Consequently, the motion is completely determined by the bending of the surface. This normal acceleration is equal to the sum of the acceleration plus the nonlinear bending term. When it doesn't vanish, it is equal to the rate of working: the force times the speed at which work is being done. Such is the case with the Lorentz force. Since EM is linear, the nonlinear bending term vanishes and the acceleration becomes proportional to the gradient of a scalar potential and the time derivative of a vector one. The latter is proportional to the velocity, and it is for this reason that Maxwell termed it a kinetic potential.


The Lorentz force, in general, does not vanish. But what vanishes in free space is the divergence of the force. Taking the divergence and introducing the Lorentz gauge condition results in a linear wave equation whose speed is c, the speed of light. Can we do the same for gravity? At least in the weak-field limit we should expect a linear wave equation to "pop" out of the woodwork. But, sadly, this does not manifest itself. Rather, Newton's law is obtained where the acceleration is found to be given by Newton law of gravitation. What happened to the time component that is necessary in order to give a wave equation? And if the analogy with EM is on track, we should expect that the vanishing of the divergence of the of the force would lead to a wave equation. Yet, it doesn't. And even the vanishing of the 4-divergence of the source densities does not translate into a Lorentz gauge condition [Ibison et al. "The speed of gravity revisited"].



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