Is there a simple formula for the gravitational self-force (due to emission of gravitational waves) in the classical limit?

Question

UPDATE: To clarify things in response to comments below, I want to reformulate my question in a very concise and abundantly clear form. Let's consider an object of a known mass M that moves under the action of some varying external force and right now has a known velocity v (i.e., a known vector of velocity), a known acceleration a (again, a vector), a known first derivative of acceleration da/dt, and so on. What is the gravitational self-force (due to emission of gravitational waves by the object itself) acting on the object right now, in the vector form? The classical limit is implied, i.e., the velocity is small, the gravitational field is weak, and whatever else needs to be small or weak is small or weak.

ONE MORE CLARIFICATION: I saw a formula for the time-averaged power of emission of gravitational waves by a binary system, but I am interested in the momentary(!) force(!) as a function of the momentary velocity and its time derivatives of all orders. Even if you tell me the momentary, not time-averaged, power of emission, it still does not define the force, as the latter may be directed at any unknown angle with respect to the velocity.

The original version of my question, now slightly cut, is below.

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The Abraham-Lorentz force is the recoil force on an accelerating or decelerating charged particle caused by the particle emitting electromagnetic radiation and is equal to

$\frac{Q^2}{ 6 π ε_0 c^3} \frac{d\mathbf{a}}{dt}$,

where Q is the particle charge, $ε_0$ is the electric constant, c is the speed of light, and $\frac{d\mathbf{a}}{dt}$ is the time derivative of the vector of acceleration. This formula is derived in the limit of non-relativistic velocities.

Is there a similar formula for the gravitational force akin to the Abraham-Lorentz force, that is, for the recoil force on an accelerating or decelerating astrophysical object caused by the object emitting gravitational waves?

I am not interested in general bulky expressions in the tensor form; I want to have a simple formula that I could use to calculate this force acting on our planet as a result of it orbiting around the sun. I believe that the general expression of the general relativity theory can be simplified for that case, be it called the classical limit or whatever else.

Please note I do not want the assumption of circular motion to be made. I want to see how the force is expressed via the momentary values of acceleration and its time derivatives of any order, similar to the above formula for the Abraham-Lorentz force.

There are a couple of questions on this SE about that gravitational force (link1, link2), but, reading the answers to them and following the links provided, I was unable to find the formula I am looking for.

To explain my motivation, I am a student from Japan studying something completely unrelated to physics, but I loved physics at school and am curious about the degree of analogy between the Abraham-Lorentz force and its gravitational analogue. I recently had a conversation about that with someone, and we got very curious as to which order of the derivative of acceleration will pop up.

So please kindly give me the formula and, ideally, a reference to the source.

Answer 1

One of my all time favorite papers is this one by Burke (1971). He works out radiative damping in GR, using a mathematical technique called matched asymptotic expansions. In the process he works out radiative damping for a simple harmonic oscillator connected to a string that extends off to infinity and the radiative damping of an oscillating charge due to EM radiation. Seeing those two results puts the gravitational result in excellent context.

The resistive force acting per unit volume on the mass density, $\rho$, as a result of the gravitational radiation emitted by the changing mass multipole moment $q_{\ell m}(t)$ is

$$F_R = - \frac{8\pi\sqrt{10}}{75} \rho\, r \sum_m Y_{21m} \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^5q_{2m},\quad\quad(\mathrm{eq.}\, 136)$$

where we have used the leading order quadrupole, $\ell=2$. $Y_{21m}$ is a vector spherical harmonic that appears as $Y_{\ell,\ell-1,m}$ in the full expression. This term arrises from the multipole expansion of the metric, in this case the current multipole (or momentum part of the metric, $g_{ti}$ for $i\in\{r, \theta,\phi\}$). This makes sense as the damping force should be connected to conservation of momentum of the radiation.

Burke makes sure to emphasize some pitfalls in calculating the radiation reaction in GR compared to EM:

Another point is that one cannot always compute the resistive force from the force law $$ F_R = \frac{1}{4}m\nabla\psi$$

(note: compare to $F_R = -q\nabla\phi$ for the EM radiation case.)

Before the gauge transformation was used to simplify the problem, the lower $\ell$ dependence of the vector and tensor potentials allowed them to compete with the scalar potential ($\psi$), and one had to use a more complete force law (derivable by writing the equations for a geodesic...).

...

The big difference between gravity and electromagnetism now appears. The electromagnetic field produces effects only through its force law. On the other hand, not only does the gravitational field affect the coordinate motion of the system, but the potentials themselves determine the clock rates and the behavior of rigid bodies. Until one knows the [tensor potential], one cannot convert coordinate differences to proper length without making errors that are $\mathcal{O}(\kappa)$,

where Burke defines the weak field parameter $\kappa \sim GM/(c^2r)$. For gravitationally bound systems $\kappa \sim (v/c)^2$ is related to the slow motion condition too.

Answer 2

The gravitational self-force is not gauge invariant quantity. It only truly makes sense averaged over a sufficient amount of time. That being said one can calculate the amount of linear momentum that an object emits (averaged over an appropriate length of time) based on the time derivatives. The formula for this was derived by Kip Thorne in 1980 (ref), to lowest orders in the post-Newtonian approximation. By conservation of linear momentum the object must encounter a radiation reaction force in the opposite (in units where $G=c=1$)

$$F_{i,RR} = -\frac{dP_i}{dt} = -\Big\{\frac{2}{63} I^{(4)}_{ijk}I^{(3)}_{jk} +\frac{16}{45}\epsilon_{ijk}I^{(3)}_{jl} J^{(3)}_{kl} \Big\}$$

Here

• $I_{ij}$ is the (symmetric trace free,STF) mass quadrupole moment,
• $I_{ijk}$ is the (STF) mass octopole moment,
• $J_{ij}$ is the (STF) current quadrupole moment,
• $\epsilon_{ijk}$ is the Levi-Civita symbol, and
• superscripts $^{(n)}$ denote the $n$th time derivative.

This expression gives the total radiation reaction force on the system to which the multipole moments apply. In particular, it will not give the force on the individual masses in a binary. In addition to this radiation reaction force, the system as a whole will experience a torque due the emission of gravitational waves,

$$T_{i,RR} = -\frac{dJ_i}{dt} = -\frac{2}{5}\epsilon_{ijk}\Big\{\ I^{(2)}_{jl}I^{(3)}_{kl} +\frac{5}{126}I^{(3)}_{jlm} I^{(4)}_{klm} +\frac{16}{9}J^{(2)}_{jl} J^{(3)}_{kl}\Big\}.$$

These two together would allow you to construct some version of the force on two objects in a mutual circular orbit. Note however, that in doing so in principle you could still add arbitrary forces for which the net force and net torque on the system cancel. These are remaining gauge ambiguities that cannot be resolved a priori without choosing some gauge condition.

In addition the emission of gravitational waves can produce stresses and shears in the system. If the system has any additional vibrational modes (e.g. eccentricity in a binary) these can be excited or damped by the radiation reaction forces.