Overall perturbation for two free-falling particles in flat spacetime

[TimWilliams87]

I am thinking about a situation in general relativity which may be in textbooks but I have not been able to find it. I appreciate that there is the geodesic deviation equation for the world line of an observer and a nearby free-falling particle, but I think I need something different.

So we have flat Minkowski spacetime and perturb it by putting in a small massive particle, this leads to a perturbation on the flat metric which can be taken of Schwarzschild form if the particle does not rotate. The metric is written as

gμν=ημν+hμν

where hμν is the perturbing piece of Schwarzschild form. This perturbation is static. If we have the particle follow a trajectory with a constant acceleration, it can still be treated as static, as one uses a coordinate system where the particle is at rest.

Can we make the perturbation time-dependent by having another particle in the same spacetime which follows a trajectory parallel to the first one? I think the tidal forces between the two particles introduces a time-dependent perturbation, but not sure how it works. I basically need to know what hμν is in this case, it must have some time-dependence as the particles become closer as time moves on.

My first thought was to look at a Schwarzschild black hole perturbed by a massive particle but the analysis is very complicated and I think does not make sense outside of the context of an actual black hole.

 

[PeterDonis]

where hμν is the perturbing piece of Schwarzschild form.

Not really. The Schwarzschild metric is not a perturbation; it's an exact curved solution.

You will find approximate solutions in textbooks that look something like this (in units where $G = c = 1$):

$$
ds^2 = - \left( 1 - \frac{2M}{r} \right) dt^2 + \left( 1 + \frac{2M}{r} \right) \left( dr^2 + r^2 d \Omega^2 \right)
$$

which is basically the small $M / r$ approximation to the Schwarzschild metric in isotropic coordinates. The terms in $2M / r$ can be viewed as perturbations on flat Minkowski spacetime, but only in a vacuum region surrounding a spherically symmetric non-rotating massive object, whose surface is at a large enough radial coordinate $r$ that $M / r$ is small at the surface.

This perturbation is static. If we have the particle follow a trajectory with a constant acceleration, it can still be treated as static, as one uses a coordinate system where the particle is at rest.

Yes, this is correct.

Can we make the perturbation time-dependent by having another particle in the same spacetime which follows a trajectory parallel to the first one?

A "particle" usually means "test particle" in GR, which means that by definition it has no effect on the spacetime geometry.

If you are considering "particles" for which that is not true, then just one of them is already a perturbation on whatever background metric you have, Schwarzschild or otherwise. You don't even need two. Having two "particles" just makes the perturbation even more complicated.

Generally speaking, such cases are treated using numerical simulation. There are no known exact solutions. For cases where you have, say, two particles with small $M / r$ at their surfaces, and nothing else in the spacetime, you can add the perturbations due to each particle in linear fashion; this amounts to assuming that $M / r$ is small enough that nonlinear terms are negligible.

My first thought was to look at a Schwarzschild black hole perturbed by a massive particle

The Schwarzschild black hole itself cannot be treated as a small perturbation on Minkowski spacetime, because it's not. $M / r$ is not small except for the region well outside the black hole horizon. That's why I said above that approximations that use perturbations on flat spacetime in vacuum are only valid in the vacuum region outside an object that is large enough that $M / r$ is small at its surface--which means there is no black hole, just an ordinary planet or star.

 

[pervect]

MTW's (Misner, Thorne, Wheeler) chapter 39 on PPN in their text "Gravitation" might do some of what the OP wants. The "paramterized" part is irrelevant to what he wants, the chapter is really overkill for the actual question. I'm sure there are better sources, but this is the one I recall.

If MTW isn't available, there's a wiki article at https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism

It's an approximation, not an exact scheme.

The Newtonian approximation sets $g_{00}$ to be -1 + 2U, where U is basically the integral of rho/r over the spatial volume element and is referred to as the "Newtonian potential". (I'm not sure if there is a minus sign I'm missing). r is basically defined by the approximately Lorentzian coordinate system that's assumed by the theory discussed in the reference.

Higher order terms (beyond the Newtonian approximation) are discussed in the reference, and I don't recall the theory well enough to attempt to present them all. Depending on the accuracy you want, just the Newtonian approximation might be good enough.

Unfortunately the generality of making the theory apply to theories other than GR makes it more complicated than what's really needed to answer the original question. For instance, the separation of rho into parts due to rest mass and parts due to internal energy is solely to accomodate theories other than GR.

 

[TimWilliams87]

I actually think the scheme described by Pervect is what I need. Is it explained in MTW how to obtain the metric when there are two particles moving slowly parallel to each other for a short period of time? Is g_{00} a superposition law in this case for the contributions from both particles?

 

[PeterDonis]

MTW's (Misner, Thorne, Wheeler) chapter 39 on PPN in their text "Gravitation" might do some of what the OP wants.

Not really. The PPN formalism is a generalized system for describing the spacetime geometry due to a single source. It is not intended for describing multiple sources.

 

[PeterDonis]

Is it explained in MTW how to obtain the metric when there are two particles moving slowly parallel to each other for a short period of time?

Not that I'm aware of.

There are no known exact solutions for multiple isolated gravitating bodies in GR.

Is g_{00} a superposition law in this case for the contributions from both particles?

If you are going to use the linearized approximation, in which we assume that all sources are weak enough that nonlinear terms in the Einstein Field Equation can be ignored, then you would add contributions from both particles to all metric coefficients, not just $g_{00}$.

I actually think the scheme described by Pervect is what I need.

It's probably overkill for what you appear to be trying to do. If you are assuming that nonlinear effects are negligible, then most of the terms in a PPN expansion for a given source will be negligible. You would just be adding, for example, the terms in $M / r$ from each source in the relevant metric coefficients in an expression like the one I gave at the start of post #2.

 

[pervect]

Not really. The PPN formalism is a generalized system for describing the spacetime geometry due to a single source. It is not intended for describing multiple sources.

I'll have to disagree here. For instance, high order post newtonian methods are used to study black hole insprals. See for instance Clifford M Willis, https://arxiv.org/abs/1102.5192, "On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics", where he comments on the effectiveness of the methods in studying black hole inspirals. (See the abstract I post below for more details).

The "paramaterized" part is probably not of interest, but I think the post newtonian part is. Personally, I've only played with ppn methods for situations such as the solar system, but they have greater applicability than that per Willis' paper.

The post-Newtonian approximation is a method for solving Einstein’s field equations for physical systems in which motions are slow compared to the speed of light and where gravitational fields are weak. Yet it has proven to be remarkably effective in describing certain strong-field, fast-motion systems, including binary pulsars containing dense neutron stars and binary black hole systems inspiraling toward a final merger. The reasons for this effectiveness are largely unknown. When carried to high orders in the post-Newtonian sequence, predictions for the gravitational-wave signal from inspiraling compact binaries will play a key role in gravitational-wave detection by laser-interferometric observatories.

Essentially, the black holes in the inspiral are conceptually very similar to 'big particles'.

So I'd say it's worth checking out, though I don't quite understand what the OP's motivations are. I doubt that GR effects are going to be significant for most situations I can think of that would be described as a "pair of particles" other than the aforementioned binary inspirals. And if they were significant, I'd have to wonder if one might wind up needing quantum gravity (I don't have a handle on that topic, sadly). If one is looking for a formal argument why Newtonian gravity is a good approximation for a pair of particles, though, it might fit the bill. And in that case one would probably only need the theory to Newtonian order, as I mentioned. The results there are easy - bascially you only need to compute $g_{00}$. If we apply the approximation to one particle, we can see that it's an approximation and not exact, because it doesn't match the Schwarzschild solution which has terms other than $g_{00}$ in the metric tensor, terms that result in effects such as the extra deflection of light around the Sun.

I'd expect at the end one would get Newtonian orbits around the center of mass, but I haven't carried out the calculation. I am suspecting that this may be all the OP wants to do, though.

I don't know where the best treatment of post newtonian methods might be found, my only familiarity is from MTW. But it's a place to start looking (even though I must say I personally didn't find their treatment all that satisfying, though it's a good reference for the results).

 

[PeterDonis]

high order post newtonian methods are used to study black hole insprals.

Yes, I see I was interpreting the term "post newtonian" too narrowly. There has been a lot of development in this area since MTW was published. The Will paper you give is an excellent reference that is more up to date.

 

[pervect]

The topic made me think about the problem of N-body post-newtonain GR simulation. I found a rather interesting series of articles about a rather advanced approach, https://journals.aps.org/prd/abstract/10.1103/PhysRevD.43.3273, "General-relativistic celestial mechanics....." (several parts), by Thibault Damour, Michael Soffel, and Chongming Xu.

We present a new formalism for treating the general-relativistic celestial mechanics of systems of 𝑁 arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies. This formalism is aimed at yielding a complete description, at the first post-Newtonian approximation level, of (i) the global dynamics of such 𝑁-body systems ("external problem"), (ii) the local gravitational structure of each body ("internal problem"), and, (iii) the way the external and the internal problems fit together ("theory of reference systems").

The actual papers are paywalled, and are likely too complex me to evaluate even if I could get access , but I thought it was interesting that someone had actually had a published solution to this problem. Apparently, it describes simulating N bodies, which are all able to change shape due to tidal effects.

I tracked this down from an interesting article I found at https://files.igs.org/pub/resource/pubs/workshop/2017/W2017-PS08-03 - Roh.pdf, Numerical Simulation of the full 1st order Post Newtonian
Corrections for the GPS and GLONASS satellites by Kyoung-Min Roh, Sung-Moon Yoo, Byung-Kyu Choi, and Jung-Ho Chowhich. This paper has some graphs that mostly shows how insignificant some of these corrections are. From the graph, it appears the PN effects on Earth satellites are below the effects of both solar radiation and earth radiation on the orbits, by a couple of orders of magnitude. There wasn't any data on other situations that might be more demanding, such as Mercury's orbit or Jupiter's moons though.

 

[renormalize]

The actual papers are paywalled, and are likely too complex me to evaluate even if I could get access , but I thought it was interesting that someone had actually had a published solution to this problem. Apparently, it describes simulating N bodies, which are all able to change shape due to tidal effects.

A 1990 preprint version of at least the first Damour et al. paper is available at Institut des Hautes Etudes Scientifiques.