Multibody system in general relativity

[ajw1]

Having only a limited knowledge of general relativity I would like to ask the following question:

In Newtonian mechanics one just needs to know the mass, velocity and the combined gravitational force from other bodies in a system to calculate the position of a body after a small interval. The computer then does a fine job calculation the movement of multiple objects over longer periods, like that of orbiting planets.

But what would be the approach for this in general relativity? Is there a tensor that is compatible with Newtonian force that can be calculated for each individual body for a given position of a test object from which one could calculate the objects next position after a small proper time slice?

 

[bcrowell]

If you already know the metric, and you want to find the motion of test particles in that metric, then there's something called the geodesic equation that is indeed closely analogous to Newton's laws, and reduces to Newton's laws in the appropriate limit. Here is an example of such a calculation: http://lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.2 (subsection 6.2.6)

In more complicated systems, it's basically not well understood whether general relativity has a property analogous to Newtonian determinism, or even how to define such a property. For example, if the chronology protection conjecture (which is itself not a totally well defined statement) fails, then you can have problems with causality, which make it impossible to even define anything that plays the role of "initial conditions." Also, general relativity has sort of a generic tendency to develop singularities. If these singularities are not hidden behind an event horizon (as stated in the cosmic censorship hypothesis), then you have problems with determinism, because the laws of physics break down at a singularity. The famous statement of this by John Earman is that anything could pop out of a singularity, including green slime or your lost socks.

 

[ajw1]

Thank you for the good reference. As for now I am not so much interested in extreme conditions (relativistic speed, black holes etc). But can one calculate the metric at the position of a testparticle A from the Schwarzschild metrics of two heavy bodies B and C, or is this already difficult?

 

[bcrowell]

Thank you for the good reference. As for now I am not so much interested in extreme conditions (relativistic speed, black holes etc). But can one calculate the metric at the position of a testparticle A from the Schwarzschild metrics of two heavy bodies B and C, or is this already difficult?

Yes, for weak fields general relativity is approximately a linear theory. I believe in a case like the one you're talking about, it's the quantity g-\eta that's approximately additive, where g is the metric and \eta is the flat-space (Minkowski) metric. This is what's known as linearized gravity. Linearized gravity is simple to calculate with, but it's missing a lot of physics. For example, it can't describe a bound system, like the Earth and moon orbiting around their common center of mass.

 

[D H]

This document, http://iau-comm4.jpl.nasa.gov/XSChap8.pdf (see page 3) describes how the JPL Development Ephemerides model gravity, including post-Newtonian terms, to compute the behaviors of the sun, the planets, and the Earth's Moon over time. Note that there is no frame dragging term in this expansion. (The Sun is rotating, but very slowly, and the planets are fairly well-separated from one another.)
Chapter 10 of the http://www.iers.org/MainDisp.csl?pid=46-25776" models the post-Newtonian (linearized gravity) gravity in terms of a perturbations on top of Newtonian mechanics for an artificial Earth-orbiting satellite. Note that this includes frame dragging due to the Earth's rotation.

Presumably the only reason one would want to model relativistic effects on an Earth-orbiting satellite is because of a need for high precision. This means the Earth cannot be treated as a point mass. The start of chapter 6 briefly covers that topic. If you want high precision, you not only need to model the Earth as a mass distribution, you have to model it as having a time-varying mass distribution (e.g., Earth tides). The bulk of chapter 6 discusses the time-varying potential due to the Earth tides, the pole tide, and the ocean tides. All that potential stuff is calculated in an Earth-fixed frame; you want to do your work in an inertial frame. Chapters 2-4 and 8 discuss the time-dependent relation between these frames.

Note: The effects of general relativity on a spacecraft at the altitude of LAGEOS are of about the same magnitude as the gravitational effects of the ocean tides in the satellite. In other words, extremely tiny.

 

[ajw1]

..., but it's missing a lot of physics. For example, it can't describe a bound system, like the Earth and moon orbiting around their common center of mass.

I would expect that with a numerical approach the approximation for a bound system would be reasonable. If I want the model to correctly describe the precession of Mercury, would you say linearized gravity would do the job?

This document, http://iau-comm4.jpl.nasa.gov/XSChap8.pdf (see page 3) describes how the JPL Development Ephemerides model gravity, including post-Newtonian terms, to compute the behaviors of the sun, the planets, and the Earth's Moon over time. Note that there is no frame dragging term in this expansion. (The Sun is rotating, but very slowly, and the planets are fairly well-separated from one another.)



Chapter 10 of the http://www.iers.org/MainDisp.csl?pid=46-25776" models the post-Newtonian (linearized gravity) gravity in terms of a perturbations on top of Newtonian mechanics for an artificial Earth-orbiting satellite. Note that this includes frame dragging due to the Earth's rotation.

Presumably the only reason one would want to model relativistic effects on an Earth-orbiting satellite is because of a need for high precision. This means the Earth cannot be treated as a point mass. The start of chapter 6 briefly covers that topic. If you want high precision, you not only need to model the Earth as a mass distribution, you have to model it as having a time-varying mass distribution (e.g., Earth tides). The bulk of chapter 6 discusses the time-varying potential due to the Earth tides, the pole tide, and the ocean tides. All that potential stuff is calculated in an Earth-fixed frame; you want to do your work in an inertial frame. Chapters 2-4 and 8 discuss the time-dependent relation between these frames.

Note: The effects of general relativity on a spacecraft at the altitude of LAGEOS are of about the same magnitude as the gravitational effects of the ocean tides in the satellite. In other words, extremely tiny.

The citations looks very promising, I'll have to check them in more detail, thanks!

 

[bcrowell]

I would expect that with a numerical approach the approximation for a bound system would be reasonable. If I want the model to correctly describe the precession of Mercury, would you say linearized gravity would do the job?

You don't need linearized gravity for that calculation. You can just use the Schwarzschild metric. There aren't two sources, just one: the sun.

Linearized gravity can't describe the binding of this system, because it can't describe the fact that the Sun accelerates due to Mercury's gravitational force.

 

[D H]

Linearized gravity can't describe the binding of this system, because it can't describe the fact that the Sun accelerates due to Mercury's gravitational force.

You've said this a couple times. Could you explain what you mean by this? The various planetary ephemerides (e.g., JPL's Developmental Ephemerides, the Institute of Applied Astronomy's EPM series of ephemerides) do an excellent job modeling the motions of objects in the solar system, including the Sun. To do this with precision they need to account for general relativistic effects -- and they do just that.

 

[bcrowell]

You've said this a couple times. Could you explain what you mean by this? The various planetary ephemerides (e.g., JPL's Developmental Ephemerides, the Institute of Applied Astronomy's EPM series of ephemerides) do an excellent job modeling the motions of objects in the solar system, including the Sun. To do this with precision they need to account for general relativistic effects -- and they do just that.

There's a discussion of this, for example, on p. 186 of Misner, Thorne, and Wheeler: "The fact that, in [linearized gravity], gravitating bodies cannot be affected by gravity, also holds for bodies made of arbitrary stress-energy (e.g., rubber balls or the Earth)... The Earth cannot be attracted by the sun; it must fly off into interstellar space!"

I don't know about the specific JPL calculations you're referring to, but they definitely can't use linearized gravity as a complete, self-consistent theory to describe the whole solar system. Most likely they use some kind of hybrid theory, in which some aspects are put in by hand. For an example of this type of approach, see this link: http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll6.html Take a look at the calculation he gives near the sentence "We will treat the motion of the stars in the Newtonian approximation, where we can discuss their orbit just as Kepler would have." He can't get the motion of the stars from linearized gravity. He has to put it in by hand, then use linearized gravity to calculate the gravitational radiation.

 

[ajw1]

Yes, for weak fields general relativity is approximately a linear theory. I believe in a case like the one you're talking about, it's the quantity g-\eta that's approximately additive, where g is the metric and \eta is the flat-space (Minkowski) metric. This is what's known as linearized gravity. Linearized gravity is simple to calculate with, but it's missing a lot of physics. For example, it can't describe a bound system, like the Earth and moon orbiting around their common center of mass.

The model I am looking for should at least give better results in most conditions compared with the Newtonian calculations. So I guess that rules out pure linearized gravity.
I would expect it for example to describe both the Newtonian as the relativistic part of the precession of Mercury, but also have some predictive power for other solar type systems.

 

[espen180]

Thank you for the good reference. As for now I am not so much interested in extreme conditions (relativistic speed, black holes etc). But can one calculate the metric at the position of a testparticle A from the Schwarzschild metrics of two heavy bodies B and C, or is this already difficult?

In the weak field, slow speed limit, gravity can be lineraly approximized to the four field equations of gravitomagnetism:

\oint \vec{g}d\vec{A}=-4\pi G m_{ins}
\oint \vec{g_m}d\vec{A}=0
\oint \vec{g}d\vec{l}=-\frac{\partial \vec{g_m}}{\partial t}
\oint \vec{g_m}d\vec{l}=-\frac{4^\pi G}{c^2}J+\frac{1}{c^2}\frac{\partial \vec{g}}{\partial t}

Where \vec{g} is the gravitational field, \vec{g_m} is the gravitational magnetic field and J is mass current.

In the weak field, slow speed limit, these equations describe how mass moves in in the presence of other masses.

The approximized gravitational Lorentz force becomes \vec{F}=m\left(\vec{g}+\vec{v}\times 2\vec{g_m} \right)

 

[D H]

I think we're talking past one another here. The pages bcrowell referenced, including the linearized gravity, are being used to explain gravitational radiation. Gravitational radiation is a non-issue when it comes to explaining the motions of planets orbiting a star. You want a parameterized post-Newtonian expansion for that, which is what the two references in post #5 address.

Or you just want good old Newtonian mechanics. If, for example, you are dealing with observed exoplanets about some other star, the accuracy in the data simply aren't good enough to merit the use of anything but Newtonian mechanics.

 

[bcrowell]

I think DH is right that we're talking past one another. I think it might help if ajw1 could clarify what he/she is trying to accomplish. There are lots of different approximations to GR, and we don't know what criteria to use to find one that would be useful to ajw1.

 

[ajw1]

In the weak field, slow speed limit, gravity can be lineraly approximized to the four field equations of gravitomagnetism:

\oint \vec{g}d\vec{A}=-4\pi G m_{ins}
\oint \vec{g_m}d\vec{A}=0
\oint \vec{g}d\vec{l}=-\frac{\partial \vec{g_m}}{\partial t}
\oint \vec{g_m}d\vec{l}=-\frac{4^\pi G}{c^2}J+\frac{1}{c^2}\frac{\partial \vec{g}}{\partial t}

Where \vec{g} is the gravitational field, \vec{g_m} is the gravitational magnetic field and J is mass current.

In the weak field, slow speed limit, these equations describe how mass moves in in the presence of other masses.

The approximized gravitational Lorentz force becomes \vec{F}=m\left(\vec{g}+\vec{v}\times 2\vec{g_m} \right)

From the explanation in Wiki I understand this is a good approximation for small objects near large ones, but cannot be used for describing planet orbits (there is also a small mass limit).

I think we're talking past one another here. The pages bcrowell referenced, including the linearized gravity, are being used to explain gravitational radiation. Gravitational radiation is a non-issue when it comes to explaining the motions of planets orbiting a star. You want a parameterized post-Newtonian expansion for that, which is what the two references in post #5 address.

Or you just want good old Newtonian mechanics. If, for example, you are dealing with observed exoplanets about some other star, the accuracy in the data simply aren't good enough to merit the use of anything but Newtonian mechanics.

Having no accurate data of exo planets doesn't mean we can't calculate their motions .
I think the first reference in post #5 comes close to what I was looking for, although I had hoped to somehow be able to get a more or less exact solution for the metric at a certain point in space-time in a multi body system, and from that the geodesic of an celestial object at that point.
As I understand the n-body equation from Estabrook is Newton with a relativistic correction.