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Grav. potential between arbitrary groups of mass in the universe?
Is there a very general equation(s?) for calculating the gravitational
potential experienced by each member of a pair of roughly equal masses (stars or galaxies, say) at any separation R, whether or not they are in an actual bound orbit with each other? Eg: two different stars separated by thousands of light years or two different galaxies separated by megaparsecs in widely separate parts of the universe? I suppose I am asking how to calculate the gravitational potential set up by any two arbitrary spherical volumes containing mass in the universe at arbitrary distance R. I know how to calculate the g-potentials felt by each member of a binary star pair in circular orbit, but those equations don't seem appropriate in the more general problem. Thanks, Gene |
#2
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Grav. potential between arbitrary groups of mass in the universe?
In article , stargene
writes: Is there a very general equation(s?) for calculating the gravitational potential experienced by each member of a pair of roughly equal masses (stars or galaxies, say) at any separation R, whether or not they are in an actual bound orbit with each other? Eg: two different stars separated by thousands of light years or two different galaxies separated by megaparsecs in widely separate parts of the universe? Unless I'm missing something, neither the distance nor the question whether they are bound makes much difference. I know how to calculate the g-potentials felt by each member of a binary star pair in circular orbit, but those equations don't seem appropriate in the more general problem. Why not? (To be sure, the finite speed of propagation needs to be taken into account, at least in a non-static situation, but this is, perhaps somewhat surprisingly, only a higher-order effect, since to first order it is cancelled by other effects. Simulations of the gravitational interaction of the universe on large scales is something many people have done. As far as I know, the standard Newtonian formula is used (though within the framework of an expanding space).) |
#3
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Grav. potential between arbitrary groups of mass in the universe?
"stargene" wrote in message
... Is there a very general equation(s?) for calculating the gravitational potential experienced by each member of a pair of roughly equal masses (stars or galaxies, say) at any separation R, whether or not they are in an actual bound orbit with each other? Eg: two different stars separated by thousands of light years or two different galaxies separated by megaparsecs in widely separate parts of the universe? Just use the center of mass for the two objects and use the standard newtonian equation for gravity. Statically, you can calculate the gravity due to any number of stationayr objects. After yo uget to 3 or more , especially many, you need to use models developed for many body theories with long range forces. |
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Grav. potential between arbitrary groups of mass in the universe?
In article ,
Phillip Helbig---undress to reply writes: (To be sure, the finite speed of propagation needs to be taken into account, at least in a non-static situation, Are you sure about that? I'm no GR expert, but I thought you can calculate the potential by considering the positions at a given instant in a given reference frame. Of course the potential is frame-dependent, but that's as it ought to be. The reason this matters is that in a simple two-body system, the force has to be directed along the line separating the two bodies at each instant, not along the line where they were one light-time ago. -- Help keep our newsgroup healthy; please don't feed the trolls. Steve Willner Phone 617-495-7123 Cambridge, MA 02138 USA |
#5
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Grav. potential between arbitrary groups of mass in the universe?
In article ,
Phillip Helbig---undress to reply writes: (To be sure, the finite speed of propagation needs to be taken into account, at least in a non-static situation, Steve Willner wrote: Are you sure about that? I'm no GR expert, but I thought you can calculate the potential by considering the positions at a given instant in a given reference frame. Of course the potential is frame-dependent, but that's as it ought to be. The reason this matters is that in a simple two-body system, the force has to be directed along the line separating the two bodies at each instant, not along the line where they were one light-time ago. Actually, it's more complicated than either Phillip or Steve suggests: (a) In full GR, the "gravitational potential" is velocity-dependent and has other complexities, so it isn't a scalar. That is, you can't describe the gravitational field by any *single* number-at-each-event. You need to know the full spacetime metric (10 numbers-at-each-event) to describe the gravitational field. Restricting consideration to a bound 2-body system [which is the case for which the vast majority of the theoretical analysis has been done], ... (b) You do indeed need to consider the finite speed of propagation (i.e., in a 2-body system, body A "feels" the effects of where body B was one light-travel-time ago). This is clearly a v/c correction ot the Newtonian force, where v is the typical velocity of the orbiting bodies. In fact, it's easy to see that this effect [viewed in isolation] results in an effective drag on the bodies proportional to v/c. *But*, ... (c) Considering this as a "post-Newtonian" power series in v/c, in turns out that other GR effects cancel out the v/c term. And still other GR effects cancel out the (v/c)^2 term. And still other GR effects cancel out the (v/c)^3 term. And still other GR effects cancel out the (v/c)^4 term. [Alas, I don't know just what those GR effects are.] Just to be clear, I'm saying that *I* don't know; I'm confident that people who specialize in this research area do indeed know.] So, the actual leading order correction to the Newtonian equations of motion is actually at (v/c)^5, describing the lowest-order (quadrupole) gravitational-radiation emission. (d) To actually prove what I've written above [starting from the Einstein equations] is rather tricky. To go to still higher orders in v/c is much much harder still -- it requires solving some really tricky mathematical problems *and* doing very lengthly algebraic computations. Fortunately, there are about 3 research groups in the world who specialize in this, and they have published extensive results. (The different groups' results agree beautifully with one another once you take into account the different coordinates in which they're expressed.) The current state-of-the-art is that the full equations of motion of a GR compact binary system are known up through (v/c)^7 [often called the "3.5 post-Newtonian or 3.5PN order"], and many, but not all, of the (v/c)^8 [4PN] terms are known. There's a great not-very-technical survey of this field (and I say "great" despite it being over 20 years old) in @incollection{ Damouor-1987-in-300-years-of-gravitation, author = "Thibault Damour", title = "The problem of motion in Newtonian and Einsteinian gravity", pages = "128--198", editor = "Stephen W. Hawking and Werner Israel", booktitle = "Three Hundred Years of Gravitation", publisher = "Cambridge University Press", address = "Cambridge (UK)", year = 1987, isbn = "0-521-34312-7", X-note = "++good discussion of how to go from Einstein eqns to N-body equations of motion; the effacement of the internal structure of a freely falling subsystem", } Another old-but-great article (again fairly non-technical) is: @article{ Nordtvedt-1999-lunar-laser-ranging-vs-GR, author = "Kenneth Nordtvedt", title = "30 years of Lunar Laser Ranging and the Gravitational Interaction", journal = "Classical and Quantum Gravity", volume = 16, X-number = "12A", pages = "A101--A112", year = 1999, month = "December", ADScite = "http://adsabs.harvard.edu/abs/1999CQGra..16A.101N", doi = "10.1088/0264-9381/16/12A/305", } For more details on recent work, a good starting point would be @article{Futamase-Itoh-2007:PN-review, title = {The Post-Newtonian Approximation for Relativistic Compact Binaries}, author = {Toshifumi Futamase and Yousuke Itoh}, journal = {Living Reviews in Relativity}, year = {2007}, number = {2}, volume = {10}, keywords = {post-Newtonian approximations, post-Newtonian expansion, equations of motion, relativistic binary systems, binary dynamics}, url = {http://www.livingreviews.org/lrr-2007-2}, } -- -- "Jonathan Thornburg [remove -animal to reply]" Dept of Astronomy, Indiana University, Bloomington, Indiana, USA "Washing one's hands of the conflict between the powerful and the powerless means to side with the powerful, not to be neutral." -- quote by Freire / poster by Oxfam |
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Grav. potential between arbitrary groups of mass in the universe?
In article , Steve Willner
writes: In article , Phillip Helbig---undress to reply writes: (To be sure, the finite speed of propagation needs to be taken into account, at least in a non-static situation, Are you sure about that? I'm no GR expert, but I thought you can calculate the potential by considering the positions at a given instant in a given reference frame. Of course the potential is frame-dependent, but that's as it ought to be. I think that's basically equivalent. In practice, I think people use a Newtonian framework and the finite speed of propagation doesn't matter since it cancels out at first (or even higher?) order in a full-GR treatment. If one thinks of a huge N-body simulation, there would probably be some "universal frame" (e.g. that of the CMB) and one would use that as a starting point, not that of some observer moving relative to it. Think of something like simulating a sky as seen by an observer, including gravitational lensing. One has to follow a light ray as it is deflected by an evolving universe on its way to the observer. In this case, one has to take more than one instant into account. This is all irrelevant to the OP's question, though. |
#7
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Grav. potential between arbitrary groups of mass in the universe?
Thus spake Steve Willner
In article , Phillip Helbig---undress to reply writes: (To be sure, the finite speed of propagation needs to be taken into account, at least in a non-static situation, Are you sure about that? I'm no GR expert, but I thought you can calculate the potential by considering the positions at a given instant in a given reference frame. Of course the potential is frame-dependent, but that's as it ought to be. The reason this matters is that in a simple two-body system, the force has to be directed along the line separating the two bodies at each instant, not along the line where they were one light-time ago. You are right. Given only two bodies in inertial motion the equations of general relativity ensure that the Newtonian approximation is correct as you have described - the equivalent Newtonian force is directed to the point where the second body is now (in the appropriate frame). If there is some change to the two body inertial motion (precluded in this question), then that change will be propagated in the gravitational field at lightspeed from the point where it took place. Regards -- Charles Francis moderator sci.physics.foundations. charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and braces) http://www.rqgravity.net |
#8
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Grav. potential between arbitrary groups of mass in the universe?
In article , "Jonathan
Thornburg [remove -animal to reply]" writes: (c) Considering this as a "post-Newtonian" power series in v/c, in turns out that other GR effects cancel out the v/c term. And still other GR effects cancel out the (v/c)^2 term. And still other GR effects cancel out the (v/c)^3 term. And still other GR effects cancel out the (v/c)^4 term. So, the actual leading order correction to the Newtonian equations of motion is actually at (v/c)^5, describing the lowest-order (quadrupole) gravitational-radiation emission. This is of course a HUGE advantage to people doing N-body simulations and the like: just use Newtonian physics and any errors due to using the wrong theory will be much less than other sources of error (finite resolution (in time and space), simplified astrophysics etc). |
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