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Grav. potential between arbitrary groups of mass in the universe?



 
 
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  #1  
Old July 26th 10, 09:01 AM posted to sci.astro.research
stargene
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Default 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  
Old July 27th 10, 08:00 AM posted to sci.astro.research
Phillip Helbig---undress to reply
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Default 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  
Old July 28th 10, 12:27 AM posted to sci.astro.research
Joseph Warner
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Posts: 5
Default 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.
  #4  
Old August 3rd 10, 10:56 AM posted to sci.astro.research
Steve Willner
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Posts: 1,172
Default 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.
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Cambridge, MA 02138 USA
  #5  
Old August 4th 10, 10:27 AM posted to sci.astro.research
Jonathan Thornburg [remove -animal to reply][_3_]
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Posts: 137
Default 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},
}

--
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  #6  
Old August 4th 10, 10:28 AM posted to sci.astro.research
Phillip Helbig---undress to reply
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Posts: 629
Default 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  
Old August 4th 10, 10:29 AM posted to sci.astro.research
Oh No
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Posts: 433
Default 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  
Old August 4th 10, 11:43 AM posted to sci.astro.research
Phillip Helbig---undress to reply
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Posts: 629
Default 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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