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Addition of grav'l potentials?



 
 
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  #1  
Old November 17th 17, 08:27 AM posted to sci.astro.research
stargene
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Posts: 38
Default Addition of grav'l potentials?

An impossible question?

I would like to know if there is a straightforward way of calculating
the sum of two different gravitational potentials U and V at some point
x.
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  #2  
Old November 18th 17, 09:48 AM posted to sci.astro.research
John Heath
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Posts: 8
Default Addition of grav'l potentials?

On Friday, November 17, 2017 at 2:27:13 AM UTC-5, stargene wrote:
An impossible question?

I would like to know if there is a straightforward way of calculating
the sum of two different gravitational potentials U and V at some point
x.

To simplify as much as possible, I imagine, say, that two neutron stars
n1 and n2 are in circular orbit around their mutual center of gravity
(*) and that the point x is always outside their orbit but on a line
joining their centers...Something roughly like:

x _ _ _ _ _ _(n1) _ _ _(*) _ _ _(n2)

and r1 is distance between x and n1; r2 is the distance between x and n2.=



Separately and classically, U, for n1 might be:
U = -G M(n1)/r1 and V, for n2 might be: V = -GM(n2)/r2 . But
what would GR say about this?

So, assuming that r1 and r2 are easy to define, would the sum U+V be
analogous to SR's addition of velocities:

w=(u+v)/(1+uv) ,

..where U=-u^2 , V=-v^2 and their sum W = -w^2 ?

Or am I wildly off base? I just realized an added wrinkle he n1 and
n2 are possibly in relativistic motion and my `straight line' thinking
might be naive.

Thanks


Ouch , you have a way of taxing the noodle. To be clear if (*) were
a light then X would never see (*) light as (n1) would always cast
a shadow hiding it from , X , view. You did say

quote

point x is always outside their orbit but on a line
joining their centers...Something roughly like:


end quote.

If N1 mass were a hint less than N2 to compensate for X then I could
see this as a stable orbit for a few months where N1 will always
shade the light of (*) from X position. Assuming these conditions
are met then special relativity could be ignored at these slow
speeds leaving general relativity. GR tracks Newtonian physics of
M * M / r^2 for force or GR shape of space depending on ones preferred
cup of poison. Keep in mind there are few options if a given math
model is used. What happens in real life is where the real beef is.
I will venture a guess that you were thinking in terms of gravitational
shading to see if it would come out in GR. If this was the case
then no it will not to the best of my understanding. Then again
this is a math simulation not real life.
  #3  
Old November 19th 17, 08:41 AM posted to sci.astro.research
Jonathan Thornburg [remove -animal to reply][_3_]
external usenet poster
 
Posts: 128
Default Addition of grav'l potentials?

stargene wrote:
I would like to know if there is a straightforward way of calculating
the sum of two different gravitational potentials U and V at some point
x.

To simplify as much as possible, I imagine, say, that two neutron stars
n1 and n2 are in circular orbit around their mutual center of gravity
(*) and that the point x is always outside their orbit but on a line
joining their centers...Something roughly like:

x _ _ _ _ _ _(n1) _ _ _(*) _ _ _(n2)

and r1 is distance between x and n1; r2 is the distance between x and n2.

Separately and classically, U, for n1 might be:
U = -G M(n1)/r1 and V, for n2 might be: V = -GM(n2)/r2 . But
what would GR say about this?

So, assuming that r1 and r2 are easy to define, would the sum U+V be
analogous to SR's addition of velocities:

w=(u+v)/(1+uv) ,

..where U=-u^2 , V=-v^2 and their sum W = -w^2 ?

Or am I wildly off base? I just realized an added wrinkle he n1 and
n2 are possibly in relativistic motion and my `straight line' thinking
might be naive.


In Newtonian mechanics, it is indeed correct to add the two potentials,
so the potential at x due to the two neutron stars is
-G M(n1)/r1 - GM(n2)/r2

In general relativity (GR) things are a lot more complicated, but
the short answer is that there's no simple formula for combining the
gravitational effects of multiple bodies (here the two neutron stars)
analogous to special relativity's addition of velocities.

And, in GR r1 and r2 aren't easy to define any more. More precisely,
there are several different plausible definitions. For example
(a) we could (gedanken) measure r1 and r2 by putting a radar set at x
and bouncing radar signals off the two neutron stars (and somehow
correcting for the neutron stars' finite radia, but we won't worry
about that here), or
(b) we could (gedanken) measure r1 and r2 by laying down a bunch of meter
sticks end-to-end between x and n1 and n2 and counting how many meter
sticks it takes to span each of these distances, or
(c) we could (gedankey) measure r1 and r2 by putting theodolites at two
different places off to the sides of the x-n1-n2 line, and using
standard surveying (basically trigonometry based on measured angles)
In GR, (a), (b), and (c) will typically all give slightly *different*
results. (They'll differ by amounts on the order of the neutron stars'
Schwarzschild radia, i.e., on the order of 10% of their physical radia.)
Another way to state this is that (a), (b), and (c) must necessarily
agree *if* we're in a flat spacetime (one where, e.g., Euclid's axioms
of geometry hold). But in GR, spacetime is generally non-flat (Euclid's
axioms are violated, so (a) differs from (b) differs from (c).

r1 and r2 here are examples of coordinates. GR deals with the
non-uniqueness of coordinates by basically saying we can use any
coordinates we want (so long as we do so consistently). The GR object
which describes the gravitational field (the spacetime metric) encodes
not just the gravitational field but also the choice of coordinates,
[note to experts: I'm deliberately fudging the
distinction between a tensor as an abstract geometric
object, and as a matrix of its coordinate components]
and the GR analog of Newton's 1st and 2nd law (the geodesic equation)
is defined in such a way that it gives the same physical results
regardless of what coordinates we choose.

But what this means in practice is that in order to solve for the
spacetime metric, we have to *choose* a coordinate system. This
amounts to specifying 4 free functions everywhere in spacetime.
And we have to do this carefully, because if we do it wrong, it's
very easy to get coordinates which are singular (like latitude/longitude
near the Earth's poles) even in a perfectly well-behaved spacetime,
and that causes no end of confusion.

[[
In Newtonian mechanics the gravitational potential is a
scalar field, i.e., it's a single number at each point
in spacetime. But in GR the spacetime metric is a
*tensor* field, which has 10 independent components at
each point in spacetime. So in a sense there are 10
gravitational potentials in GR.

It turns out there are really only 2 dynamical degrees
of freedom in the spacetime metric, but in general
there's no easy way to separate those out -- we have to
choose the coordinates, then solve for the full
spacetime metric. This is complicated, both
conceptually, mathematically, and computationally.
]]

Another complication worth noting is that in GR the two orbiting
bodies necessarily radiatiate gravitational waves (GWs), and those
GWs themselves carry energy and thus gravitate. So to really determine
the gravitational effects at point x we need to calculate all the GWs
that are around as well.

Fortunately, in many practical situations (including the one we
asked about) there are useful approximations which greatly simplify
the problem:

For example, we can expand all the GR equations in powers of the
velocities of the two neutron stars. More precisely, we expand in
powers of v/c, where v is the velocity of one of the neutron stars.
If we assume that the neutron stars haven't (yet) physically collided,
and that they were slowly moving when they were far apart, then their
velocities now are at most a few tenths of the speed of light, i.e.,
v/c is no more than 0.2 or so, so a power series in v/c should converge
pretty quickly. Doing such a "post-Newtonian approximation" consistently
and correctly is still a big job, but it's easier than solving the
full Einstein equations. The result of such a calculation is a set
of series expansions in v/c for all the interesing dynamical quantities.

If we take just the leading-order terms in the post-Newtonian series,
the result is "just" Newtonian mechanics and gravitation.

Another approximation is to assume that GWs are negligible. There
are a couple of different mathematical ways to make this assumption
(the most common way is to assume that *space* (not spacetime) is
"conformally flat"), but the result is to greatly simplify the
mathematical structure of the Einstein equations (from 10 hyperbolic
PDEs down to 4 elliptic PDEs). This approximation turns out to be
pretty good for orbiting neutron stars, or even for orbiting black
holes until they get very close to each other.

So, to summarize, in GR the original poster's nice simple question
becomes harder to pose (we have to specify precisely how we're defining
r1 and r2), and *much* harder to answer.

ciao,
--
-- "Jonathan Thornburg [remove -animal to reply]"
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
currently on the west coast of Canada
"There was of course no way of knowing whether you were being watched
at any given moment. How often, or on what system, the Thought Police
plugged in on any individual wire was guesswork. It was even conceivable
that they watched everybody all the time." -- George Orwell, "1984"

  #4  
Old November 20th 17, 10:33 PM posted to sci.astro.research
external usenet poster
 
Posts: n/a
Default Addition of grav'l potentials?

On Friday, November 17, 2017 at 2:27:13 AM UTC-5, stargene wrote:
An impossible question?

I would like to know if there is a straightforward way of calculating
the sum of two different gravitational potentials U and V at some point
x.

To simplify as much as possible, I imagine, say, that two neutron stars
n1 and n2 are in circular orbit around their mutual center of gravity
(*) and that the point x is always outside their orbit but on a line
joining their centers...Something roughly like:

x _ _ _ _ _ _(n1) _ _ _(*) _ _ _(n2)

and r1 is distance between x and n1; r2 is the distance between x and n2.

Separately and classically, U, for n1 might be:
U = -G M(n1)/r1 and V, for n2 might be: V = -GM(n2)/r2 . But
what would GR say about this?

So, assuming that r1 and r2 are easy to define, would the sum U+V be
analogous to SR's addition of velocities:

w=(u+v)/(1+uv) ,

..where U=-u^2 , V=-v^2 and their sum W = -w^2 ?

Or am I wildly off base? I just realized an added wrinkle he n1 and
n2 are possibly in relativistic motion and my `straight line' thinking
might be naive.

Thanks


I once posed this type of addition concerning a gravity anomaly
in Denver Colorado. One mass was to be a spheroid earth with
all of the Rocky Mountains removed. Basically a plain with a
sea level altitude of Denver's, about 5000 feet. This is Newtonian
geometry.

The Rocky Mountains were to be a type of planar disk as a first
approximation. So the equation will be non Newton's, but
approximately as a set of spheres inside the disk. Just add up
distance outcomes for the disk interior.

The intention was to interpret the anomaly as an disk asteroid sitting
on a plain.

If course I was ridiculed, only.

[[Mod. note -- Your proposal is in fact not implausible. Actual gravity
models need to consider lots of density variations in the solid Earth
as well (actual mountains also have "fundations" which project below the
spheroid-earth-with-all-mountains-removed). See
https://en.wikipedia.org/wiki/Gravity_anomaly
https://en.wikipedia.org/wiki/Bouguer_anomaly
for a bit more information.
-- jt]]
  #5  
Old November 27th 17, 10:06 PM posted to sci.astro.research
[email protected]
external usenet poster
 
Posts: 47
Default Addition of grav'l potentials?

On Monday, November 20, 2017 at 4:33:30 PM UTC-5, wrote:
On Friday, November 17, 2017 at 2:27:13 AM UTC-5, stargene wrote:
An impossible question?

I would like to know if there is a straightforward way of calculating
the sum of two different gravitational potentials U and V at some point
x.

To simplify as much as possible, I imagine, say, that two neutron stars
n1 and n2 are in circular orbit around their mutual center of gravity
(*) and that the point x is always outside their orbit but on a line
joining their centers...Something roughly like:

x _ _ _ _ _ _(n1) _ _ _(*) _ _ _(n2)

and r1 is distance between x and n1; r2 is the distance between x and n2.

Separately and classically, U, for n1 might be:
U = -G M(n1)/r1 and V, for n2 might be: V = -GM(n2)/r2 . But
what would GR say about this?

So, assuming that r1 and r2 are easy to define, would the sum U+V be
analogous to SR's addition of velocities:

w=(u+v)/(1+uv) ,

..where U=-u^2 , V=-v^2 and their sum W = -w^2 ?

Or am I wildly off base? I just realized an added wrinkle he n1 and
n2 are possibly in relativistic motion and my `straight line' thinking
might be naive.

Thanks


I once posed this type of addition concerning a gravity anomaly
in Denver Colorado. One mass was to be a spheroid earth with
all of the Rocky Mountains removed. Basically a plain with a
sea level altitude of Denver's, about 5000 feet. This is Newtonian
geometry.

The Rocky Mountains were to be a type of planar disk as a first
approximation. So the equation will be non Newton's, but
approximately as a set of spheres inside the disk. Just add up
distance outcomes for the disk interior.

The intention was to interpret the anomaly as an disk asteroid sitting
on a plain.

If course I was ridiculed, only.

[[Mod. note -- Your proposal is in fact not implausible. Actual gravity
models need to consider lots of density variations in the solid Earth
as well (actual mountains also have "fundations" which project below the
spheroid-earth-with-all-mountains-removed). See
https://en.wikipedia.org/wiki/Gravity_anomaly
https://en.wikipedia.org/wiki/Bouguer_anomaly
for a bit more information.
-- jt]]


Thanks the issue is in the sciences. I forget the reference,
but around 1982 a researcher used a spinning top in an
aircraft to look at the gravity field along the New Jersey
cliff system. He saw a definite precession change in the
gyro.

This might be the only way to measure the Bouguer Anomaly
I suggest.

The NIST responded by making a top and measured the mass change
of it. They compared the mass at two altitudes and saw
no mass change. Implying no altitude gravity field. This
was done at the Boulder Lab of the NIST.

In my way of thinking the NIST was at fault for not defining
the lower limit of sensitivity of their answer. I would
demand the use of a spring lever balance equal to a gravity
survey detector system. They used a common electronic strain
gauge laboratory scale.

 




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