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#1




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




Addition of grav'l potentials?
On Friday, November 17, 2017 at 2:27:13 AM UTC5, 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




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 endtoend 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 xn1n2 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 nonflat (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 nonuniqueness 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 wellbehaved 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 "postNewtonian 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 leadingorder terms in the postNewtonian 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




Addition of grav'l potentials?
On Friday, November 17, 2017 at 2:27:13 AM UTC5, 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 spheroidearthwithallmountainsremoved). See https://en.wikipedia.org/wiki/Gravity_anomaly https://en.wikipedia.org/wiki/Bouguer_anomaly for a bit more information.  jt]] 
#5




Addition of grav'l potentials?
On Monday, November 20, 2017 at 4:33:30 PM UTC5, wrote:
On Friday, November 17, 2017 at 2:27:13 AM UTC5, 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 spheroidearthwithallmountainsremoved). 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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