the speed of gravitational field propagation

[blaksheep423]

I am working on some interesting topics which relate to the speed at which gravity propagates. My question is this: within the frameworks of special and general relativity (or any other widely accepted theory, for that matter), is it necessary for gravitons/gravity waves to propagate at exactly the speed of light? Is it theoretically valid to suppose that gravitational fields could propagate at speeds of 0.9c, or 0.99c, etc.?

The limited amount of online research I've done concerning this has told me that, using a binary star system, the speed of gravity has been experimentally measured to within 0.95c, which, when coupled with a 0.25 experimental error, led to the assumption that gravity propagates at exactly c. Is there any theoretical work or research which refutes this, or any other reason to intelligently doubt it?

Thank you in advance for your time and your responses

[rathat]

I always kind of wondered how planets could keep a stable orbit around the sun while the suns gravity takes 500 seconds to reach the earth and the sun goes around the milky way at 486,000 miles per hour, i would figure that delay would stop things from orbiting but what do i know.

[JesseM]

I always kind of wondered how planets could keep a stable orbit around the sun while the suns gravity takes 500 seconds to reach the earth and the sun goes around the milky way at 486,000 miles per hour, i would figure that delay would stop things from orbiting but what do i know.
There's a good discussion of how certain effects of gravity "compensate" for the propogation delay here:

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

[rathat]

wow, that can't be a more perfect article on my question.

[blaksheep423]

this article sums up what i already knew, but is there any indication that the speed of gravity may not be exactly c? and is it absolutely essential to any standing theory that gravity propagate at the speed of light, or is it conceiveable that it could be slightly less?

[Creator]

There's a good discussion of how certain effects of gravity "compensate" for the propogation delay here:

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html


Jesse; thanks for the link.
I've seen this argument many times in attempts to resolve the issue of grav. propogation delay causing orbital instability ...

I appreciate Carlip, et al, and their Gen Rel. point of view claiming that the speed of gravity explanation is analogous to Electrodynamic equations, but simply saying "when you solve the gravitational eqns. ...a cancellation of retardation effects occur" really seems like handwaving. Van Flandern and others do bring up some valid counter points..... (PLEASE, folks; no flaming about Van Flandern....that is NOT the point here.)

It would be nice if a little more in depth analysis could clear up a few questions which I find Carlip has not addressed.....
So let me start by asking you, Jesse, since I see you to be exacting in many of your responses.
If you don't feel you have a handle on this issue I understand....but I'm NOT looking for another link to Carlip Vs. Van Flandern debate.. (I've read all the counter punches and am not satisfied. :smile: )


First, let me say I agree with GR in most of its predictions, and I think Carlip tries to get to the real point by admitting , " Strictly speaking, gravity is not a "force" in general relativity,..." And I think he should have left it there. So there would be no need to address the non-centrality of forces...which causes the orbital instability problem.

But since he continues (in the weak field approx.), "one finds that the "force" in GR is not quite central--it does not point directly towards the source of the gravitational field--and that it depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled..."

Again apparent handwaving...

First, let me ask you....Is not this "velocity dependent" force in Gen Rel. based upon a v^2 term, namely v^2/c^2 ??

BTW, please give me simple answers WITHOUT using latex since all latex stuff is blacked out on my computer ( maybe that's asking alot , but try me). also, I post rather infrequently, and so it may be some time before I respond....so please be patient).

Thanks.
:smile:

[JesseM]

I appreciate Carlip, et al, and their Gen Rel. point of view claiming that the speed of gravity explanation is analogous to Electrodynamic equations, but simply saying "when you solve the gravitational eqns. ...a cancellation of retardation effects occur" really seems like handwaving.
Why do you think it is handwaving? Presumably there are technical details behind that statement, which was just a nonmathematical one for an article aimed at a general audience.
So let me start by asking you, Jesse, since I see you to be exacting in many of your responses.
If you don't feel you have a handle on this issue I understand....but I'm NOT looking for another link to Carlip Vs. Van Flandern debate.. (I've read all the counter punches and am not satisfied. :smile: )
I am not very well-versed in the technical details of GR, so I can't help you with that. Perhaps someone else here can answer your questions. In the meantime you may find at least some useful info on these earlier physicsforums threads discussing the quadrupole dependence of gravity:

http://www.physicsforums.com/showthread.php?t=148954
http://www.physicsforums.com/showthread.php?t=147047
http://www.physicsforums.com/showthread.php?t=145020

First, let me say I agree with GR in most of its predictions, and I think Carlip tries to get to the real point by admitting , " Strictly speaking, gravity is not a "force" in general relativity,..." And I think he should have left it there. So there would be no need to address the non-centrality of forces...which causes the orbital instability problem.
Just because it's not a force doesn't mean we don't still need to explain why there is no orbital instability problem. GR is still after all a local theory--I'm pretty sure it would be true for example that the curvature of spacetime in a given region is determined only by points in the past light cone of that region, so for example if the Sun were split into pieces which went in different directions at nearly the speed of light, this couldn't have an effect on the orbits of the planets until the event of the splitting entered their past light cones.
But since he continues (in the weak field approx.), "one finds that the "force" in GR is not quite central--it does not point directly towards the source of the gravitational field--and that it depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled..."

Again apparent handwaving...
Do you really imagine there is no math behind it to make it precise?
First, let me ask you....Is not this "velocity dependent" force in Gen Rel. based upon a v^2 term, namely v^2/c^2 ??
Did you read this somewhere, or conclude it from some equation, or what? According to the threads I quoted earlier it sounds a bit more complicated, post #1 on this thread (http://www.physicsforums.com/showthread.php?t=147047) says "the strongest type of gravitational radiation from an isolated concentration of mass-energy occurs when the quadrupole moment tensor of the source of the field has a nonzero second derivative", and post #13 on this one (http://www.physicsforums.com/showthread.php?t=148954) adds: "Properly speaking, there can be other sources of gravitational radiation in addition to the quadrupole mass moment. For example, a source with constant quadrupole mass moment but time varying octupole mass moment will radiate, but this "octupole radiation" is far weaker than quadrupole radiation. And in addition to quadrupole mass moment there is a current quadrupole moment."

[Creator]

Jesse; thanks for your respnse but you are missing the point of my post entirely.
Carlip's answer is not about gravitational WAVES, i.e., quadrupolar gravitational radiation. Please read it again.



I am not very well-versed in the technical details of GR, so I can't help you with that. Perhaps someone else here can answer your questions.


Thanks for admitting it.
Anyone else who is familiar with the 'speed of gravity' arguments want to give it a try?....again...the question is...

In Carlip's response as to how GR explains why the speed of gravity itself can have propogation delay without causing orbital instability , he says: "the 'force' in GR...... depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled..."

This apparent handwaving cannot be denied without more thorough examination.

My first question is: Does not this velocity dependent "force" in GR actually depend upon a v^2 term, or more precisely, upon v^2/c^2 ??

Anyone?

[Ich]

simply saying "when you solve the gravitational eqns. ...a cancellation of retardation effects occur" really seems like handwaving.
Yeah, and saying that Carlip is "simply saying..." really seems like distorting the truth.

Carlip's answer is not about gravitational WAVES, i.e., quadrupolar gravitational radiation.
It is about gravitational waves also; please read it again.

Does not this velocity dependent "force" in GR actually depend upon a v^2 term, or more precisely, upon v^2/c^2 ??
Yes.

Carlip presents a rigorous derivation which is hard to follow, of course. Therefore, additionally, he presents a less involved mathematical and physical argumentation from "first principles". To make sure that everybody understands, he also describes his findigs in plain English. What more could he do?

Accusing him of handwaving seems quite dishonest to me, as you surely must know the difference between "I don't understand his derivation" and "he gives no derivation".

[JesseM]

Jesse; thanks for your respnse but you are missing the point of my post entirely.
Carlip's answer is not about gravitational WAVES, i.e., quadrupolar gravitational radiation. Please read it again.
Many of his comments are about gravitational waves. But I think I understand, your question is more about the "compensating effects" that occur even in situations where gravitational waves are not being emitted, correct?
In Carlip's response as to how GR explains why the speed of gravity itself can have propogation delay without causing orbital instability , he says: "the 'force' in GR...... depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled..."

This apparent handwaving cannot be denied without more thorough examination.

My first question is: Does not this velocity dependent "force" in GR actually depend upon a v^2 term, or more precisely, upon v^2/c^2 ??
Can you explain where specifically you get the idea that it depends on a v^2 term? I think it would help others to address your question. Again, have you seen this stated somewhere, are you deriving it from some equation, or what?

Also, I got the idea that when he said the force "depends on velocity as well as position", he meant that in the weak-field approximation, the gravitational force an object A feels from another object B which lies X light-seconds away is not directed at the position B was X seconds ago, but rather is directed towards where the object B would be if we took its velocity vector X seconds ago and used it to "extrapolate" its current position under the assumption it had moved at constant velocity for the last X seconds. So the force would be directed towards the actual present position of B if it had indeed moved at constant velocity, whereas if B was accelerated during the last X seconds than this wouldn't have any effect on A until a wave created by the acceleration reached A, until then A would continue to be pulled towards the position of an imaginary phantom version of B that had not accelerated. This would fit with what Carlip said about electromagnetism:
This cancellation may seem less strange if one notes that a similar effect occurs in electromagnetism. If a charged particle is moving at a constant velocity, it exerts a force that points toward its present position, not its retarded position, even though electromagnetic interactions certainly move at the speed of light. Here, as in general relativity, subtleties in the nature of the interaction "conspire" to disguise the effect of propagation delay. It should be emphasized that in both electromagnetism and general relativity, this effect is not put in ad hoc but comes out of the equations. Also, the cancellation is nearly exact only for constant velocities.
Am I incorrect in my understanding of what Carlip said? If my understanding is right, then what would be the corresponding physical meaning of your statement that the force depends on a "v^2 term"?

[Estif]

There's a good discussion of how certain effects of gravity "compensate" for the propogation delay here:

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

Has no one noticed how "delay canceling " is a logical paradox?

To say delay gets canceled makes sense as much as if they said distance gets canceled.



this article sums up what i already knew, but is there any indication that the speed of gravity may not be exactly c? and is it absolutely essential to any standing theory that gravity propagate at the speed of light, or is it conceiveable that it could be slightly less?

What do you mean? Article very directly states speed of gravity is almost instantaneous. That is nowhere even close to the speed of light.


Article says:
- "In the simple newtonian model gravity propagates instantaneously"

- "The net result is that the effect of propagation delay is almost exactly cancelled, and general relativity very nearly reproduces the newtonian result."


v= s/t
v= s/delay

v= s/(delay - delay_almost_exact_cancel)
v= ~s/0.000000000000000000000000000000000000000000000001


Speed of gravity seem to be adjusting itself so it can act "almost instantaneously" regardless of distance. -- Has no one considered that gravity fields could be a fixed-part of the elementary particles and not something that propagates from it. Like a rigid-body fields would then move along with its center and that would explain instantaneous action over distance. How about it?

[JesseM]

Has no one noticed how "delay canceling " is a logical paradox?

To say delay gets canceled makes sense as much as if they said distance gets canceled.
Not if you pay attention to the context. In this case, they are saying that the delay that would be present in a hypothetical theory of gravity where the force vector always points to an object's "retarded" position (i.e. the position it was at when it was emitting a signal moving at the speed of light which would just be reaching you at this moment) is cancelled out when you calculate the actual direction of the gravitational pull in the "weak field" approximation to GR (or at least, it is cancelled for objects moving at constant velocity).

[Vanadium 50]

Van Flandern and others do bring up some valid counter points..... (PLEASE, folks; no flaming about Van Flandern....that is NOT the point here.)

Then I am afraid you shouldn't have brought him up.

I've actually had lunch with Tom van Flandern. He is a true gentleman. Unfortunately, he's also an object lesson in what happens when one becomes too enamored of one's own theory. During the course of this lunch, when presented with a counter-argument told me that this proves that not only is GR wrong, but Newtonian mechanics is wrong as well.

You simply cannot take Tom van Flandern to be a reliable source.

You will not learn anything whatever about the speed of gravity by looking at orbits around a static gravity source. The source is static. Specifically, the potential is just a function of distance, and the force is just the gradient of the potential. The relevant time scale is not the 8 minutes it takes gravity to reach the earth from the sun, it's the 4.5 billion years the sun has been here.

All this talk of aberration and remarkable cancellations, while true, is simply an artifact of working in a geocentric frame, where things are a mess. In a heliocentric frame, everything looks simple - there's a static potential, which has a gradient, the force, and the earth responds to this force.

Measuring the speed of gravitational waves is difficult but straightforward. The radiated power depends on the speed of gravitational radiation, and it's straightforward to measure the energy loss in a binary pulsar system. One gets that the speed of gravity is close to and compatible with the speed of light.

Measuring the speed of static gravity (assuming it even can be different from radiation - this is far from clear) is very difficult. You need two large masses, close enough to each other that one's field doesn't dominate, but far enough apart that there is a measurable delay in propagation time - and you need something to act as a test probe to measure the local spacetime curvature. This is incredibly difficult, as few such systems exist, and placing a test probe where you want it adds even more complications. People have tried this using the sun, Jupiter (already not ideal) and EM radiation as a test probe. This has proved to be even more difficult than expected, as the use of EM radiation as the test probe mixes the speed of light in to the problem in a non-trivial way, and the interpretation of the results have been controversial.

[Estif]

... The relevant time scale is not the 8 minutes it takes gravity to reach the earth from the sun, it's the 4.5 billion years the sun has been here.

Are you saying this delay (8 minutes) does not actually get canceled, and so Earth does not orbit Sun, but the point where Sun was 8 minutes ago? What does it matter what happened in the past, is there some accumulative effect that impacts potentials and delays? I thought orbitals depend ONLY on distance, mass and velocity (initial conditions so to say).

What formula are you using instead of F= k* m1*m2/r^2 ?

[A.T.]

and so Earth does not orbit Sun, but the point where Sun was 8 minutes ago?
There is no frame of reference in which this question makes any sense:
- In the frame of the Sun, the Sun and the point where Sun was 8 minutes ago is the same.
- In the frame of the Earth, the Earth does not orbit anything.

[JesseM]

There is no frame of reference in which this question makes any sense:
- In the frame of the Sun, the Sun and the point where Sun was 8 minutes ago is the same.
- In the frame of the Earth, the Earth does not orbit anything.
Am I correct in understanding that the "weak field" approximation to general relativity treats gravity as a type of field in flat minkowski spacetime? (see p. 468 of this book (http://books.google.com/books?id=xma1QuTJphYC)). If so, can you pick an "inertial" coordinate system where the Sun has some constant velocity and figure out whether the Earth is being accelerated towards its current position at each moment?

[A.T.]

If so, can you pick an "inertial" coordinate system where the Sun has some constant velocity and figure out whether the Earth is being accelerated towards its current position at each moment?
Yes, right. Stating "There is no frame of reference in which this question makes any sense" was wrong. I just considered the two mentioned frames.

The fact that the Earth is being accelerated towards the inertially moving Sun's current position doesn't imply that the speed of gravitational field propagation is infinite. The field could have a velocity component.

[JesseM]

The fact that the Earth is being accelerated towards the inertially moving Sun's current position doesn't imply that the speed of gravitational field propagation is infinite. The field could have a velocity component.
Right...when people like Van Flandern claim the speed is infinite (or faster than c), are they actually claiming gravitational information is traveling faster than light, so if you shake one planet, observers on other planets would feel the gravitational effect earlier than the light from the shaking-event would reach them? If so, could this be debunked just by showing that in GR, the curvature in some finite region of spacetime can be determined by what's in the past light cone of that region, not by anything outside outside the past light cone?

[Vanadium 50]

If so, could this be debunked just by showing that in GR, the curvature in some finite region of spacetime can be determined by what's in the past light cone of that region, not by anything outside outside the past light cone?

But van Flandern doesn't (well, didn't) believe in GR. Or SR. So he's not going to be convinced by that argument.

Like I said, he's an object lesson in what happens when one loves one's own theories too much. When faced with a contradiction between his theories (intended to preserve Newtonian mechanics) and Newtonian mechanics, he concluded Newtonian mechanics is wrong.

[Estif]

There is no frame of reference in which this question makes any sense:
- In the frame of the Sun, the Sun and the point where Sun was 8 minutes ago is the same.
- In the frame of the Earth, the Earth does not orbit anything.

That would be true if those two masses were the only ones in the universe and the distance between them was constant.

A third mass with an orbit in a different plane should make for a very good vantage point to judge what is going on with the first two. From this planet we would still not know if the system is moving or not as a whole, position of the Sun and where Sun was 8 minutes ago would still be the "same", but we should be able to see all the other relations.

But, even if the first two masses were the only ones, without instantaneous action and orbits being exactly around their present centers they would not be able to keep a constant distance and stable orbits, according to Newtonian mechanics, as I've have heard.


However, having a sky full of distant stars it's like having a grid or millimeter paper (absolute reference frame) that can serve as a guide for plotting mathematical functions and deduce relative motion, if not absolute one.


If we can easily measure 8 minutes delay with the light, why can't we do the same with gravity? Eight minutes is quite a delay, and if you try to simulate solar system with this delay, it will fall apart, according Newtonian mechanics, as I have heard.


What formula are you using instead of F= k* m1*m2/r^2?

How do you know it's more correct?

[JesseM]

But van Flandern doesn't (well, didn't) believe in GR. Or SR. So he's not going to be convinced by that argument.
Oh, somehow I thought he was saying physicists are misunderstanding what GR says about the speed of gravity on a theoretical level. So what's his deal, he agrees GR predicts gravity travels no faster than light, but thinks there is empirical evidence that it actually does? But for any observation he might point to about how planets move in our solar system, isn't it possible to show that the movements match the predictions of GR, using a weak-field approximation or a numerical approximation or something?

[Estif]

But van Flandern doesn't (well, didn't) believe in GR. Or SR. So he's not going to be convinced by that argument.

Like I said, he's an object lesson in what happens when one loves one's own theories too much. When faced with a contradiction between his theories (intended to preserve Newtonian mechanics) and Newtonian mechanics, he concluded Newtonian mechanics is wrong.

Are scientists equally divided in their opinions about this, or is there a majority? If there is a majority, then what is their explanation, what is their _official document explaining all this?


Is there an "official" kind of document for anything in science at all? I mean, we can't even agree if we are to consider a virus living or non-living thing, and I'm pretty sure there are many even more extreme examples of the division of opinions.


So, what do we do without majority?

And what do we do if majority is wrong, like in Galileo time?

[A.T.]

But, even if the first two masses were the only ones, without instantaneous action and orbits being exactly around their present centers they would not be able to keep a constant distance and stable orbits, according to Newtonian mechanics, as I've have heard.

Instantaneous action is merely the simplest way to make orbits work like they do, so Newton went with it. This doesn't mean it is the only way or the correct way.

Even without GR, you could modify Newtons gravity field to be non-instantaneous but have a velocity component inherited from the velocity of the field's source. The field lines in such a field emitted from a inertially moving source would point towards the current position of the source, not the position of their emission. See the electric field. No information is transported instantaneously, because the information about the source's velocity was already available at emission time.

I'm not saying that this is a good law of gravity. I'm just pointing out the flaw in the simplistic argument which goes like: "If objects are pulled to the current position of a moving mass, information must travel instantaneously"

[Estif]

Instantaneous action is merely the simplest way to make orbits work like they do, so Newton went with it. This doesn't mean it is the only way or the correct way.

I must agree, somewhat reluctantly though, because as a logical principle Occam's razor would demand that scientists accept the simplest possible theoretical explanation for existing data. It is the same principle that led to abandonment of the aether theories. But yeah, I'm with you... even if all seemed to fit perfectly well, it doesn't hurt to look for an alternative answers or interpretations.



Even without GR, you could modify Newtons gravity field to be non-instantaneous but have a velocity component inherited from the velocity of the field's source. The field lines in such a field emitted from a inertially moving source would point towards the current position of the source, not the position of their emission. See the electric field. No information is transported instantaneously, because the information about the source's velocity was already available at emission time.


You lost me. I like mathematics definitions and formulas most of all, they are much harder to misinterpret than worded theories, so if you can point some link where this is explained in a bit more depth?



I'm not saying that this is a good law of gravity. I'm just pointing out the flaw in the simplistic argument which goes like: "If objects are pulled to the current position of a moving mass, information must travel instantaneously"

I agree, but I do not see any other explanation.


So, what can we do?

First we have to assume quite a bit of determinism in this universe. Then, we can measure the real-world and come up with some theories, and these theories we should be able to re-phrase in mathematical formulas.

With this math all we need to do now is to predict the real-world measurements.

Prediction in deterministic universe should be straight forward... if we only had a power of precision, infinite divisibility, parallelism and continuity universe has. In other words, we need to do kinematic simulation, we need to integrate our equation over time and calculate trajectories, but we are bound to have errors.

So, we plug "F=k* m1*m2/r^2" in our numerical integration and we plot trajectories, step-by-step, the smaller the better, and we find this simple equation works remarkably well, especially considering all the imprecision that came in, not only due to limited computer precision, but also due to measurements of initial condition.

Hence, considering all that, this equation makes me happy.

To convince me otherwise, just give me a better equation, that's all.

[A.T.]

I must agree, somewhat reluctantly though, because as a logical principle Occam's razor would demand that scientists accept the simplest possible theoretical explanation for existing data.
Yes, but instantaneous Newtonian gravity doesn't explain all existing data: amount of light bending, orbit precession.


You lost me. I like mathematics definitions and formulas most of all, they are much harder to misinterpret than worded theories, so if you can point some link where this is explained in a bit more depth?

Take the electric field of an inertially moving electron. Even at a distance the field lines always point to the current position of the electron. Yet if the electron suddenly stops you see that the information about the stop propagates only at c. At distant points the field lines still point to the extrapolated position where the electron would be, if it didn't stop.
On the electric field;
http://farside.ph.utexas.edu/teaching/em/lectures/node125.html
Similar discussion on gravity:
http://www.physicsforums.com/showthread.php?p=2270590

[Estif]

"If objects are pulled to the current position of a moving mass, information must travel instantaneously"

Imagine gravity fields as a fixed extension of elementary particles and not something that propagates from it. Like a rigid-body fields would then move along with its center (like a soccer ball), all at once, though during the interaction there would still might be some delay depending on the rigidity, flexibility or compressibility of this field, but the propagation of this force does not need to be similar to EM waves at all.

EM waves I imagine as ripples in this gravity field "material", but this medium built from gravity fields could still have its non-compressibility, like water, so it could propagate waves at one speed, but transmit a displacement or friction action at different speeds.


Does this make sense? Is there any theory based on similar idea?


[EDIT]
Hey, there you are... I see links, thanks a lot. Catch you latter.

[Vanadium 50]

Are scientists equally divided in their opinions about this, or is there a majority? If there is a majority, then what is their explanation, what is their _official document explaining all this?

There is no "Official Document of Science", and if there were, it would certainly not be devoted to refuting bad arguments. Van Flandern's argument is simply incorrect - it doesn't measure what he says it measures.

Even if one doesn't know the "right" answer doesn't mean that one can't spot a bad argument.

And what do we do if majority is wrong, like in Galileo time?

And speaking of bad arguments, this one is a doozy. Jim Lippard, I think summed it up best: "Yes, they laughed at the Wright Brothers. But they also laughed at Laurel and Hardy."

[atyy]

And speaking of bad arguments, this one is a doozy. Jim Lippard, I think summed it up best: "Yes, they laughed at the Wright Brothers. But they also laughed at Laurel and Hardy."

And Laurel and Hardy were right! :smile:

[S.Vasojevic]

I have no problem in my mind with Earth being attracted to the point where the Sun was 8 min ago. Of course frame dependent "floating" of common centre of mass arises, but it is not hard to imagine how it cancels out if you make a special case of velocity vector laying in a orbital plane.

But I have whole other issue with gravity propagation being equal to C, and that is how on the largest scales gravity is not doomed to fade away completely towards the edge of OU, just as light is. After all gravitational waves are redshifted through expansion just as light waves are, or not?

[Estif]

Yes, but instantaneous Newtonian gravity doesn't explain all existing data: amount of light bending, orbit precession.

When did they come up with the conclusion about precession, what year?

Who simulated it? What was the size of the time-step and what kind of integration algorithm were they using? Did they take the size of the planets into account? Did they integrated all the planets and all the moons in this simulation?

If they did everything right maybe they would get the correct precession result.

Bending of light might be EM interaction, rather than gravity related, but otherwise I agree with you, I can not support what I just said, though I would like to see a new attempt at simulating orbit precession with today's computers and data.


The problem is that GR produces very similar results as Newtonian gravity with much more dubious interpretation visa vi 'instantaneous action', and I don't even know what is that interpretation if it's not the one where delay gets canceled. What are the other interpretations for this "instant" action beside "delay-cancel" one?

I do not even know what formula in GR is supposed to calculate the same thing? What are the inputs, what are the outputs? Do you integrate it over time, or can you get some sort of statistical prediction from only one calculation? How do you input position, velocity and mass in GR equation? How do you setup initial conditions of all the planets and all the moons with GR? How it all works? Can it be simulated on the computer just like with classical physics? What model NASA uses for their trajectories, Newtonian or GR?



Take the electric field of an inertially moving electron. Even at a distance the field lines always point to the current position of the electron. Yet if the electron suddenly stops you see that the information about the stop propagates only at c. At distant points the field lines still point to the extrapolated position where the electron would be, if it didn't stop.
On the electric field;
http://farside.ph.utexas.edu/teaching/em/lectures/node125.html
Similar discussion on gravity:
http://www.physicsforums.com/showthread.php?p=2270590

Why "r^3"? Where did "r^2" go?
How many dimensions are they modeling?


In any case, I have to disagree with all that.

I could just as same said this: F(electric)= k* q1*q2/r^2 ...and conclude action is instantaneous, which takes us back to the almost exactly the same formula as with gravity, and so the argument about the delay continues.


There is one thing that strikes me as very strange there. They calculate electric force, but they do not calculate magnetic force. Instead, they calculate magnetic _field, but to calculate magnetic _force you need one more cross product with the velocity, and then you get a right-hand rule and what is known as Lorentz force. But even more odd is that they do not use their magnetic field for anything. It's not part of any equation necessary for this conclusion, so it looks strangely superfluous.

[A.T.]

When did they come up with the conclusion about precession, what year? Who simulated it? What was the size of the time-step and what kind of integration algorithm were they using? Did they take the size of the planets into account? Did they integrated all the planets and all the moons in this simulation?
http://lmgtfy.com/?q=precession+mercury
If they did everything right maybe they would get the correct precession result.
Do everything right then and check it, if you think that everyone before you was unable to do it right. Come back if you have something.

Bending of light might be EM interaction, rather than gravity related

What kind of EM interaction? You have to invent new phenomena, to explain things which are already included in GR?

I could just as same said this: F(electric)= k* q1*q2/r^2 ...and conclude action is instantaneous,

But it is not instantaneous. If the electron stops, distant observers will notice it after a delay. Yet still:
E acts in line with the point which the charge occupies at the instant of measurement, despite the fact that, owing to the finite speed of propagation of all physical effects, the behaviour of the charge during a finite period before that instant can no longer affect the measurement.

[S.Vasojevic]

Bending of light might be EM interaction, rather than gravity related, but otherwise I agree with you, I can not support what I just said, though I would like to see a new attempt at simulating orbit precession with today's computers and data.


My advice to you, as it is to myself, is to acknowledge yourself not how much you know, but how much you do not know. It is clear that you have background in computers and some physics, but if you stick around PF you will learn that there are a bunch of people here whose opinion you must value, and few of them are posting in this thread.

And how, on Earth, among thousands of scientists, who among the other things, built HST, send it to orbit, took pictures of distorted distant objects, analyzed it over and over again, put it on the web for public, and named those pictures gravitational lensing, you are claiming that it may be EM interaction!?

[Estif]

My advice to you, as it is to myself, is to acknowledge yourself not how much you know, but how much you do not know. It is clear that you have background in computers and some physics, but if you stick around PF you will learn that there are a bunch of people here whose opinion you must value, and few of them are posting in this thread.

You sound emotional? This is not about favorite theories, it's about the truth, the Value and Veracity of such. I value logic, reason and arguments. No one opinion I must value, nor should you. It is not about WHO says it, but WHAT is said. So, bring your arguments here and leave emotions at home.


And how, on Earth, among thousands of scientists, who among the other things, built HST, send it to orbit, took pictures of distorted distant objects, analyzed it over and over again, put it on the web for public, and named those pictures gravitational lensing, you are claiming that it may be EM interaction!?

I said - "I can not support what I just said".

Is there anything in particular you disagree about, or you just don't like what I said because it's not a popular opinion? I find it disappointing you put trust in some authorities and majorities. -- Uzdaj se use i u svoje kljuse.

[Estif]

http://lmgtfy.com/?q=precession+mercury


You told me to Google it?

Two things are wrong with your response:

1.) None of the links actually answer the question.

2.) The question was rhetorical.



Do everything right then and check it, if you think that everyone before you was unable to do it right. Come back if you have something.


If you tried to answer the question you would understand.



But it is not instantaneous. If the electron stops, distant observers will notice it after a delay.

What are you talking about? They base their conclusion on some equations full of assumptions, it is not an experiment. It is not experimental data, you know? Did you really think you can measure some delay on microscopic scale with charges, while you can not measure 8 minutes delay with interplanetary distances?
-------------------------------------------------------------


So, can someone point out that formula used in GR to calculate trajectories?

Can it be simulated on the computer just like with classical physics?

What model NASA uses for their trajectories, Newtonian or GR?

[George Jones]

So, can someone point out that formula used in GR to calculate trajectories?

Can it be simulated on the computer just like with classical physics?

You mean like

http://physicsforums.com/showthread.php?p=1091901#post1091901?




Note that Physics Forums rules to which you agreed when you registered,

http://www.physicsforums.com/showthread.php?t=5374,

in part, state
Overly Speculative Posts: One of the main goals of PF is to help students learn the current status of physics as practiced by the scientific community; accordingly, Physicsforums.com strives to maintain high standards of academic integrity. There are many open questions in physics, and we welcome discussion on those subjects provided the discussion remains intellectually sound. It is against our Posting Guidelines to discuss, in most of the PF forums, new or non-mainstream theories or ideas that have not been published in professional peer-reviewed journals or are not part of current professional mainstream scientific discussion.

Discussion of non-published non-mainstream personal theory is not allowed.

[Estif]

You mean like

http://physicsforums.com/showthread.php?p=1091901#post1091901?


Thanks, but I don't see any equations there. Can you explain how do you calculate trajectories with GR? What are the equations necessary to simulate solar system?



Note that Physics Forums rules to which you agreed when you registered,

http://www.physicsforums.com/showthread.php?t=5374,

in part, state Discussion of non-published non-mainstream personal theory is not allowed.


I want to discuss mainstream theory.

So, what does mainstream theory say - does the delay get canceled or not?

Does Earth orbit Sun, or the point where Sun was 8 minutes ago?

Why can we measure this delay with light, but not with gravity?


Thank you.

[Jonathan Scott]

Thanks, but I don't see any equations there. Can you explain how do you calculate trajectories with GR? What are the equations necessary to simulate solar system?




I want to discuss mainstream theory.

So, what does mainstream theory say - does the delay get canceled or not?

Does Earth orbit Sun, or the point where Sun was 8 minutes ago?

Why can we measure this delay with light, but not with gravity?


Thank you.

I think you've missed the fact that the force on a test object due to the overall field of a moving source, both in electromagnetism and in gravity, includes components due to the motion of the source as well as static components. In electromagnetism, this is of course magnetism. For gravity, there are multiple components due to motion, but the important one here is closely analogous to magnetism and is often called "gravitomagnetism".

The static fields (electric or conventional gravitational) point to where you see the source to be, at the time that light left it. However, the dynamic fields mean that the overall force points to where the source will be right now. [Edit: That's not really technically accurate - that would be the "static fields as calculated without taking the motion into account"; it depends how you define it, but that's roughly how it works.]

In both cases, for a slow-moving source, you can look at it from the reference frame of the source instead, and see that the overall force on the test object (in terms of rate of change of momentum) points to the source in that frame, and hence that it must in other inertial frames too. This is in fact what we find; when you transform both the combined fields and the force law to find out how a test particle will move as seen from another inertial frame, you find that the force is directed towards the extrapolated location of the source at the current time.

Obviously, if the motion of the source was disturbed, the extrapolated location may no longer match the actual location. However, for gravity the conservation laws of energy and momentum mean that it's very difficult to create a "gravitational surprise", because the center of mass of any complete system always moves with constant velocity. Even if for example the sun could explode into two equal parts which moved apart at relativistic speed, the total energy and center of mass of the combined system would still be unaffected, and it would take some time for the parts to separate far enough for the overall gravitational field to change.

[A.T.]

You told me to Google it?
Two things are wrong with your response:

1.) None of the links actually answer the question.

2.) The question was rhetorical.


1) So you read all the Google results? Wow!

2) Why complain about the insufficient answer then?


If you tried to answer the question you would understand.

Understand what? You have shown nothing to understand so far.

They base their conclusion on some equations full of assumptions, it is not an experiment. It is not experimental data, you know?

The propagation speed of changes in the electric field is experimentally well determined to be c. There are many exmaples of that. Take the the x-ray tube:
http://en.wikipedia.org/wiki/X-ray_tube
Fast electrons are suddenly stopped at the anode. The information about that sudden change of the E-field propagates as x-rays at a finite speed. If the propagation of the E-field was instantaneous, you could not create EM-waves propagating at finite speed by accelerating charges. There would be actually no EM-waves at all.

[Estif]

The propagation speed of changes in the electric field is experimentally well determined to be c. There are many exmaples of that. Take the the x-ray tube:
http://en.wikipedia.org/wiki/X-ray_tube

Fast electrons are suddenly stopped at the anode. The information about that sudden change of the E-field propagates as x-rays at a finite speed. If the propagation of the E-field was instantaneous, you could not create EM-waves propagating at finite speed by accelerating charges. There would be actually no EM-waves at all.

That link does not support what you said. Any other links?

[A.T.]

That link does not support what you said. Any other links?
http://en.wikipedia.org/wiki/Bremsstrahlung
http://en.wikipedia.org/wiki/Electromagnetic_radiation#Speed_of_propagation

Do you at least understand that if the propagation so of the E-field was instantaneous, you could not have EM-waves propagating at a finite speed?

[Estif]

http://en.wikipedia.org/wiki/Bremsstrahlung
http://en.wikipedia.org/wiki/Electromagnetic_radiation#Speed_of_propagation

Do you at least understand that if the propagation so of the E-field was instantaneous, you could not have EM-waves propagating at a finite speed?

No, I do not understand that, and your links do not talk about anything like it at all. You seem to be confusing EM fields and EM waves. Perhaps I'm confusing something, but you are not explaining it. I also don't understand why is it hard to point some paper or article that makes these claims you do, and explains it.


I want to understand how GR explains its results closely match newtonian mechanics, which makes calculations based on instantaneous gravity. There was only one link presented here that says this delay gets canceled, but it turned out majority does not agree with it.

So, what is the explanation then that majority do agree with?


I also want to understand how do you simulate solar system with GR. What equations do we use, and how. Without being able to that I do not see how GR can make better predictions about planetary trajectories than classical physics.

How did they calculate Mercury precession, what equations did they use?

[Estif]

I think you've missed the fact that the force on a test object due to the overall field of a moving source, both in electromagnetism and in gravity, includes components due to the motion of the source as well as static components. In electromagnetism, this is of course magnetism. For gravity, there are multiple components due to motion, but the important one here is closely analogous to magnetism and is often called "gravitomagnetism".

I like the sound of that, thanks. But, that does not seem to be a standard part of GR. It actually seem to be completely independent calculation maybe even able to produce results all on its own, without GR, so to say.

Can you show some equation where we can see how parameters vary due to velocity? Shouldn't it be the same if some system, say a binary star, is moving 9000m/s or 90m/s as a whole? It seems that theory implies different relative orbits depending on some absolute speed through the metric of spacetime or something. Did I get that right?



The static fields (electric or conventional gravitational) point to where you see the source to be, at the time that light left it. However, the dynamic fields mean that the overall force points to where the source will be right now. [Edit: That's not really technically accurate - that would be the "static fields as calculated without taking the motion into account"; it depends how you define it, but that's roughly how it works.]

In both cases, for a slow-moving source, you can look at it from the reference frame of the source instead, and see that the overall force on the test object (in terms of rate of change of momentum) points to the source in that frame, and hence that it must in other inertial frames too. This is in fact what we find; when you transform both the combined fields and the force law to find out how a test particle will move as seen from another inertial frame, you find that the force is directed towards the extrapolated location of the source at the current time.

Obviously, if the motion of the source was disturbed, the extrapolated location may no longer match the actual location. However, for gravity the conservation laws of energy and momentum mean that it's very difficult to create a "gravitational surprise", because the center of mass of any complete system always moves with constant velocity. Even if for example the sun could explode into two equal parts which moved apart at relativistic speed, the total energy and center of mass of the combined system would still be unaffected, and it would take some time for the parts to separate far enough for the overall gravitational field to change.

Is this based on some experiments? Have you got some links that explain the math and show equations for that? Are you talking about the delay cancellation? If yes, then let me repeat that is a paradox, it breaks continuity. You can not cancel time, you can not cancel distance. You can only traverse them both in a continuous manner, and the time it takes will tell you what was the speed; velocity= distance/delay. Equations can cancel time and space, sure, but I'm skeptic those equations actually correspond to reality.

[D H]

If so, can you pick an "inertial" coordinate system where the Sun has some constant velocity and figure out whether the Earth is being accelerated towards its current position at each moment?
The Sun does not have a constant velocity in an "inertial" frame; Jupiter's mass is about 1/1000 that of the Sun. The International Celestial Reference Frame with origin at the solar system barycenter is a better bet. See http://www.usno.navy.mil/USNO/astronomical-applications/astronomical-information-center/icrs-narrative for a description.

As far as equations of motion, see section 8.3 (page 3) of chapter 8 of The Explanatory Supplement to the Astronomical Almanac (to be published), http://iau-comm4.jpl.nasa.gov/XSChap8.pdf. These are what JPL uses as the basis for its Development Ephemerides.

[pervect]

To calculate the radiation from an electric charge, the standard thing to do is to use Larmor's formula:

http://en.wikipedia.org/wiki/Larmor_formula

The gravitational case is discussed, using approximations (weak field, nearly Newtonian, velocities << c ) in MTW, pg 989. MTW = "Gravitation" by Misner, Thorne, and Wheeler.

Various people have already posted links to past discussion on the topic, so I won't repeat it.

There was no formalism at the time MTW was written to describe the general case of high velocities/strong fields. Perhaps things have changed since then, though I doubt it. However, for the situation of the Earth orbiting around the sun, the approximations in MTW should be just fine. They wouldn't be so good for more extreme situations, where more detailed calculations specific to the problem at hand would be required. Some such calculations have been done, for instance for the pulsar inspiral that's been experimentally measured.

In any event, the useful way to approach the problem has already been described. To figure out the amount that the force appears as "non-central" for the electromagnetic case, you calculate the momentum that's radiated away by electromagnetic and gravitational radiation.

For the GR case, I'll direct you to the discussion in MTW.

You won't find anything that's detailed that's a "simple web page" on the topic. Not all technical subjects are reducible to "simple web pages". Sorry.

As far as interpreting "forces" in GR, I would assume that Carlip is using Newton-Cartan theory to perform the conversion. See http://en.wikipedia.org/wiki/Newton%E2%80%93Cartan_theory, or MTW chapter 12 pg 289.

What Newton-Cartan theory does is to translate Newtonian gravity into the geometric language of GR. By performing the translation "backwards", you can sometimes turn GR concepts into Newtonian ones. The answers you get will be highly coordinate dependent, though, they won't be nicely coordinate independent as they are in GR.

To get any meaningful results, the contribution of the space curvature components (by curvature components I mean the Christoffel symbols) to the motion of the body in the coordinate system you choose have to be negligible. This turns out to be the case for low velocities. As objects approach the speed of light, the contribution of the space curvature components can't be ignored, and the conversion process of viewing things as a "force" fails.

But you should really ask him (Carlip) - politely - if you need more details. Cartan theory is the approach I'd take to the problem, though.

[Estif]

As far as equations of motion, see section 8.3 (page 3) of chapter 8 of The Explanatory Supplement to the Astronomical Almanac (to be published), http://iau-comm4.jpl.nasa.gov/XSChap8.pdf. These are what JPL uses as the basis for its Development Ephemerides.

Can someone explain how does "r^3" come about in those formulas?

How does 'distance squared' get replaced by 'distance cubed' and why?


--------------------------------
pervect,

I'm not sure if you were answering my question as to how GR can simulate solar system or something else, so can you confirm if that is the "easiest" way to actually practically implement GR? And if so, why isn't that conversion included in GR equations to start with?

[A.T.]

Do you at least understand that if the propagation so of the E-field was instantaneous, you could not have EM-waves propagating at a finite speed?

No, I do not understand that, and your links do not talk about anything like it at all. You seem to be confusing EM fields and EM waves.

An EM wave is a change of the EM-fields propagating at finite speed. If changes of the E-field would be instantaneous the waves would propagate at infinite speed. Well, they wouldn't be waves anymore, just instantaneous changes of the E-field, felt immediately in the entire universe.

[Estif]

An EM wave is a change of the EM-fields propagating at finite speed.


Electromagnetic waves CONSIST of electric and magnetic fields which oscillate. Wave-particles do propagate at or below their terminal velocity of lightspeed, but this has nothing to with field propagation. You should imagine a moving particle describing a helicoid path and you get your wave and your change of EM fields regardless of their propagation speed.

http://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Onde_electromagnetique.svg/350px-Onde_electromagnetique.svg.png



If changes of the E-field would be instantaneous the waves would propagate at infinite speed. Well, they wouldn't be waves anymore, just instantaneous changes of the E-field, felt immediately in the entire universe.



I think you are picturing it the wrong way.


For example, we can have two kinds of waves, say in water.

Here a light would correspond to slow transverse waves on water surface, depending on viscosity, while field propagation would correspond to longitudinal waves, depending on compressibility, and these longitudinal waves would be spherical, much faster and pretty much how you should imagine fields, as opposed to transverse propagation of EM waves.


To take this illustration further, you can imagine these waves would have their different terminal velocity, depending on some "drag coefficient" which would further be influenced by density, that represents gravitational potentials in this example. Interesting enough, this illustration can even explain some peculiar properties of magnetic fields in analogy to vorticity in fluid-dynamics.


http://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Airplane_vortex_edit.jpg/250px-Airplane_vortex_edit.jpg

You can see a right-hand rule just like with Lorentz force, that we were talking about earlier, and funny enough Biot-Savart law ('magnetic field' from your previous example) can be equally used in electromagnetics and aerodynamics. -- Hope this helps with visualization of these otherwise very abstract concepts.


- "Imagination is more important than knowledge." (Albert Einstein)

[D H]

Can someone explain how does "r^3" come about in those formulas?

How does 'distance squared' get replaced by 'distance cubed' and why?

To start with, Newton's law of gravitation for a system of point masses is

\ddot {\mathbf r}_i = \sum_{j\ne i}\frac{\mu_j (\mathbf r_j-\mathbf r_i)}{r_{ij}^3}

The majority of the other terms in that equation are first-order post-Newtonian corrections to Newton's law.

[A.T.]

while field propagation would correspond to longitudinal waves,
See, it's you who is confusing waves and fields here, not me. The analogy to the change of the E-field is not some longitudinal waves, but simply the change of amplitude of the transverse wave.

[Vanadium 50]

The problem is that GR produces very similar results as Newtonian gravity with much more dubious interpretation visa vi 'instantaneous action', and I don't even know what is that interpretation if it's not the one where delay gets canceled. What are the other interpretations for this "instant" action beside "delay-cancel" one?

You are getting dangerously close to crackpottery here - claiming a theory is wrong when you yourself don't understand it.

There are no instantaneous forces in either GR or in EM.

If you have questions, please ask them. If you want to push an agenda that a particular theory is wrong, I would suggest that you devote the effort into understanding the theory (at the level of being able to do the calculations yourself, and not relying one what someone else - even me - tells you it says) before advancing that claim.

[Jonathan Scott]

The effect that "predicts" the location of the source is very simple, and a close non-relativistic equivalent can be seen in the way ripples spread from an object moving with a slow constant velocity on the surface of water. Circular waves spread out from each instantaneous location, but the source location moves. If an observer some distance away runs a straight line through a series of points on nearby wave fronts where the front is aligned in the same direction, the direction of the line points to where the object is expected to be at the current time.

To put it another way, if you plot those circular waves horizontally with the time at which they were emitted time as a vertical axis, the result for a static source is a circular cone, but for a moving source it is a skewed cone, where the center line is sloped according to the velocity. The straight lines up the sides of the cone for the static case remain straight lines but now point to the shifted apex for the moving case. The projections of those lines on to the base correspond to the projected direction of the source in the plane of the water.

A trivial calculation of the "static field" without taking motion into account is like assuming that the field is perpendicular to the local wave front. However, when motion is taken into account the effective field is like the line that joins parallel parts of each wave front, pointing to the extrapolated future position.

[Disclaimer: Please check this for yourself as this is from my head rather than any textbook]

[Estif]

You are getting dangerously close to crackpottery here - claiming a theory is wrong when you yourself don't understand it.

There are no instantaneous forces in either GR or in EM.

If you have questions, please ask them. If you want to push an agenda that a particular theory is wrong, I would suggest that you devote the effort into understanding the theory (at the level of being able to do the calculations yourself, and not relying one what someone else - even me - tells you it says) before advancing that claim.

I'm not pushing agenda, I'm simply interpreting the article that was given here by someone else and is actually supposed to represent mainstream GR opinion, as far as I can figure.

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

I want to know if that is the "official" explanation of how GR works, and if not I want to know what is. I'm doing exactly what you want me to do, I am devoting the effort into understanding the theory by coming here and asking questions, here they are again:


1.) Article says the delay gets canceled, is that true?

2.) Does Earth orbit Sun, or the point where Sun was 8 minutes ago?

3.) Why can we measure this 8 minutes delay with light, but not with gravity?

4.) Do gravity waves propagate as transverse or longitudinal waves and how do you know?

[Estif]

To start with, Newton's law of gravitation for a system of point masses is

\ddot {\mathbf r}_i = \sum_{j\ne i}\frac{\mu_j (\mathbf r_j-\mathbf r_i)}{r_{ij}^3}

The majority of the other terms in that equation are first-order post-Newtonian corrections to Newton's law.

Can you explain that equation?
Why inverse-CUBED, where did inverse-SQUARE law go? Where are the masses, m(i) * m(j)?


Newton's law of universal gravitation
http://en.wikipedia.org/wiki/Newton%27s_law_of_gravitation

http://upload.wikimedia.org/math/b/6/5/b65000f8f887a68545ce63eb1cada232.png

Vector form:
http://upload.wikimedia.org/math/1/d/1/1d133119bde0ace20718ce274db8bf84.png

[D H]

Can you explain that equation?
Why inverse-CUBED, where did inverse-SQUARE law go? Where are the masses, m(i) * m(j)?
Start with the vector form you quoted from wikipedia:

\mathbf F_{12} = -G\frac{m_1m_2}{|\mathbf r_{12}|^2}\hat{\mathbf r}_{12}

This is "the force applied on object 2 due to object 1". Dividing this by the mass of object 2 yields the acceleration of object 2 induced by the gravitational interaction between the two objects.

\mathbf a_{12} = -\,\frac{Gm_1}{|\mathbf r_{12}|^2}\hat{\mathbf r}_{12}

The unit vector \hat{\mathbf r}_{12} is the vector from object 1 to object 2 scaled by the distance between the objects. Thus

\mathbf a_{12} = -\,\frac{Gm_1}{|r_{12}|^3}\mathbf r_{12}

Now denote |\mathbf r_{12}|\equiv r_{12} = r_{21} and using \mathbf r_{12} = - \mathbf r_{21} yields

\mathbf a_{12} = \frac{Gm_1}{r_{21}^3}\mathbf r_{21}

Compare this to the form in the link,

\mathbf \ddot{\mathbf r}_i =
\sum_{j\ne i} \frac{\mu_j}{r_{ji}^3}\mathbf r_{ji}(1 + \text{relativistic terms})

The only differences in these two forms are
The former represents the acceleration due to the one interaction while the latter represents the acceleration due to the totality of mass in the solar system,
The latter expression incorporates "relativistic terms", and
The former expression uses Gm_1 while the latter uses \mu_j.

Astronomers can assess the standard gravitational parameters (http://en.wikipedia.org/wiki/Standard_gravitational_parameter) \mu_j to a high degree of accuracy (more than 10 decimal places for the Sun, almost 9 for the Earth). In comparison, G is know to a paltry 4 decimal places.

[Estif]

Start with the vector form you quoted from wikipedia:

\mathbf F_{12} = -G\frac{m_1m_2}{|\mathbf r_{12}|^2}\hat{\mathbf r}_{12}


Let m(i) = 100 kg
Let m(j) = 999,999 kg

Let m(i) have its center at coordinates [0, 0, 0] meters
Let m(j) have its center at coordinates [500, 0, 0] meters

m(i)= mass 1, m(j)= mass 2, G= constant, R= unit vector
r= distance from the center of mass1 to the center of mass 2
=================================================

r= sqrt(dx*dx + dy*dy + dz*dz)
r= sqrt(500*500 + 0*0 + 0*0)
r= sqrt(250,000)
r= 500 meters


Solve for mass 1:
-----------------
F(i,j)= -G * m(i)*m(j) * R /r^2

F(i,j)= -G * 100 * 999,999 * [(0 - 500)/500, 0/500, 0/500] / 500^2
F(i,j)= -G * 99,999,900 * [-1, 0, 0] / 250,000
F(i,j)= -G * [-400, 0, 0]
F(i,j)= [G * 400, 0, 0]

F = m * a
a(i)= F(i,j) / m(i)
a(i)= [G * 400, 0, 0] / 100

vector_a(i)= [G * 4, 0, 0]



Solve for mass 2:
-----------------
F(j,i)= -G * m(j)*m(i) * R /r^2

F(j,i)= -G * 999,99 * 100 * [(500 - 0)/500, 0/500, 0/500] / 500^2
F(j,i)= -G * 99,999,900 * [1, 0, 0] / 250,000
F(j,i)= [-G * 400, 0, 0]

F = m * a
a(j)= F(j,i) / m(j)
a(j)= [-G * 400, 0, 0] / 999,999

vector_a(j)= [-G * 0.0004, 0, 0]


If you do this for all the planetary pairs in the solar system, all the planets, moons and the sun, it works very well. Depending on the precision and time you want to invest you can get extremely precise results (past or future predictions), not only for days, months and years but the system stays stable for millions of years and errors you get are largely due to floating point precision, size of delta-time and the measurement data you set as initial conditions.



\mathbf F_{12} = -G\frac{m_1m_2}{|\mathbf r_{12}|^2}\hat{\mathbf r}_{12}

This is "the force applied on object 2 due to object 1".


This is probably just a matter of phrasing in the context of what you gonna say next, but let me say that the force is not relative in that way, it's the same from object 2 to object 1 and from object 1 to object 2, only direction is opposite and that is why we have unit vector in that equation, so to give us the direction vector and the same equation works for both objects equally well.



Dividing this by the mass of object 2 yields the acceleration of object 2 induced by the gravitational interaction between the two objects.

\mathbf a_{12} = -\,\frac{Gm_1}{|\mathbf r_{12}|^2}\hat{\mathbf r}_{12}

The unit vector \hat{\mathbf r}_{12} is the vector from object 1 to object 2 scaled by the distance between the objects. Thus

\mathbf a_{12} = -\,\frac{Gm_1}{|r_{12}|^3}\mathbf r_{12}


Unit vector has scalar value of 1, it is only there so to gives us correct sign, correct direction, but I'm afraid that equation will not come up right with three dimensional vectors, as you just lost two dimensions with that conversion. Anyway, for the example given above, with both masses on the same axis, that equation does work, but it produces scalar not vector value.

a(j)= -G * m(i) * r / r^3
a(j)= -G * 100 * 500 / 125,000,000

X_axis_a(j)= -G * 0.0004


To make this correct it needs to be noted that it works with individual vector components or just with two objects on the same axis, so if you want to make it work in 3-D, then "r" is not the real distance "r", but it's either r[X], r[Y] or r[Z] component, and "r^2" is then really equal to (dx*dx + dy*dy + dz*dz).



Now denote |\mathbf r_{12}|\equiv r_{12} = r_{21} and using \mathbf r_{12} = - \mathbf r_{21} yields

\mathbf a_{12} = \frac{Gm_1}{r_{21}^3}\mathbf r_{21}

Compare this to the form in the link,

\mathbf \ddot{\mathbf r}_i =
\sum_{j\ne i} \frac{\mu_j}{r_{ji}^3}\mathbf r_{ji}(1 + \text{relativistic terms})

The only differences in these two forms are
The former represents the acceleration due to the one interaction while the latter represents the acceleration due to the totality of mass in the solar system,
The latter expression incorporates "relativistic terms", and
The former expression uses Gm_1 while the latter uses \mu_j.
Astronomers can assess the standard gravitational parameters (http://en.wikipedia.org/wiki/Standard_gravitational_parameter) \mu_j to a high degree of accuracy (more than 10 decimal places for the Sun, almost 9 for the Earth). In comparison, G is know to a paltry 4 decimal places.

1.) I take it that "r(i)" stands for acceleration then, but why not "a(i)"? Anyway, if you put the "sum sign" in front of the first equation and instead of the index (1,2) place (i,j), then you would also get the "totality of the mass". It's just a way to denote an algorithm, it's nothing more but vectors addition where you cycle through all the pairs of the system.

It means that if you have Sun, Venus, Mars, Earth and Moon, you need to calculate the force between all the pairs, like this: F(Sun-Venus), F(Sun-Mars), F(Sun-Earth), F(Sun-Moon), F(Venus-Mars), F(Venus-Earth), F(Venus-Moon), F(Mars-Earth), F(Mars-Moon), F(Earth-Moon)... then calculate acceleration for each side and then sum all those acceleration vectors for each mass.

2.) I wonder how much different result you get with all the relativistic terms. Could you solve that equation for the example given above so we can compare?

3.) You have to have both masses and so I suspect that variable 'mu' contains the mass of the second body m(j). Therefore we can use it just the same in classical equations without any relativistic terms. In other words, I'm pretty sure that if we divide 'mu' by m(j) we get constant G with whatever precision 'mu' has.


I'm not sure what to conclude... equations do seem to be very similar, if not identical, but your equation is not 3-D friendly, it throws away that vector and if not careful that can lead to wrong implementation in three dimensional space.

[Jonathan Scott]

I'm not pushing agenda, I'm simply interpreting the article that was given here by someone else and is actually supposed to represent mainstream GR opinion, as far as I can figure.

http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

I want to know if that is the "official" explanation of how GR works, and if not I want to know what is. I'm doing exactly what you want me to do, I am devoting the effort into understanding the theory by coming here and asking questions, here they are again:


1.) Article says the delay gets canceled, is that true?

2.) Does Earth orbit Sun, or the point where Sun was 8 minutes ago?

3.) Why can we measure this 8 minutes delay with light, but not with gravity?

4.) Do gravity waves propagate as transverse or longitudinal waves and how do you know?

You asked that before, and were given lots of answers, but if you want the points answered specifically:

According to standard GR (and SR in the weak field limit):

1) The effect is as if the delay is cancelled. However, that is not how it actually works.

2) Earth orbits the point where the Sun is expected to be now given what it was doing 8 minutes ago.

3) The 8 minutes delay is the same for light and gravity.

4) Gravitational waves only occur when there are changes in a gravitational field. They propagate at c. They are transverse quadrupole. See the Wikipedia article on Gravitational Wave (http://en.wikipedia.org/wiki/Gravitational_wave) for a picture.

[D H]

[Example of computing gravitational acceleration via force]
If you do this for all the planetary pairs in the solar system, all the planets, moons and the sun, it works very well.
No, it doesn't work very well. The reason is that the gravitational constant G is one of the, if not the, least well known of the fundamental physical constants. Moving from force to acceleration eliminates one of the masses in Newton's law. Replacing G*Mj with μj eliminates the other mass and makes the resulting equation much more accurate. The only way scientists know to estimate the masses of a planet is to divide the gravitational parameter for the planet by G. A planet's gravitational parameter is observable; it's mass is not. Why bring G into the equation when the product G*M is directly observable, and in many cases, accurately observable. For example, the Sun's gravitational parameter is known to 10+ decimal places, Earth's to almost 9 places, Jupiter's to 6 places, Saturn's to 5 places.

Depending on the precision and time you want to invest you can get extremely precise results (past or future predictions), not only for days, months and years but the system stays stable for millions of years and errors you get are largely due to floating point precision, size of delta-time and the measurement data you set as initial conditions.
The uncertainties in the gravitational parameters alone limits the time span over which predictions remain close to accurate. Add in the uncertainties in planetary states (positions+velocities at some epoch) and its game over in terms of accurately predicting the state of the solar system for hundreds of thousands of years or more.


Unit vector has scalar value of 1, it is only there so to gives us correct sign, correct direction, but I'm afraid that equation will not come up right with three dimensional vectors, as you just lost two dimensions with that conversion.
The way to compute a unit vector in the direction of some known vector \mathbf v in any cartesian space is \mathbf v = \hat{\mathbf v}/||\mathbf v||. Whether the space is a one dimensional cartesian space or a 10,000 dimensional cartesian space doesn't matter.

Anyway, for the example given above, with both masses on the same axis, that equation does work, but it produces scalar not vector value.
It is a vector and it is pointing in the right direction. Look at the equation.

1.) I take it that "r(i)" stands for acceleration then, but why not "a(i)"? Anyway, if you put the "sum sign" in front of the first equation and instead of the index (1,2) place (i,j), then you would also get the "totality of the mass". It's just a way to denote an algorithm, it's nothing more but vectors addition where you cycle through all the pairs of the system.
That is exactly what that sum is doing. It computes the second time derivative of position for each body in the solar system. Only it does so very accurately.

It means that if you have Sun, Venus, Mars, Earth and Moon, you need to calculate the force between all the pairs, like this: F(Sun-Venus), F(Sun-Mars), F(Sun-Earth), F(Sun-Moon), F(Venus-Mars), F(Venus-Earth), F(Venus-Moon), F(Mars-Earth), F(Mars-Moon), F(Earth-Moon)... then calculate acceleration for each side and then sum all those acceleration vectors for each mass.
There is no need to compute forces. The end goal is to compute accelerations, and this can be done without computing forces. All that computing forces does is to add a source of error.

2.) I wonder how much different result you get with all the relativistic terms. Could you solve that equation for the example given above so we can compare?
That general relativity solved a well-known problem, the anomalistic precession of Mercury, is one of the reasons general relativity gain fairly quick acceptance in the scientific community. Even after accounting for the interactions of the planets, 19th astronomers could not fully explain Mercury's anomalistic precession. Theory and observation differed by 43 arcseconds per century. General relativity fully explains that difference.

3.) You have to have both masses and so I suspect that variable 'mu' contains the mass of the second body m(j). Therefore we can use it just the same in classical equations without any relativistic terms. In other words, I'm pretty sure that if we divide 'mu' by m(j) we get constant G with whatever precision 'mu' has.
You are assuming we know the masses of the planets. We don't. We do however know the product μ=G*M for many of the planets by observations of the planets' moons.


I'm not sure what to conclude... equations do seem to be very similar, if not identical, but your equation is not 3-D friendly, it throws away that vector and if not careful that can lead to wrong implementation in three dimensional space.
Look at it this way. There are two primary sources of planetary ephemerides: The Jet Propulsion Laboratory in Pasadena, California and the Institute of Applied Astronomy in St. Petersburg, Russia. Both use equations of the cited form. You are in essence arguing with the world renowned experts in this domain.

[Estif]

Look at it this way. There are two primary sources of planetary ephemerides: The Jet Propulsion Laboratory in Pasadena, California and the Institute of Applied Astronomy in St. Petersburg, Russia. Both use equations of the cited form. You are in essence arguing with the world renowned experts in this domain.

Look at it this way. NASA is using the same equations I do, and they landed on the Moon, believe it or not. You are in fact arguing with the world renowned expert in this domain, me.


It's really easy to resolve every single point of this argument, probably in less than 5min.

Do you think you're up to it?


Let m1 = 100 kg
Let m2 = 999 kg

Let m1 have its center at coordinates [100, 700, 400] meters
Let m2 have its center at coordinates [500, 200, 300] meters
================================================== ===========

acceleration1= ?
acceleration2= ?

[A.T.]

Look at it this way. NASA is using the same equations I do, and they landed on the Moon, believe it or not.
What does landing on the Moon has to do with exact predictions of the planet's movement over long periods of time? The Moon-Earth distance is tiny compared to the solar system, the involved masses are tiny compared to Sun's mass, the flight time is only a few days and most important: you have humans on board to correct the course.

You are in fact arguing with the world renowned expert in this domain, me.

An world renowned expert in the domain of silly arguments like: You don't need GR to fly to the Moon so GR must be wrong.

[D H]

Look at it this way. NASA is using the same equations I do, and they landed on the Moon, believe it or not. You are in fact arguing with the world renowned expert in this domain, me.
Oh, please! That you are arguing this is prima facia evidence that you are not. And believe it or not, NASA used (and still uses)

\frac{d^2\mathbf r}{dt^2} =
\frac{\mathbf F_{\text{ext}}}{m} +
\sum_j \frac{\mu_j}{||\mathbf r_j - \mathbf r||^3}(\mathbf r_j - \mathbf r)

to propagate the state of an interplanetary spacecraft, where Fext is the total of the non-gravitational forces acting on the vehicle and where the sum is over the gravitational bodies. On the time scale that it takes a vehicle to go from the Earth to Mars, for example, relativistic effects are very small, and are much smaller than the uncertainties in those external forces.

It's really easy to resolve every single point of this argument, probably in less than 5min.

Let m1 = 100 kg
Let m2 = 999 kg

Let m1 have its center at coordinates [100, 700, 400] meters
Let m2 have its center at coordinates [500, 200, 300] meters
================================================== ===========

acceleration1= ?
acceleration2= ?

\begin{aligned}
\mathbf r_{12} &\equiv \mathbf r_2 - \mathbf r_1 =
\left[ 500 , 200, 300 \right]\,\text{m} -
\left[ 100, 700, 400 \right] \,\text{m}=
\left[ 400, -500, -100 \right]\,\text{m} \\
\mathbf r_{21} &\equiv \mathbf r_1 - \mathbf r_2 =
-\mathbf r_{12} = \left[-400, 500, 100\right]\,\text{m} \\
r_{12} &= r_{21} = ||\mathbf r_{12}|| = 100\sqrt{42}\,\text{m} \\
\mathbf a_1 &= \frac{Gm_2}{r_{12}^3}\mathf r_{12} =
\left[97.96, -122.45, -24.49\right]\times 10^{-15}\,\text{m}/\text{s}^2 \\
\mathbf a_2 &= \frac{Gm_1}{r_{21}^3}\mathf r_{21} =
\left[-9.806, 12.26, 2.452\right]\times 10^{-15}\,\text{m}/\text{s}^2
\end{aligned}

Compare using forces,

\begin{aligned}
F &= \frac{Gm_1m_2}{r^2} = 15.872\times 10^{-12}\,\text{kg}\,\text{m}/\text{s}^2 \\
a_1 &= \frac{F}{m_1} = 158.72 \times 10^{-15}\,\text{m}/\text{s}^2 \\
\hat{\mathbf r}_{12} &= \frac{\mathbf r_{12}}{||\mathbf r_{12}||}
= \left[0.617213, -0.771517, -0.154303 \right] \\
\mathbf a_1 &= a_1 \hat{\mathbf r}_{12}
= \left[97.96, -122.45, -24.49\right]\times 10^{-15}\,\text{m}/\text{s}^2 \\
a_2 &= \frac{F}{m_2} = 15.888 \times 10^{-15}\,\text{m}/\text{s}^2 \\
\hat{\mathbf r}_{21} &= -\hat{\mathbf r}_{12}
= \left[-0.617213, 0.771517, 0.154303 \right] \\
\mathbf a_2 &= a_2 \hat{\mathbf r}_{21}
= \left[-9.806, 12.26, 2.452\right]\times 10^{-15}\,\text{m}/\text{s}^2
\end{aligned}

The same result as before.

[S.Vasojevic]

Things to do for today:

1. switch on GPS, see if it works.

2. Look at the picture below, and describe in terms of mass and force what is happening.

[George Jones]

Locked for moderation.