Weak equivalence principle and GR

[TrickyDicky]

An interesting discussion about the EP can be seen here:
https://www.physicsforums.com/showthread.php?t=311097

There DH for instance seems to be saying the same things I'm saying but with much less opposition by PF posters, including the distinction about relative acceleration I mentioned to DrGreg.
One difference is that there the particles test term that here seems to be utilized to obfuscate matters is not used in the sense of objects at the limit of vanishing mass(which is the restricted sense used by many quantum filed theorists, and that as shown in many references is plagued with infinities and other mathematical problems, but then again these field theorists are trying to fit GR into global Lorentz invariance which is impossible apparently) but in the more appropriate sense of objects with any mass (independence of mass of the EP).

 

[Physics Monkey]

... the COM of bodies of any mass will pass an arbitrary point at the same time exactly, not just approximately true for small masses.

This is not true in Newtonian gravity as a simple example will illustrate. Perhaps writing down some equations will be helpful.

Consider a body consisting of two masses moving in the z direction glued together by a light inextensible rod of length d. Let the masses move in an inhomogeneous gravitational field g(z). Mass $$m_1 = m \lambda$$ has position $$z_1$$ and mass $$m_2 = m (1-\lambda)$$ has position $$z_2 = z_1 + d$$. There will be some internal forces due to the rod, but we can forgot about those by studying the motion of the center of mass.

The equation of motion is $$m_1 \ddot{ z_1} + m_2 \ddot{ z_2} = m_1 g(z_1) + m_2 g(z_2)$$
The left hand side is by definition of the center of mass given by $$(m_1 + m_2 ) \ddot{z}$$ where $$z = \lambda z_1 + (1-\lambda) z_2 = z_1 + (1-\lambda) d$$
Thus writing everything in terms of $$z$$ we have the equation of motion $$\ddot{ z} = \lambda g(z - (1-\lambda) d) + (1-\lambda ) g(z+ \lambda d)$$
Since this equation manifestly depends on the body parameters $$d, \, \lambda$$, it is clear that the motion of the center of mass depends on them as well.

We can simplify matters by considering motion in the limit of a slowly varying field. Expanding the terms on the right hand side of the equation of motion we find $$\ddot{z} = g(z) + \frac{1}{2} \lambda (1-\lambda) d^2 g''(z)$$
Thus we have a deviation due to the second derivative term which for a spherical Earth would give a correction on the order of $$(d/R_E)^2 \approx 10^{-14}$$ for an object of size 1 meter. Naturally the correction is small, but it is there, and thus the COM of different objects will follow slightly different trajectories.

A vivid but sillier example comes by considering the extreme opposite limit. Suppose the field g(z) varies so rapidly that it actually changes sign between $$z_1$$ and $$z_2$$ For some choices of body parameters the body will actually move up, for others it will move down, and for still others it will not move at all (of course, the equilibrium may be unstable).

Hope this helps.

 

[TrickyDicky]

This is not true in Newtonian gravity as a simple example will illustrate. Perhaps writing down some equations will be helpful.

Consider a body consisting of two masses moving in the z direction glued together by a light inextensible rod of length d. Let the masses move in an inhomogeneous gravitational field g(z). Mass $$m_1 = m \lambda$$ has position $$z_1$$ and mass $$m_2 = m (1-\lambda)$$ has position $$z_2 = z_1 + d$$. There will be some internal forces due to the rod, but we can forgot about those by studying the motion of the center of mass.

The equation of motion is $$m_1 \ddot{ z_1} + m_2 \ddot{ z_2} = m_1 g(z_1) + m_2 g(z_2)$$
The left hand side is by definition of the center of mass given by $$(m_1 + m_2 ) \ddot{z}$$ where $$z = \lambda z_1 + (1-\lambda) z_2 = z_1 + (1-\lambda) d$$
Thus writing everything in terms of $$z$$ we have the equation of motion $$\ddot{ z} = \lambda g(z - (1-\lambda) d) + (1-\lambda ) g(z+ \lambda d)$$
Since this equation manifestly depends on the body parameters $$d, \, \lambda$$, it is clear that the motion of the center of mass depends on them as well.

We can simplify matters by considering motion in the limit of a slowly varying field. Expanding the terms on the right hand side of the equation of motion we find $$\ddot{z} = g(z) + \frac{1}{2} \lambda (1-\lambda) d^2 g''(z)$$
Thus we have a deviation due to the second derivative term which for a spherical Earth would give a correction on the order of $$(d/R_E)^2 \approx 10^{-14}$$ for an object of size 1 meter. Naturally the correction is small, but it is there, and thus the COM of different objects will follow slightly different trajectories.

A vivid but sillier example comes by considering the extreme opposite limit. Suppose the field g(z) varies so rapidly that it actually changes sign between $$z_1$$ and $$z_2$$ For some choices of body parameters the body will actually move up, for others it will move down, and for still others it will not move at all (of course, the equilibrium may be unstable).

Hope this helps.

Thanks for this correct analysis, it sure helps to clarify that point and further allows me to explain better my POV.
You are right about that hard to measure time deviation, I wrote that in a hurry. This dependence on $$d, \, \lambda$$ is easy to understand just by noticing that the Earth being spherical doesn't have a homogenous grav. field as you point out in your set up, that is the cause of tidal forces.
So what I was trying to stress was that the two trajectories are in fact geodesic in GR. The switching back and forth from GR to Newtonian is making me a little dizzy :)
This is seen also in the classic example of the cofee grounds relaeased on Earth and the effect of Weyl curvature on their shape.These are relative acceleration examples that as I explained have to do with the curvature of the manifold, rather than with what is discussed in the OP.

Edit: but if we somehow gave the rod in the thought experiment the appropriate curvature there'd be no time difference.

 

[TrickyDicky]

Actually Newtonian gravity, being set up in a flat absolute space is not the best place to discuss about geodesic motion, that in purity requires a curved space. the word geodesic loses its meaning in Euclidean space. Perhaps is better to stick to GR.

 

[PAllen]

To talk about the scope of validity of some principle you have to specify the definition as precisely as possible. My initial statement leading to the last couple pages of discussion on this thread started by repeating the relevant WEP definition from post #1 on this thread. I repeated to avoid mis-understanding, but this seems to have been ignored:

"A more modern definition: "The world line of a freely falling test body is independent of its composition or structure""

*This* definition is perfectly good for test bodies, as it says, with the ordinary concept of test particles as non-perturbing of the background. *This* definition would need many changes and much unnecessary additional complexity to deal with arbitrarily massive bodies (let alone bodies of large extent). This definition, with its limited applicability (but still true for any number of orders of magnitude below a size related to your desired precision, and any composition or structure) is still sufficient to specify how gravity couples to matter - which is all it needs to accomplish. Note, that with limitation to test bodies, as ordinarily understood, this definition applies to arbitrarily complex source configuration and motion of sources - which is nice.

Let's look at how this definition would complexify if you want it to accommodate arbitrary test bodies. First, if you try:"The world line of a freely falling test body is independent of its composition or structure, or mass (without limit)"

It is trivially false, as I have demonstrated. The evolution of the system as a whole would change for massive test bodies. Trying to extend by introducing a center of mass, in the GR context, runs into the issue that COM is a difficult issue in GR. Much more seriously, if the background consists of multiple sources, some closer to the massive test body, you get different evolutions that are impossible to compare in any simple way. There is no way to give meaning to 'world line independent of mass' for such a system for arbitary mass test bodies.

A better approach would be to try:

"The world line of a freely falling test body of given mass is independent of its composition or structure"

This works well for arbitrarily massive 'pointlike' masses. (However, it is in a significant way worse than the simple definition: it loses that the world line is mass independent over any range of masses that are 'non-perturbing'). However, it fails (as Physicsmonkey has shown in detail) for extended objects. So now you could try something like:

"The world line of a freely falling test body of given mass is independent of its composition or structure, as long the body deos not span significant curvature over the potion of world line of interest; and we use the COM of the body do define its world line." (Of course, we must define spanning curvature. For example: the Fermi-Normal coordinates extended from the COM world line are arbitrarily close to the Minkowski metric over the extent of the body's world tube, for any small time period along the world COM world line).

The curvature constraint serves to remove the difficulty of defining COM in GR, as well as allowing one to speak of world line of an extended body.

So if we accept 'test bodies' we can keep a simple, useful, practical principal, strictly true only in the limit. If we refuse to limit it to test bodies we are forced to ever more cumbersome definitions.

 

[atyy]

In Newtonian theory, the "universality of freefall" applies only relative to the centre-of-gravity of the Earth+object system.

The acceleration of the object relative to the C-of-G is independent of the object's mass, but does depend on the mass of the Earth.

The acceleration of the Earth relative to the C-of-G is independent of the Earth's mass, but does depend on the mass of the object.[/indent]

I'm now trying to check if this holds only for the initial accelerations. Could you clarify if this refers to the centre of mass or the centre of gravity?

 

[TrickyDicky]

"A more modern definition: "The world line of a freely falling test body is independent of its composition or structure""

*This* definition is perfectly good for test bodies, as it says, with the ordinary concept of test particles as non-perturbing of the background. *This* definition would need many changes and much unnecessary additional complexity to deal with arbitrarily massive bodies (let alone bodies of large extent). This definition, with its limited applicability (but still true for any number of orders of magnitude below a size related to your desired precision, and any composition or structure) is still sufficient to specify how gravity couples to matter - which is all it needs to accomplish. Note, that with limitation to test bodies, as ordinarily understood, this definition applies to arbitrarily complex source configuration and motion of sources - which is nice.

Let's look at how this definition would complexify if you want it to accommodate arbitrary test bodies. First, if you try:


"The world line of a freely falling test body is independent of its composition or structure, or mass"

It is trivially false, as I have demonstrated. The evolution of the system as a whole would change for massive test bodies. Trying to extend by introducing a center of mass, in the GR context, runs into the issue that COM is a difficult issue in GR. Much more seriously, if the background consists of multiple sources, some closer to the massive test body, you get different evolutions that are impossible to compare in any simple way. There is no way to give meaning to 'independent of mass' for such a system for arbitary mass test bodies.

A better approach would be to try:

"The world line of a freely falling test body of given mass is independent of its composition or structure"

This works well for arbitrarily massive 'pointlike' masses. However, it fails (as Physicsmonkey has shown in detail) for extended objects. So now you could try something like:

"The world line of a freely falling test body of given mass is independent of its composition or structure, as long the body deos not span significant curvature over the potion of world line of interest; and we use the COM of the body do define its world line." (Of course, we must define spanning curvature. For example: the Fermi-Normal coordinates extended from the COM world line are arbitrarily close to the Minkowski metric over the extent of the body's world tube, for any small time period along the world COM world line).

The curvature constraint serves to remove the difficulty of defining COM in GR, as well as allowing one to speak of world line of an extended body.

So if we accept 'test bodies' we can keep a simple, useful, practical principal, strictly true only in the limit. If we refuse to limit it to test bodies we are forced to ever more cumbersome definitions.

First, I would ask you again what do you think they refer to in the definition by composition of a test body?, a test body according to your definition of strictly non-perturbing the background can't have any composition, nor structure so that would make the definition useless.
Second, I think you are still confused about what my point is. I'm not saying that all bodies must follow the same geodesic regardless their mass. Nor that the worldlines are totally independent of the mass of the body, actually the geodesic is indirectly dependent of the mass of the body thru the non-linear contribution it may have on the background curvature that determines what geodesic the body will follow. The very fact that the curvature of spacetime is inhomogeneous, due to geometrical reasons and the non-linearity of GR makes bodies of different masses follow different geodesic paths, but they are still geodesic. Depending on the location of the sources of curvature in the manifold the curvature varies, and therefore they follow different geodesic trajectories (in the absence of other forces like EM forces...).
All bodies subject only to the curvature of spacetime are obliged to follow freefall paths or geodesic trajectories. And their proper acceleration is exactly canceled by the gravitational field they are subjected to. That is why GR is considered a geometrical theory. Do you not agree?

So here there is a problem with the vague use of the term test body or test particle in many instances of GR papers about this, which makes it easy to confuse the matter.
Also there is some serious sloppiness with the multiple definitions of the various Equivalence Principles.

 

[TrickyDicky]

Perhaps another source of confusion comes from the usual statement that the EP is only valid locally. This is just the trivial fact that for the formulation of the EP in terms of an object in an accelerating frame equivalent to being subjected to a gravitational field, this is explained with SR terms (logically because at the time this formulation of the EP was stated by Einstein he only had SR), and SR spacetime is flat, but in a curved spacetime its is obvious that this formulation is only valid locally since gravitational field are inhomogeneous.
But this has nothing to do with having to restrict the WEP to test bodies considered as points at the limit of vanishing mass so that they don't perturb the background. I think some people just take things by the wrong end here.

 

[bcrowell]

"The world line of a freely falling test body of given mass is independent of its composition or structure, as long the body deos not span significant curvature over the potion of world line of interest; and we use the COM of the body do define its world line." (Of course, we must define spanning curvature. For example: the Fermi-Normal coordinates extended from the COM world line are arbitrarily close to the Minkowski metric over the extent of the body's world tube, for any small time period along the world COM world line).

Actually, I don't think even this list of conditions suffices. You need an energy condition as well: arxiv.org/abs/gr-qc/0309074v1

So if we accept 'test bodies' we can keep a simple, useful, practical principal, strictly true only in the limit. If we refuse to limit it to test bodies we are forced to ever more cumbersome definitions.

You can get deviations from geodesic motion even for a *test* particle if the particle has spin:

MTW, p. 1121
Papapetrou, Proc. Royal Soc. London A 209 (1951) 248

But if the particle satisfies an energy condition, then its spin has to scale down as you scale down its size.

 

[TrickyDicky]

I see the misterious "test particle" swamp holds a powerful sway over some people. However the first point of the Strong Equivalence principle says:
"1. WEP is valid for self-gravitating bodies as well as for test bodies."

And most relativists would say GR follows the Strong Equivalence Principle. I don't want to use terms like Kook, that is commonly used here by one posters but that is actually what they call those that are not in the mainstream.

 

[PAllen]

I see the misterious "test particle" swamp holds a powerful sway over some people. However the first point of the Strong Equivalence principle says:
"1. WEP is valid for self-gravitating bodies as well as for test bodies."

And most relativists would say GR follows the Strong Equivalence Principle. I don't want to use terms like Kook, that is commonly used here by one posters but that is actually what they call those that are not in the mainstream.

(Un)fortunately, in a structured document, authors tend introduce the main idea first, and later expand upon it. Later in the same document (which was first linked in a post of mine, pointing out the critical later sections with fuller treatment), the following is noted:
----
4.1.2 Compact bodies and the strong equivalence principle

When dealing with the motion and gravitational wave generation by orbiting bodies, one finds a remarkable simplification within GR. As long as the bodies are sufficiently well-separated that one can ignore tidal interactions and other effects that depend upon the finite extent of the bodies (such as their quadrupole and higher multipole moments), then all aspects of their orbital behavior and gravitational wave generation can be characterized by just two parameters: mass and angular momentum. Whether their internal structure is highly relativistic, as in black holes or neutron stars, or non-relativistic as in the Earth and Sun, only the mass and angular momentum are needed. Furthermore, both quantities are measurable in principle by examining the external gravitational field of the bodies, and make no reference whatsoever to their interiors.
----

So we have:

- angular momentum as well as mass must be considered (Bcrowell pointed this out a couple of posts ago)
- The extent of the object must be small enough not to experience significant tidal effects (the same thing I was trying to capture in my 'spanning curvature' condition in a definition mainly meant as a reductio ad absurdum)
- There can't be significant effect from changing quadrupole moments within the body (which Bcrowell has highlighted a few times).

Clifford Will is well aware of the limiting nature of various EP formulations, but does have a tendency to present the basic idea first, and get into the details later.

 

[TrickyDicky]

I can't see much more of any interest in this exchange.
Let's just agree to disagree. Surely I'm not here to convince anyone, and I feel I made my point clear. Hope someone finds it interesting.

 

[atyy]

When an EP is obeyed, it means that the statement is true to first order (or some low order), not to all orders.

The EPs are not sufficient to determine the structure of GR. Nordstrom's second theory and GR both obey the strong EP.

 

[Physics Monkey]

Thanks for this correct analysis, it sure helps to clarify that point and further allows me to explain better my POV.
You are right about that hard to measure time deviation, I wrote that in a hurry. This dependence on $$d, \, \lambda$$ is easy to understand just by noticing that the Earth being spherical doesn't have a homogenous grav. field as you point out in your set up, that is the cause of tidal forces.
So what I was trying to stress was that the two trajectories are in fact geodesic in GR. The switching back and forth from GR to Newtonian is making me a little dizzy :)
This is seen also in the classic example of the cofee grounds relaeased on Earth and the effect of Weyl curvature on their shape.These are relative acceleration examples that as I explained have to do with the curvature of the manifold, rather than with what is discussed in the OP.

Edit: but if we somehow gave the rod in the thought experiment the appropriate curvature there'd be no time difference.

I appreciate your comments, but I'm afraid I still can't agree with your point of view. What geodesics are we talking about? Nothing in this problem moves on geodesics. Mass 1 and mass 2 certainly don't as they are acted on by tension forces due to the rod as well as gravity. The center of mass doesn't as I demonstrated above. So I ask, who is moving on a geodesic? And your comment about switching between Newton and GR is I think not relevant since GR reduces to Newton in the limit we consider here. The geodesic equation in GR in the weak field small velocity limit is nothing but Newton's law (the Christoffel symbols simply give you gradients of the Newtonian potential). If you can demonstrate that I am violating these assumptions in some way, I'll be happy to generalize things, but I don't think I am.

And the silly example I gave is still there. If g(z) has a zero somewhere, then the "dumbbell" I considered above will move in opposite directions depending on whether more mass is in the g < 0 region or the g > 0 region. Thus the internal structure not only effects the time to pass a point but the whether a point is passed at all.

 

[TrickyDicky]

I appreciate your comments, but I'm afraid I still can't agree with your point of view. What geodesics are we talking about? Nothing in this problem moves on geodesics. Mass 1 and mass 2 certainly don't as they are acted on by tension forces due to the rod as well as gravity. The center of mass doesn't as I demonstrated above. So I ask, who is moving on a geodesic? And your comment about switching between Newton and GR is I think not relevant since GR reduces to Newton in the limit we consider here. The geodesic equation in GR in the weak field small velocity limit is nothing but Newton's law (the Christoffel symbols simply give you gradients of the Newtonian potential). If you can demonstrate that I am violating these assumptions in some way, I'll be happy to generalize things, but I don't think I am.

And the silly example I gave is still there. If g(z) has a zero somewhere, then the "dumbbell" I considered above will move in opposite directions depending on whether more mass is in the g < 0 region or the g > 0 region. Thus the internal structure not only effects the time to pass a point but the whether a point is passed at all.

I probably wasn't precise enough in my answer.
When I say "the two trajectories are in fact geodesic in GR" I was referring to atty's post, where my slip about time originated, not to your Z1 and Z2 that obviously are acted by the rod.
If you release 2 marbles separated a certain distance d at a certain distance from Earth forming a triangle with the Earth's COM, in vacuum, they acquire a differential relative acceleration towards each other due to the inhomogeneous gravitational field of the Earth characteristic of tidal forces. Now the two masses are in freefall and drawing 2 different geodesic trajectories, and this behaviour is independent of their masses.

 

[TrickyDicky]

To summarize this thread I'll cite again (see post #31) something that has been conveniently ignored from the reference gently provided in post #3,
http://arxiv.org/abs/0707.2748

on page 4 it says:

"It is important to stress that the WEP only says that there exist some preferred
trajectories, the free fall trajectories, that test particles will follow and these curves
are the same independently of the mass and internal composition of the particles
that follow them (universality of free fall)."

 

[PAllen]

To summarize this thread I'll cite again (see post #31) something that has been conveniently ignored from the reference gently provided in post #3,
http://arxiv.org/abs/0707.2748

on page 4 it says:

"It is important to stress that the WEP only says that there exist some preferred
trajectories, the free fall trajectories, that test particles will follow and these curves
are the same independently of the mass and internal composition of the particles
that follow them (universality of free fall)."

And, showing the problem with selective quotation, rather than reading complete content, here is further discussion from the same page of the same source, saying the same thing as all the science advisors on this thread:

"The second subtle
point is the reference to test particles in all the EP formulations. Apart from the
obvious limitation of restricting attention to particles and ignoring classical fields
(such as, e.g., the electromagnetic one), apparently no true test particles exist,
hence the question is how do we know how “small” a particle should be in order
to be considered a test particle (i.e., its gravitational field can be neglected)?"

 

[TrickyDicky]

And, showing the problem with selective quotation, rather than reading complete content, here is further discussion from the same page of the same source, saying the same thing as all the science advisors on this thread:

"The second subtle
point is the reference to test particles in all the EP formulations. Apart from the
obvious limitation of restricting attention to particles and ignoring classical fields
(such as, e.g., the electromagnetic one), apparently no true test particles exist,
hence the question is how do we know how “small” a particle should be in order
to be considered a test particle (i.e., its gravitational field can be neglected)?"

You haven't bothered to check #31, have you?

 

[PAllen]

You haven't bothered to check #31, have you?

Admittedly, I did not look at #31, just disputing the isolated quote as a summary. However, looking at your concluding statements in #31, I can comment a little:

"So it is plain to see that the concept of "test" body or particle can be used in a deliberately confusing way (in a theory-dependent way at the least), so that it can be made to mean different things for different authors as it most convenes to their purposes. And while it is often well used to simplify certain problems, this doesn't seem to be the case here as the authors of this paper admit that it rather confuses than simplifies.
Precisely what the WEP (and the EEP) assert is that the gravitational field of a body can be neglected for its own motion in the absence of non-gravitational forces, how can then the same principle imply that self- gravitation alters that motion?
Hopefully some GR expert will clarify this important issues. "

1) Nobody is being deliberately confusing in discussing test bodies, and (as PhysicsMonkey explained at the beginning of this thread, the concept of test particles among physicists is old and established). Talking about authors having 'purposes' or 'agendas' is sociology, not physics. Despite theoretical conundrums in the 'fine print', the concept has long and ongoing utility.

2) Your other questions here are more complex. My knowledgeable amateur (not expert) opinon on them is: With the normal understanding of the WEP (and EEP), the test bodies own gravity can be ignored if it doesn't perturb 'sources'. However, finer distinctions vary by theory. Some non-GR theories will bind to self gravitation of even 'small' test particles; GR will not (by SEP). Other points are whether a test particle is allowed to have significant spin. The impact of this will be theory dependent (none for Newton, relevant for GR).

 

[TrickyDicky]

Thanks for the constructive contribution.

 

[Physics Monkey]

I can't see much more of any interest in this exchange.
Let's just agree to disagree. Surely I'm not here to convince anyone, and I feel I made my point clear. Hope someone finds it interesting.

Perhaps I'll give my own "closing argument".

I am trying to convince people. I am not doing this just to be argumentative but because I want everyone reading this thread to be able to appreciate the stunning power and subtlety of GR. Instead of quibbling about the precise meaning and history of the equivalence principle, we can accept it as a very useful approximation and move on understand the incredible richness of gravitational phenomena in the universe.

We can follow the evolution of the universe from the hot plasma that existed 13 billion years ago to the stark and empty desert we now find ourselves in. We can calculate the minute deflection of distant star light as it passes the gravitational field of our own Sun. We can study the gravitational dynamics of colliding supermassive black holes. We can predict the orbital decay of binary pulsars due to the slow emission of gravitational radiation. I could obviously go on.

In my opinion, readers of this thread can choose between at least two points of view. On one side you have vague complaints about the idea of a test body, lots of quotations about the equivalence principle, and a point of view that finds it hard to acknowledge the role of approximation in science. On the other side, you have equations and derivations that anyone with a background in calculus and a reasonable study of GR can verify, a careful confrontation with experiment, and a willingness to accept approximation and uncertainty.

 

[TrickyDicky]

Perhaps I'll give my own "closing argument".

I am trying to convince people. I am not doing this just to be argumentative but because I want everyone reading this thread to be able to appreciate the stunning power and subtlety of GR. Instead of quibbling about the precise meaning and history of the equivalence principle, we can accept it as a very useful approximation and move on understand the incredible richness of gravitational phenomena in the universe.

We can follow the evolution of the universe from the hot plasma that existed 13 billion years ago to the stark and empty desert we now find ourselves in. We can calculate the minute deflection of distant star light as it passes the gravitational field of our own Sun. We can study the gravitational dynamics of colliding supermassive black holes. We can predict the orbital decay of binary pulsars due to the slow emission of gravitational radiation. I could obviously go on.

In my opinion, readers of this thread can choose between at least two points of view. On one side you have vague complaints about the idea of a test body, lots of quotations about the equivalence principle, and a point of view that finds it hard to acknowledge the role of approximation in science. On the other side, you have equations and derivations that anyone with a background in calculus and a reasonable study of GR can verify, a careful confrontation with experiment, and a willingness to accept approximation and uncertainty.

Yeah, I bet you'll convince many people with such a well-balanced summary, you forgot to say you were not able to refute anything from the OP, ignored most of the arguments offered and recurred to inventing "personal attacks" to hide the fact you couldn't cope with the argumets given.
As a reader of this thread I don't have any problem with both of the sides you mention, no need to choose, they are compatible,with the only caveat that certainly I have seen no one else but you finding hard to acknowledge the role of approximation in science. But I guess you find hard science in general.
Thanks for your constructive contribution too.

 

[cosmik debris]

As long as the bodies are sufficiently well-separated that one can ignore tidal interactions and other effects that depend upon the finite extent of the bodies (such as their quadrupole and higher multipole moments), then all aspects of their orbital behavior and gravitational wave generation can be characterized by just two parameters: mass and angular momentum.

This statement confuses me, if you ignore quadrupole and higher moments how can the mass and angular momentum describe gravitational wave generation?

 

[PAllen]

This statement confuses me, if you ignore quadrupole and higher moments how can the mass and angular momentum describe gravitational wave generation?

The quote referenced was from Clifford Will. The way you quoted it made it seem like my words.

The explanation is that you have two bodies of given mass and angular momentum in mutual orbit. You compute the gravitational waves on that basis, no other information needed (given the approximating conditions describe in Will's quote are met). The quadrupole moment generating the gravitational waves comes from the mutually orbiting bodies.

Maybe something else needs clarification: the idea is that if the bodies are far enough apart, you can ignore any quadrupole moment (changes) of the body itself (e.g. due to internal pulsations) for the purpose of calculating gravitational waves due to their mutual orbit.

 

[Physics Monkey]

Yeah, I bet you'll convince many people with such a well-balanced summary, you forgot to say you were not able to refute anything from the OP, ignored most of the arguments offered and recurred to inventing "personal attacks" to hide the fact you couldn't cope with the argumets given.
As a reader of this thread I don't have any problem with both of the sides you mention, no need to choose, they are compatible,with the only caveat that certainly I have seen no one else but you finding hard to acknowledge the role of approximation in science. But I guess you find hard science in general.
Thanks for your constructive contribution too.

Naturally my "closing argument" used rhetorical devices, it's not meant to be a unbiased presentation, just a fun attempt at debate. The equations and arguments given earlier already provide a relatively unbiased point of view without any help needed from me.

And of course, my post is hardly worse than an out of context quote highlighting a definition (of test bodies and their paths) that would only paragraphs later be acknowledged as impossible to realize using physical particles (as I and others have pointed out $$n \rightarrow \infty$$ times). This is true even though you had the larger quote buried earlier in the thread.

I propose the following. If I understand your claim correctly, you maintain that all objects follow geodesics in GR if acted only by gravitation forces. So I ask you once more straight up, where are the geodesics in the two mass problem I gave? As I see it, you can either:
1) Show to the readers here the geodesic.
2) Otherwise tell us why the formulation I gave is wrong.
3) Complain about geodesics in Newtonian gravity, even though we know Newton is a limit of GR and that geodesics satisify $$\ddot{x} = g(x)$$
4) Ignore the question
5) Clarify for us your actual position so that I can repose the question.

 

[TrickyDicky]

I propose the following. If I understand your claim correctly, you maintain that all objects follow geodesics in GR if acted only by gravitation forces. So I ask you once more straight up, where are the geodesics in the two mass problem I gave? As I see it, you can either:
1) Show to the readers here the geodesic.
2) Otherwise tell us why the formulation I gave is wrong.
3) Complain about geodesics in Newtonian gravity, even though we know Newton is a limit of GR and that geodesics satisify $$\ddot{x} = g(x)$$
4) Ignore the question
5) Clarify for us your actual position so that I can repose the question.

I already explained in a previous post that we can't speak strictly about geodesics in a flat Newtonian space, and also explained your formulation of the thought experiment with a rod is fine with me, and how it had little to do with my claim, you must have missed those posts.
You are of course entitled to opine otherwise, that is fine with me, once again I'm not trying to convince anyone, nor do I think I hold the TRUTH as you seem to, but at this point I guess if you didn't grasp what I'm saying is due to any of these:
1)You are not willing to, and are trying to engage in gratuitous dispute
2)You are not able to

I'll be delighted with any kind of serious debate though.

 

[TrickyDicky]

The explanation is that you have two bodies of given mass and angular momentum in mutual orbit. You compute the gravitational waves on that basis, no other information needed (given the approximating conditions describe in Will's quote are met). The quadrupole moment generating the gravitational waves comes from the mutually orbiting bodies.

Maybe something else needs clarification: the idea is that if the bodies are far enough apart, you can ignore any quadrupole moment (changes) of the body itself (e.g. due to internal pulsations) for the purpose of calculating gravitational waves due to their mutual orbit.

But how do you separate the quadrupole moment which is proportional to the momentum of inertia for a particular orbital shape, from the angular momentum of the system?
You seem to forget that in the Hulse-Taylor pulsar the calculations of the GW energy is derived from the quadrupole moment tensor, and it is a detached binary system (bodies far enough apart). So you are saying that precisely what is used for the purpose of calculating gravitational waves must be ignored. Maybe you didn't explain yourself well enough, or else I (and maybe cosmik debris), am misunderstanding you.

 

[TrickyDicky]

About test particles:
First of all, let's remember again, test bodies are an idealization. They don't exist. Bodies of different masses do exist, at leat last time I checked.
I was doubting whether or not quoting any more relevant references , because curiously, even though in this site citing well known and relevant texts and papers to back one's claims is apparently officially encouraged (if not mandatory), everytime I cite some author even if that reference is provided by someone else I'm harshly criticized. And when I use my own words they're rather ignored. Not sure what's better.
But here they go, the authors are 't Hooft and Sean Carroll, hope it is fine to quote their public notes on GR.
Actually 't Hooft, don't even use the term "test particle" or "test body",neither in his brief treatment of the EP,nor on his whole notes about GR, soI just use it as an example that "test particles" are just a useful approximation for solving problems, but given the fact they can't be defined rigorously, or rather that they can be used for many purposes so they are better not used in formal definitions if the try to be specific.

http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2010.pdfCarroll does name test particles in his explanation of the EP and does it precisely in the sense I've used (which as I said it is not the only one possible, thus the formal vagueness of the concept, and its usefulness in solving problems in the approximative, linear regime),
when he says on p. 97:
"the behavior of freely-falling test particles is universal, independent of their mass (or any other qualities they may have)"

According to this, precisely what the EP does is allowing us is to use test bodies as another way to say, bodies of any mass.
Of course the use of the term meaning bodies at the limit of low mass is also valid in the right context, and it is usually used by FTheorists as I explained on some other post.(But then again the quantum space is flat so it makes sense)http://arxiv.org/PS_cache/gr-qc/pdf/9712/9712019v1.pdf

 

[Physics Monkey]

I already explained in a previous post that we can't speak strictly about geodesics in a flat Newtonian space, and also explained your formulation of the thought experiment with a rod is fine with me, and how it had little to do with my claim, you must have missed those posts.
You are of course entitled to opine otherwise, that is fine with me, once again I'm not trying to convince anyone, nor do I think I hold the TRUTH as you seem to, but at this point I guess if you didn't grasp what I'm saying is due to any of these:
1)You are not willing to, and are trying to engage in gratuitous dispute
2)You are not able to

I'll be delighted with any kind of serious debate though.

How does the motion of the balls+rod system have little to do with your claim when you claim all bodies move on geodesics? Of course, I'd be happy to hear if this is not your claim, but if not, can you please state your claim clearly and precisely once and for all. Also, you didn't "explain" that we can't talk about geodesics in Newtonian space, you simply declared it.

But that's fine. Here is the Newtonian limit metric:
$$ds^2 = - (1 + 2 \phi ) dt^2 + (1 - 2 \phi ) (dx^2 + dy^2 + dz^2)$$

Show us that the massive body consisting of two massive balls connected by a light rod of fixed length follows a geodesic. Otherwise, please state you claim clearly and precisely so that we can adjust the problem to discuss it.

 

[PAllen]

But how do you separate the quadrupole moment which is proportional to the momentum of inertia for a particular orbital shape, from the angular momentum of the system?
You seem to forget that in the Hulse-Taylor pulsar the calculations of the GW energy is derived from the quadrupole moment tensor, and it is a detached binary system (bodies far enough apart). So you are saying that precisely what is used for the purpose of calculating gravitational waves must be ignored. Maybe you didn't explain yourself well enough, or else I (and maybe cosmik debris), am misunderstanding you.

I think I explained fine, for some reason you are not following. I said, for the purpose of calculating GW from the mutual orbit, you can ignore the contrubution due finite extent and shape changes of each body, if they are far enough apart - treating them as point mass (possibly with angular momentum from their spin). So the only quadropole moment you worry about is due to the mutual orpit of spinning point masses.

This, of course, is just a 'very good approximation', if the separation is large enough (the larger the separation, and the more compact the bodies are to begin with, the better the approximation).

(Please note, I am not the source for any of this analysis: it is Clifford Will summarizing his and other's analysis; he provides pointers to the primary research papers justifying the approximations).

 

[Q-reeus]

This thread has greater longevity than anticipated so am butting in again. Physics Monkey presented an argument in #72 (mentioned again in #84, and I see just now in #99) that seems open and shut case. For an extended body sampling a non-uniform field it is only to be expected net motion is not generally the result of assuming a COM applicable for a perfectly rigid mass immersed in a perfectly uniform field. But is this truly pointing to the limited validity of WEP, or rather the limited validity of a particularly simple definition of COM? Why wouldn't one define an effective COM that took proper and sensible account of things like tidal deformation, non-uniform 'sampling effects in a tidal field, non-uniformity of energy density owing to gravitational interaction, and non-uniformity of the metric defining COM? In short, COM in the general setting is properly a dynamical quantity. So are we to believe that when all of the above is correctly incorporated, path of free-fall of effective COM still follows a non-geodesic? Depends on convention here surely - what is to be the yardstick for defining what. And I note this extended body matter is departing from the OP's query which centers around mass independence of free-fall, not spatial extent as factor.
[EDIT: Darn it - on second thoughts one will always find with extended rigid-body systems that inertial and passive gravitational COM will generally differ (as per Physics Monkey's extreme example). But in this setting is a 'pathology' of an extended composite entity. So I will stick to the matter of mass as determining factor, and thus below remarks.]

On a similar vein: PAllen in responding to the example of two co-orbiting neutron stars in an otherwise flat background metric, admitted there was no generally agreed on position as to whether a geodesic made sense or could be well defined. But a read of the article raised in #42
"New limits on the strong equivalence principle from two long-period circular-orbit binary pulsars" http://arxiv.org/abs/astro-ph/0404270
makes it clear there are dynamical consequences if WEP/SEP fails that cut right through any ambiguities about defining geodesic motion. Namely that the combined system will move in ways not consistent with the momentum conservation principle - and that would unambiguously show up on the canvas of a flat background metric - ie astronomical observations. There is no such observed effect. My conclusion: mass-independent free-fall consistent with WEP/SEP is fact, and 'departures' from that are artifacts of adopting simplifying definitions (eg rigid, invariant COM, excising contribution of test mass from total metric curvature). Now you fellas can run rings around me as far as mathematical grasp of GR goes. But looked at just as matter of logical consistency of founding principles (elevator in free-fall etc), seems clear the OP's premise necessarily holds once proper (as opposed to what looks to be purely 'consensus') definitions are made and adhered to. If this is all wrong-headed then please explain where and how exactly! :zzz:

 

[TrickyDicky]

How does the motion of the balls+rod system have little to do with your claim when you claim all bodies move on geodesics? Of course, I'd be happy to hear if this is not your claim, but if not, can you please state your claim clearly and precisely once and for all. Also, you didn't "explain" that we can't talk about geodesics in Newtonian space, you simply declared it.

From the WP:"In mathematics, a geodesic (pronounced /ˌdʒiːɵˈdiːzɨk/, /ˌdʒiːɵˈdɛsɨk/ JEE-o-DEE-zik, JEE-o-DES-ik) is a generalization of the notion of a "straight line" to "curved spaces"
Excuses for not presenting the definition before, I was under the belief that it was a moreless known concept for the participants in this thread.

My claim has been stated many times and actually is not my claim but the definition of the WEP.
Q-reeus has answered any remaining doubts about the specific problem at hand.

I'm starting to suspect the cause of your misunderstanding is 2) in the above post, in which case I would recommend you to read some basic text on GR (Ryder's is a good intro), being aware of the limitations of the concept of "test body" mentioned in this thread.

 

[PAllen]

This thread has greater longevity than anticipated so am butting in again. Physics Monkey presented an argument in #72 (mentioned again in #84, and I see just now in #99) that seems open and shut case. For an extended body sampling a non-uniform field it is only to be expected net motion is not generally the result of assuming a COM applicable for a perfectly rigid mass immersed in a perfectly uniform field. But is this truly pointing to the limited validity of WEP, or rather the limited validity of a particularly simple definition of COM? Why wouldn't one define an effective COM that took proper and sensible account of things like tidal deformation, non-uniform 'sampling effects in a tidal field, non-uniformity of energy density owing to gravitational interaction, and non-uniformity of the metric defining COM? In short, COM in the general setting is properly a dynamical quantity. So are we to believe that when all of the above is correctly incorporated, path of free-fall of effective COM still follows a non-geodesic? Depends on convention here surely - what is to be the yardstick for defining what. And I note this extended body matter is departing from the OP's query which centers around mass independence of free-fall, not spatial extent as factor.
[EDIT: Darn it - on second thoughts one will always find with extended rigid-body systems that inertial and passive gravitational COM will generally differ (as per Physics Monkey's extreme example). But in this setting is a 'pathology' of an extended composite entity. So I will stick to the matter of mass as determining factor, and thus below remarks.]

On a similar vein: PAllen in responding to the example of two co-orbiting neutron stars in an otherwise flat background metric, admitted there was no generally agreed on position as to whether a geodesic made sense or could be well defined. But a read of the article raised in #42
"New limits on the strong equivalence principle from two long-period circular-orbit binary pulsars" http://arxiv.org/abs/astro-ph/0404270
makes it clear there are dynamical consequences if WEP/SEP fails that cut right through any ambiguities about defining geodesic motion. Namely that the combined system will move in ways not consistent with the momentum conservation principle - and that would unambiguously show up on the canvas of a flat background metric - ie astronomical observations. There is no such observed effect. My conclusion: mass-independent free-fall consistent with WEP/SEP is fact, and 'departures' from that are artifacts of adopting simplifying definitions (eg rigid, invariant COM, excising contribution of test mass from total metric curvature). Now you fellas can run rings around me as far as mathematical grasp of GR goes. But looked at just as matter of logical consistency of founding principles (elevator in free-fall etc), seems clear the OP's premise necessarily holds once proper (as opposed to what looks to be purely 'consensus') definitions are made and adhered to. If this is all wrong-headed then please explain where and how exactly! :zzz:

I will comment only on a few aspects of this.

The reference paper makes no statement as to whether the pulsars may be treated as following geodesics of the complete, dynamic, spacetime. This interesting question is complicated by several factors: lack of universally accepted definition of COM in GR context (though this is not very significant for the case of compact, nearly sperical objects); and the fact that all solutions to the two body problem are numeric, making it hard to accurately decide if some world line is precisely following a geodesic. It definitely appears that the answer to this question is not well known (maybe known by some experts, but not well known; I have no idea of the answer).

Note, also, a fact not mentioned in the paper because it is 'obvious background understanding': If the assemblage of matter into a compact object releases energy (which, of course, it does), the resultant mass of the object declines, the difference representing the gravitational binding energy of the object. In this sense, self gravitation clearly affects the mass of an object. However, the point of the paper and relevant experiments is that to the extent 'finite size' can be ignored, self gravitation has no other impact beyond its affect on mass (specifically, the Nortveldt effect has never been observed).

None of this is really relevant to the WEP, as most commonly stated. Its most common statement is *specifically* to highlight the fact that for test bodies small enough in mass enough not to perturb other 'sources' of gravity, and small enough in extent for finite size effects to be insignificant, that the the trajectory is independent of mass, composition, internal structure. It is is not trying to probe the most general conditions under which an object follows a spacetime geodesic.

 

[Physics Monkey]

From the WP:"In mathematics, a geodesic (pronounced /ˌdʒiːɵˈdiːzɨk/, /ˌdʒiːɵˈdɛsɨk/ JEE-o-DEE-zik, JEE-o-DES-ik) is a generalization of the notion of a "straight line" to "curved spaces"
Excuses for not presenting the definition before, I was under the belief that it was a moreless known concept for the participants in this thread.

My claim has been stated many times and actually is not my claim but the definition of the WEP.
Q-reeus has answered any remaining doubts about the specific problem at hand.

I'm starting to suspect the cause of your misunderstanding is 2) in the above post, in which case I would recommend you to read some basic text on GR (Ryder's is a good intro), being aware of the limitations of the concept of "test body" mentioned in this thread.

Thanks for this, I laughed out loud when I saw that you had included pronunciations in your response.

I guess you aren't aware that the geodesics of the metric I wrote above are, in the Newtonian limit, simply identical to solutions of Newton's 2nd law with potential $$\phi$$.

In any event, since it's clear you are unwilling and unable to seriously discuss the issues, I shall not waste anymore time here.

 

[TrickyDicky]

Thanks for this, I laughed out loud when I saw that you had included pronunciations in your response.

I'm glad you did, that was the purpose of including it, to keep a relaxed and humorous tone when treating these sometimes dry issues.

I guess you aren't aware that the geodesics of the metric I wrote above are, in the Newtonian limit, simply identical to solutions of Newton's 2nd law with potential $$\phi$$.

Sure, I am aware of that, and guess what makes possible that identity: the WEP in the way I'm formulating it.
The fact remains that the Newtonian limit is the metric of a space at the limit of being flat, and geodesics in strict sense apply to curved spaces.
I've already explained how the WEP allows recovering the Newtonian limit in GR, and linear approximations for calculations such as the precession of Mercury and deflection of light. But IMO it doesn't allow to generalize features intrinsic to the linear solutions to the non-linear theory. (see doubt about GW thread).

In any event, since it's clear you are unwilling and unable to seriously discuss the issues, I shall not waste anymore time here.

I regret you get that impression, I can only assure it doesn't correspond with reality.
Don't consider it a total waste of time though: without any sarcasm, I really think you might learn something. I certainly have.