Weak equivalence principle and GR

[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.

So far so good, this basically agrees with what I've been saying.

I suspect this holds only for infinitesimal time if both masses are finite.

 

[TrickyDicky]

Galileo never used Jupiter as a test particle.

Good one-liner.
I guess he would have had some trouble putting it at the top of the Pisa tower.

 

[TrickyDicky]

I think the point is that it doesn't have to be a test particle. It can be a body of any mass.

Exactly.

 

[TrickyDicky]

I suspect this holds only for infinitesimal time if both masses are finite.

Nope, to describe a trajectory you need finite time.

 

[TrickyDicky]

PAllen is correct.

By the way DrGreg , in this post you seem to disagree with what PAllen is saying now:
https://www.physicsforums.com/showpost.php?p=2444886&postcount=7

Have you changed your mind?

 

[TrickyDicky]

g = \frac{G(M + m)}{r^2}​

I forgot to mention that formula is of course referring to relative acceleration.
All this thread has dealt with absolute acceleration, in lay terms what a body "feels" or what an accelerometer in the COM of the body measures, and what the EP states, and was the insight from which Einstein developed GR is precisely that this absolute acceleration is canceled in freefall so that the subject doesn't feel its own weight when describing geodesic motion in curved spacetime. This is so basic that I'm amazed that almost everybody in this forum seems to disagree with it. It is obvious that objects in free fall without the influence of non-gravitational forces have exact geodesic motion no matter their own mass.
Maybe what confuses people is that it is also obvious that since the object is also acting as a source in the spacetime curvature, the geodesic it draws is different depending on its mass and the mass of the other objects that act as sources of curvature, that is where the non-linearity comes in and what makes so hard to deal with 2-body or n-bodies problems in GR and the reason that in Newtonian gravity it can be done since the background is Euclidean space and the masses of the bodies can be treated linearly plus the fact that in Newtonian physics the WEP also applies so that the mass of one of the bodies wrt the other can always be neglected.

Let's remember some textbooks (I believe one is the Eyvind Gron one but not sure so don't quote me on this) outline GR by saying it is basically the sum of SR+WEP and the only way to reconcile SR with WEP and recover the results at the Newtonian limit is a curved spacetime with general covariance.

 

[atyy]

Good one-liner.
I guess he would have had some trouble putting it at the top of the Pisa tower.

Well, he tried to, but changed his mind once it started leaning.

 

[atyy]

In Newtonian gravity,
1) gravitational mass=inertial mass (always true)
2) two objects of different mass and composition dropped from the same height will take the same time to reach the ground (approximately true for small masses)
3) an object in a two body system in which both objects have finite mass has the same acceleration relative to the centre of gravity of the system, regardless of its mass, but is dependent on the mass of the second object (I'm not sure, but I think it's true only for an infinitesimal time)

 

[TrickyDicky]

In Newtonian gravity,
1) gravitational mass=inertial mass (always true)
2) two objects of different mass and composition dropped from the same height will take the same time to reach the ground (approximately true for small masses)
3) an object in a two body system in which both objects have finite mass has the same acceleration relative to the centre of gravity of the system, regardless of its mass, but is dependent on the mass of the second object (I'm not sure, but I think it's true only for an infinitesimal time)

I agree except with what is between parenthesis in your points 2 and 3.
Unless what you mean is that if we throw a trailer and a marble, due to the much bigger size of the trailer it will touch the ground before the marble, that is trivially true, but remember that I've been alway referring to geodesic paths and the COM of bodies of any mass will pass an arbitrary point at the same time exactly, not just approximately true for small masses.

Here is a post where you quote as valid what I'm considering the WEP and that is in the definition as the first requisite of the strong equivalence principle. BTW if you agree with PAllen I guess you don't accept the SEP either, which is normally accepted in mainstream GR. (not by all authors tha's true, but the ones that don't are usually considered crackpot by mainstreamers which IMO doesn't follow necessarily).

https://www.physicsforums.com/showpost.php?p=2492126&postcount=3

"These ideas can be summarized in the strong equivalence principle (SEP), which states that:
1. WEP is valid for self-gravitating bodies as well as for test bodies.
2...
3..."

 

[TrickyDicky]

Well, he tried to, but changed his mind once it started leaning.

...lol

 

[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.