Weak equivalence principle and GR

[PAllen]

Thanks for joining. One definition that has been used in the thread is:"The world line of a freely falling test body is independent of its composition or structure". By which I understand that they also mean to be independent of mass.

I agree with mass also, up to some upper limit where the mass is large enough to perturb other sources. For example, Jupiter orbiting the sun, Jupiter can be considered a test particle for reasonable precision. However, it is not intended to say anything about, e.g. a binary system of comparable masses.

All this is true. So what it was proposed is to use the path followed by the center of mass of the massive object as the one representing the body's geodesic motion.

And a problem discussed in this thread is that COM in general in GR is ill defined. I think, for a compact, nearly spherical object, it may be taken to reasonably well defined.

 

[PAllen]

I haven't had a chance to read the long review paper on motion in GR, but I probably will later.

FYI: That review paper covers motion of a body big enough to have have back reaction from GW (gravitational wave) emission, and perturb the metric in its vicinity, but still small enough compared to a massive central body to allow specialized methods to be used. Under these conditions, a nice geodesic result follows (the small body follows a geodesic of the background metric plus the perturbation (including GW) of its own motion). I am very curious to know if anything similar can be said for binary system of comparable masses.

 

[TrickyDicky]

I don't understand this question. Given a theory, you go in with initial conditions, boundary conditions, and possibly symmetry condtions that corrrespond to reasonable hypotheses about the system you want to study (up to the whole observable universe). Then you look for a solution. There is only a problem if reasonable conditions lead to ambiguous or unreasonable solutions. I am not aware of this being the case for GR.

Yes, but for instance you have several diffrent solutions of the EFE that describe different spacetimes but are all considered part of the theory in practice and used for different purposes or problems. From the static Schwarzschild solution for vacuum, to the FRW non-static cosmological solution, to Kerr's geometry or the linearized equations, etc, they are different solutions with some of their symmetry conditions incompatible with each other, that respond to different hypothesis about the systems they deal with, with totally different outcomes and consequences. But their common denominator is they all use the same field equations, I just don't know how something axiomatic can give rise to so many different solutions, an axiom is a principle of universal application, I don't think the field equations by themselves can qualify as an axiom.

 

[PAllen]

Yes, but for instance you have several diffrent solutions of the EFE that describe different spacetimes but are all considered part of the theory in practice and used for different purposes or problems. From the static Schwarzschild solution for vacuum, to the FRW non-static cosmological solution, to Kerr's geometry or the linearized equations, etc, they are different solutions with some of their symmetry conditions incompatible with each other, that respond to different hypothesis about the systems they deal with, with totally different outcomes and consequences. But their common denominator is they all use the same field equations, I just don't know how something axiomatic can give rise to so many different solutions, an axiom is a principle of universal application, I don't think the field equations by themselves can qualify as an axiom.

Maybe the word axiom bothers you (don't know why). So say, instead, the field equations are whole content of the theory. Nothing else is needed besides differential geometry and what I call correspondence rules: how to relate mathematical objects to natural objects. Each of these solutions covers different problems. I don't see any ambiguity about which corresponds to given situation in nature, e.g. the final collapsed state of a rotating star, use Kerr; non (or minimally) rotating, use Schwarzschild; overall evolution of the universe, FRW (depending on your assumptions about what the universe is like).

The linearized equations are (I'm sure you agree) are not a solution but crude approximation method. There are now high order post-Newtonian equations, as well as direct numerical solutions available.

 

[TrickyDicky]

I agree with mass also, up to some upper limit where the mass is large enough to perturb other sources. For example, Jupiter orbiting the sun, Jupiter can be considered a test particle for reasonable precision. However, it is not intended to say anything about, e.g. a binary system of comparable masses.

It might be like you say, the problem is that this is implicitly assumed, as an assumed code, or so it has been called in this thread.
The thing is if that was the case, that the mass independence is always referred to the body's mass compared with other sources, why isn't that included in the Principle explicitly?
Certainly the low-mass code is true in linearized GR, could it be that it has become a habit to think of the EP and geodesic motion in the terms of linearized GR, since most problems are tackled in this regime, and from the habit of thinking in these kind of setting, this low mass code has become the most prevalent interpretation?

 

[TrickyDicky]

The linearized equations are (I'm sure you agree) are not a solution but crude approximation method. There are now high order post-Newtonian equations, as well as direct numerical solutions available.

Yes, I meant they are derived from the EFE under certain conditions that might be different from the conditions assumed for the other situations mentioned.

 

[PAllen]

It might be like you say, the problem is that this is implicitly assumed, as an assumed code, or so it has been called in this thread.
The thing is if that was the case, that the mass independence is always referred to the body's mass compared with other sources, why isn't that included in the Principle explicitly?
Certainly the low-mass code is true in linearized GR, could it be that it has become a habit to think of the EP and geodesic motion in the terms of linearized GR, since most problems are tackled in this regime, and from the habit of thinking in these kind of setting, this low mass code has become the most prevalent interpretation?

The wording as stated is obviously false for high enough mass (we are repeating ourselves; another star introduced in place of Jupiter will not follow the same world line as Jupiter). There is nothing hidden about this.

 

[TrickyDicky]

The wording as stated is obviously false for high enough mass

I know we get stuck here, still to me is not obvious, at least from the wording.

(another star introduced in place of Jupiter will not follow the same world line as Jupiter).

I agree with this, but I think the wording refers to a single body, it doesn't mean that all bodies no matter their mass have the same worldline, since the worldline depends on the total curvature of the manifold at that point and that is determined by all the sources, not just the body in question. I think there is too much ambiguity in the definition anyway, so we might go on forever in this loop. Hopefully not.

 

[PAllen]

I know we get stuck here, still to me is not obvious, at least from the wording.


I agree with this, but I think the wording refers to a single body, it doesn't mean that all bodies no matter their mass have the same worldline, since the worldline depends on the total curvature of the manifold at that point and that is determined by all the sources, not just the body in question. I think there is too much ambiguity in the definition anyway, so we might go on forever in this loop. Hopefully not.

Yes, this is one of the places we keep getting stuck. Almost everyone except you sees 'test body' and says oh, I know that code word. Note that an atom, me, the moon, earth, and Jupiter introduced at the same place and velocity relative to the sun will fill follow the same world line (to pretty high precision). This gives a huge constraint on how gravity couples to matter. However, another star will not follow the same world line. Almost everyone else is happy to say, oh fine, another star is too massive to be a test body for the Sun's gravity.

 

[atyy]

The EP in full GR is usually thought to be:
1) Lorentzian signature of the metric
2) Minimal coupling between non-metric fields and the metric
3) Ability to state fundamental laws of non-metric fields using first derivatives

1) means that locally Minkowskian coordinates exist
2) & 3) mean that the "fundamental" laws of physics reduce to those of special relativity at a point, and don't probe the curvature of spacetime.

The curvature of spacetime can still be probed because the "derived" laws of physics.

 

[TrickyDicky]

Yes, this is one of the places we keep getting stuck. Almost everyone except you sees 'test body' and says oh, I know that code word. Note that an atom, me, the moon, earth, and Jupiter introduced at the same place and velocity relative to the sun will fill follow the same world line (to pretty high precision). This gives a huge constraint on how gravity couples to matter. However, another star will not follow the same world line. Almost everyone else is happy to say, oh fine, another star is too massive to be a test body for the Sun's gravity.

Nope, even I can see things that way, the difference is I seem to be the only one aware that this weak field, linearized approximation, is just that, an approximation, and maybe we shouldn't extrapolate it to the full non-linear GR in certain cases.

 

[PAllen]

Nope, even I can see things that way, the difference is I seem to be the only one aware that this weak field, linearized approximation, is just that, an approximation, and maybe we shouldn't extrapolate it to the full non-linear GR in certain cases.

And I can't understand what this means. None of the discussion about the WEP we've been having has any connection at all to linearized GR, that I can see.

The WEP, as I understand it, is limiting principle true to exceeding precision for applicable situations in the real world, and is also consistent with several theories of gravity, including full GR. Where does the linearized approximation come into this?

 

[PAllen]

Nope, even I can see things that way, the difference is I seem to be the only one aware that this weak field, linearized approximation, is just that, an approximation, and maybe we shouldn't extrapolate it to the full non-linear GR in certain cases.

I'm wondering, are you thinking that my example of a star not following the same trajectory as Jupiter is related to gravitational waves? That's not what I'm referring to at all. I simply mean, literally, a star would follow a radically different trajectory because it and the sun would mutually orbit; and it doesn't matter whether you compare Jupiter and a star in a center of mass frame or sun centered frame, the trajectories would be completely different. This is true in reality, in Newton, and in any plausible theory of gravity. Thus, the literal wording of this variant of WEP would simply be false (for all theories of gravity). This simply means that this situation is not intended be covered because a star cannot be test body for the sun's gravity.

And we've been here before... and I suspect the circle will not end. Not only has everything been said on this thread, it has been said too many times.

 

[TrickyDicky]

And I can't understand what this means.

Maybe this quote from Ryder's book on relativity helps to see what I'm referring to:
"So in the linearised theory the gravitational field has no influence on the motion of matter that produces the field...
...It is therefore possible in principle, as pointed out by Stephani,1 that an exact solution, provided it could be found, could differ appreciably from the linearised solution. So we must beware, especially since the linear approximation may be used in cases where an exact solution is not known; and therefore the conclusions drawn may not be reliable."

None of the discussion about the WEP we've been having has any connection at all to linearized GR, that I can see.

From the first moment you have been claiming that the WEP, and test partcles must be understood in the weak field limit and as an approximative approach (low mass code) and now you say none of the discussion has any connection with linearized GR, I truly find hard to understand you too.
All the examples used thru the thread refer to Newtonian limit, weak field, linearized approximation.

 

[PAllen]

Maybe this quote from Ryder's book on relativity helps to see what I'm referring to:
"So in the linearised theory the gravitational field has no influence on the motion of matter that produces the field...
...It is therefore possible in principle, as pointed out by Stephani,1 that an exact solution, provided it could be found, could differ appreciably from the linearised solution. So we must beware, especially since the linear approximation may be used in cases where an exact solution is not known; and therefore the conclusions drawn may not be reliable."



From the first moment you have been claiming that the WEP, and test partcles must be understood in the weak field limit and as an approximative approach (low mass code) and now you say none of the discussion has any connection with linearized GR, I truly find hard to understand you too.
All the examples used thru the thread refer to Newtonian limit, weak field, linearized approximation.

I don't understand how you are reading what I write to say any of this. I have never mentioned linearized theory except perhaps in direct response to something you brought up about. Nothing I have said in this thread, as I wrote it, and read as written, has anything to do with linearized theory.

The statement that the WEP be considered for test particles is not, in my mind, connected in any way to the linearized theory. It is, in fact, connected (in the case of GR) with being able to use a geodesic of the background geometry - that is, a calculation in the full exact theory (in, e.g. the strongest field section of Kerr geometry), but limited to test particles that follow background geodesics. It is truly mind boggling how much your interpretation differs from the words I wrote.

In a few places, I have referred to the obvious fact that there is no known exact, non-static two body solution. However, the alternatives I've proposed for dealing with this have nothing to do with linearized GR - the papers I've linked (for the two body problem) involve using numeric solution of the full field equations, with sophisticated convergence control, to compute corrections to 3.5 order post Newtonian approximation, which is already way beyond linearized theory.

It really sometime feels like you have separate dictionary for English when reading what I write.

 

[TrickyDicky]

I don't understand how you are reading what I write to say any of this. I have never mentioned linearized theory except perhaps in direct response to something you brought up about. Nothing I have said in this thread, as I wrote it, and read as written, has anything to do with linearized theory.

The statement that the WEP be considered for test particles is not, in my mind, connected in any way to the linearized theory...

In my mind there is some connection, in the sense that if test particles are considered as only those bodies with not enough mass to perturb the background source(s), that is exactly linearized GR, a perturbative approach valid as long as the test body doesn't perturb the Minkowski background too much (weak field limit), in that sense in my opinion you have referred to the linearized approximative approach to GR most of the time when you have mentioned test particles and more explicitly:
In post #15 when referring to GW which are derived from the linearized equations.
In posts #37, #41, #44 and#89 when describing the low mass limit for test particles in terms of not perturbing the background metric.
In posts #94 and #106 when talking about gravitational radiation situations.

I think the english dictionary might not be the problem here after all.

 

[PAllen]

In my mind there is some connection, in the sense that if test particles are considered as only those bodies with not enough mass to perturb the background source(s), that is exactly linearized GR, a perturbative approach valid as long as the test body doesn't perturb the Minkowski background too much (weak field limit),

I agree that the linearized approach means ignoring non-linear corrections to Minkowski background. What has that got to do with computing geodesics in exact Kerr geometry (for example)?

A related usage is to perturb a background metric other than Minkowski, e.g. a simple approach to the two body problem. However, in discussing WEP my view is you simply limit it to cases where the geodesic equation of motion is valid. For cases where it isn't, nowadays, you needn't rely on linearized theory if you are concerned about its validity - you can use numeric solution of the full field equations.

Many of the calcualtions (even very old ones) showing that the goedesic equation of motion is true only in the limit use the full field equiations, not linearized equations. And to try to limit confusion, by geodesic equation of motion, I mean a geodesic of the background geometry.

As an aside, there are now numerous derivaitons of GW that make no use of perturbative approaches at all. I linked two of them (equal mass neutron stars, equal mass black hole papers; these use numeric solution of the full field equations). There are also arguments based not even on any numeric simulations, e.g. based on the 'relaxed' form of field equations which are exact as long as you are dealing with non-singular spacetimes (roughly; more precisly, you have to be able to cover the spacetime in harmonic coordinates (introduced by Vladimir Fock)).

 

[TrickyDicky]

It is obvious that given the ambiguous nature of all the versions of the EP there is little chance to reach an agreement.

But to me is enough with what is said in posts #106 to #109 by atty and PAllen in which a certain consensus is reached, at least I can agree with most of what is said in those.
So I guess it's been worthy. There's been some mutual misundertandings on both sides of the question, but at least I can get something out of it all. Thanks everyone.