# Is the equivalence principle good for anything?

1. Feb 23, 2010

### CuriousKid

I heard that there is now consensus in literature that a charged particle will take a DIFFERENT path around a neutral object than an uncharged particle. The reason is that the charged particle will radiate.

I have even heard physicists try to wave this away as "well a charged particle needs to carry its "fields" around with it" so that is not a local object. If you take that stance, then the equivalence principle is completely worthless except for interactions which are purely point/contact interactions ... of which there are none. All four known fundemental forces can be described with fields.

So what good is the equivalence principle?

Alternative phrasing to make this more constructive:
Can anyone completely mathematically/rigorously define the equivalence principle, and NOT have it be violated by the differring trajectories of a charged and neutral particle?

EDIT: Just checked. According to the definitions in wikipedia, this would indeed violate the equivalence principle.

2. Feb 23, 2010

### jfy4

I don't understand what the violation is? or where the equivilence principle fails?

why should a charged and neutral particle have the same paths?

please state it for the forum and me .

3. Feb 24, 2010

### atyy

Aren't all the particles described by fields too? So if you have no more particles, do you still have a problem?

The EP only applies to freely falling particles. A particle in its own field is acted on by its own field, so it is not freely falling.

The EP suggests that gravity is geometric. Newton's theory obeys an EP, and can be reformulated as geometry (Newton-Cartan theory). Nordstrom's theory, which was the first relativistic theory of gravity can also be reformulated as geometry. General relativity is also in some sense a geometric theory.

The EP suggests a simple way of generalizing special relativistic laws to general relativistic laws (minimal coupling or the "comma goes to semi-colon rule").

When all is said and done, the EP is neither a single principle nor a logical necessity. Chapter 24 of Blandford and Thorne has an amusing discussion of when the EP fails.

Last edited: Feb 24, 2010
4. Feb 24, 2010

### bcrowell

Staff Emeritus
Keep in mind that even if you think of this as violating the EP (which is partly a matter of interpretation), the violation is ridiculously small -- way too small to be measured in any practical experiment.

5. Feb 24, 2010

### TcheQ

Who were these physicists, and how many papers have they had published in refereed journals?

6. Feb 24, 2010

### CuriousKid

How is it a matter of interpretation? Or better yet, how can a mathematical principle even be left to interpretation: is the EP really defined that poorly?
These are not rhetorical questions, I really am asking and really am confused.

jfy4,
in answer to your question, here is the definition of the weak EP from wikipedia
The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition.

That is why I would expect a neutral particle and charged particle to have the same trajectory around a neutral gravitating body. But the EP misleads us here (as is why apparently there was disagreement in literature for awhile), as the two will not have the same trajectory.

I'm not sure what you are suggesting here. Is the point that the EP can never be used?

The point is that a free falling frame is supposed to be locally equivalent to an inertial frame.

Can a particle at rest in an inertial frame feel a proper force due to its field? No. So I would expect the same from a free falling field. That fact that it isn't confuses me deeply. I don't even understand how this can be violated since I though the EP along with some other conditions could be used to derive GR.

I don't have that book. But from people's comments here, it is sounding like there is no real equivalence principle in any rigorous sense. It is just a thought that is useful, but hard to define, like "inertia"?

7. Feb 24, 2010

### CuriousKid

Was that a really non-mainstream opinion? Why I am being asked to provide references?

I'm new here, so I'm sorry if I was supposed to supply references for comments (other questions didn't seem to, so I thought it was okay).

Parrott
Found.Phys. 32 (2002) 407-440.
"since the field extends throughout all spacetime, no measurements on the particle can
be considered truly local"
http://arxiv.org/pdf/gr-qc/9303025
To be fair, the full quote shows that he is quoting yet other physicists, and it references another published article. I couldn't find a free copy of the other article.

But I've heard that "non-local due to fields" comment from other physicists in person (I'm a student at a large research university), and yes they are well published. So I thought it was, at the least, a not-unheard-of-opinion.

Basically it looks like the EP fails for electrodynamics, and this is a reason some people give.

Last edited: Feb 24, 2010
8. Feb 24, 2010

### TcheQ

You're adept at quoting and researching your sources, just keep in mind that when you quote something from an old source, it is your imperative to make sure that it hasn't been re-researched of refuted in the meantime.

The reason I brought this up is that the phrase "carries it's fields" implies the person who wrote that line doesn't understand what they talk about (which is common when you try to describe Quantum mechanics using classic physics).

So for your question "what good is the equivalence principle", we would say "because it allows us to conduct experiments with acceleration as simulating gravitational fields"

I am unsure where the confusion is also... charged particles have different mass and charge than neutral particles...so we would expect them to behave differently.

9. Feb 24, 2010

### Altabeh

Proving every spacetime is locally flat does completely rely on a mathematical basis (see for example Schutz B. A first course in general relativity, p.p. 158-160 or Weyl H. Space-Time-Matter, 1922, p. 80.) So it's not a matter of interpretation at all!

I don't know why EP confuses and misleads you that much! EP is purely geometrical and as atty said, it only occurs for those fields that can curve spacetime, including gravitational field and very small affects of electromagnetic field (which are neglected if gravity is strong enough to make their affect be ruled out). EP does always happen to exist for freely falling particles having mass or being point masses! But the charged particles don't follow any kind of geodesics, though $$d\tau^2>0$$ for them and this of course does not make EP worthless to rest on it for support because there are so many particles that have no charge but we can apply them to EP as test particles for simplicity of calculations (here the testing properties let us ignore all measuring aspects of particles [except photons or neutrinos]
such as mass, charge which can be assumed zero, and size.) Nevertheless, even if we take a particle to have electron's mass and be of its size but uncharged, yet the geodesic equations around any gravitating mass are still valid because we are just trying to study gravitational affect on particles that don't experience any force but gravitational so if charge was involved, then one would expect other perturbative forces acting on particle due to an electrical field it carries along so that this stuff is not GR anymore!

AB

10. Feb 24, 2010

### Ich

I think so.
But I think you already formulated the resolution:
You omitted "locally" in your second statement. Include it, and everything is fine again.
The EP is valid locally, that means, if there is some backreacktion with the field from inside the region you are considering, that's ok. You just make the region smaller if you want less deviation, and it goes to zero for a point like region. If you include backreaction from large distance, well, that's not local, so it doesn't concern the EP.
We would expect so, if gravity were a force. It's the whole point of the EP that all objects behave exactly the same way, independent of composition. That the gravitational mass of the internal EM field energy is exactly the same as the inertial mass.
But you mustn't include any influence whatsoever from the outside world, be it self-made or not. You have to restrict yourself to truly local penomena.

11. Feb 24, 2010

### atyy

http://relativity.livingreviews.org/Articles/lrr-2004-6/ [Broken]
"In the scalar and electromagnetic cases, the picture of a particle interacting with a radiative field removes any tension between the nongeodesic motion of the charge and the principle of equivalence."

http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html
"What is the minimum amount of nonlocality that can produce curvature-coupling modifications in physical laws? As a rough rule of thumb, the minimum amount is double gradients:"

http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll4.html
"In fact, let us be honest about the principle of equivalence: it serves as a useful guideline, but it does not deserve to be treated as a fundamental principle of nature. From the modern point of view, we do not expect the EEP to be rigorously true."

http://relativity.livingreviews.org/Articles/lrr-2006-3/ [Broken]
"Empirically it has been found that almost every metric theory other than GR introduces auxiliary gravitational fields, either dynamical or prior geometric, and thus predicts violations of SEP at some level (here we ignore quantum-theory inspired modifications to GR involving “R2” terms). The one exception is Nordström’s 1913 conformally-flat scalar theory [195], which can be written purely in terms of the metric; the theory satisfies SEP, but unfortunately violates experiment by predicting no deflection of light."

http://arxiv.org/abs/0707.2748
"There is no precise definition of “gravitational” and “non-gravitational” field. One could say that a field non-minimally coupled to the metric is gravitational whereas the rest are matter fields. This definition does not appear to be rigorous or sufficient and it is shown in the following that it strongly depends on the perspective and the terminology one chooses. ............... following Will’s book one can argue that the EEP can only be satisfied if there exists some metric and the matter fields are coupled to it not necessarily minimally but through a non-constant scalar"

Last edited by a moderator: May 4, 2017
12. Feb 24, 2010

### Altabeh

No it is not okay! Even in small regions, one can't make a charged particle obey the general geodesic equations so in GR they use something like a test particle to ignore qualities like charge and large sizes of particles to make the theory geometrically compatible with EP and taking the particle to have charge, due to Coulomb' law, would impose a very large electrical force compared to Newtonian gravitational force (if we are, for instance, freely falling towards earth's surface) which leads to a large deviation from the path described by the geodesic equations around the target gravitating body! The general definition of EP only applies to particles idealized in some way so as to get the theory to work well while keeping things in agreement with empirical viewpoints! From this loophole, I'd also be tempted to say that EP is not well defined in the context of physics!

13. Feb 24, 2010

### Altabeh

Atty, those links are amazing! Thanks.

AB

14. Feb 24, 2010

### Mentz114

I think this is a storm in a tea-cup. I have always understood that the EP applies only to gravitational phenomena. It starts with the equivalence of inertial and gravitational mass which means we can say that the effect of a gravitational field on a body is independent of the mass and composition of the body ( not test particles only, anything ).

The difference between gravitational fields and EM is that there is free-fall in gravity but not in EM. So it's blindingly obvious that electrodynamics has no equivalent of the gravitational EP !

It isn't possible to invoke the EP in situations where force fields are present and it's bound to lead to the confusion that is apparent in this thread.

[edit : I just noticed the references given by atyy. I will check them but I can see at least one solecism in the the summaries. I suspect there's at least some hot-air in there.]

15. Feb 24, 2010

### Altabeh

Unfortunately you are (sort of) wrong! Charged particles carry electric fields which can affect their motion along a path if in particular this path is a geodesic and the deviation is not negligible at all; then they won't follow geodesic and so the definition of "freely falling bodies" is not applicable in which case where charge is also involved unless you omit the field blindly! Though EP has a completely gravitational origin and thus geometrical, this doesn't mean it is in a completely safe zone when it comes to the charged particles involved in the situation! But I think you are also correct, if a comparison of EM and gravitation is considered when studying EP! In such easygoing case, where mostly photons are involved, EP isn't damaged unless the electromagnetic force is also dominant so it can curve the photon's path! I don't think this has a very influential impact on EP as in small regions such deviation due to EM could be safely neglected!

AB

16. Feb 24, 2010

### Ich

Of course it is ok. Maybe you overlooked "from inside the region you are considering".
As an example: The EP would predict a capacitor to fall at the same rate, whether it's charged or not. But only as long as there are no stray fields that reach beyond the finite region that you consider, there's no additional error. There is some error of course due to the finite extension of the region.
??
Are you saying that a treatment of EM in inertial (=free falling) frames is invalid? Or that acceleration in an EM field is not free fall (true, of course)?
The EP includes of course all conceivable interactions - but only locally. When the fields go through the whole universe, well, that's not exactly local.

17. Feb 24, 2010

### Mentz114

I think you've missed my point completely. Applying the gravitational EP ( what other kind is there ? ) to any situation where force fields are acting is not even wrong ! The EP has nothing to say about 'deviations from geodesics due to EM forces' ! Electrodynamics is written entirely in terms of inertial mass - so how can the EP have any relevance since it is concerned with the equality /non-equality of inertial and gravitational mass. The latter concept is missing entirely from electrodynamics.

No it doesn't. It is only applicable to gravitational fields.

Last edited: Feb 24, 2010
18. Feb 24, 2010

### Ich

I can't make sense of this statement. Could you please answer my request for clarification?

19. Feb 24, 2010

### Mentz114

Only the parts that depend on the ratio of gravitational mass to inertial mass.

What does that mean ? Of course it's not free fall.

20. Feb 24, 2010

### Ich

Hmm, didn't help either.

I mean, it's obvious that the presence of external fields may lead to a force on the test particle, therefore it's no longer free falling.
It's less obvious when we consider the (charged) test particles own field. Still, if the field extends far longer than the region around the test particle that we regard as "pointlike" in the context of the experiment, there may be deviations from a geodesic, too.
But if all fields of the test particle are constrained to said "pointlike" region around it, and there are no external fields, the EP predicts the particle to move on a geodesic. This prediction is nontrivial, it is what is meant by "independent of the constitution of the test particle".

Agreed?

21. Feb 24, 2010

### Mentz114

Hi Ich,

Yes, but the EP has nothing to say about this. The test particle is experiencing proper acceleration while moving in curved spacetime. The covariant curl ( wedge derivative) of the potential is identical to the flat space curl, so the two effects are completely independent. I believe that interaction betweem EM and gravity have almost been ruled out by experiment ( I may be mis-remembering this ).

Hmmm. I think this is where we part company. I don't think that statement was intended to include anything other than mass. Charge does not have mass, or rather it is not mass, so it does not gravitate per se ( I'm sticking my neck out here because obviously some energy gravitates ...).

1. For these reasons I conjecture that in the absence of other fields, a charged particle will follow the same geodesic as an uncharged particle.
2. a charged body freely falling in the absence of any other fields will not radiate.

Are we getting anywhere ?

Last edited: Feb 24, 2010
22. Feb 24, 2010

### Ich

EM fields have energy content, thus mass. Same with the strong force. Eötvös-type experiments check different materials where - mostly on the nucleon level - a different part of the mass is contributed by the different fields. So that's exactly what the EP means.

I think so.

23. Feb 24, 2010

### Mentz114

The binding energy that holds bodies together may be taken as part of the gravitational and inertial mass. But if we have an isolated charged body, does the field that permeates the space around have mass ? I think not.

I seem to remember now that there is an argument that some kind of energy will contribute to the gravitational mass but not to inertia ( or vice-versa), which would violate the EP, but those arguments are not supportable. I think the references atyy gave are of those type.

I think I've made my position clear, but I have to take a break now.

I'll be back.

24. Feb 24, 2010

### bcrowell

Staff Emeritus
Yes, it really is defined that poorly. It's not a mathematically well defined principle. It's more like a guiding philosophical precept for evaluating candidate theories of gravity.

The classic paper on the topic of the free-falling charge and the e.p. is: C. Morette-DeWitt and B.S. DeWitt, "Falling Charges," Physics, 1,3-20 (1964). It's available here: http://www.physics.princeton.edu/~mcdonald/examples/EM/dewitt_physics_1_3_64.pdf [Broken] Note the last sentence of the abstract, where they point out that the effect is "far too small to be detected experimentally." When they say far too small, they really mean it. For a calculation of just how small it is, see ch. 1, problem #7 in this book http://www.lightandmatter.com/genrel/ . The solution is given in the back of the book. For a proton and a neutron orbiting the earth, the relative violation of the e.p. (difference in acceleration divided by acceleration) turns out to be $10^{-34}$.

There are various versions of the equivalence principle. For a detailed discussion of why the e.p. is not a single, mathematically well defined statement, see this paper: "Theory of gravitation theories: a no-progress report," Thomas P Sotiriou, Valerio Faraoni, Stefano Liberati, http://arxiv.org/abs/0707.2748 On p. 3, they lay out various versions of the e.p. (WEP, EEP, SEP). Your characterization of the e.p. as a statement of local flatness is one that you often hear stated loosely, but if you take a look at the forms of the e.p. given in the Sotiriou paper, you'll see that it's not one of them. I think the basic reason for that is that local flatness only works as a way of characterizing the e.p. within GR. The real answer to the OP's question ("Is the equivalence principle good for anything?") is that it's useful for comparing candidate theories of gravity.

If all you want to do is work within GR, then you could certainly throw away the e.p. and never mention it again. To the extent that the e.p. holds in GR, you don't need it because the results of GR calculations will obey they e.p., whether or not you explicitly know or invoke the e.p. To the extent that the e.p. fails in GR (e.g., very weak violations for charged particles), you don't want the e.p. because it gives the wrong answer.

If you want to use the e.p. for the one thing that it's really useful for, which is testing competing theories of gravity, then just saying that it's equivalent to local flatness is not useful. It's only within GR that it's (approximately) equivalent to local flatness. I think this is the reason that the Sotiriou paper's statements of three versions of the e.p. all refer to actual experiments. Unless you talk about actual measurements, you can't state the e.p. in a model-independent way.

Last edited by a moderator: May 4, 2017
25. Feb 24, 2010

### bcrowell

Staff Emeritus
Yes, that's a good way of looking at it.

C. Morette-DeWitt and B.S. DeWitt, "Falling Charges," Physics, 1,3-20 (1964), http://www.physics.princeton.edu/~mcdonald/examples/EM/dewitt_physics_1_3_64.pdf [Broken]

Parrott, http://arxiv.org/abs/gr-qc/9303025

Harpaz and Soker, http://arxiv.org/abs/physics/9910019

Gralla, Harte, and Wald, http://arxiv.org/abs/0905.2391

Grøn and Næss, http://arxiv.org/abs/0806.0464

Bryce DeWitt and Wald are both extremely well known relativists. I know that Grøn is also a well known relativist who publishes frequently in refereed journals. I'm not as familiar with the reputations of the others, but I think they're all mainstream physicists. If you read this sequence of papers, I think you'll see very clearly that although there are some disagreements, what they're all clarly agreeing on as the topic of discussion is exactly the issue that CuriousKid pinpointed.

I don't think this attempt to sidestep the problem really works. The e.p. violation we're talking about is not a violation due to an externally applied field, it's a violation due to radiation. The big picture of GR is that it's a geometrical theory of spacetime, and it's perfectly happy to be coupled to any matter field you like. You tell it, "Hey, I want you to tell me about the gravitational field of a cloud of axions," and it says, "Sir, yes, sir!" The whole reason it's been such a useful and durable foundational theory is that it's completely agnostic about what sources are creating the stress-energy tensor. The e.p. is an important part of the interpretation of GR, and the ability to insert any matter fields you like is an important feature of GR. The fact that you can't have both of these things at once is actually a potentially important issue. It just turns out to be a numerically negligible effect for practical cases, like a neutron and a proton orbiting the earth.

I think the problem with this statement is that you need to add the important distinction made in Ich's #16. For example, see the statement of the Weak Equivalence Principle on p. 3 of this paper http://arxiv.org/abs/0707.2748 :

The qualifier "uncharged" is equivalent to Ich's statement that the object's field can't extend far away (as in his example of the capacitor). This is the distinction that needs to be made, not something stronger or weaker. If you strengthen it by forbidding electromagnetic phenomena completely, then, e.g., every test of the e.p. done by the EotWash group in the last decade is useless, because they used material objects that were held together by electromagnetic interactions. If you weaken it by allowing test bodies with nonzero net charge, then you get the extremely small violations of the e.p. that are too small to measure, but possibly philosophically important.

Last edited by a moderator: May 4, 2017