Talk:Gravitoelectromagnetism/Archive 3

Factor of 2
I see in the discussion thread others have questions about the factor of 2. It looks like someone eliminated the 1/2 in the GEM equations redefining the gravitomagnetic field by a factor of 2 and didn't correctly adjust the Lorentz force equation. If you define it so the Lorentz force equation has a factor of 2 then 1/2 shows up on the B terms in the GEM equations. If you want the GEM equations to match Maxwell's not having the factor, then the factor in the Lorentz force equation is a 4. If you want no factor in the Lorentz force equation, then 1/4 shows up on the B terms in the GEM equations. That it why people are noticing that the page was inconsistent with the derivations in the resources. The modern relativity site external link shows the complete math derivation of both the GEM equations and the Lorentz force equation and you'll see defining it so the GEM equations match Maxwells leads to the factor of 4 in the Lorentz force. — Preceding unsigned comment added by Waitedavid137 (talk • contribs) 19:34, 14 August 2011 (UTC)
 * Can you provide a source says so that would qualify as a reliable source? VQuakr (talk) 20:21, 14 August 2011 (UTC)

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 * I did. You just keep deleting it. If you can't understand the actual math explicitly deriving it there, why do you care? Someone told you about this error in this discussion in 2008 and you still haven't fixed it? Others noted it isn't consistent with your own resources derivations and you still haven't fixed it? I give you a link for an external link explicitely deriving both the GEM equations and the Lorentz force and you still haven't fixed it? Really?

Waitedavid137 (talk) 23:41, 14 August 2011 (UTC)David Waite


 * Your own personal website does not quality as a Reliable source. Publish this in a peer-reviewed journal such as Classical and Quantum Gravity, and then this can be included in the article. Headbomb {talk / contribs / physics / books} 00:05, 15 August 2011 (UTC)

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This IS consistent with publications in Journals. YOUR site here was NOT. You were told this 3 years ago by someone else, and told your site was not consistent with YOUR OWN RESOURCES. In this discussion by others. Why do you want people to have wrong equations? Waitedavid137 (talk) 00:22, 15 August 2011 (UTC)David Waite
 * No, the math there is a reliable resource wether you like it or not. You can't make 2+2 = 5
 * Please read WP:RS to understand what we mean by a reliable source. Please do not re-add this link to the article; the external link is a separate issue from the math question. Thanks! VQuakr (talk) 01:36, 15 August 2011 (UTC)

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I confirm that there is a factor of 2 wrong in the current version of the article. Absorbing the 1/2 factor in Bg to make the GEM equations look similar to Maxwell equations implies a 4 factor in the Lorentz force. This is not the usual convention in academic publications. Following the equations given in this Wikipedia article leads to a wrong prediction for the precession of a gyroscope in a gravitomagnetic field by a factor of 2. Quetchka (talk) 10:44, 23 February 2012 (UTC)


 * Exactly how much of this article has been calculated by editors rather than summarized from reliable sources? I had a look at another physics article recently where people had been calculating equations themselves and sticking them in. By far he safest and best thing to do is follow the damn policy on WP:ORIGINAL research. If two sources say differently then there can be a ding dong here but having editors saying equations are wrong is just not the way to go. The question should be do they correspond to the WP:RS sources in published books and journals? Dmcq (talk) 11:21, 23 February 2012 (UTC)


 * The references seem difficult to check – the article is pretty poor with regard to indicating which facts are sourced from which sources. Reference 11 uses the concept of two different effective masses (one for the gravitoelectric force, and one for the gravitomagnetic force); not a promising approach.  Reference 12 seems to agree with the equivalents of the Maxwell equations (with the time derivative of Bg omitted!) provided that one rescales both the Eg and Bg fields by a factor of 4.  One then seems to obtain the Lorentz equation wih a factor of 4 when rescaling.  The question of course is where one can find a source that is both reliable and matches the scaling of Eg and Bg.  I would love to see some references on any of the equations presented that I can actually check.   — Quondum☏✎ 11:54, 23 February 2012 (UTC)
 * Sounds like time for Failed verification on some of those equations then. 'Sources should directly support the information as it is presented in an article,' according to WP:RS. If people are arguing about it on the talk page and we don't have source we can actually check against what's here then the equations should be removed after some reasonable grace period to allow whoever put them in to provide reliable sources. Dmcq (talk) 14:09, 23 February 2012 (UTC)


 * From Mashhoon, et. al., Gravitomagnetism and the Clock Effect:
 * The fact that the magnetic parts of equations (8) - (11) always appear with a factor of 1/2 as compared to standard electrodynamics is due to the circumstance that the effective gravitomagnetic charge is twice the gravitoelectric charge. That is, QE = M and QB = 2M are the effective gravitoelectric and gravitomagnetic charges of the source.
 * The linear approximation of general relativity involves a spin-2 field. This field, once its spatial components are neglected, can be interpreted in terms of a gravitoelectromagnetic vector potential. To sustain the electromagnetic analogy, however, we need to require that the gravitomagnetic charge be twice the gravitoelectric charge. This factor of 2 is a remnant of the spin-2 character of the original field, while for a pure spin-1 field (i.e. the electromagnetic field) the ratio of the magnetic charge to the electric charge is unity.
 * From Mashhoon, Gravitoelectromagnetism (This should be the primary title of the Wikipedia article):
 * There are different approaches to GEM within the framework of general relativity [11]. For the sake of simplicity and convenience, we adopt a convention that results in as close a connection with the formulas of the standard electrodynamics as possible [12]. In this convention, a test particle of inertial mass m has gravitoelectric charge qE = −m and gravitomagnetic charge qB = −2m. If the source is a rotating body of inertial mass M, the corresponding gravitational charges are positive, i.e. QE = M and QB = 2M, in order to ensure that gravity is attractive. Thus, we always have qB/qE = 2; this can be traced back to the spin-2 character of the gravitational field. Hence for a spin-1 field qB/qE = 1, as in Maxwell’s theory.
 * 70.109.190.231 (talk) 14:58, 23 February 2012 (UTC)
 * How you think that "This is often referred to as gravity being a spin-2 field and electromagnetism being a spin-1 field" is a paraphrase of "This factor of 2 is a remnant of the spin-2 character of the original field" and "this can be traced back to the spin-2 character of the gravitational field" is beyond me. The wording needs to change if reference to the spin-2 field is to remain.
 * Mashoon's equations also show a factor of 4 in the Lorentz force equation, once you rescale the B field to match its use in the article. If you don't rescale it and present the equations as given by Mashoon (in reference 13), there is a factor of 1/2 in the Maxwell-like equations and there is a factor of 2 in the Lorentz equation.  But Mashhoon is rather difficult to interpret; there seems to be no clear definition of Qm and Qe.  Are these to be considered as field source "charges", or only rather one-sidedly in terms of the force experience by a charge in an external field (i.e. of relevance in the Lorentz equation, but not in Maxwell's equations – this actually appears to be the intent)?  Since the article does not quote these equations with these quirks, it seems strange to be basing anything on them.  Are there any references that the equations do in fact reflect?  — Quondum☏✎ 17:50, 23 February 2012 (UTC)
 * I haven't actually been able to find a counterpart to the Lorentz force equation in the Mashhoon articles or the one from Clark et.al. I don't have a problem with someone rewording it to be more faithful to the lit.  I don't have a problem with changing the 2 to a 4 (or putting half of the 4 factor back into the GEM equations like Mashhoon has it).  I just don't think that the reference to the spin-1 vs. spin-2 fields should be removed and replaced with white space. 70.109.190.231 (talk) 04:53, 24 February 2012 (UTC)
 * I've not looked into this for a while, but for now I'll simply respond to some points above:
 * I agree that the title of the article should be Gravitoelectromagnetism and not Gravitomagnetism. This is clearly Mashhoon's intent and usage (I have not checked other authors); wherever his papers refer to gravitomagnetism it is the analogue of magnetism, and not of electromagnetism.  It feels like an article on classical electromagnetism being titled Magnetism.
 * Mashhoon gives the analogue of the Lorentz equation in reference 11 of the article, equation (27):
 * $$m\frac{d^2\mathbf{X}}{d\tau^2}=q_E\mathbf{E}+q_B\mathbf{V}\times\mathbf{B}$$, where $$q_E=-m, q_B=-2m$$ [in the context of B/2 appearing in the Maxwell-like equations].
 * In putting back the lost additional factor of 2, I would like to see how this looks in the stress-energy tensor before we do so; anyone know where to find this in terms of the vectors E and B?
 * — Quondum☏ 15:54, 7 May 2012 (UTC)


 * I'd be happy to leave the confirmation of the correct scaling to you. I just don't think we should leave out the spin-1 vs. spin-2 particle issue, since it seems to speak directly to this factor of 2 thing and is cited.  I would also be happy if you changed the primary title back to Gravitoelectromagnetism and made Gravitomagnetism redirect to that.  Being an IP don't have the permissions to do a name change (although I could switch the article content, but that is not the proper way to do it).  I thought that both Mashhoon and Clark et. al. derived these GEM equations from the Einstein field equation, no?  71.169.190.151 (talk) 03:38, 8 May 2012 (UTC)


 * Quondum, would you please move the article from Gravitomagnetism to Gravitoelectromagnetism? It's something that I cannot do as an IP, but you can do it.  I think that the redirect will switch around so that Gravitomagnetism will redirect to Gravitoelectromagnetism, but if the move doesn't do that automagically, any of us can fix that. 71.169.190.151 (talk) 14:17, 9 May 2012 (UTC)


 * I'm afraid Quondum can't do it either. In view of the fact that the topic has been discussed earlier (see /Archive 1), a renaming must be considered controversial, and you will have to follow the procedure explained in Requested moves to initiate a new discussion. Favonian (talk) 15:12, 9 May 2012 (UTC)

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The table in Gravitomagnetism is now incorrect if the big 4 gets moved from the Lorentz force equations to the GEM equation involving B. Are you guys sure that's where you want the factor to be? I think I'm gonna bring this up to sci.physics.research and see what those guys say about the article and this issue. Feel free to drop in there and voice your opinion and read what some of the leading physicists might say about it. 70.109.176.173 (talk) 18:46, 1 June 2012 (UTC)


 * I think that wherever the scaling factors are placed in the equations, we must be explicit about the scaling of each field with respect to sources. The recent changes in this respect may have made the article inconsistent (at least in respect of the statement about scaling constants used); I have not checked consistency of these changes against the sources.  A quadratic term in the velocity has also been introduced, which seems to be inconsistent with the idea of linearization at low velocities.  — Quondum☏ 09:19, 2 June 2012 (UTC)


 * Linearity in h&alpha;&beta; is not the same as linearity in vi. JRSpriggs (talk) 12:08, 2 June 2012 (UTC)


 * Point taken. I'm displaying my lack of familiarity with the subject... — Quondum☏ 13:33, 2 June 2012 (UTC)

Page move proposal: Gravitomagnetism → Gravitoelectromagnetism

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: moved. Reliable sources use the longer name. Aervanath (talk) 19:38, 31 May 2012 (UTC)

Gravitomagnetism → Gravitoelectromagnetism – This article has a history of contention relating to the name, e.g. Talk:Gravitomagnetism/Archive_1. My own (somewhat limited) reading of authors such as Mashhoon suggests that the usage is consistent with the intuitive analogy
 * electromagnetism ↔ gravitoelectromagnetism
 * electric field ↔ gravitoelectric field
 * magnetic field ↔ gravitomagnetic field

In particular, this article should be considered as an analogy of electromagnetism and not of only magnetism, and since I've seen no reference using the word gravitomagnetism for the analogy to electromagnetism as a whole as opposed to specifically magnetism, I consider it appropriate to move this article as indicated. Previous contention appears to have been based on Google hit rates for the respective terms, with very few editors involved. It seems to me that such an argument would in general lead to rather inappropriate article renaming, since it omits to consider what is being referred to in each instance (e.g. it would suggest we should rename the article Birth control to Carrot – or vice versa, but you get the point). Barring contradicting input by experienced physicists etc., I think this move should be made. — --Relisted JHunterJ (talk) 11:32, 17 May 2012 (UTC) Quondum☏ 15:40, 9 May 2012 (UTC)
 * Support If you look at the archive, such an attempt was twice made to properly name the article.  User:Hillman (whom I know is a physicist and has a presence on the USENET group sci.physics.research) and User:Rbj (whom also has posted to sci.physics.research but is not a physicist) have moved it, and both times the move was reversed because of some outspoken opposition, namely from User:Nixer who cited a more frequent occurrence of Google hits on "gravitomagnetism" than on "gravitoelectromagnetism".  Hillman has since quit Wikipedia and both Rbj and Nixer have since been banned.  But the primary name of the article should be Gravitoelectromagnetism as long as it is about this analogy between EM and what GR becomes in reasonably flat space-time, which is what Mashhoon and Clark et.al.  The article is mistitled as it present exists. 71.169.190.151 (talk) 15:57, 9 May 2012 (UTC)
 * Question - Do scholarly references use the term "gravitoelectromagnetism" to any significant degree? What terms _are_ used extensively? My understanding was that use in literature trumped most other considerations for article names. If there is no widely-used term covering all aspects of this subject, the umbrella-article should be titled something along the lines of "electromagnetic analogies to gravitation", rather than coining a neologism or giving undue weight to a term not in widespread use. --Christopher Thomas (talk) 08:45, 11 May 2012 (UTC)
 * Comment - It's without question true that "gravitoelectromagnetism" is a term that more completely describes the analogy between general relativity and electromagnetism than "gravitomagnetism". However, there are two (related) potential problems with the proposed change:
 * I suspect (but haven't checked) that gravitomagnetism is a much more common usage in reliable sources. Wikipedia should probably stick with the common usage.  Still, if enough reliable sources can be found that use gravitoelectromagnetism, an argument can perhaps be made in favor based on enduring notability.
 * The electric part of gravity is simply the usual (Newtonian) gravitational field. So in a sense the new thing in general relativity was the magnetic part, and by the same token that's also the part that gives rise to the most non-intuitive and surprising effects (such as frame dragging).  That's why gravitomagnetism is more commonly discussed than gravitoelectric effects.
 * So for now I remain neutral, pending more evidence on usage etc.  Waleswatcher  ( talk ) 13:13, 11 May 2012 (UTC)
 * This explains why the term gravitomagnetism used specifically in analogy to magnetism would be so prevalent, bolstering my position; this should not be confused with an argument for an article devoted specifically to gravitomagnetism. — Quondum☏ 14:21, 11 May 2012 (UTC)
 * And yet, despite including all four "GEM" equations, references like http://arxiv.org/abs/gr-qc/9912027 are titled "gravitomagnetism". As I said, this is because the magnetic aspects are much more interesting than the electric aspects.  Waleswatcher  ( talk ) 11:22, 13 May 2012 (UTC)
 * Ah, yes, this is getting close to the heart of it. This is because that article is addressing the effects of gravitomagnetism, and does so within the context of GEM.  You will note that this article does use the word "gravitelectromagnetism", and other articles by the same authors are entitled "Gravitoelectromagnetism".  If the article is to focus on the dynamic modifications to Newtonian gravity, then the title is correct.  Yet its focus is not this; it is the similarity with the EM equations generally.  I'm not saying that there should not be two articles - having an article on gravitoelectomagnetism and another on gravitomagnetism makes sense; each is a worthwhile and notable topic.  My concern is that too many people seem to think that the two terms have the same meaning, and that the one should take precedence as being more widely used.  — Quondum☏ 11:33, 13 May 2012 (UTC)
 * Still without taking a position on the proposed move, the article should spend more time discussing the magnetic aspects than the electric aspects, because the electric aspects are trivial (they are simply Newtonian gravity).  Waleswatcher  ( talk ) 11:51, 13 May 2012 (UTC)


 * Comment My first thought when seeing 'gravitoelectromagnetism' was that gravitomagnetism (I have heard the term gravitomagnetism before, I just can't remember where) was somehow combined with electric effects. In other words, rather then reading as gravito-(electromagnetism) I read it as (gravito/electro)-magnetism. My feeling is that the more common used term is gravitomagnetism for that reason. But, I am not an expert on the subject so I will leave this as only a comment. TStein (talk) 18:48, 11 May 2012 (UTC)
 * Support – My own vote as original proposer, but here substantiated with a little work. I did a Google book search (on the principle that published books are more authoritative than general Google-visible texts), looking primarily at the usage of the term "gravitomagnetism" or "gravitomagnetic".  It took the first approximately 20 books.  Some authors were repeated.  The picture that emerges is clear: the term "gravitoelectromagnetism" occurs significantly, "gravitomagnetism"/"gravitomagnetic" in the vast majority (i.e. even when the terms "gravitoelectromagnetism"/"gravitoelectromagnetic" and "gravitoelectric" do not occur) unambiguously refer to only the B-like field, and in no instance is its use inconsistent with this interpretation.   I therefore challenge anyone to locate a single instance of an authoritative text that uses "gravitomagnetism" to mean the gravitational analogue of the whole electromagnetic field, i.e. of both E and B rather than of only B.  The authors involved were: B. Mashoon, F. Gronwald, H.I.M Lichtenegger; Jairzinho Ramos Medina, Robert Gilmore; Clovis Jacinto de Matos; I. Ciufolini; Christoph Schiller; B. Bertotti, Paolo Farinella; Remo Ruffini, Costantino Sigismondi; Øyvind Grøn, Sigbjørn Hervik; B. Bertotti, Paolo Farinella, David Vokrouhlický; Sergei A. Klioner, P. Kenneth Seidelmann, Michael H. Soffel; Paolo Carini; Bernard F. Schutz; Peeter Joot; [C. Adam Reynolds – unknown as online text unavailable].  — Quondum☏ 19:55, 11 May 2012 (UTC)
 * If the term "gravitoelectromagnetism" is not widely used, then we can't use it as an article title, per WP:NEO/WP:SYN. Use a descriptive title instead (per my suggestion). --Christopher Thomas (talk) 20:43, 11 May 2012 (UTC)
 * First of all, "Gravitoelectromagnetism" is the descriptive title. It is the sole descriptive title for the content of this article, which is about the analogous behavior of gravitation, in the context of reasonably flat space-time, to classic electromagnetism.  It is about the GEM equations, which look just like Maxwell's equations with charge density replaced by mass density (or charge replacing mass) and the scaling factor used in Coulomb's law, 1/(4πϵ0), replaced by the scaling factor used in Newton's law of universal gravitation, -G.  "Gravitoelectromagnetism" not merely the most descriptive title, it is the only fully descriptive title of an article with the content of this article.
 * "Gravitomagnetism" is also a descriptive title, but it is descriptive of a phenomenon that necessarily excludes the "gravitostatic" field (and this article clearly does not exclude the gravitostatic field). "Gravitomagnetism" is about the additional effect of gravitation that occurs because of masses in motion.
 * The discussion that occurred years ago in the talk archive about the primary name clearly missed the point. It appears that a bunch of non-physicists imposed their ignorance upon the decision and based the primary title upon the more widespread used word that applied to a subset of the topical content of what this article presents.  This article is about Gravitoelectromagnetism, the analogy with Electromagnetism, which includes both static and magnetic effects.  It does not emphasize gravitomagnetic effects nor omit gravitostatic effects (if it did, it would be about frame dragging).
 * This is about a particular line of research (from authors like Mashhoon and others cited). They say it's "Gravitoelectromagnetism".  It's not about frame dragging, although half of it is, but the mathematical language looks more like EM than it does GR.  I do not understand what the controversy is.  It's obvious that the primary title is missing something.  It's mistitled and a mistake from years ago should not be perpetuated indefinitely. 71.169.190.111 (talk) 22:04, 11 May 2012 (UTC)
 * I think we can also comfortably say that the term is notable and not a neologism: it occurs in over 600 *books* (by Google-count). — Quondum☏ 22:23, 11 May 2012 (UTC)


 * Oppose Newtonian gravity is already the analogue of the electric part &mdash; compare Gauss's law for gravity with Electrostatics. So putting "electro" in is redundant and makes the name longer which is undesirable. Also the factor of 4 in the force law puts emphasis on the magnetic effects. JRSpriggs (talk) 21:53, 11 May 2012 (UTC)
 * Hmm. I can see a stalemate developing here for the wrong reasons.  The term gravitomagnetism has more in common with Ampère's circuital law than Maxwell's equations as a whole.  If the title stays, the focus of the  article must change to the Bg correction to Newton's law of gravitation.  If the title stays, and I edit it to address specifically this "new" component of gravitational interaction, are you going to object?  — Quondum☏ 11:21, 13 May 2012 (UTC)


 * To Quondum: I think that that would be taking the title way too seriously. JRSpriggs (talk) 11:36, 13 May 2012 (UTC)
 * Before my edit, the lead implied that the two terms were synonymous, which is incorrect. If the title remains unchanged, it would be appropriate, at the very least, to change the introductory sentence to define both terms correctly.  — Quondum☏ 13:03, 13 May 2012 (UTC)


 * Comment: It seems possible from the above and the article that there is no agreed and authoritative definition for either term. Would it therefore be preferable to use a descriptive phrase instead? I'm skeptical that two articles is a solution; The current article scope seems a good one, and covers the possible meanings of both terms. Andrewa (talk) 10:53, 17 May 2012 (UTC)
 * On the contrary, it seems the two terms have unambiguous meanings, but that some editors feel that the one (gravitomagnetism) is generally used, whereas there are alternatives/synonyms to the other (gravitoelectromagnetism) and is thus less often used. Point me to a source if I'm wrong in this.  — Quondum☏ 16:05, 17 May 2012 (UTC)
 * The onus of proof is on those who state that there is an unambiguous definition, not on those who question this. I note there is currently no citation for the lead Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles. That to me is most unclear, I will admit that I dropped university physics after first year, but from a semantic point of view it seems explicitly vague and rings a very loud OR bell. Andrewa (talk) 17:27, 18 May 2012 (UTC)


 * Comment I think that one article is better than two. Although I prefer to keep the existing name, I am not going to make a fuss if the majority decide to move it. JRSpriggs (talk) 14:20, 17 May 2012 (UTC)


 * Question I hope this is not too off topic, but why do the sources just not use the term 'gravitodynamics'? Is it being used for something else? For reasons I mentioned above I think that from a naming point of view 'gravitoelectrodynamics' is a horrible term because it is ambiguous. I am NOT suggesting we use 'gravitodynamics' since it most certainly would be either a neologism or a less popular name. I was just curious. TStein (talk) 16:53, 17 May 2012 (UTC)


 * "Gravitodynamics" is already being used for purposes that have nothing to do with this analogy to electromagnetism. JRSpriggs (talk) 18:08, 17 May 2012 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Relationship to general relativity
Could we say something about the what values for the Christoffel symbols would give the same gravitational force as the force law here; and perhaps something about how this approximation to GR was derived from Linearized gravity? JRSpriggs (talk) 11:36, 13 May 2012 (UTC)
 * Indeed - an absolutely crucial aspect of this that's entirely absent from the article is a discussion of exactly what sort of approximation this is, when it's valid, etc. "Weak field" is very vague and looks wrong.  For example, these equations look like they predict radiation from a time-varying dipole, which is wrong (you need a time-varying quadrupole in GR).   So I guess you can't trust the wave solutions, which means it's not just weak field, it's also some kind of non-relativistic limit.  Waleswatcher  ( talk ) 12:07, 13 May 2012 (UTC)
 * I agree that it makes sense to have more detail on how and when GR leads to this approximation.
 * On the dipole and radiation, remember that it is impossible to have a dipole (not even an isolated mass being vibrated laterally), and hence only quadrupole radiation is emitted. This does not mean the equations predict something that does not occur.  — Quondum☏ 13:22, 13 May 2012 (UTC)
 * It's certainly not impossible to have a dipole. It might be impossible to have just a dipole, but those equations look to me like they predict radiation from a time-varying dipole moment, and that's inconsistent with GR.  I suspect that's because they are invalid when applied to matter or stress-energy moving at relativistic speeds, and you probably need that for my dipole to matter.  The article needs to discuss under what circumstances the solutions to these equations are physically meaningful.   Waleswatcher  ( talk ) 00:32, 14 May 2012 (UTC)


 * There is no such thing as negative mass. So a static gravitational dipole consisting of a positive mass and a negative mass is impossible.
 * Even waving a mass back and forth cannot be done without moving another mass in the opposite direction. So dipole gravitational radiation is impossible. JRSpriggs (talk) 02:35, 14 May 2012 (UTC)
 * Take any distribution of matter you like. Is its dipole moment zero?  Almost certainly not.  Even a single point mass has a non-zero dipole moment if you don't put it at the origin.  So no, that's not the explanation.  The true explanation is that gravity waves are always polarized in such a way that they cannot be sourced by a time-varying dipole.  Anyway, that will be my last comment on this as I find physics debates on wiki pages a total waste of time unless they're directly related to the article.  I think we all agree the article needs to explain what precisely is the approximation that gives rise to these equations, and equivalently which solutions of them mean something.   Waleswatcher  ( talk ) 03:34, 14 May 2012 (UTC)

There is a conserved quantity which does not have any official name (as far as I know) which bears the same relationship to angular momentum that the electric field bears to the magnetic field. For a finite set of particles it is
 * $$\Sigma_k ( M_k \vec{r}_k - \vec{p}_k t ) \,$$

where Mk is the relativistic mass of the kth particle, etc.. The conservation of that quantity (together with the conservation of linear momentum and energy) is the reason why waving a mass without anything else being waved is impossible and why dipole gravitational radiation is impossible. JRSpriggs (talk) 04:49, 14 May 2012 (UTC)
 * Unless someone intends to introduce this into the article (relating to applicability conditions of GEM), I agree that it is pointless pursuing this here. More pertinent would be a different question of applicability, namely whether gravitational radiation carries negative or positive energy with it.  GR indicates that it carries away positive energy, but AFAICT, GEM predicts negative energy carried away by gravitational waves.  But I stand to be corrected.  — Quondum☏ 05:45, 14 May 2012 (UTC)


 * According to Linearized gravity, "then one ignores all products of h (or its derivatives) with h or its derivatives". Since Eg and Bg each already include one factor of a derivative of h in each of their terms, their product (the Poynting vector) is beyond the limit of the validity of our approximation. So we cannot infer anything about the sign or magnitude of the energy in gravitational radiation. JRSpriggs (talk) 18:38, 14 May 2012 (UTC)
 * The quantity you refer to is the "charge" of a Lorentz boost, and it is indeed conserved. But it's not the dipole, and its conservation does not imply that the dipole moment cannot vary (a trivial counterexample is a particle moving with constant velocity, which conserves the boost charge but obviously has a time-varying dipole moment).  Still, your argument has the right flavor - you can probably use that charge to prove the lack of dipole radiation, just not quite like that.  Regarding the Poynting vector, I think you are correct - anything quadratic in h will receive contributions of the same size from non-linearities that have been ignored here.  It's not clear to me that any of the wave solutions to these equations have any meaning.  It would be nice to have better sources for this article - the ones given are pretty low-grade, and that's part of the problem.   Waleswatcher  ( talk ) 19:36, 14 May 2012 (UTC)
 * To add to this: at least according to, the gravitational E and B are determined by derivatives of the time-space and time-time components of the metric fluctuation (if we can find a reliable source, we should add that).  That of course makes sense, since by analogy to E&M there should be a scalar and a vector potential, and those can only be those components.  But in GR, those components are pure gauge and do not represent physical waves in vacuum.  The physical polarizations correspond to the space-space components of the metric fluctuation, as you can easily see by using a coordinate transformation.  So my suspicion that the wave solutions to these equations are meaningless is getting stronger - at least for wave solutions in vacuum.   Waleswatcher  ( talk ) 19:53, 14 May 2012 (UTC)

Comparison of force equations
If we compare two equations for the gravitational force (derivative of contravariant linear momentum):
 * $$\frac{d}{d t} \left( m_0 \frac{d x^\lambda }{d \tau} \right) = - m_0 \frac{d t}{d \tau} \, \Gamma^{\lambda}_{\mu \nu } \frac{d x^\mu }{d t} \frac{d x^\nu }{d t} \,$$

from general relativity with
 * $$\mathbf{F} = m \left( \mathbf{E}_\text{g} + \mathbf{v} \times 4 \mathbf{B}_\text{g} \right) \,$$

from GEM, then we see that:
 * m in the GEM force equation must be taken to be the relativistic mass
 * $$ m = m_0 \frac{d t}{d \tau} \,;$$


 * the GEM electric field is
 * $$ \mathbf{E}_\text{g}{}^{i} = - \Gamma^{i}_{0 0} \,;$$


 * the GEM magnetic field is
 * $$ 4 \varepsilon_{i j k} \mathbf{B}_\text{g}{}^{k} = - \Gamma^{i}_{j 0} - \Gamma^{i}_{0 j} \,$$
 * which is only possible if
 * $$ \mathbf{B}_\text{g}{}^{k} = - \frac{1}{4} \varepsilon_{k i j} \Gamma^{i}_{j 0} \,;$$


 * and a term of order v2 needs to be added to the right hand side of the GEM force equation. JRSpriggs (talk) 20:47, 14 May 2012 (UTC)
 * That confirms both of my suspicions: that GEM isn't just a weak-field approximation, but also requires low velocity, and that its E and B are related to the time-time and time-space parts of the metric fluctuation, which are pure gauge at least in vacuum . The wave solutions are probably meaningless.  Waleswatcher  ( talk ) 12:09, 15 May 2012 (UTC)
 * Though I'm not following the detail here, I would be chary of jumping to this conclusion without more complete investigation. Likewise, the newly added section Lack of invariance is making a vague and possibly unjustified statement, effectively WP:OR.  At one point I started looking at a flat-space, fully covariant tensor approach to a hypothetical field with the second-order stress-energy tensor as the source in flat space-time.  Such a formulation is inherently Lorentz-invariant, and thus would be accurate at all speeds subject only to the weak field condition.  It would also give the transformation law if the components can identified with GEM.  — Quondum☏ 15:06, 15 May 2012 (UTC)


 * Speaking of gauge, we might want to consider what coordinate conditions would be appropriate to use with GEM. One of Einstein's early favorites was to set the determinant of the metric to its value in Minkowski space. This has the advantage that it wipes out the second term in the Ricci tensor; the third and fourth terms are already zero in the linear approximation. But that is just one of four conditions one would need to choose, and it might be incompatible with some other desirable conditions. JRSpriggs (talk) 17:29, 16 May 2012 (UTC)

If we set $$ 0 = h_{i i} = h_{1 1} + h_{2 2} + h_{3 3} \,$$ everywhere as one coordinate condition, this gives $$\Gamma^{i}_{i \alpha} = 0 \,.$$ So
 * $$\begin{align}

R_{\alpha 0} & = \Gamma^{\rho}_{\alpha 0,\rho} - \Gamma^{\rho}_{\rho \alpha, 0} \\ & = \Gamma^{i}_{\alpha 0, i} + \Gamma^{0}_{\alpha 0, 0} - \Gamma^{i}_{i \alpha, 0} - \Gamma^{0}_{0 \alpha, 0} \\ & = \Gamma^{i}_{\alpha 0, i} \\ & = {8 \pi G \over c^4} \left(T_{\alpha 0} - {1 \over 2}T\,\eta_{\alpha 0}\right) \,. \\ \end{align}$$

In particular,
 * $$R_{0 0} = \Gamma^{i}_{0 0, i} = {8 \pi G \over c^4} \left(\rho_g c^4 - {1 \over 2} \rho_g c^4 \right) \,$$

where stress has been ignored in T and thus
 * $$ \nabla \cdot \mathbf{E}_\text{g} = - 4 \pi G \rho_g \,$$

which is the first GEM equation. JRSpriggs (talk) 14:35, 17 May 2012 (UTC)


 * If you want to develop the detail in the form of a discussion, I'd suggest you invite such a discussion on a user talk subpage (e.g. here). Be aware that some, like myself, would need some more tutoring (e.g. on the interpretation of the components of the Christoffel symbol), and will bring their own perspectives.  Use of this page hardly seems to fit into the guidelines and will result in many watchers tuning out of this page.  — Quondum☏ 05:34, 17 May 2012 (UTC)

Since
 * $$\begin{align}

{\mathbf{E}_\text{g}}^{i} & = - \Gamma^{i}_{0 0} = {-1 \over 2} ( 2 h_{i 0, 0} - h_{0 0, i} ) \\ & = - ( {-1 \over 2} h_{0 0} )_{, i} - ( h_{i 0} )_{, 0} \,, \\ \end{align}$$ we get
 * $$ V_g = {-1 \over 2} h_{0 0} $$
 * $$ {\mathbf{A}_g}^{i} = h_{i 0} $$

are the potentials and thus
 * $$ {\mathbf{B}_\text{g}}^{i} = \varepsilon_{i j k} h_{k 0, j} \,$$

will automatically satisfy the second and third GEM equations. JRSpriggs (talk) 15:42, 17 May 2012 (UTC)


 * It could be that the reason that there are so few good references for GEM is that it does not really work.
 * I tried to derive the fourth GEM equation (the analogue of Ampere's law) as I did with the first three, but it does not seem to be working (so far). I can get some partial cancellation of terms, if we change the factor of -4&pi;G to -16&pi;G but that is it. Unless I am making some mistake in my calculation, the only other possibility would seem to be to find a still better coordinate condition than the one I am using above. In any case, the force law has no hope of being exact as I indicated above due to missing v2 terms.
 * So perhaps we should present GEM as a failed alternative to GR developed by Heavyside rather than as an approximation to GR. JRSpriggs (talk) 23:07, 17 May 2012 (UTC)
 * Well, that looks like the same factor of 4 that, in this article, was all collected and placed on the magnetic component, B, of the Lorentz force equation. 70.109.177.148 (talk) 04:26, 18 May 2012 (UTC)

The Bg field must be as indicated in my second previous message, or else the third GEM equation would have to be changed. The four in the force law should probably be a one. My thinking on the fourth GEM equation follows. Using
 * $$\begin{align}

\nabla \times \mathbf{B}_g & = \nabla \times ( \nabla \times \mathbf{A}_g ) \\ & = \nabla ( \nabla \cdot \mathbf{A}_g ) - \nabla^2 \mathbf{A}_g \, \\ \end{align}$$
 * $$\begin{align}

\Gamma^{j}_{i 0, j} & = {8 \pi G \over c^4} T_{i 0} \\ & = {8 \pi G \over c^4} ( - c^2 {\mathbf{J}_g}^{i} ) \\ & = {- 8 \pi G \over c^2} {\mathbf{J}_g}^{i} \,. \\ \end{align}$$ we get
 * $$\begin{align}

\left( \nabla \times \mathbf{B}_\text{g} + \frac{16 \pi G}{c^2} \mathbf{J}_\text{g} - \frac{4}{c^2} \frac{\partial \mathbf{E}_\text{g}} {\partial t} \right)^{i} & = h_{j 0, j i} - h_{i 0, j j} - 2 \Gamma^{j}_{i 0, j} + \frac{2}{c^2} ( 2 h_{i 0, 0 0} - h_{0 0, i 0} ) \, \\ & = h_{j 0, j i} - h_{i 0, j j} - ( h_{j 0, i j} + h_{j i, 0 j} - h_{i 0, j j} ) + \frac{2}{c^2} ( 2 h_{i 0, 0 0} - h_{0 0, i 0} ) \\ & = - h_{j i, 0 j} + \frac{2}{c^2} ( 2 h_{i 0, 0 0} - h_{0 0, i 0} ) \, \\ \end{align}$$ which is not apparently zero. I also applied the extra factor of four to the derivative of the gravitoelectric field also in order to retain the conservation of mass-energy which should hold at this level of approximation since
 * $$\begin{align}

0 & = T^{0 \alpha}_{; \alpha} \\ & = T^{0 \alpha}_{, \alpha} + T^{\beta \alpha} \Gamma^{0}_{\beta \alpha} + T^{0 \beta} \Gamma^{\alpha}_{\beta \alpha} \\ & \approx T^{0 \alpha}_{, \alpha} \, \\ \end{align}$$ since T and &Gamma; are both of the first order, so their product should be negligible. I am bothered by ignoring the trace of the stress in the Einstein equation for 0 0 since it should be at least
 * $$\frac{J_g \cdot J_g}{\rho_g} \,$$

which would require us to change the first GEM equation to
 * $$ \nabla \cdot \mathbf{E}_\text{g} = - 4 \pi G \left( \rho_g + \frac{J_g \cdot J_g}{\rho_g c^2} \right) \,$$

which would be incompatible with the conservation of energy law. If we tried to get around this by saying that the velocity of the source mass is of the first (rather than zeroth) order, then that would wipe out Jg which would make the whole GEM exercise pointless. What do you-all think? JRSpriggs (talk) 06:42, 19 May 2012 (UTC)


 * I think you're right. The linearized Einstein's equations certainly include terms proportional to the space-space components of the metric fluctuation - but those have no place in "GEM", and therefore, the equations as written in this article are manifestly incomplete.  One needs a reason or reasons to ignore all the extra terms.  Perhaps slow motion of source and test mass and no gravity waves and the right gauge choice?  Waleswatcher  ( talk ) 12:12, 19 May 2012 (UTC)
 * It's one thing for Jg to be small and it's another thing for Jg2 to be small. In addition that second term can be considered much smaller than the first term ρg because of what is in the denominator.  And since Jg = ρg vg, perhaps the case can be made for ignoring/eliminating the second term solely when vg << c.  Put it all over ρg c2 and see what you get. The other way to get to the bottom of this is to compare what JR is doing, step-by-step, to what Mashhoon or Clark et.al. have done.  I believe both have started with EFE and ended up with GEM equations. 70.109.177.148 (talk) 16:21, 19 May 2012 (UTC)
 * That's not the only term missing. Mashoon explicitly ignores all but a subset of the terms, with no justification (or at least I didn't see one).  When I have time I plan to re-write the article following Carroll.   Waleswatcher  ( talk ) 17:18, 19 May 2012 (UTC)
 * Well, my understanding of immediate issue is not that of missing terms, but of getting rid of a single term. I thought that JR wanted to ditch the Jg2/(ρg c2) and felt that it was inconsistent to do so and have any Jg left in the M part of the GEM equations.  I don't see a problem
 * $$\begin{align}

\nabla \cdot \mathbf{E}_\text{g} & = - 4 \pi G \left( \rho_g + \frac{J_g \cdot J_g}{\rho_g c^2} \right) \\ & = - 4 \pi G \rho_g \left( 1 + \frac{v_g^2}{c^2} \right) \\ & \rightarrow - 4 \pi G \rho_g \quad \quad \mathrm{if} \ v_g \ll c \\ \end{align}$$
 * I thought there were two essential assumptions to get GEM out of EFE: flat space-time and non-relativistic speeds. Do we not have enough to make this work with what JR verified? 70.109.177.148 (talk) 20:54, 19 May 2012 (UTC)
 * It's not flat spacetime that's needed - if the spacetime is exactly flat, all these GEM fields and their sources are zero.  It's small perturbations (i.e. weak field) around a Minkowski background, so that you can linearize Einstein's equations.  As for low velocity, at really low velocity and weak fields you just get Newton - i.e. the "electric" part, with  J=0.  So the question (for me at least) is whether you can keep J and B but neglect the space-space components of the metric.  I suspect the answer is actually yes - somehow there's another factor of 1/c for each space component - but I'd like to see a cogent argument.  There's also the problem that many of the solutions of these GEM equations - like all the wave solutions, I think - are meaningless, presumably because they propagate at c and hence invalidate the approximation used to derive them.   Waleswatcher  ( talk ) 01:18, 20 May 2012 (UTC)
 * I meant small perturbation from flat. I just thought that it still made sense to ditch the Jg2 in Gauss's Law, but not ditch Jg in whatever counterpart to Ampere's Law comes out of doing this to the EFE.  And I don't see how, dimensionally, we could expect another 1/c in there.  Also, why would gravity waves propagating at c make the GEM approximation to the effects of EFE from the point-of-view of euclidean space be invalid? I thought that's what's supposed to happen. 70.109.177.148 (talk) 03:18, 20 May 2012 (UTC)
 * It might in fact make sense, but one needs an actual argument (and it's not just that term, it's all the space-space metric fluctuations, which are completely ignored in GEM). As for gravity waves, the wave solutions to the GEM equations are pure gauge.  They can be removed by a coordinate transformation.  Real gravity waves are tensor modes, not vector modes.   Waleswatcher  ( talk ) 14:45, 21 May 2012 (UTC)

Acceleration
From special relativity and this article, we get
 * $$ \gamma^3 m_0 \, \mathbf{a}_{\parallel} + \gamma m_0 \, \mathbf{a}_{\perp} = \mathbf{F} = \gamma m_0 \left( \mathbf{E}_\text{g} + \mathbf{v} \times \mathbf{B}_\text{g} \right) \,.$$

So we have
 * $$ \frac{\mathbf{a}_{\parallel}}{1 - \frac{v^2}{c^2}} + \mathbf{a}_{\perp} = \mathbf{E}_\text{g} + \mathbf{v} \times \mathbf{B}_\text{g} \,.$$

To get an actual formula for the acceleration itself, let us define a temporary variable b
 * $$ \mathbf{b} = \mathbf{E}_\text{g} + \mathbf{v} \times \mathbf{B}_\text{g} \,.$$

Then
 * $$ \mathbf{b}_{\parallel} = \frac{ ( \mathbf{b} \cdot \mathbf{v} ) \mathbf{v} }{ \mathbf{v} \cdot \mathbf{v} } = \frac{\mathbf{a}_{\parallel}}{1 - \frac{v^2}{c^2}} \,$$
 * $$ \mathbf{b}_{\perp} = \mathbf{b} - \mathbf{b}_{\parallel} = \mathbf{a}_{\perp} \,$$
 * $$ \mathbf{a} = \mathbf{a}_{\parallel} + \mathbf{a}_{\perp} = \mathbf{b}_{\parallel} \left( 1 - \frac{v^2}{c^2} \right) + ( \mathbf{b} - \mathbf{b}_{\parallel} ) \,$$
 * $$ \mathbf{a} = \mathbf{b} - \mathbf{b}_{\parallel} \frac{v^2}{c^2} = \mathbf{b} - \frac{ ( \mathbf{b} \cdot \mathbf{v} ) \mathbf{v} }{c^2} \,.$$
 * $$ \mathbf{a} = \mathbf{E}_\text{g} + \mathbf{v} \times \mathbf{B}_\text{g} - \frac{ ( \mathbf{E}_\text{g} \cdot \mathbf{v} ) \mathbf{v} }{c^2} \,.$$

However, notice that this new term is not the term of order v2 about whose absence I was complaining above. That one is still missing. JRSpriggs (talk) 08:42, 21 May 2012 (UTC)

I am having trouble with ...
... this edit diff:. It doesn't seem congruent to either the Clark et. al. or Mashoon references. A factor of 4 on B in the Lorentz, or translating that back to the GEM equations (where it only makes a difference to the last one) is not the problem, but it is the extra term that neither Clark nor Mashoon have. 70.109.182.12 (talk) 06:49, 25 May 2012 (UTC)


 * Perhaps you are having a problem with my identification of the "gravitational charge" with relativistic mass rather than rest mass.
 * First, &rho;g is understood to be from the T0 0 component of stress-energy which means that it is the density of energy divided by c2, i.e. the density of relativistic mass, not the density of rest mass (if that can even be said to have a meaning). So as in Newtonian gravity, the active gravitational charge (source of the field) and the passive gravitational charge (sensitivity to the field) must be the same or else Newton's third law of motion would be violated. Thus the gravitational charge in the force equation must be relativistic mass rather than rest mass.
 * Second, if this force law is to be an approximation to the correct general relativistic force law then it must contain a factor of dt/d&tau; = &gamma; on the right hand side because such a factor is in the general relativistic force equation which is equivalent to the geodesic equation (notice that t is used in that article for what I call &tau; here). If we are going to use the ordinary velocity vector in the force law, then the &gamma; factor can only be placed in the mass making it relativistic mass. JRSpriggs (talk) 10:05, 25 May 2012 (UTC)


 * The derivation of the third term on the right hand side of the formula for acceleration is explained at special relativity and just above here. It is unavoidable, if you do not want to accelerate past the speed of light.
 * I suppose that Clark & Mashoon left it out because they assumed slow movements of masses including the test particles. However, that seems like an unreasonable approximation to me given that the whole point of this exercise is to determine the effect of motion on the gravitational force and that effect would only show up near the speed of light. Also the bending of light itself is one of the methods we use to detect gravity. JRSpriggs (talk) 10:19, 25 May 2012 (UTC)


 * Specifically I am dubious of the change from


 * $$ \mathbf{a} = \mathbf{E}_\text{g} + \mathbf{v} \times 2 \mathbf{B}_\text{g} \,.$$


 * to


 * $$ \mathbf{a} = \mathbf{E}_\text{g} + \mathbf{v} \times 2 \mathbf{B}_\text{g} - \frac{ ( \mathbf{E}_\text{g} \cdot \mathbf{v} ) \mathbf{v} }{c^2} \,.$$


 * The former fits with the references and the latter does not. It might be cited as WP:OR if it cannot be supported by a reference.  I don't think that these GEM approximations are supposed to word with anything approaching c to begin with.  This is about the analogy of gravitation to EM and that extra term really breaks the analogy.  JR, can you find a good citation for this addition? If so, then it belongs.  If not, I don't think it does belong.   And the factor of 4 (and where it's going to end up) needs to be discussed. 70.109.182.12 (talk) 17:06, 25 May 2012 (UTC)
 * Do your sources mention acceleration (rather than force) at all? If not, then your argument would be to take out the acceleration equation entirely.
 * Anyone who understands special relativity (let alone general relativity) should be able to tell you that the formula as I modified it is a better approximation to the truth than the previous formula. If you want to insist on ignoring what we can figure out for ourselves and only copying sources slavishly (a copyright violation), then there would be no article left.
 * By the way, just to be really clear, I am talking about acceleration as defined in classical non-relativistic physics, that is
 * $$ \mathbf{a} = \frac{ d }{ d t } \left( \frac{ d \mathbf{x} }{ d t } \right) \,.$$
 * If you mean something else (such as proper acceleration or 4-acceleration), then please say so. JRSpriggs (talk) 09:20, 26 May 2012 (UTC)

I did not fully consider the effects of the variation in the metric on the left hand side of the acceleration equation. But now I have reconsidered this. Please see User:JRSpriggs/Force in general relativity.
 * $$ a^{\beta} = - \Gamma^{\beta}_{\alpha\mu} v^{\alpha} v^{\mu} + \Gamma^{0}_{\alpha\mu} v^{\alpha} v^{\mu} v^{\beta} + \frac{1}{m \gamma} g^{\beta\sigma} f_{\sigma} - \frac{1}{m \gamma} g^{0\sigma} f_{\sigma} v^{\beta} \,$$

JRSpriggs (talk) 07:06, 2 June 2012 (UTC) Since this appears superficially to be different from the formula from special relativity, I will show that this reduces to that when gravity is eliminated. First, without gravity the Christoffel symbols are zero, so one has
 * $$ a^{\beta} = + \frac{1}{m \gamma} g^{\beta\sigma} f_{\sigma} - \frac{1}{m \gamma} g^{0\sigma} f_{\sigma} v^{\beta} \,$$

next $$ \gamma = \frac{d t}{d \tau} = \frac{1}{\sqrt{1 - {v^2 \over c^2}}} \,,$$ so
 * $$ a^{\beta} = \frac{1}{m} \left( g^{\beta\sigma} f_{\sigma} - g^{0\sigma} f_{\sigma} v^{\beta} \right) \sqrt{1 - {v^2 \over c^2}} \,.$$

If &beta;=0, this reduces to 0=0, so we need only consider
 * $$ a^{i} = \frac{1}{m} \left( g^{i\sigma} f_{\sigma} - g^{0\sigma} f_{\sigma} v^{i} \right) \sqrt{1 - {v^2 \over c^2}} = \frac{1}{m} \left( f_{i} - {-1 \over c^2} f_{0} v^{i} \right) \sqrt{1 - {v^2 \over c^2}} \,.$$

Now notice that $$ f_0 = - \mathbf{v} \cdot \mathbf{f} \,,$$ that is, power is the product of velocity and force (or in 4-D terms, the 4-velocity is orthogonal to the 4-force), so
 * $$ \mathbf{a} = \frac{1}{m} \left( \mathbf{f} - \frac{ ( \mathbf{v} \cdot \mathbf{f} ) \mathbf{v} }{c^2} \right) \sqrt{1 - {v^2 \over c^2}} \,$$

which is just what one expects in special relativity. JRSpriggs (talk) 11:14, 2 June 2012 (UTC)

That extra c factor in the Poynting vector
The reason why there is some confusion with the c vs. c2 in the Poynting vector is because some sources might use Gaussian units.


 * $$\nabla \cdot \mathbf{E} = 4\pi\rho $$
 * $$\nabla \cdot \mathbf{B} = 0$$
 * $$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}$$
 * $$\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} + \frac{4\pi}{c}\mathbf{J} $$

Here the Lorentz force equation is


 * $$\mathbf{F} = q(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B})$$

I sorta like Guussian units because B is the same dimension as E. I think, in that case, it should be c not c2 in the Poynting vector.

So we need to make sure we are consistent in usage. Even though I sorta like B and E being the same dimension, I don't really care which convention the article uses. 71.169.188.236 (talk) 16:30, 13 May 2012 (UTC)


 * I think we are following SI units because that is the accepted modern standard. At least, that is my intention. JRSpriggs (talk) 16:51, 13 May 2012 (UTC)


 * Yes, the section explicitly starts in SI units, and should remain consistent in this respect. However, the observation that the source may have been using Gaussian units is a good call: the source's (i.e. Mashhoon's) equations do fit with the interpretation that they are using Gaussian units, with no normalization of G.  And it does explain the c in the equation for the Poynting vector.  An aside: Gaussian units are the last thing we should be using here: they make dimensional analysis and determination of multiplying constants more difficult.  — Quondum☏ 17:55, 13 May 2012 (UTC)

New, reliable source
Carroll's book on general relativity discussed this at some length, see eq. 7.25 on p. 281. That's a much higher quality source than what is in the article now, so I will add it when I have time. Among other things it confirms my comments in the discussion above. These GEM equations as written in this article are explicitly incomplete - there are additional terms that appear at the same order (linear) in the metric perturbation. In fact, in vacuum E and B are pure gauge, and can be set to zero by a coordinate transformation. So as I thought they cannot describe gravity waves, and the wave solutions to those equations are more or less meaningless.

These are crucial facts about this formalism, and the article needs to discuss them. Carroll also gives explicit formulas for E and B in terms of the metric perturbation.  Waleswatcher  ( talk ) 21:12, 15 May 2012 (UTC)


 * I assume that you are talking about Sean M. Carroll's book, "Spacetime and Geometry: An Introduction to General Relativity". Right?
 * Unfortunately, I do not have it at this time. JRSpriggs (talk) 02:49, 16 May 2012 (UTC)


 * Yes, that one.  Waleswatcher  ( talk ) 02:57, 16 May 2012 (UTC)


 * If you recommend this as one of the best comprehensive and authoritative references on this subject, I'd like to purchase it (some difficulty here though: I have no local book source that stocks it; importing it also presents problems). What you are saying (in effect that some linear terms have been selected and others ignored) in effect comes down to saying that the GEM equations are, in any event, not what they claim to be: a valid linearization of the GR equations.  If they have any applicability (and I understand that in earth-like gravity they match GR pretty well), then as you say, the range of applicability will have to be sharply defined.  If GEM is significantly inaccurate for waves, though, it would seem that the Maxwell's equation equivalents must also be significantly in error; rather a conundrum.  — Quondum☏ 12:33, 16 May 2012 (UTC)
 * It's certainly a reliable source - but whether it's the best or most comprehensive, I can't say. I think the GEM equations are probably a valid linearization when one: (1) ignores or sets to zero gravity waves, and (2) considers only slowly moving sources and test particles.   Waleswatcher  ( talk ) 13:06, 16 May 2012 (UTC)
 * One cannot simply do your point 1: that invalidates the equations. A more appropriate way forward would perhaps be to say that the time-derivative of the fields must be very small compared to the space derivatives (I think I've seen something along the lines of static distribution of the stress-energy tensor, i.e. the time-derivative of this tensor must be (very nearly) zero, but this does not imply low speeds of either the source or the test particle). This would achieve much the same thing, but in a more formal fashion.  Whatever the case, the actual constraints should be made clear by the sources because they know exactly what approximating assumptions are made, and they should not be surmized by us.  — Quondum☏ 13:45, 16 May 2012 (UTC)
 * If time derivatives of fields are small compared to space derivatives (in units of c), that immediately and trivially excludes all wave solutions. If we want to go by Carroll, the articles needs some re-writing.  Carroll writes the full linearized equation, and then defines E and B and points out how their part of the full equation looks like Maxwell.     Waleswatcher  ( talk ) 14:42, 16 May 2012 (UTC)
 * That certainly sounds promising. But it will be some time before I can lay my hands on a copy.  — Quondum☏ 14:54, 16 May 2012 (UTC)

Exchanging charge and distance (mass as an inductance)
What is it called if you exchange charge and distance in Maxwell's equations? Instead of gravitoEM would it be "inertialEM"? This comes from the parallels between -F=d(mv)/dt and -V=d(Li)/dt. Inertia (-F) would result from the relativistic interaction of "accelerating meters" into a rest mass in the same way voltage results from the relativistic interaction of accelerating charges in a given inductance. (Magnetism is the relativistic result of moving charges as other wiki articles discuss, and acceleration causes the voltage inertia.) The equation for inductance would be an equation for mass simply by replacing meters with charge. The inductor stays still in space while the charges accelerate in time and thereby increase the mass of the inductor by stored energy in the "magnetic" field (relativistic effects of charges at higher velocity*number). Conversely then, this use of Maxwell's equations for meters would insist mass stays still relative to charges while meters going into it increase as velocity*number. In trying to figure out what that means, I realized it's just the inverse of the spring constant where velocity spring end = 1/k*dF/dt in perfect parallel with I=C*dV/dt (capacitance in Maxwell's EM). So the increase in meters going into the mass is compressing it like a spring, storing energy. Compressing it in the direction of travel, just like relativity, increasing it's energy but not its rest mass. Since I just followed the math and it makes things simpler without doing anything new, it seems like it should be a known topic with a name. This article is the closest I could find. Ywaz (talk) 12:21, 30 June 2012 (UTC)

Getting rid of negative sign difference
The negative sign difference in the Plank units section goes away if you use Einstein's meters=i*c*seconds (which gives E= -mc^2 as c then carries a sqrt(-1)). G has units of 1/s^2 inside of it which is 1/i^2 = -1. To use it, you replace all instances of seconds in equations and constants with i*c*seconds and call it meters, letting the meters cancel wherever possible and not losing the i, -i, -1, or 1 instead of letting it's hidden nature force more complicated equations (as 4D space time?). Getting rid of the negative sign will make the parallel exact instead of approximate. The speed of light is a conversion factor that does not have real dimensions since space and time are the same thing (excepting a sqrt(-1)), so Plank or Natural units are messed up the way people on Wikipedia are using them, because "c" still shows up in these equations. It's a human-made conversion factor, not a natural number. But the sqrt(-1) it should be carrying is a dimension-less universal thing that should *not* get thrown away. So people are keeping the human part, and throwing way the natural part (this article was lucky in that the c^2 cancelled anyway). See Einstein's "Relativity" Appendix 2.Ywaz (talk) 18:47, 29 June 2012 (UTC)
 * Physicists stopped using ict as a space-like dimension more than half a century ago, for good reason. JRSpriggs (talk) 19:44, 30 June 2012 (UTC)
 * Do you have a general short explanation as to why? Ywaz (talk) 00:08, 1 July 2012 (UTC)
 * There are far better approaches today that are also more general. The imaginary number i happened to have convenient properties in this extremely limited context but is not as good as many of the approaches used today in which an inner product is defined on the four-dimensional spacetime, such as with Minkowski space and geometric algebra.  — Quondum☏ 08:25, 1 July 2012 (UTC)

The Minkowski inner product not rigorously following the definition of inner products seems to me to be the result of avoiding the use of a split-complex vector space. Einstein explicitly said meters=i*c*t defines the Minkowski space-time. "We can regard Minkowski's 'world' in a formal manner as a four-dimensional Euclidean space (with an imaginary time coordinate)..." Can you give me a real world example calculation (for example, use the Earth and Sun masses for the numbers) where you can do a calculation more efficiently than using meters=i*c*t? All I know is that it reduces the complexity of the equations in this article, and many others. The ability to compress algorithms to a higher degree is usually a sign of greater understanding and following Occam's razor. I think Wikipedia articles such as this one which reference Plank units as if they were unitless should take the mathematical equivalence between meters and seconds seriously, which means getting rid of either seconds or meters in all constants and equations, recognizing c as a unitless dimensional ratio rather than a constant. Even the S.I. definition recognizes the redundancy. This reminds me that this equivalence places E and B on the same playing field by equating eo and uo (I think insertion of a negative sign is needed since c^2=(1/eo)/uo, but it might be -i), so that's another reduction in the complexity of these equations. In other words, the fact that the B is the relativistic equivalence of E might be "predictable" from the reduced "maxwell's" equation(s) if the "i" were kept where Einstein's relativity insists. That gravity is a negative (attractive) force compared to charge repulsion is also predicted by the use of "i" without prior knowledge of reality, which means better mathematical reduction to fewer terms for higher Occam compression of reality. Ywaz (talk) 10:29, 1 July 2012 (UTC)
 * The modern approach actually generalizes and combines complex numbers and split-complex numbers into the spacetime algebra and thus has all of their advantages. When expressed properly, compressibility is pretty good.  Take Maxwell's equations in four dimensions (as in geometric algebra): ∇F=μ0cJ.  The Wikipedia policy is also to express the mainstream approach on any topic.  — Quondum☏ 12:26, 1 July 2012 (UTC)
 * That compressibility is not as high as, nor as enlightening as ∇F=J/i. (edit: I'm wrong because you could have said ∇F=J by using plank constants like me.  So I say they're the same complexity since "i" like a negative sign.   Also, this covers only two of the 4 maxwell equations. ∇G=0 cover the other two.  )   meters=i*c*seconds is used in some current quantum gravity theories so it is not far afield from this article.  My main complaint could be addressed in the Plank article, or I think there is another one dedicated to unitless quantities that doesn't have such a heavy weight of tradition.

( edit: I see F would need to be redefined too. It's F=E+icB. I mentioned equating E and B but that sort of makes Maxwell's equations and this article moot.  But the insertion of natural meter "i" units in place of time should still be beneficial.  The F=E+icB complicates things since that "i" is of a different nature, so I'll use "j" which gives me F=E-jB/i and  ∇F=J/i where J is in C/m^2 instead of C/(s*m^2).  )  Ywaz (talk) 13:09, 1 July 2012 (UTC)

Gravity is to electricity what inertia is to magnetism?
When a mass passes another mass the force between them is greater (than if they were still) from a relativistic increase in the "moving" mass. Likewise when a charge passes another, the force between them is greater than if they were sitting still. Unlike the first case, we gave it a name (magnetism) because we observed it early on because charges are 10^42 times stronger than gravity. We still do not usually point out, like we do in the first case, that it is just a relativistic effect (see Feynman Vol2, Ch13 for example, or Schwartz's EM Dover text). Furthermore, we view kinetic energy of a mass to be the result of an increase in its relativistic mass, but we still think there is "magnetic field energy" surrounding a moving charge. So to insert mass into Maxwell's equations seems like it should be insisting there is no increase in the relativistic mass, nor kinetic energy, but to insist there is an "inertial field" perpendicular to and surrounding the moving mass, storing energy, and increasingly resisting an increase in the mass velocity. So a moving mass's gravity and inertial fields would feed back on each other, just as with with the electric and magnetic fields. This is back-tracking from relativity, but it seems to give a perfectly valid way to calculate the connection between gravity and inertia without having to use a morphing space-time. Maxwell's equations accurately specify the relationship of magnetism to electricity and I would think inertia to gravity, but relativity more compactly describes magnetism and inertia as the result of the scalar equivalence between space and time. I would think inertia is what gravitomagnetism is supposed to be about, but I can't determine if this perspective is part of the article. Ywaz (talk) 21:29, 2 July 2012 (UTC)


 * Indeed, if one charge is at rest and one is in motion, then the charge at rest has no magnetic field and so affects the moving charge only through the electric field. And the magnetic field of the moving charge has no effect on the charge at rest because the charge at rest has no velocity.
 * Magnetism only changes the resulting force when both charges are moving. When they are moving in the same direction, the force is reduced. When they are moving in opposite directions, the force is increased. Perpendicular motion is irrelevant.
 * However, gravitational effects are not simply similar to the electromagnetic effects. There is a difference which goes beyond the general difference in strength and the sign of the coupling constant. JRSpriggs (talk) 09:28, 3 July 2012 (UTC)


 * In the example I gave of a charge passing another, the magnetic field is changing as the point charges approach. But now I realize the analogy must work only for "explaining" inertia and does not address a relativistic mass increase. The reason it doesn't work as well is that mass increases in addition to length contraction and time dilation, whereas relativistic charge experiences only the latter two.
 * As my Feynman reference shows, both charges do not have to be moving to create a magnetic force between them. Otherwise, you could make any moving charge your frame of reference, and thereby F=q*v x B forces would disappear, when you know they are still there.  But this does not mean Maxwell's equations are wrong under different frames of reference.  The Feynman example is complex, so I'll do a simpler one: take static charge distributed on a long rod, with the rod moving in the direction of its length past a stationary charge (a current "in a wire" without the complexity of positive charges moving at a different speed). From Maxwell we have only an E force that measures charge per length that takes relativity into account because of our q/l measurement is "at velocity".  So both EMAG and relativity say it's an electrostatic charge force and agree.  Now as you change your frame of reference to make the stationary charge move and the moving rod slow down, EMAG will see there is a decreasing charge per length (as measured) which will decrease the electrostatic contribution, but there is also a decrease in the time relative to the point charge, which increases ANY forces on it because momentum has to be conserved in relativity and F=dp/dt (as the Feynman example informed me).  So the the time change in the point charge, offsetting the q/l change in the rod, is viewed as a magnetic force in EMAG, but is relativity affecting the electrostatics. Since the length change is in the rod and the opposing time change is in the point, they both work to decrease E and increase B.  So sqrt(1-(v/c)^2) is multiplied and the v^2 is the change in v of the rod current increasing B and an increase in v of the point, so this works out to why there is only one v in F=q*v x B.  Ywaz (talk) 13:50, 3 July 2012 (UTC)


 * Your writing is not clear. For example, you said "but there is also a decrease in the time relative to the point charge" but do not make clear to which time you are referring &mdash; a time period beginning with which event? and ending with which event? as measured by whom? There are many other instances of your lack of clarity and specificity. Thus I do not understand what you are saying and I doubt that you yourself understand what you are saying.
 * Here are a few of the many ways in which gravity differs from electromagnetism:
 * The source and the target of electromagnetism are (q, qv) a 4-vector; but the source and target of gravity are (m&gamma;, m&gamma;v, m&gamma;vv) a tensor.
 * Electromagnetism is linear; but gravity is non-linear.
 * In electromagnetism, issues of gauge can be avoided by working with E and B rather than &phi; and A; in gravity, gauge cannot be ignored as the force-field itself (the Christoffel symbol) is modified by a gauge transformation.
 * The equations of gravitation are horrendously more complicated than the equations of electromagnetism.
 * JRSpriggs (talk) 16:14, 3 July 2012 (UTC)
 * You can refer to the Feynman text if you do not trust that my text is at least as clear. Since it's relativity "event" beginning and end do not need not be specified, just as Feynman did not in the example. As I said, it does not help in terms of general relativity and all that that implies.  Ywaz (talk) 20:09, 3 July 2012 (UTC)

Derivation from linearized general relativity
From linearized gravity, we get
 * $$ 16 \pi G c^{-4} (T_{\beta \delta} - T_{\alpha \gamma} \eta^{\alpha \gamma} \eta_{\beta \delta}/2) \, = h_{\rho \delta, \beta \sigma} \eta^{\rho \sigma} + h_{\rho \beta , \delta \sigma} \eta^{\rho \sigma} - h_{\rho \sigma , \beta \delta} \eta^{\rho \sigma} - h_{\beta \delta , \rho \sigma} \eta^{\rho \sigma} \,.$$

Let us adopt a coordinate condition
 * $$ h_{\alpha i ,i} = \frac12 h_{i i ,\alpha} \,,$$

(implied summation over i=1,2,3) which is not Lorentz invariant (since it lacks the time derivatives) but is otherwise similar to the harmonic coordinate condition. Because stress is ignored in GEM, it has no hope of being Lorentz invariant anyway, so we might as well use a coordinate condition which also lacks that property.

If we take &beta;=0 and &delta;=0 and assume that the trace of stress is zero (Ti i=0), then
 * $$ 16 \pi G c^{-4} (T_{0 0} - T_{0 0}/2) \, = h_{i 0, 0 i} - c^{-2} h_{0 0 , 0 0} + h_{i 0 , 0 i} - c^{-2} h_{0 0 , 0 0} - h_{i i , 0 0} + c^{-2} h_{0 0 , 0 0} - h_{0 0 , i i} + c^{-2} h_{0 0 , 0 0} \,$$

which simplifies to
 * $$ 8 \pi G \rho = 8 \pi G c^{-4} T_{0 0} = h_{i 0, 0 i} + h_{i 0 , 0 i} - h_{i i , 0 0} - h_{0 0 , i i} = - h_{0 0 , i i} \,$$

where the first three terms on the right hand side cancelled due to the coordinate condition (with &alpha;=0).

If we take &beta;=j and &delta;=0, then
 * $$ 16 \pi G c^{-4} T_{j 0} = h_{i 0, j i} - c^{-2} h_{0 0 , j 0} + h_{i j , 0 i} - c^{-2} h_{0 j , 0 0} - h_{i i , j 0} + c^{-2} h_{0 0 , j 0} - h_{j 0 , i i} + c^{-2} h_{j 0 , 0 0} \,$$

which simplifies to
 * $$ - 16 \pi G c^{-2} \mathbf{J}_{j} = h_{i 0, j i} + h_{i j , 0 i} - h_{i i , j 0} - h_{j 0 , i i} = - h_{j 0 , i i} \,$$

where we have again used the coordinate condition.

If we use the identifications of GEM potentials
 * $$ V_g = {-1 \over 2} h_{0 0} $$
 * $$ {\mathbf{A}_g}^{i} = h_{i 0} $$

made in an earlier section of talk above, then the equations obtained here can be rewritten in vector notation as follows
 * $$ 4 \pi G \rho = \nabla^2 V_g \,$$
 * $$ 16 \pi G c^{-2} \mathbf{J} = \nabla^2 \mathbf{A}_g \,.$$

If we take &beta;=j and &delta;=k and assume that stress is zero, then
 * $$ 8 \pi G c^{-2} \rho \, \delta_{j k} = h_{i k, j i} - c^{-2} h_{0 k , j 0} + h_{i j , k i} - c^{-2} h_{0 j , k 0} - h_{i i , j k} + c^{-2} h_{0 0 , j k} - h_{j k , i i} + c^{-2} h_{j k , 0 0} \,$$

which (using the coordinate condition) becomes
 * $$ 8 \pi G c^{-2} \rho \, \delta_{j k} = c^{-2} ( - h_{0 k, j 0} - h_{0 j , k 0} + h_{0 0 , j k} + h_{j k , 0 0} ) - h_{j k , i i} \,.$$

< >

JRSpriggs (talk) 08:24, 12 July 2012 (UTC)

A NEW FORCE
I´ve got this link about something related with Gravitoelectromagnetism:

http://gsjournal.net/Science-Journals/Essays/View/1110

here appears a new force. — Preceding unsigned comment added by 201.157.31.151 (talk) 15:17, 27 August 2012 (UTC)

Useful Paper...
Message received at OTRS 2012082910009196 - "I wrote an article in 2008 about this subject and I´d like they read this information. This is the link: http://gsjournal.net/Science-Journals/Essays/View/1110"  Ron h jones (Talk) 21:37, 29 August 2012 (UTC)


 * Late to the party, but the General Science Journal is not a reliable sources. They're a bunch of relativity deniers that can't get published anywhere else. The same person likely posted in the above section too. Mostly replying for people that see this years later and think it needs to be addressed. Headbomb {t · c · p · b} 19:40, 25 March 2019 (UTC)

Dubious
In the section Gravitoelectromagnetism, added the sentence "In 2013 there appeared an article that shows, there exists Lorentz invariant version of GEM equations that reproduce General Relativity formula, what opens new areas for research." referencing "Maxwell-like picture of General Relativity and its Planck limit" by Piotr Ogonowski and Piotr Skindzier. This appears to contradict the fact, explained in the rest of the section, that no GEM theory can be Lorentz invariant because it uses only a fragment of a tensor as the source of the field. So I added a "dubious" template to start a discussion here about that situation. JRSpriggs (talk) 10:32, 25 August 2013 (UTC)


 * I feel that the added sentence might be inappropriate, but primarily because this addition virtually constitutes publication/spamming of a fresh primary source and has not been reviewed. To claim that "no GEM theory can be Lorentz invariant" is however almost certainly false, depending on how one constrains a theory to be a "GEM theory". If restricted to the particular form of the equations given in the article with mass density and current density interpreted as a 4-vector, it inherently cannot be Lorentz invariant because this vector is not, and if you have verified this applies with the paper, I'd support you (a glance suggests that this might be the case).  On the other hand, it is not difficult to start with Newton's law of gravitation for a static point source (relative to some inertial frame) and relativistic mass from special relativity, to derive a fully Lorentz-invariant theory of "GEM" in a flat spacetime background and that uses the full stress–energy tensor. I think that such a theory, though its equations would differ a little from those presented in this article, would nevertheless qualify as a GEM theory: a classical formalism similar to Maxwell's equations. — Quondum 13:05, 25 August 2013 (UTC)


 * Hi, I have added the sentence and the reference. As you may read in the refereed article, in section "3.1 Gravity as wave equation in Minkowski spacetime" authors have introduced their own Lorentz invariant version of GEM. I agree with Quondum, that it depends what we define as GEM: e.g. authors do not even call their equations "GEM". However it is plain GEM idea behind it, if we understand GEM the way mentioned at the beginning of this Wiki article: "Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity". — Preceding unsigned comment added by 83.6.56.248 (talk) 15:30, 25 August 2013 (UTC)


 * If one is going to make a claim about another version of GEM, substantially different from the one described in this article, then that version should also be described in this article. Otherwise, it will appear that the claim is being made about the version described in the article. In particular, how can one relate a rank-two tensor source (the stress-energy tensor) with a field based on a vector (rank-one tensor) potential???? JRSpriggs (talk) 16:55, 25 August 2013 (UTC)


 * I agree that if distinct versions are to be addressed, the article should describe each – quite a challenge, working from limited primary sources. What falls inside the scope of this article would of course be up for debate.
 * In answer to the question, I'm not too sure why one should assume a vector potential in GEM; my guess is that a rank-2 tensor would take its place. The brief exploration that I did a while back produced a rank-3 tensor for the "GEM" field, which could not be fully described by only two vectors as in the EM case. WP:OR of this nature is of course not relevant to the article, but might act as an aid to what might lie within the scope of the article.  — Quondum 03:01, 27 August 2013 (UTC)


 * Of course, one can relate the stress-energy tensor to a linear field, if one follows the path of linearized gravity. But that is not "Maxwell-like". The source for Maxwell's equations is a 4-vector tensor (density).
 * In the absence of a reliable source which we can use to describe an Lorentz invariant version of GEM, I think we should comment out the sentence which claims that there is one. OK? JRSpriggs (talk) 18:21, 27 August 2013 (UTC)


 * Agreed - the claim should go. The paper does seem to have some Maxwell-like equations (though in places noticeably elaborated), but does not seem to claim Lorentz invariance (contrary to 83.6.56.248's comment above, using a Minkowski spacetime as a background does not imply Lorentz invariance). It might be an interesting GEM variation (though I'm not qualified to comment), so keeping the reference to it may make sense.  Perhaps under "Further reading" until someone can make an informed characterization of what it says?  — Quondum 23:37, 27 August 2013 (UTC)

It seems to be using the Schwarzschild metric as a background. In any case, a link to it is available here in my first post in this section of talk so it should not be necessary to put the link in the article before we can verify that it is a reliable source. JRSpriggs (talk) 13:31, 28 August 2013 (UTC)


 * The paper looks like a preprint from January this year, reviewed recently in July, and still not yet submitted to a journal. As it is not published that would mean it probably isn't reliable by WP standards, when it is then it could be used. M&and;Ŝc2ħεИτlk 06:49, 29 August 2013 (UTC)
 * Then I'd suggest that the sentence, including reference, may be removed as per JRSpriggs's suggestion, leaving this thread as the link to the paper, until such time as its reliability is better established somehow. — Quondum 06:59, 29 August 2013 (UTC)


 * Done. JRSpriggs (talk) 15:00, 29 August 2013 (UTC)