Unification of gravitation and electromagnetism

Take a look at equation 7.1.17 at the end of section 1 of chapter 7 at Modern Relativity and the surrounding discussion. This is a classical theory unification of the electric and gravitational fields for arbitrary static matter distributions of extremal charge. I do think unification is going to turn out to be something a lot simpler than strings.

 

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Since most people don't have that book it's unlikely you're going to get much discussion. Can you provide an overview of what precisely you're referring to. I think I know what you're referring to, because the unification of classical gravity and classical electromagnetism isn't terribly difficult, it was developed in the 1910s by Kaluza and Klein. But I might be thinking of something different from you.

 

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Intensity similarities between Gravity and Electromagnetic force

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courtesy: "Inverse Square Law" (hyperphysics.phy-astr.gsu.edu)

 

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Inverse square law for gravity and e-m. What is the point of the comment?

 

Modern Relativity is my online textbook on relativity. I just don't have the option to post links. I'm not referring to Kaluza-Klein theory. I am talking about treating general relativity as an already unified classical theory of gravity and electromagnetism. Lets say hypothetically you were to know the exact time dependent solution to Einstein's field equations for all the electric and magnetic field sources it contained. Since it is the solution to his field equations, it is a completely geometric description of the time developement of all matter contained in relation to the stress-energy tensor, completely a general relativistic solution for gravitation. However you could in principle figure out how the electromagnetic field tensor looks consistent with that metric solution, in which case the general relativistic version of maxwell's equations turns out to be just an equivalent description of the behavior of the solution to the metric.
Some time ago I found the exact solution to Einstein's field equations for arbitrary static matter distributions of extremal charge. It wasn't to hard to find the rank two electric field tensor corresponding to the solution. What I recently realised, which equation I was referring to in the post, is that for this solution I could equate the rank two electric field tensor to a product of the metric with components of the Christoffel symbols. That equation is a classical unification of the electric field and gravitational field for my solution to Einstein's field equations. The equation will be more complicated for time dependent solutions and solutions containing magnetic fields, but it was a simple relation for this case and shows that the fields are indeed already unified in general relativity. We just need to find the more general solutions to Einstein's field equations to show how the electromagnetic field tensor relates to the metric and Christoffel symbols in general.
I didn't discuss this next step at the site, but I'll mention it here. Once the complete time dependent solution to Einstein's field equations is found for arbitrary electromagnetic fields one then also knows the Lagrangian of spacetime L^2 = gUU and one could then treat the Lagrangian mechanics opperations that yield conserved parameters of motion as quantum opperators, opperate them on the wave function of the universe to yield the conserved parameters as Eingenvalues, at which point one has unification theory of electromagnetism with a means of applying quantum mechanics.

 

http://www.modernrelativitysite.com/chap7.htm

Equation 7.1.17, right?

I'm so nice.

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Right, thanks. Seeing if I'm allowed tex here:
\(ds^{2}=\frac{dct^{2}}{(1-\frac{\Phi }{c^{2}})^{2}} - (1-\frac{\Phi }{c^{2}})^{2}(dx^{2}+dy^{2}+dz^{2})\)
\(F^{\mu \nu }=\pm \frac{c^{2}}{\sqrt{4\pi G\varepsilon _{0}}}g_{\mu \kappa }\Gamma ^{\kappa } _{0\nu }\)

 

Okay, so you're referring to the Reissner-Nordstrom black hole solution, ie the space-time metric and EM field for a non-rotating charged black hole.

This is a very interesting metric, as it's pretty much the simplest non-trivial metric which includes the EM contribution to the field equations. However, I don't quite see how this represents a unification, it's just an application of electromagnetism to general relativity. It's clear from the field equations that if you put charged objects into the space-time their mass and EM energies will warp it and you have to account for that. The R-N is an example where you can get the metric analytically.

It isn't a unification in the sense of having a single description which spits out both gravity and electromagnetism. Kaluza-Klein models are because you start with just gravity in 5 dimensions, compactify the 5th direction onto a circle and you end up with a 4d effective theory containing gravity and electromagnetism (and a scalar field too). In that case you don't put in the electromagnetic field by hand and stipulate it obeys Maxwell's equations, you actually get Maxwell's equations from the field equations via the compactification! That is an example of unifying them.

Doing it classically isn't terribly difficult, the problem is getting a single quantum description.

That's somewhat circular. How would you get the metric's behaviour? If by observation then you do it by observing the material in the space-time, which would give you the EM field's behaviour, from which you can derive Maxwell's equations. If you do it by calculation then you need to include the EM field's contributions, which needs Maxwell's equations.

The starting place to any problem would be to have the initial configuration of all the relevant matter, charged or not, including initial velocities. You then want to model it. A unified model wouldn't require you to put in seperately Maxwell's equations and the Einstein field equations but just a single equation. KK methods do just that, using just the EFEs. Your method involves somehow being given the answer to how all the matter etc behaves and knowing how part of the system (ie gravity) behaves and then deducing the dynamics of the remaining part (electromagnetism). That isn't a unified model, it's how any bit of science works, ie you strip away the known parts until you get to the unknown then you try to develop a model for it. Where is a unified description then? At the end of it you'd have the EFEs and Maxwell's equations separately.

By that do you mean a set of charged point masses (ie R-N black holes) positioned in such a way that their EM repulsion and gravitational attraction balance and they are stationary? I would be interested to see your derivation, though if you might not be aware but that is a known result in the literature. Or did you mean you found the result in a book. It's a fairly straight forward method to be walked through but it's not something I'd call easy to derive off the bat.

Not really, it's a statement that in the very simple, very symmetry case of the R-N black hole the very simple EM field (which has the same spherical symmetry) can be easily substituted into the metric expression.

I happen to have done a black holes course which covered exactly this topic and I was nerdy enough to LaTeX the notes with pictures so I've just read back through a few pages so I can be a bit more specific.

By spherical symmetry and the fact nothing is moving for the R-N black hole we get that \(A_{0} = \frac{Q}{r}\) where Q is the charge. This gives \(F_{r0} = E_{r} = -\frac{Q}{r^{2}}\) as expected. All other unrelated components are zero. Note that we've really implicitly used Maxwell's equations already. We know the charge distribution, \(\rho(\mathbf{x}) = Q\delta(\mathbf{x})\). If you solve Maxwell's equations for that you get the stated EM field, which has a possible gauge potential of the one given (gauge potentials are not unique). We obviously don't need to go through all of that as everyone knows the electric field for a point charge from high school via Gauss's law but that really is a consequence of \(\nabla \cdot E \propto \rho\) so if you were to claim deriving the R-N solution doesn't need Maxwell's equations you're really palming a card. Anyway, when you crunch through the metric derivation you get the following :

\(ds^{2} = g_{ab}dx^{a}dx^{b} -\Lambda \, dt^{2} + \frac{1}{\Lambda}dr^{2} + r^{2}d\Omega_{2}^{2}\) where \(\Lambda = 1-\frac{2M}{r} + \frac{Q^{2}}{r^{2}}\)

Given the very simple form of the vector potential we can now say that \(\Lambda = 1-\frac{2M}{r} - E_{r} = 1-\frac{2M}{r} + \frac{A_{0}}{r} = g_{tt} = \frac{1}{g_{rr}}\).

In this case it's literally possible to guess the form of the EM field and thus its vector potential, which gives its contribution to \(T_{ab}\), which gives the field equations, which gives the metric. But the only reason you can actually guess the EM field directly from the charge is you already know how electromagnetism behaves and since the space-time you're looking into is, by definition, spherically symmetric the metric doesn't alter the EM field in this case. The reason it generalises to multiple extremal points is related to them forming a supersymmetric bound state.

So I don't see how any of this is a 'unification' of gravity and electromagnetism. It's an analytic solution for multiple charged black holes consistent with both but that isn't the same as a unified model. It's obtained by solving two different but related sets of equations simultaneously, rather than a single overarching lot of equations. As the KK example shows, it's possible to have both 4d gravity and 4d electromagnetism drop out from the same set of equations (5d gravity). That's a unification method.

 

Actually, no. The extremely charged Reissner-Nordstrom solution is one special case of the solution I found. It is the case for a 1/r potential. My solution actually yeilds exact electric field solutions for any laplacian potential. For any potential that obeys the poisson equation it yeilds exact solutions for extremely charged matter.

No, it allows the matter distribution to have an arbitrary static distribution.

But surprise, this potential IS. Yeah it surprised me too.

It allows one to write the equation of motion of a charge in terms of spacetime geometry alone because you can trade out the rank 2 electric field tensor in the four-vector force equation of motion for the Christoffel symbols expression.

But yes it does. Both gravity and the electric field tensor are now written as the resulting Christoffel symbols.

What would be easier for to go though to check it and I suggest you do is download grtensorII, put in my metric and calculate the Einstein tensor to verify that it cooresponds to the stress-energy tensor for the electric field that I say it does at the site. (Make sure to calculate G(up,dn), not G(up,up) for the comparison because it was T(up,dn) that I give that you want to compare to.)

 

*edit* You replied why I was typing my second set of thoughts so I haven't addressed them. It's 1.05am here so I'm going to bed. I'll look in the morning *edit*

I was trying to work out whether your website was presenting mainstream work or your own, because it isn't terribly clear, given how much of it seems to be reviewing the mainstream but your Chapter 7 talks about your work.

The derivation of a multi R-N extremal black hole solution is not too difficult if you go through the method of considering one black hole, then adding other isolated black holes. By 'not too difficult' I mean it was lectured to students, it's still something I wouldn't call obvious by any stretch of the imagination.

I think I get what you're trying to say about unification though, on further thought. However I still don't agree. The fact you can express the space-time metric for a collection of extremal black holes (which are point charges so you might be tempted to think of normal matter in terms of them, though that's a mistake as I'll explain shortly) in terms of a field \(\Phi\), which itself is defined purely in terms of the charge distribution does indeed mean that if someone says "This system is composed of extremal non-moving R-N black holes, here's the metric and the Einstein field equations" then you can derive Maxwell's equations but that isn't a unification between EM and gravity.

The main problem is you're considering a very very special restricted case. Normal matter is not composed of extremal non-rotating black holes so even though you could argue a point charge in EM is a R-N black hole it is not necessarily extremal as that would mean there's a specific mass for a given charge (M=|Q| in natural units). Then there's the other problem, you don't get all of Maxwell's equations. In the derivation you consider a spherically symmetric point charge which isn't moving. You only need to solve \(\nabla \cdot E = \rho = Q\delta(\mathbf{x})\) for that. The other three you just ignore because you explicitly set \(\partial_{t}E=0\), regardless of its true equation of motion and without motion (or magnetic monopoles) you don't have any magnetic conditions. Without them you cannot derive their equations of motion from the metric and the field equations. So your requirement of stationarity is a problem there, as well as the fact real systems are generally dynamical and you don't have that.

So yes, you've used Maxwell's equations and the EFEs to construct a particular setup where you can get the EM field from the metric and the one electric field equation which doesn't care about time but that isn't a unifying model. It's a massively restricted intertwining of two different models such that there's a 1-1 relationship between a part of one model and a part of the other.

I still don't know why you say you 'found' this, because you've obviously read plenty of the literature and its in the literature. And if I know about it isn't not buried particularly deep in the literature.

 

You're a little too stuck on black holes to see what I'm saying I found. I have 5 exact solutions in that section. Only 1 of which is the Reissner-Nordstrom solution in isotropic coordinates already known. The other 4 as far as I know don't exist in any other literature as I was the one that found them. The first which is what I've been talking about is the solution for extremely charged matter of arbitrary static configuration. The only things that are restricted about it is that the matter is static and that it is extremely charged. That's the one I wrote the relation for the electromagnetic field tensor to the Christoffel symbols for. Theres no reason to restrict your thinking of it to black hole applications. In fact what I was actually looking for when I found it was the solution to Einstein's field equations for a lifter to see if there is any gravitational effect associated with the electric field. The other three that are mine are a charged matter sheet with extra uncharged matter perterbing the metric like a domain wall but without the unrealistic pressure state, an infinite line charge of arbitrary extra mass which is the correct charged cosmic string, and a solution for a uniform electric field at right angle to a uniform magnetic field which may turn out to be a frame transformation of the case of just one inducing the other field. There is nothing really not mainstream about anything prior to the fringe physics unit. I include exact solutions and results I personally found, but its ordinary general relativity.

 

But this family of solutions (often referred to as multi-black hole solutions) are well known, I was even taught about them in University. For example, see 'Black Hole Uniqueness Theorems' by Heusler, page 162.

 

Not really, because your argument rests specifically on the stationary nature of the system.

For example, suppose you have 2 extremal R-N black holes and you plonk them down such that they aren't moving and won't move. Using the methods discussed you can compute the electric field from the geometry or the geometry from the electric field.

But what if we make the two black holes move relative to one another? They'll now be producing magnetic fields. The magnetic energy and the kinetic energy will alter the metric, which is now also time dependent. This means it cannot be represented in the form we're discussing (ie you just work out \(\Phi\) via Laplace's equation).

A unification is a general result, a result which applies to all situations. What you have is a massively restricted case. You cannot phrase an arbitrary system in terms of extremal charges which are stationary.

For example, could you use this method to give the proper full GR metric for two non extremal Kerr-Newman black holes of, lets say, equal masses, charges and spins, orbiting their common center of mass (ie a binary system)? The black holes produce both electric and magnetic fields on their own, as well as due to their orbital motion.

I don't deny the result, I would like to see your derivation. As I said, it's in the literature already and it's sufficiently 'nice' a result to be taught to pre-PhD students.

I have in my lecture notes from 2006 the general result for finitely many disjoint point particles because the lecture course was called 'Black Holes' and the points are black holes. The generalisation to arbitrary configurations is straight forward as the sum becomes an integral in the usual manner. The form of the metric, the EM potential and Laplace's equations all are in my notes.

So it isn't a general result.

To give an example, quantum field theory unifies special relativity and quantum mechanics. It's a single formulation which reduces to special relativity mechanics if you turn off the quantum properties (\(\hbar \to 0\)) and non-relativistic quantum mechanics if you turn off relativity (\(c \to \infty\)). That is a unification, it applies in all scenarios. Similarly electric and magnetic phenomena were originally thought to be separate and then people like Faraday realised differently and Maxwell developed the single formulation to combine them. Thus we don't have electric equations and magnetic equations, we have a single set of electromagnetic equations. Again, valid in all cases.

What you have is you've removes all additional degrees of freedom (in some sense). By considering a sufficiently restricted case you has a direct conversion between the allowable degrees of freedom in electromagnetism and those in the metric. The extremal condition is an example, M=|Q|. If you know one you know the other, while in a more general setup just because you know the charge of an object doesn't mean you know it's mass. The absolute value of the proton's charge is the same as that of the electron but they have different masses. Heck, the muon and electron has exactly the same charge, sign and all, but very different masses.

We know that in general EM+GR formulations you can't just pick any metric and gauge fields, they have to simultaneously satisfy Maxwell's equations and the EFEs. Hence, in principle, some of the variables can be expressed in terms of the others, that's generally what equations result in (thermodynamics being a primary example). If you shut off or equate enough of the metric and gauge field by saying "Metric is stationary and static and all masses are extremal" then you have removed loads of possible configurations. Actually, come to think of it how could you even describe a single Kerr-Newman black hole? It's metric is explicitly non-stationary because \(g_{ti} \neq 0\) in some cases. The charge held in its ring singularity is rotating, which makes a magnetic field too.

If you could demonstrate how to build such a metric using your method I'd be interested in seeing it. A quick consistency check would be whether or not your expression relating the EM tensor F with the metric and connection terms is true for a K-N black hole. Does it?

My second set of thoughts wasn't, in the sense you can build other objects than just point clouds. However, remember that your own condition is that the point charge is extremal, which is a very special kind of black hole. More elaborate black objects are common in string theory, including extremal ones because they often represent supersymmetric objects because, if you're familiar with any of this, they saturate a BPS bound so are good for examining strongly coupled phenomena. They are all still black 'holes' but they have complicated, non-point, singularities. You're talking about arbitrary configurations of extremal charges. If these configurations are supersymmetric and yet arbitrary it would mean nature is supersymmetric but obviously nature doesn't have intact supersymmetric.

Have you submitted any of this to a journal? If not, why not? If so what did they say?

 

Skipped the wrong stuff already addressed.
The way I did derive it in hindsight wasn't the most straight forward easiest way to do it so I'll tell you how it can be derived in an easier manner. Start with a trial solution the same way you would derive for example Schwarzschild geometry with the line element
\(ds^{2}= (f(x,y,z))^{n}dct^{2} - (g(x,y,z))^{m}(dx^{2}+dy^{2}+dz^{2}) \)
Calculate the Einstein tensor (up,dn) and the Ricci-scalar. You're looking for a solution for an electric field so if the functions are laplacian the Einstein tensor must be proportional to the special relativistic electric field's stress energy tensor at locations where the metric reduces to that of special relativity and if the functions are laplacian the Ricci-scalar must be zero everywhere. You'll see in the space-space cross terms of the Einstein tensor a mixed partial derivative of f and a mixed partial derivative of g with m and n coefficients that does not look like any term in the electric field stress energy tensor for special relativity so they must add to zero. Because of the products of f and g and f^2, g^2 that are on them as well you'll realise that the only way they can add to zero is if g=f and n=-m. So you find what the Einstein tensor and the Ricci-scalar simplify to with those insertions. The Ricci-scalar will now have some terms in the numerator that look like (m-2) times the sqare of first order partial derivatives of f. Since you're looking for an electric field solution the Ricci-scalar must be zero for a laplacian f so these must be zero and the only way they can be is if m=2. So you make that insersion.
What you have left now already is an exact solution for a static electric field of arbitrary laplacian potential, however you probably want it to meet a boundary condition that the potential goes to zero at infinity, the metric reduces to the metric of special relativity at infinity. All of the remaining terms in the numerators are first and second order partial derivatives of f so you can actually add any constant to f so that if the potential goes to zero at infinity, the metric will then also reduce to the metric of special relativity at infinity meeting your boundary condition. So you add 1 to f. Letting f be Phi/c^2 just gives the potential units of gravitational potential.

 

You must have missunderstood what they taught. There is NO exact many black hole solution any more than there is a nonrelativistic exact many body orbital mechanics solution. The reason the my solution describes many bodies is specifically because the extreme charge case allows the matter to remain static in any configuration.

 

Well, you know best.

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The extremal static many black hole solution IS known and is exact. I have it in my lecture notes from when I did the aforementioned course in black holes. The person lecturing it has several theorems/results in GR named after him so I am inclined to think he knows the literature pretty well.

As I said, generalising from one extremal point charge to many to extended objects and general configurations is not difficult and not new. Extremal black branes, be they points, strings, sheets etc, have received large amounts of attention in the literature because of their ability to probe strongly coupled regimes of string theory. In fact, stacking multiple black branes is how you go about deriving S duality in Type II string theory as well as how you construct certain gravity/gauge duality models for QCD-like constructs. I know, I have worked with both of them.

Nothing you said in the reply to me addresses what I said. I have said I don't deny the result, in fact both Guest and I are telling you it's in the literature! What I deny is the claim it's a unification. No, it's just a particular family of solutions which includes Maxwell's equations. It doesn't apply to arbitrary setups, such as even simple ones like the Kerr-Newman metric. In fact it's even easier than that, a valid solution to the Einstein-Maxwell equations is a non-extremal R-N black hole. Clearly you can't build such a solution from your extremal case so there are solutions to the Einstein-Maxwell equations your results can't describe so it isn't a unification. QED.

Another thing you completely failed to respond to was whether you'd sent any of this to a journal. You've had enough time to make web pages and YouTube videos so it's no like you haven't been able to get around to it. Have you? If no, why not? If you have, what did they say? This isn't a rhetorical question, I would like you to answer it. Presently a personal theory which isn't found in peer reviewed journals requires that you present a good case for it and you answer questions, else it goes to the alternative theories subforum, or even the pseudoscience subforum. Here you're making claims about results in GR so it's a little bit different but you should still be willing to answer relevant direct simple questions. It's also different because there's a few people here who consider themselves at least familiar with GR, at the very least capable when it comes to tensors and differential geometry. Guest's precise area of expertise I'm not sure about but I'm fairly sure it includes things like applications of Laplace's equations to things like GR and fluid mechanics. My own includes dynamics of black branes geometries and the implications of extremal D0-brane physics (specifically S duality) so if you would like to reword your last post describing your methodology to be much more mathematically formal then please go ahead, as it's my experience discussions of matters pertaining to this level of physics either go mathematically formal or go nowhere.

 

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Hi waitedavid137, AlphaNumeric.

Interesting and informative exchange.

A quick third party observation regarding the various 'manipulations' involved in the arguments presented on both sides.

It seems that both the relativity and the quantum scenario/mathematical 'manipulations' stray back and forth between 'absolute states' and 'relative states' domains when exploring the 'cross-boundary' validity of the 'solution' derivations presented on either side.

I read your respective arguments as jumping back and forth between relative and non-relative domains, and so I cannot accept either approach as being 'the last word' on the matter under discussion.

Just an onlooker's comment for your joint consideration (or not) as you see fit, guys. Nothing more.

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I don't have the time to say any more now. I will read with interest, though.

Thanks for a very intriguing and informative exchange, guys! Cheers.

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My theory for unifying gravity and electromagnetism is based on a relationship between positive charge and mass, then negative charge to positive charge. The reason is based on the preponderance of the observable data, or protons and electrons, where the positive charge associates with the bigger mass far more likely that does negative charge. This is not to say the opposite cannot occur, but the majority of the time positive gets the mass.

Beyond this basic observation we have to assume and theorize which should get less leverage that the preponderance of the data in the universe.

What this suggests for a BB theory is energy to mass and then mass to positive charge for a repulsive inflation and expansion, then negative charge.

 

Corrected.