Unifying Gravity and EM

**sweetser** · Dec20-05, 05:55 AM

Hello Careful:

** I would kindly request you to study tensor calculus in more detail (I am sorry, but I have no time to answer all your questions concerning tensor calculus). **

The request has been kindly noted. In no way do I expect you to answer all my questions on tensor calculus. I have tried to make clear I was reading background material, and that did change my views.

I know I have tested your patience, but there is method to the madness. Linux Pauling was asked how he came up with so many good ideas, and it was by having so many bad ones. In biology, the things we understand best have the shortest lifespans, so more experiments can be made in a day. I'd rather make a clear but incorrect mathematical statement than a fuzzy claim. By rapid rough approximations, a solution can be converged to quickly.

A casual reader to this thread would realize that you were an expert on general relativity as promised, and had issues with the proposal. It is important to demarcate these issues. Focus on the positives first.

Claim 1. The GEM action as written below is a well-formed, covariant action:

$LaTeX Code: S_{GEM}=\\int \\sqrt{-g} d^4 x (\\frac{-\\rho_{m}}{\\gamma}-\\frac{1}{c}(J_{q}^{\\mu}-J_{m}^{\\mu})A_{\\mu} -\\frac{1}{2c^{2}}\\nabla_{\\mu}A^{\\nu}\\nabla^{\\mu}A_{ \\nu}) $

Claim 2. The GEM action as rewritten is also well-formed:

$LaTeX Code: S_{GEM}=\\int \\sqrt{-g} d^4 x (\\frac{-\\rho_{m}}{\\gamma}-\\frac{1}{c}(J_{q}^{\\mu}-J_{m}^{\\mu})A_{\\mu} -\\frac{1}{4c^{2}}(\\nabla_{\\mu} A^{\\nu}-\\nabla^{\\nu} A_{\\mu})(\\nabla^{\\mu} A_{\\nu}-\\nabla_{\\nu} A^{\\mu}) - \\frac{1}{4c^{2}}(\\nabla_{\\mu} A^{\\nu}+\\nabla^{\\nu} A_{\\mu})(\\nabla^{\\mu} A_{\\nu}+\\nabla_{\\nu} A^{\\mu})) $

Biggest problem: Well-formed statements about gauge and other symmetries.

As to what I will do with the GEM action, I see little choice. The field to vary is the 4-potential. Folks that are good with actions can look at the action in claim one and get to the field equations as a one liner, $LaTeX Code: J_{q}^{\\mu}-J_{m}^{\\mu}=(\\frac{1}{c}\\partial^{2}/\\partial t^{2}-c\\nabla^{2})A^{\\mu}$ . In the static case, if one chooses to work with a flat Minkowski metric, the solution is charge/distance. If one chooses to work with the exponential metric,
$LaTeX Code: g_{\\mu\\nu}=\\left(\\begin{array}{cccc} exp(-2\\frac{GM}{c^{2}R}) & 0 & 0 & 0\\\\ 0 & -exp(2\\frac{GM}{c^{2}R}) & 0 & 0\\\\ 0 & 0 & -exp(2\\frac{GM}{c^{2}R}) & 0\\\\ 0 & 0 & 0 & -exp(2\\frac{GM}{c^{2}R})\\end{array}\\right). $
the potential is static. It is a fun exercise to show this metric solves the field equations for a static potential.

doug

**Careful** · Dec20-05, 06:05 AM

Hi,

I am not going to repeat here the things I said before (apart from the fact that Pauling probably meant that at least the math of your theory should be clear). You should really take a look at the work of Masreliez and others as well who have roughly the same ideas...

Good Luck,

Careful

**sweetser** · Dec20-05, 08:07 AM

Much on Masreliez work can be found here: www.estfound.org. A Google site search did not reveal any words on "Lagrange" or "Lagrangians", which is the starting place for field theories, and the fundamental way to compare two field theories. At first glance, Maxreliez looks like a variation on GR to deal with problems in cosmology that has an exponential as part of the metric. I suspect the differences are greater than the similarities, but I will look into it further.

doug

**Careful** · Dec20-05, 08:41 AM

Originally Posted by sweetser

Much on Masreliez work can be found here: www.estfound.org. A Google site search did not reveal any words on "Lagrange" or "Lagrangians", which is the starting place for field theories, and the fundamental way to compare two field theories. At first glance, Maxreliez looks like a variation on GR to deal with problems in cosmology that has an exponential as part of the metric. I suspect the differences are greater than the similarities, but I will look into it further.

doug

That's true: Masreliez does not work with a Lagrangian but that is no problem (you can equally well start from the field equations) - almost any theory can be reformulated in terms of a covariant Lagrangian but that is not the issue. I referred you to this work since he does more or less the same to GR than you seem to do (I recall you that the way you get out the metric is not satisfactory because of the problems with EM gauge transformations). His ideas about cosmology and quantum mechanics are not relevant for this thread.

Cheers,

Careful

**sweetser** · Dec23-05, 01:51 PM

Hello Careful:

One can certainly have this perspective:

** Masreliez does not work with a Lagrangian but that is no problem (you can equally well start from the field equations) - almost any theory can be reformulated in terms of a covariant Lagrangian but that is not the issue. **

I will give two examples why I do not adopt it in my own outlook.

Rosen was the first person to work with an exponential metric exactly like the one I use in this thread (equation 67 in GRG, vol. 4, No 6, 1973, p 435). The metric is consistent with all weak field tests of GR done to date, and will be slightly different at the next level of precision for tests of gravity. Why is there not more interest in his approach?

Let's look at the action for GR. Hilbert deserves much more credit than he gets for finding this piece of the GR puzzle - Einstein guessed the field equations. The action is austere in its simplicity:
$LaTeX Code: S_{Hilbert}=\\int \\sqrt{-g} d^{4} x R$
The square root of g is needed to get volumes correctly in curved spacetime and

is the Ricci scalar, a contraction of a contraction of the Riemann curvature tensor. Vary this action with respect to the metric, and one gets the second rank, nonlinear Einstein field equations of GR.

For an isolated source, the only way to generate waves is through what I like to call the water-balloon wobble: sides come in, the top and bottom blob out. The wobble is a quadrupole kind of thing. We have experimental data from binary pulsars that indicates that the rate of gravity wave emission is consistent with a quadrupole momentum, not a dipole emitter. If a binary pulsar could emit as a dipole, we would expect more energy loss from gravity waves than is seen.

The Lagrange density for Rosen's proposal adds in another field. That field is for a flat metric, so the proposal is known as the bi-metric theory of gravitation. The additional term in the action creates a problem for strong field tests of gravity. The other metric could store energy and momentum. This would make dipole gravity wave emission possible. The experimental data for quadrupole emissions of gravity waves is why the Rosen's approach has not attracted much interest. It can be seen by looking at the Lagrange density.

It is quite the challenge to construct a Lagrange density so simple it will not emit dipole gravity waves. Here is one candidate, the Einstein-Maxwell equations, which is just the sum of the two separately:

$LaTeX Code: \\mathcal{L}_{Einstein-Maxwell}=\\int \\sqrt{-g} d^{4} x (R-\\frac{1}{4 c^{2}}(\\nabla^{\\mu}A^{\\nu}-\\nabla^{\\nu}A^{\\mu})(\\nabla_{\\mu}A_{\\nu}-\\nabla_{\\nu}A_{\\mu}))$

[note to self: it would be wrong to use $LaTeX Code: \\partial^{\\mu}$ instead of $LaTeX Code: \\nabla^{\\mu}$ because spacetime here is curved even though this is the contractions of an antisymmetric tensor.] The Einstein-Maxwell equations cannot be quantized with our current techniques. Vary the metric, one gets GR. Vary the 4-potential, Maxwell. There is no unity.

I am skeptical that Masreliez's Lagrange density is so simple. If the action was available, it would be possible to think about in detail.

The second story is a personal one. Back in 2000, I had an audience with one of the most well known physicists in Boston. I said here are my field equations:
$LaTeX Code: J_{q}^{\\mu}-J_{m}^{\\mu}=(\\frac{1}{c}\\partial^{2}/\\partial t^{2}-c\\nabla^{2})A^{\\mu} $
See how wonderful they are?. If the mass current density in the same units as electric charge are thirteen orders of magnitude smaller than electric charge, one has the Maxwell equations in the Lorenz gauge. If the system is electrically neutral and static, one has Newton's law of gravity. If the system is neutral and not static, the field equations transform like a 4-vector, and thus gets along with SR, the motivation for GR being disolved.

He replied that a field theory requires more than field equations. One needs a Lagrange density, one needs to vary the Lagrange density so that it generates the field questions, one needs a solution to the field equations that is consistent with all current data, and one needs a solution to the field equations that makes it different from our current field theories. Then one can claim they have a field theory.

I thanked him and departed. I accepted his assessment. I was frightened. At that time, although I had hear the word Lagrangian, I had never worked with them. I had never varied an action to generate field equations. But I had no choice, I had to figure these things out that I did not understand. I was scared that I would never be able to do so. I reconnect with that fear when messing up on mixed tensor derivatives and being too liberal with partial derivatives instead of covariant derivatives. It took about a year and a half, but I now have a field theory because that list of requirements has been met. The GEM Lagrange density is simpler than Einstein-Maxwell, because I am about to cut and paste Einstein-Maxwell, then delete a few things:
$LaTeX Code: \\mathcal{L}_{GEM}=\\int \\sqrt{-g} d^{4} x (-\\frac{1}{2 c^{2}}\\nabla_{\\mu}A^{\\nu}\\nabla^{\\mu}A_{\\nu})$
The Ricci scalar was dropped, the source of problems with quantization. The antisymmetric tensor was made into an asymmetric tensor. With certain choices of basis vectors, the asymmetric tensor can be viewed as a symmetric tensor for gravity and an antisymmetric tensor for EM. I am spending time pondering the apparent lack of gauge symmetry for the 4-potential: is this good, bad, or what? I don't know. It is something worth thinking about, which I am. It is an important open question at this time for the GEM proposal. The issue of gauge symmetry arises because I have the Lagrange density worked out.

Like when one admires art, one can see different things from different angles. It is my own personal option that should you write out a field equation, you are obligated to figure out the Lagrange density. I appreciate this is not a common view, but at least my work is consistent with that view.

Happy vacation days,
doug

**sweetser** · Dec31-05, 07:31 AM

Hello all:

I have decided to ditch the mixed field strength tensor $LaTeX Code: \\nabla_{\\mu}A^{\\nu}$ for $LaTeX Code: \\nabla^{\\mu}A^{\\nu}$ : mixed tensors confused me and lead to technical errors Careful pointed out. This is a change in representation, meaning that the GEM Lagrange density is unaltered because:
$LaTeX Code: \\nabla_{\\mu}A^{\\nu}\\nabla^{\\mu}A_{\\nu}=\\nabla^{\\mu }A^{\\nu}\\nabla_{\\mu}A_{\\nu}$
I am combing though my seb site, making the appropriate changes. The main benefit is that the symmetric and antisymmetric tensors tensors, $LaTeX Code: \\nabla^{\\mu}A^{\\nu}+\\nabla^{\\nu}A^{\\mu}$ and $LaTeX Code: \\nabla^{\\mu}A^{\\nu}-\\nabla^{\\nu}A^{\\mu}$ are symmetric and antisymmetric no matter what the manifold is. I prefer to think of these as the average amount of change in the potential, and the deviation from the average amount of change respectively because it sounds more physical, less like a math exercise.

I have enjoyed thinking about gauge symmetry over the last week. I'll write up more later, but some things are clear to me.
1. The GEM Lagrange density breaks the gauge symmetry of EM.
2. Because of 1, the potential must be physically measurable. I believe that mass charge may be the measure of A.

Have a good new year,
doug

**Careful** · Dec31-05, 08:01 AM

Originally Posted by sweetser

I'll write up more later, but some things are clear to me.
1. The GEM Lagrange density breaks the gauge symmetry of EM.
2. Because of 1, the potential must be physically measurable. I believe that mass charge may be the measure of A.
Have a good new year,
doug

Good ! As a far as we know the potential is not measurable apart from some topological winding numbers such as in the Bohm Aharonov effect. So, I am afraid your theory is incorrect

Have a good new year (and keep on learning

)

Cheers,

Careful

**sweetser** · Dec31-05, 10:21 AM

** As a far as we know the potential is not measurable apart from some topological winding numbers such as in the Bohm Aharonov effect.

In the longer writeup I was thinking about, I was going to point this out, so we are in complete agreement. This is the EM 4-potential of Maxwell's theory which is exclusively about EM, the metric must be supplied as part of the background for the theory.

There is no charged particle that does not have a mass. Mass is a measurable property of every particle with an electric charge. My proposal with the potential being responsible for both electric charge and a measurable mass charge still looks like a plausible way to unify gravity and EM, something the Maxwell equations do not try to do. The mass charge for a proton is 13 orders of magnitude smaller than the electric charge of a proton, and we know electric charge to only 10 orders of magnitude. I don't know quite how to say it, but that may make the symmetry breaking by mass charge decouple from EM in a way consistent with our current approach to the EM potential (yeah, I know that sentence was garbled, need to think some more).

Will keep learning. Enjoy the moment.
doug

**sweetser** · Jan3-06, 06:01 AM

Hello:

I have attended regional APS/AAPT meetings, but have yet to go to a big APS meeting. The discussions here have help refine my proposal. Writing an abstract is a game of word choice efficiency since it is limited to 1300 characters with all the other stuff like the title. Here is my current 1295 character draft:

Title: Unifying Gravity and EM: A Riddle You Can Solve

Abstract: Apply three rules to this riddle:
1. Start from standard theory
2. Work with quantum mechanics
3. No new math
Start from the vacuum Hilbert-Maxwell action:
$LaTeX Code: S_{H-M}=\\int\\sqrt{-g}d^4x(R-\\frac{1}{4c^2}(\\nabla^{\\mu} A^{\\nu}-\\nabla^{\\nu}A^{\\mu})(\\nabla_{\\mu}A_{\\nu}-\\nabla_{\\nu}A_{\\mu}))$
The Hilbert action cannot be quantized, so drop the Ricci scalar. To do more than EM, use an asymmetric tensor:
$LaTeX Code: S_{GEM}=\\int\\sqrt{-g}d^4x\\frac{1}{4c^2}\\nabla^{\\mu}A^{\\nu}\\nabla_{\\mu }A_{\\nu}$
The metric is fixed up to a diffeomorphism. With a constant potential, the Rosen metric solves the field equations, is consistent with current tests, but predicts 0.7

arcseconds more bending around the Sun than GR. Gauge symmetry is broken by the mass charge of particles.

doug

**Lawrence B. Crowell** · Jan10-06, 09:58 PM

The Lagrangian
$LaTeX Code: L~=~ (J^a_q~-~J^a_m)A_a~-~1/2(\\nabla_aA^b\\nabla^aA_b $
involves the combination of a mass current and a charge current. Further the four potential is defined as being associated with both gravity and EM. Electromagnetism is a U(1) gauge theory. Gravity is an $LaTeX Code: SO(3,1)~\\sim~SL(2, C)$ theory. So this theory appears to have some analogy with the standard model of electroweak interactions. Yet in that case the gauge potential is

LaTeX Code: <BR>A_a = A(em) + A(weak),<BR>

for the $LaTeX Code: SU(2)\\times U(1)$ theory. For an $LaTeX Code: SL(2,C)\\times U(1)$ theory one might consider a similar construction, which is a twisted bundle. However, this is not apparent from your Lagrangian. I am presuming that the mass current is defined as
$LaTeX Code: J_a~=~T_{ab}e^b, $
or by some similar means. However, $LaTeX Code: \\nabla^aJ_a$ , a term which would emerge from the Euler-Lagrange equation, does not transform homogenously as the connection term emerges. This is related to the so called nonlocalizability of mass-energy in general relativity. So this would indicate that the field equations which emerges from the Lagrangian are not gauge covariant. This can only be recovered if there is are Killing vectors in the direction of this mass current. So without some special considerations the theory appears not to be covariant under the transformations of the theory.

This approach might best be extended to consider a theory that is $LaTeX Code: SO(3,1)\\times SO(4)$ with,
$LaTeX Code: SO(4)~\\simeq~SU(2)\\times SU(2). $
One of the

might be split on a singularity in its moduli space to give $LaTeX Code: U(1)\\times U(1)$ , where one of these can play the role of the electromagnetic field. The other

would then correspond to some massive field that is irrelevant to physics if the mass is large enough. The other

is then the weak interactions.

This might be started by considering a tetrad of the form
$LaTeX Code: E_a^b~=~\\gamma^be_a, $
where $LaTeX Code: \\gamma^b$ is a Dirac matrix in some representation. One then would have

where

is the gauge potential for the Yang-Mills gauge field. Similarly by $LaTeX Code: tr(\\gamma_a\\gamma_b)~=~4g_{ab}$ , if the representation of the Dirac matrices is local (changes from chart to chart) the differential on the tetrad gives
$LaTeX Code: \\partial_c(E_a^b)~=~\\gamma^bA_ae_c~+~{\\Gamma^b}_{c d}E_a^d. $
From here your general gauge potential, call it $LaTeX Code: {\\cal A}$ , would be
$LaTeX Code: {{\\cal A}^b}_{ac}~=~\\gamma^bA_ae_c~+~{\\Gamma^b}_{cd}E_a^d $
The field tensor for this theory would be defined from
$LaTeX Code: \\partial_d{{\\cal A}^b}_{ac}~-~\\partial_c{{\\cal A}^b}_{ad}, $
which if one were to work out the bits should result in the gravity and EM sectors.
Lawrence B. Crowell

**sweetser** · Jan11-06, 06:15 AM

Hello Lawerence:

A small note on how to post LaTeX at physics forums: the $ does not play the expected role. Instead one needs to use square brackets [] and the word tex to start, and /tex to end it. If you ever want to "borrow" an equation, just click on it and a pop up shows the tex needed for this site. To drop an equation into the middle of a sentence, use [] with itex to start, /itex to finish. Best of all, you can edit a post until the equations are correct. I do that a dozen times until all the parts look right.

A good reply will take me a few hours to compose, so I'll save that for this evenings activities. Thanks for your comments.
doug

**Doc Al** · Jan11-06, 08:10 AM

Lawrence:

I took a crack at inserting the appropriate Latex delimiters into your post (See Doug's last post). You may need to tune it up and repost. You can find more Latex info, should you need it, here: http://www.physicsforums.com/showthread.php?t=8997

- Doc

**sweetser** · Jan11-06, 09:59 PM

Hello Lawrence:

The post deals with gauge symmetry issues.

My proposal breaks U(1) gauge symmetry. Let's be clear for readers what that means. This is the transformation we have all seen before:
$LaTeX Code: A^{\\mu} \\rightarrow (\\phi,\\vector{A})single-quote=(\\phi-\\frac{\\partial \\Lambda}{\\partial t},A+\\nabla \\Lambda)$
The antisymmetric field strength tensor $LaTeX Code: \\nabla^{\\mu}A^{\\nu}-\\nabla^{\\nu}A^{\\mu}$ can be represented by the fields E and B defined as follows:
$LaTeX Code: E=-\\frac{\\partial A}{\\partial t}-\\nabla \\phi$
$LaTeX Code: B=\\nabla \\times A$
Plug in the U(1) gauge transformation into those definitions:
$LaTeX Code: E \\rightarrow Esingle-quote = -\\frac{\\partial A}{\\partial t}-\\frac{\\partial \\nabla \\Lambda}{\\partial t}-\\nabla \\phi+\\nabla \\frac{\\partial \\Lambda}{\\partial t}=E$
$LaTeX Code: B \\rightarrow Bsingle-quote=\\nabla \\times A+\\nabla \\times \\nabla \\Lambda=B$
For the E field, the mixed time/space derivatives cancel. For the B field, the curl of curl of a scalar is zero.

The GEM proposal has exactly these two fields E and B. But there are also fields to represent the symmetric tensor. I call them small e and small b, the symmetric analogues to EM's big E and big B. There is also a field for the four along the diagonal. Here are the definitions for the 5 fields in the GEM field strength tensor:
$LaTeX Code: E=-\\frac{\\partial A}{\\partial t}-\\nabla \\phi$
$LaTeX Code: B=\\nabla \\times A$
$LaTeX Code: e=\\frac{\\partial A}{\\partial t}-\\nabla \\phi-\\Gamma_{\\sigma}{}^{0u}A^{\\sigma}$
$LaTeX Code: b=(-\\frac{\\partial A_{z}}{\\partial y}-\\frac{\\partial A_{y}}{\\partial z}-\\Gamma_{\\sigma}{}^{yz}A^{\\sigma}, -\\frac{\\partial A_{x}}{\\partial z}-\\frac{\\partial A_{z}}{\\partial x}-\\Gamma_{\\sigma}{}^{xz}A^{\\sigma}, -\\frac{\\partial A_{y}}{\\partial x}-\\frac{\\partial A_{x}}{\\partial y}-\\Gamma_{\\sigma}{}^{xy}A^{\\sigma})$
$LaTeX Code: g=(\\frac{\\partial \\phi}{\\partial t}-\\Gamma_{\\sigma}{}^{tt}A^{\\sigma}, -\\frac{\\partial A_{x}}{\\partial x}-\\Gamma_{\\sigma}{}^{xx}A^{\\sigma}, -\\frac{\\partial A_{y}}{\\partial y}-\\Gamma_{\\sigma}{}^{yy}A^{\\sigma}, -\\frac{\\partial A_{z}}{\\partial z}-\\Gamma_{\\sigma}{}^{zz}A^{\\sigma})$
Apply the U(1) gauge symmetry, and it becomes apparent that the E and B fields are fine, but the fields I think deal with gravity, g, e, and b, are not. Gravity and breaking gauge symmetry are linked in the GEM proposal.

Gauge theory is very powerful. Starting from the U(1) symmetry in 4D, people good at this sort of thing can derive the Maxwell equations. That is a reason why if one states their proposal breaks U(1) gauge symmetry, it is reasonable to think the theory cannot regenerate the Maxwell equations. I am trying to do something more, to fundamentally include mass.

Look at one limitation of gauge theories. Let me quote extensively from Michio Kaku's "Quantum Field Theory: A Modern Introduction" p. 106:

Because of gauge invariance, there are also complications when we quantize the theory. A naive quantization of the Maxwell theory fails for a simple reason: the propagator does not exist. To see this let us write down the action in the following form:
$LaTeX Code: \\mathcal{L}=1/2 A^{\\mu}P_{\\mu \\nu}\\partial^{2}A^{\\nu}$
where:
$LaTeX Code: P_{\\mu \\nu}=g_{\\mu \\nu}-\\partial_{\\mu}\\partial_{\\nu}/(\\partial)^2$
The problem with this operator is that it is not invertible, and hence we cannot construct a propagator for the theory. In fact, this is typical of any gauge theory, not just Maxwell's theory. This also occurs in general relativity and in superstring theory. The origin of the noninvertibility of this operator is because $LaTeX Code: P_{\\mu \\nu}$ is a projection operator, that is, its square is equal to itself:
$LaTeX Code: P_{\\mu \\nu}P^{\\nu \\lambda}=P_{\\mu}^{\\lambda}$
and it projects out longitudinal states:
$LaTeX Code: \\partial^{\\mu}P_{\\mu \\nu}=0$
The fact that $LaTeX Code: P_{\\mu \\nu}$ is a projection operator, of course goes to the heart of why Maxwell's theory is a gauge theory. This projection operator projects out any states with the form $LaTeX Code: \\partial_{\\mu}\\Lambda$ , which is just the statement of gauge invariance.

Physicists understand exactly how to deal with this issue: pick a gauge. With the GEM proposal, this choice is not available. That may be a good thing for quantizing the theory.

There is the problem of mass in the Standard Model. The symmetry $LaTeX Code: U(1) \\times SU(2)\\times SU(3)$ justifies the number of particles needed for EM (one photon for

, the weak force (three W+, W-, and Z for

), and the strong force (8 gluons for

). Straight out of the box, the Standard Model works only if all the masses of particles are zero. Something else is needed to break the symmetry. Readers here know the standard answer: the Higgs mechanism uses spontaneous symmetry breaking to introduce mass into the standard model. As far as I know, there is no compelling connection between the Higgs and the graviton.

Let's think on physical grounds about how mass and charge relate to each other. Consider a pair of electrons and a pair of protons, each held 1 cm apart from each other. Release them, and the electrons repel each other, as do the protons. Measure the acceleration. The electrons accelerate more for two distinct reasons. First, there is the difference in inertial mass because an electron weighs 1800x less than a proton, good old

. Second, the gravitational masses will change the total net force, more attraction for the heavier protons, good old $LaTeX Code: F=-Gmm/R^{2}$ , which would be too subtle to measure directly. One could say that both inertial and gravitational mass break the symmetry of the standard model. In the GEM proposal, the 3 fields (10 total components) of g, e, and b make up the symmetric field strength tensor $LaTeX Code: \\nabla^{\\mu}A^{\\nu}+\\nabla^{\\nu}A^{\\mu}$ that could do the work of the graviton, while the trace of that matrix could do the work of the Higgs. I am no where near good enough to make those connections solid. I am just pointing out what looks like a duck might be a duck.

Lawrence has pointed out several ways to be a good gauge theory proposal, but I think GEM proposal is heading a different direction. There is a need to break gauge symmetry in a way consistent with gravity and quantum field theory.

doug

**sweetser** · Jan11-06, 10:29 PM

Hello Lawrence:

This post is in reply to the nonlocalizable issue.

Let me explain for folks why mass-energy in GR is nonlocalizable. Pick a point in spacetime, any point in spacetime. You are free to choose whatever coordinates you want. Riemann normal coordinates set the connection to zero at that point (they cannot set the connection to zero everywhere). Since gravity in GR depends only on the metric, the energy density of the gravitational field is zero there. This is one way to see that the energy density of the gravitational field is nonlocalizable. The three forces in Nature we know how to quantize using gauge theories, EM, the weak force, and the strong force, are localizable. No coordinate choice can make the fields zero at a point.

With the GEM proposal, go ahead, pick the Riemann normal coordinates. The energy density of the gravity field is not zero because the gravitational field depends on both the connection and the changes in the potential. Riemann normal coordinates may set the connection to zero but the energy density could still be in the change of potential. One could in fact choose to work in entirely flat spacetime background - I am often accused of this - and all would be explained by the potential. There is nothing wrong with doing everything with the potential. But I could also decide to work with a dead dull potential, and do all of gravity with the connection (see the definitions of g, e, and b in the preceding post).

In GR, mass-energy density in the gravity field is nonlocalizable.
In GEM, mass charge - strictly similar to electric charge - is localizable.

Which is better? You make the call,
doug

**Blackforest** · Jan12-06, 03:40 AM

Originally Posted by Lawrence B. Crowell

The Lagrangian
$LaTeX Code: L~=~ (J^a_q~-~J^a_m)A_a~-~1/2(\\nabla_aA^b\\nabla^aA_b $
involves the combination of a mass current and a charge current. Further the four potential is defined as being associated with both gravity and EM. Electromagnetism is a U(1) gauge theory. Gravity is an $LaTeX Code: SO(3,1)~\\sim~SL(2, C)$ theory. So this theory appears to have some analogy with the standard model of electroweak interactions. Yet in that case the gauge potential is

for the $LaTeX Code: SU(2)\\times U(1)$ theory. For an $LaTeX Code: SL(2,C)\\times U(1)$ theory one might consider a similar construction, which is a twisted bundle. However, this is not apparent from your Lagrangian. I am presuming that the mass current is defined as
$LaTeX Code: J_a~=~T_{ab}e^b, $
or by some similar means. However, $LaTeX Code: \\nabla^aJ_a$ , a term which would emerge from the Euler-Lagrange equation, does not transform homogenously as the connection term emerges. This is related to the so called nonlocalizability of mass-energy in general relativity. So this would indicate that the field equations which emerges from the Lagrangian are not gauge covariant. This can only be recovered if there is are Killing vectors in the direction of this mass current. So without some special considerations the theory appears not to be covariant under the transformations of the theory.
This approach might best be extended to consider a theory that is $LaTeX Code: SO(3,1)\\times SO(4)$ with,
$LaTeX Code: SO(4)~\\simeq~SU(2)\\times SU(2). $
One of the might be split on a singularity in its moduli space to give $LaTeX Code: U(1)\\times U(1)$ , where one of these can play the role of the electromagnetic field. The other would then correspond to some massive field that is irrelevant to physics if the mass is large enough. The other is then the weak interactions.
This might be started by considering a tetrad of the form
$LaTeX Code: E_a^b~=~\\gamma^be_a, $
where $LaTeX Code: \\gamma^b$ is a Dirac matrix in some representation. One then would have

where is the gauge potential for the Yang-Mills gauge field. Similarly by $LaTeX Code: tr(\\gamma_a\\gamma_b)~=~4g_{ab}$ , if the representation of the Dirac matrices is local (changes from chart to chart) the differential on the tetrad gives
$LaTeX Code: \\partial_c(E_a^b)~=~\\gamma^bA_ae_c~+~{\\Gamma^b}_{c d}E_a^d. $
From here your general gauge potential, call it $LaTeX Code: {\\cal A}$ , would be
$LaTeX Code: {{\\cal A}^b}_{ac}~=~\\gamma^bA_ae_c~+~{\\Gamma^b}_{cd}E_a^d $
The field tensor for this theory would be defined from
$LaTeX Code: \\partial_d{{\\cal A}^b}_{ac}~-~\\partial_c{{\\cal A}^b}_{ad}, $
which if one were to work out the bits should result in the gravity and EM sectors.
Lawrence B. Crowell

These explainations sound good to me; I have a hope to be on the right road. Thanks

**Blackforest** · Jan12-06, 04:02 AM

The present GEM discussion is one possible way to consider the problematic of the connections between EM and gravitation. The (E) approach (see other proposed discussion on this subforum) is another one. The "general gauge potential" in the approach proposed by Lawrence B. Crowell could perhaps find an equivalence in my approach under the label of what I have called a local "cube" defining the extended vector product. So far I understand this difficult discussion, we are looking for mechanismus able to explain the symmetry breaking. I don't know if my point of view is relevant, but couldn't we see the begining of an explaination in the not necessary coherent behavior of two mathematical operations defined in any frame: the scalar product and of the extended vector product. Can't we relate this eventually incoherence to the so-called Palatini's principle?