I agree, any time the group is "bigger" in some sense, there must be a symmetry breaking to get things down to the observed level.
We have observed photon and the three particles of the weak force. The gluons are massless mediators of force between quarks. We have never seen quarks in isolation. I don't think we have ever been able to form a beam of gluons due to quark confinement.
At this time I don't know what all the consequences of that happen to be. I'd love to speculate about other mysteries this proposal my address. I will refrain and keep my eye on what it does, which is give a visual justification for why there are three, and these three in particular, if one wants to develop a complete and smooth collection of any event that an observer at 0,0,0,0 happens to see.
Your position is both widely accepted and reasonable. Mathematically, it is clear that U(1) and SU(2) are subgroups of SU(3). The way I showed this was by the programming in the animation. To make U(1), I took a thousand quaternions at random, calculated q/|q|, and plotted that. To calculate SU(2), a thousand random quaternions were fed into exp(q-q*). To generate electroweak symmetry, U(1)xSU(2), these two were the products of each other: q/|q| exp(q-q*).
Up to this point, I am playing by the book, and have no disagreements.
Now I try to represent SU(3). Sounds like it might require two independent quaternions. Multiplying two randomly assigned quaternions together will not generate a new group, it will generate another member of the group q/|q| exp(q-q*).
Instead of multiplying the two together, I first took the conjugate of one, and then formed the product:
(q/|q| exp(q-q*))* q'/|q'| exp(q'-q'*)
A question I don't know how to answer is how to write this. At first I made a silly mistake, and jotted down U(1)xSU(2)xSU(3). That has 12 elements of its Lie algebra, and I only used 8. In terms of the calculation, I know I calculated q/|q| exp(q-q*) twice for two randomly select quaternions, about 12,000 times for the youtube animation.
> electric charge would be easily violated by some strong force interaction
If true, the proposal would be wrong. Working with the nuts and bolts as I do, I see no reason why the q/|q| - the basic symmetry of EM and thus charge conservation - are in any danger by taking a conjugate and forming a product with another.
> We probably shouldn't see discrete electric charges under such a scenario.
If true, the proposal would be wrong. Again, I see no reason to suspect this will be the case. I can see the U(1) symmetry doing its thing whether inside SU(3) or not.
As I understand it, chromodynamics is such that it is nearly impossible to calculate anything. I doubt I'll ever understand any of that work. This definitely feels like it is a new direction, so there is hope.
I most like the "ontology", a fancy word for "why". There is no reason why these three groups of all the possible ones out there should be relevant to three of the four forces of nature. Nor is there any idea of how to bring in the forth. The animations give a visual explanation: to be able to describe any possible event in a smooth, continuous way from an observer at 0, 0, 0, 0 one needs the symmetries U(1)xSU(2) times the conjugate of another U(1)xSU(2). Make the ball a different size if you want to include gravity. A simple and visual explanation.
I respect your critique that much more needs to be done to make the case persuasive, but at least I can carry my work around on my ipod and show it to friends, and advancement of a sort.
Thanks for this explanation, I think I get your point.
There is a way this proposal uses the word "gauge" in a different way than is usually done. Gauge means to measure. What is normally meant by this is you can add in a gauge field and it won't change a darn thing. EM is a gauge field theory, as is GR.
I want to use gauge to mean how a measurement is made, with the choice being how the spacetime manifold changes (the connection) or how a 4-potential changes or any combination of the two. So I could decide to work in a completely flat spacetime and do all the work with a 4-potential. Or I could decide to work with a boring (constant) 4-potential, and have the manifold bend as need be. I am making a choice on how the measurement is being made. No field is being added in. I can choose to do a mix and match, using a Newtonian potential to get the g_00 part right, and a dynamic metric with the g_11 a bit bigger than one so the light bending around the Sun works with this combo meal.
Talking with folks schooled in GR, I emphasize the case where the 4-potential is there in spirit, not in the calculation. I solve a differential equation, and out pops an exponential metric:
I almost never write the G^(1/2)q part because unified field theory is not a good thing to claim to do. So for an electrically neutral system, it is zero.
Now to your point, this could be approximated using the Taylor series like so:
This has the local Lorentz symmetry, and a first order perturbation. Cool. I've known that for a long time, but this is a different way to think about it. Exponentials are very math friendly, so in my case it probably would not help in a calculation. It does connect to what other people do with gravity proposals.
The short answer: Jesus, those are tough technical questions that I don't have the answer to.
The creation and annihilation operators are in your quantum field theory book, in the section on Lorentz invariant quantization of a 4D wave. I was unable to understand how to write a spin 2 propagator (I gave some papers from Wienberg in the 1960s to a Ph.D. physics friend of mine, and he also did not figure it out).
I do know that the force equations that arise by varying the action with respect to the 4-velocity have like charges attract. I do know that the field equations that arise by varying the action with respect to the 4-potential have like charges attract. I do know the particles travel at the speed of light c if the trace of the field strength tensor is zero.
> Can you identify your rank-2 symmetry generators with the Lorentz group
You are correct that people have tried to make bigger gauge groups. So far that has not worked. I am trying to make things smaller. We know there is EM and its symmetry is U(1). We know there is the weak force and its symmetry is SU(2). We know there is the strong force and its symmetry is SU(3).
With programs, the problems often appear at the interfaces. The tensor product of these three groups is presented as simple. Yet it means that you have to add a special flag onto both the groups U(1) and SU(2). The reason is that they are subgroups of SU(3), so any element of U(1) times SU(3) will be in SU(3). Identical particles don't like labels.
The animations are a visual justification of the 4 known forces in a smaller group, so I am in conflict with current research. Oh well.
Your impression sounds correct to me. The vacuum field equations I use are linear, and NOT a linear approximation of GR. Instead of binding gravity to the second rand field strength tensor T^uv, the coupling is between a 4-potential and the 4-momentum density. The difference between these two is not big. The only difference is that gravity fields are not sources. This is consistent with EM, were EM fields are not sources. Consistency is good.
The trick here is to be like EM when necessary, and different where necessary. There are no photon-photon interactions (unless things get odd), and likewise, there are no graviton-graviton interactions. That does not prove they theory is renormalizable. I don't have the skills to prove or disprove that point. I am hopeful, because it is basically the manifestly Lorentz invariant quantization of a 4D wave, but with 2 spin fields, a spin 1 for EM, a spin 2 for gravity. In text books, two of the modes of emission for the manifestly Lorentz invariant quantization of a 4D wave have to be made virtual. I put those virtual modes to work.
On point of contention is EM tensor_product Weak tensor_product Strong. That leads to a nice, neat, 12 degrees of freedom. That also reflects the history of particle physics, where we learned about EM first, then the weak force in the 1930s, then then eight-fold way in the 1960s. Things is, U(1), SU(2), and U(1)xSU(2) are subgroups of SU(3). I didn't read that, I saw it, and that was confirmed by talking with a math guy from Mathematica at an APS meeting. I think the standard model should more tightly reflect what goes on with the math, not the history of our discoveries in particle physics. I know this makes EM and the weak force feel "too close" to the strong force, but the math side of my brain says too bad, that's the way it goes. This is not a high energy theory. It is an odd assault on the second x of U(1)xSU(2)xSU(3).
> * I don't understand your concept of inertial mass breaking gauge symmetries.
Great question. The Standard Model is only about EM, the weak and the strong forces. The Higgs is there to bring in inertial mass. And gravity as everyone knows, doesn't get invited to the party. We all know that is wrong. We need gravity right there. The graviton also must be gauge invariant to travel at the speed of light. So do I want it both ways? Of course I do!
Here's how it is done. Use an asymmetric tensor, d^u A^v. This tensor is reducible, so it cannot represent fundamental forces. Cleave it in 2, to an antisymmetric tensor, d^u A^v - d^v A^u, to do the work of EM with a spin 1 field, and a symmetric tensor d^u A^v + d^v A^u to do the work of gravity with a spin 2 field. The antisymmetric tensor will always be gauge invariant, cruising at the speed of light. The symmetric tensor will also be gauge invariant if and only if the trace happens to be zero. This is the house where the graviton lives. The graviton does the work of gravitational mass. When the trace is not zero, that breaks the gauge invariance of the model. The trace of d^u A^v + d^v A^u is a scalar field that does the job the Higgs is suppose to do. The job is done better because now there is a particle explanation of the equivalence principle: tr(d^u A^v) = 0 is for the graviton and gravitational physics, and tr(d^u A^v != 0) is for the scalar field needed for interial mass.
I'll ramble to anyone over a beer. I bought an IPod just to demo these animations:-)
U(1) is a unit circle in the complex plane. SU(2) is a unit quaternion which is easy to animate if you have software for the job (barf out thousands of exp(q-q*), sort by time, drive through POVRay). Electroweak is the product of the first two. The animation of SU(3) tells you what the standard model is about, namely the ability to smoothly describe any event seen by an observer at 0,0,0,0. Gravity is about the sizes of things, so scale the ball to different sizes in a smooth way, and that is the symmetry behind gravity.
It is inertial mass that breaks the symmetry of standard model, not some phony Mexican hat dance around a false god of a vacuum.
Here's my explanation. Quantum mechanics results from the collision of the 2 biggest ideas in physics: calculus and 4D spacetime. The way to handle these mathematically is with quaternions, which most nerds are not aware of. It is like a complex number, but with 4 parts, one for time, 3 for space. They can be added, subtracted, multiplied or divided like a real number.
Take the derivative of a quaternion function. Oops, not going to work. If you recall the definition of a derivative using the limit process, there is a small differential element that goes to zero. Because quaternions do not commute, writing that on the left is different from writing it on the right. There are papers where people work with such a buggy definition.
Steal from L'Hospital's playbook (which was stolen from Bernoulli). Take 2 limit processes, letting the 3-vector go to zero first, then the scalar. Technically that would be a directional derivative along the real axis. This is the world of classical physics because everything is order in time: a comes first, then b, then c. Imagine a movie with 10 frames, and each one has a clear place in a time line.
Now reverse the limit processes. Oops, same problem again. This time the fix is to take the norm of the derivative: that will be the same whether the differential element is on the left or the right. This is as much as can be known. This is the land of quantum mechanics. Instead of 10 frames of an animation played one after another, the 10 frames are possible states the system can be in. You could superimpose the 10 frames, saying this is all the states the system could be in. Doing a measurement picks out one of the 10 frames, nothing more, nothing less.
All the folks using Command Line Quaternions over at quaternions.sf.net do 4D work, from, well, the command line. Any other interface is for wimps. The result is an animated gif. 60 downloads already!
Time + 3D = 4D. This is the way to do analytic animations.
doug
See the symmetries of the standard model
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Hello:
The standard model has the symmetries U(1)xSU(2)xSU(3). The one in the middle, SU(2), is a unit quaternion, where a quaternion is like a real or complex number, but has four parts. I have developed the software to visualize quaternions at http://quaternions.sf.net/ using one number for time, three for space. SU(2) can be represented by the quaternion function exp(q-q*). Feed a thousand random quaternions into exp(q-q*), and get POVRay to make a nice animation. Do the same for q/|q| exp(q-q*), and you have a visual representation of the electroweak symmetry. Smash two of these together, and you get the symmetry of the standard model.
Visually, there is a clear message: if you want to smoothly represent all possible events in spacetime as quaternions, the group description must be U(1)xSU(2)xSU(3). You won't read that in a journal because it has to be done with animations.
I've posted clips from TheStandUpPhysicist.com, the education arm of my unified gravity and EM field theory using quaternions. I got a nice OK from YouTube, no questions. At Revver, they said thanks, but are you sure you have permission to use that song? It was "Math Prof Rock Star" by Jim's Big Ego, a perfect tune for the most nerdly narrowcast theoretical physics show ever. The song is licenced under the creative commons license attributions, non-comercial, share-alike license. Hoping to make billions off of the shows, I had negociated with JBE's business manager, and had agreed they get a percentage.
I told the revver folks that story. Sorry, that wasn't good enough. They wanted to see either the contract or get an email from JBE. A few emails later, and the clips are up on Revver.com. So I think revver.com is investing time and money in the process of due dilegence on copyrights.
I'd certainly like to play Texas hold'em against you. Remember, probability was developed to win more often at cards, and to make more money. There is nothing spiritual about it. There is a lot of diversity out there, whether you are a boy, a girl, or intersex (and the odds are not 50-50-50 fortunately). The odds are better to be conceived as a boy baby, but girl babies survive better. Sorry, not enough stats on intersexuals.
I am working to be analytical about animations using quaternions. The brain SUCKS at remembering visual stuff. Instead, the brain is great at shop & compare. That is why artists use easels by the way - it's not just to hold up the canvas, but because visual memory is so bad, but comparison is so good, so the artist's work can quickly be compared to "the real thing".
> That is a situation [hair standing up due to a wee bit of static electricity] where forces cancel out.
There are two ways to look at what gravity and EM do: as fields or as forces. If one takes the Lagrangian, and does the variation with respect to the potential, one gets the field equations. If one takes the same Lagrangian, and does the variation with respect to the velocity, one gets the force equations. I still remember exactly where I read that in Landau and Lifshitz! So sure, one can view the issue as forces canceling (which really only makes technical sense for my GEM or gravitomagnetism, not GR). One can describe the exact same thing using a field equation approach. The Maxwell equations are logically consistent, so the story generated by looking at forces must be the same as the one for fields.
> It is demonstrably different from this.
Even if I concede this point, it does not address my reply that we know darn well how to characterize a vacuum, whose algebra is the same as Jq - Jm = 0, Jq != 0.
> And when you square that term you don't get the E&M action.
The asymmetric tensor gets contracted, and the result is
Take the derivative of L_GEM with respect to A^mu, and the resulting field equations should have a familiar look to it as far as EM is concerned. This Lagrangian will not look familiar with expectations based on linearized GR or GR itself. The proposal works differently, using only one connection, not the divergence of two connections as happens in the Riemann curvature tensor.
Any well-trained person will remain skeptical until with good old paper and pencil they calculate for themselves the divergence of the Christoffel of the exponential metric. I would bet you did not do that, so your stance is reasonable.
At the end of the day, I only care about technical issues, they last. doug
>For instance, in equation (2), if you make the mass density equal to the charge density, then you get nothing.
I'll presume you find discussion of a vacuum is a sensible thing, where J=0. People study the propagation of waves in a vacuum which can also be done here. You have pointed out that there is another way to have J=0. The same solutions will apply to the case.
Note that the balance of mass and electric charge cannot happen for fundamental particles because mass charge is so far smaller than electric charge. For example, an electron's mass charge is sixteen orders of magnitude smaller than the electric charge of the electron. This is a macroscopic situation you propose looking at. There are classical situations where gravity is balanced by EM: think of hair standing up due to a wee bit of static electricity.
> But with opposite charge, you do get something.
The total charge, electric charge minus mass charge, can be positive or negative, depending almost entirely on what is going on electrically. If J is positive, the particles will repel. If J is negative, they will attract. Electric charge can have 2 signs, but mass charge can only have 1 (the difference is due to the properties of their respective field strength tensors, which I am about to come to...).
> Your action, equation (1), contains neither E&M nor linearized gravity.
I don't want linearize GR, so I am glad you cannot spot it:-) Seeing EM requires a little algebra. The source of the GEM unified field proposal is a reducible second rank asymmetric tensor, d^mu A^nu. Any asymmetric tensor can be represented as the sum of a symmetric and an antisymmetric tensor:
The second term is F^mu;nu (and I am sophisticated enough to care about the semicolon, since the theory is about having the choice to work in curved spacetime).
I have done no work with macroscopic media, which is why there is nothing on D or H. Only if I were to establish these equations are sufficient to do gravity and EM would such an investigation be warranted.
>I'd suggest that if this is something you are really interested in, you take some courses and learn the fundamentals before you start putting together a theory.
There is nothing wrong with this suggestion, but I do think it is meant to be a put down. As to fundamentals I know how to work with, that would include what a Lagrange density is, how to generate the field equations by the calculus of variations, finding perturbation solutions to the field equations, and working with Christoffel symbols.
I want to thank you for at least flipping through the paper and citing equations. Those equations do not map to either linearized GR or gravitomagnetism. That does not make them right or wrong, just alien. I suspect that you did not calculate the divergence of the Christoffel of the exponential metric. That sound like a run on sentence full of hard math!
For those in the listening audience that are unfamiliar with what a Christoffel symbol is (a topic that only comes up in GR courses), it involves taking 3 different derivatives of a 4x4 metric tensor, and contracting that with another metric tensor. Scary, even for hard core nerds. Thing is, for the metric I work with is WAY EASY. The metric is static, no time dependence, so that drops one of the three derivatives. The metric is also diagonal, so another one of those derivatives drop out. One is then taking the derivative of an exponential, which is the exponential times the derivative of the exponent. One ends up with charge over distance!
del_u Gamma_v^0u A^v = Del^2 GM/c^2 R
Doing that calculation for the exponential metric, and getting charge/R was one of the most amazing calculations I have done in my life.
Thanks for the reference. Gravitomagnetism is an example of starting from the field equations, kind of what Einstein did with GR. The more proper thing to do is to start from the action (which is the sum of every type of interaction that can happen inside a box), which is what Hilbert did. In my proposal, there is a antisymmetric tensor in the action that does the work of spin 1 photons, and a symmetric tensor that does the work of spin 2 gravitons.
The field equations are generated by varying the action with respect to the 4-potential. One ends up with a Gauss' law and an Ampere's law. The vector identities for EM, no monopoles and Faraday's law, are still true, but they do not apply to the gravity analogues in my approach.
A significant difference is that I use covariant derivatives (I'm sure the authors who work on gravitomagnetism formally are writing covariant derivatives, but they are not actually putting them to use). A covariant derivative is like the derivatives we all learned, plus a connection term which has derivatives of the metric inside it. Any measurement of change needs to account for both the change in the thing being measured, and a change in the rulers used to measure the changes. So one ends up with something that looks much like Gauss' law, which for a 4-potential looks like:
Del^2 phi = G^(1/2) rho (I have chosen to make my units all like electric charge)
Now the challenge is to find a solution to this differential equation when phi is constant. With normal derivatives, only a vacuum will work. There is a non-trivial solution if the Del^2 is a covariant derivative. One has to look for a metric, calculate the Christoffel symbol for that metric, and then take the divergence of that, and show it equals 4 pi G^(1/2) rho, in other words, to solve this equation:
del_u Gamma_v^0u A^v = G^(1/2) rho
The metric that solves this divergence of the Christoffel is known in the literature as the Rosen metric. What I wrote earlier was the Taylor series expansion of the exponential metric.
Thanks again for the reference. This is not an attempt to simplify GR or to copy directly off of a Maxwell cheat sheet, but it will be hard to convince anyone who doesn't calculate the divergence of the Christoffel of the exponential metric that is the case.
The measurement is still in the range of first order parametrized post-Newtonian accuracy. What the Donkey Kong that means is that these are the coefficients to the metric that are being tested:
It is the 5 integers there (1, -2, +2, -1, -2) that are confirmed by this experiment. That is NOT NEWS, because it is not new. Shapiro got the same results. What would be news is if the experiment got to second order parameterized post Newtonian accuracy. I asked Prof. Clifford Will an expert on experimental tests of GR when where the data hunters going to gather that data. He said he knew of no one even discussing it. The reason is that the data must for 2nd order PPN effects must be a million fold more accurate, so we need data that is 99.99995% accurate.
I care a lot about 2nd order PPN tests, since that is were my proposal to unify gravity and EM using a 4D wave equation differs. GR says the metric should go here:
At first order PPN accuracy, the coefficients (1, -2, 2, -1, -2) are the same. At second order, they are different. That's the data I need. I'll probably be dead before it shows up.
The point of gravity is not to do work. If the Universe was completely empty, then to measure the distance between a pair of events would require this metric:
dtau^2 = 1^2dt^2 - 1^2/c^2(dx^2 + dy^2 + dz^2)
Why include the 1^2? It is to emphasize that they are constants, the same for everyone, no matter where they are in the Universe. Yet there is nothing else in the Universe to check on it. Add a blob into the Universe, and those constants no longer work. They are very good. Only nerds with telescopes and atomic clocks could tell it was a wee bit different from one.
General relativity has a way to determine the functions that take the place of the 1^2. They start with a connection, which has three derivatives of a metric. Then they work with the Riemann curvature tensor, which is the divergence of two connections. This means now there is a second derivative of a metric. It is the second derivative of a metric that leads to an equation that can be solved. The problem with GR in my opinion is that one has to compare 2 paths, and that breaks the math machinery of quantum mechanics.
I take a different approach. Gravity is about doing nothing. Once there is something else in the Universe, you must do something, but the very least amount of something possible. That has a name: a simple harmonic oscillator, better known as a slinky. The math gets a little scary because it is a 4D slinky, with two modes for EM (the transverse ones) and two modes for gravity. Still, one is trying to do close to nothing, and that is a slinky.
One gets into 4D kinky slinky physics with a 4D wave equation. That itself has two covariant derivatives. If you look at two convariant derivatives acting one after the other, the trained eye can spot a second order derivative of a metric. That leads to a differential equation that can be solved to yield a metric.
The metric from my work ain't the one for GR (the Schwarzschild solution), it is prettier. Beauty always wins in physics, just like in the movies. It has exponentials in the place of the 1^2, and exponential appear again and again in physics. Why? Well when the exponent is super small, as it is for gravitational systems, it basically is 1. Only the first, and sometimes second terms matter. Those terms are identical for the exponential metric and GR. At second order parametrized Post Newtonian accuracy, the metrics are different, so unlike silly string theory, the proposal can be accepted or rejected based on an experiment.
Gravity is the least you can do because there is other crap in the Universe.
The reason several billion dollars are going into the electromagnetic crop circle known as the Large Hadron Collider is to detect the Higgs particle. The standard model of physics predicts all the detected particles we have seen out there. The model can be represented by the symmetry groups U(1) for EM, SU(2) for the weak force, and SU(3) for the strong force. Well, it can do that so long as none of the none forces has a mass. Oops, that's not what the experimentalist tell us.
So what do theoretical physicists do? They try to sneak in something without breaking it. That is what is known as Higgs mechanism, or the false vacuum Mexican hat dance. Instead of saying the vacuum is a vacuum is the home field of zero, it is claimed that vacuum is utterly false, there is a Higgs field everywhere there can be a where, so that fundamental particles can get some mass when they need it (always). This mathematical trick works because it doesn't mess up the symmetry in the Lagrangian (a fancy way of writing all the interactions that can happen in a volume), but does get the vacuum to add the mass back in.
You may have noticed gravity is not in the standard model. Guess what is going to happen when gravity gets in? Gravity will break the nice symmetry, and no Higgs boson will be needed.
The uber-sophisticated will complain (whine) that gravity is done by a spin 2 particle, but the Higgs is all about inertia, so it must be spin 0. This is not a flaw, it is a sign from Einstein, specifically the equivalence principle that gravitational mass (the spin 2 thing) must be cow-tied to inertial pass (the spin 0 thing). There is no way to wrestle a steer to the ground unless those two are expressions of one and the same thing.
The way I do it, because of course I have my own personal unified field theory, is to use a second rank symmetric field strength tensor for gravity and the spin 2 stuff, and then use the trace of that very same tensor for the spin 0 Higgslike stuff.
A little story... When I came up with this idea a few years ago, I showed it to a friend who recently got a Ph.D. in physics. He pointed out that as a vector expression, it was in error, because the V dm/dt term does not have to point in the R_hat direction, and neither does the V c dm/dR. Bummer. Actually, I felt more like drinking (seriously). The correction was to put a new directional vector on the other side, the V_hat. In a galaxy, the mass is moving around in the direction of V_hat. Vector algebra problem solved.
It was also forgotten, until this recent result with the bullet galaxy. This recent result indicates the gravitational source looks separate from the light source. With MOND that plays with the exponent, that is a puzzle. With dark matter, you can put the dark matter where you like, so it is easy. Why there should be any correlation between dark matter and light sources is now a mystery.
With the chain rule, we have two contributions, one that has to do with things speeding up, the other with where stuff is in spacetime. It is possible for these two effects to not align. Like a rocket ship, the center of mass doesn't have to be where the rocket is. Relativistic rocket science is tough!
The relativistic 4-force law is d mV^u/dtau, where V^u has four places, and tau is the spacetime interval, sqrt(t^2 - R^2). Nothing is going fast, so we get classical laws. In the case of gravity, the road from d/dtau goes to d/dt. Simple, and standard enough.
d/dtau is asking about changes with respect to spacetime intervals. We know the changes with respect to time work for our little solar system. What I am suggesting is that a change in spacetime may in the classical limit also be seen as a change in space. That would require c*d/dR to have the same units.
There are limits to what can be done in ASCII, so they appear as derivatives.
There is nothing radical about the V dm/dt, which people sometimes mention does not amount to squat. There is nothing radical about saying the "truer" force law must be a 4-force law. There is nothing wrong with the units in the switch from d/dt to c*d/dR. Don't worry, I do think it is a darn strange thing to do, but the data is forcing us in an odd direction, and at least the math here is far more constrained, as there are NO new factors or mass distributions, just relativistic rocket science.
It doesn't work for galaxies, it doesn't work for the big bang, it is broken for almost anything BIG. It also has a tiny bit of error that GR corrects, but that is minor. The problems with this law are HUGE. So we have two schools of thought. One wants to stuff the big M box with dark matter:
d mV/dt = - G (M + Dark_M) m/R^2 R_hat
These folks get to put Dark_M wherever it needs to go to get the answer right. Then there the MOND folks who want to mess with the R:
d mV/dt = - G (M + Dark_M) m/R^2 or if dV/dt is small, d m V/dt = - a_0 sqrt(G M/R^2) m R_hat
where a_0 is a new constant in nature that changes the form of gravity's law if tiny. I got my own proposal. Remember the chain rule from calculus?
d mV/dt = m dV/dt + V dm/dt
That V dm/dt is the stuff of rocket science. We know it is not relevant for stars cause those big star things and galaxies don't change. But we could, just for the fun of it, do a relativistic swap-out, and consider:
d mV/dt = m dV/dt + V dm/dt + V c dm/dR
Force is a change in momentum, which can be seen either as the usual acceleration, the rocket-ship effect, or as where stuff is distributed in space. That sounds like what is going on. So my proposed modification is this one:
d mV/dt = m dV/dt + V dm/dt + V c dm/dR = - G M m/R^2 (R_hat + V_hat)
Too bad I suck at numerical integration or I'd try and see if it could match real data sets. I like it because it uses stuff we know is true (the chain rule) with a fun twist to make an old law point in a new direction.
m^2 c^4 = E^2 - P^2 c^2. We cannot actually say what the signs of any of these are relative to each other since all the terms are squared. You do quote the predominant view that everything is positive. Too bad the simple algebra does not support the case.
I get what 0 Kelvin means - no vibrations. This issue is different. I know that it takes energy to lift something, and I can get energy back by letting it fall. Therefore I have experience with positive energy (the lifting) and negative energy (the falling). If I were to film the lifting, then play it in reverse, it would look like the falling. So a particle that I decided to define as a positive energy system would - if I ran the film backward - would be its antiparticle. This is what happens on a technical level with Feynman diagrams to calculate probability densities.
A zero-energy plane does not make sense to me. I have no experience with them.
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I agree, any time the group is "bigger" in some sense, there must be a symmetry breaking to get things down to the observed level.
We have observed photon and the three particles of the weak force. The gluons are massless mediators of force between quarks. We have never seen quarks in isolation. I don't think we have ever been able to form a beam of gluons due to quark confinement.
I am claiming something like
SU(3) = (U(1) tensor SU(2))conjugate tensor (U(1) tensor SU(2))
At this time I don't know what all the consequences of that happen to be. I'd love to speculate about other mysteries this proposal my address. I will refrain and keep my eye on what it does, which is give a visual justification for why there are three, and these three in particular, if one wants to develop a complete and smooth collection of any event that an observer at 0,0,0,0 happens to see.
doug
Your position is both widely accepted and reasonable. Mathematically, it is clear that U(1) and SU(2) are subgroups of SU(3). The way I showed this was by the programming in the animation. To make U(1), I took a thousand quaternions at random, calculated q/|q|, and plotted that. To calculate SU(2), a thousand random quaternions were fed into exp(q-q*). To generate electroweak symmetry, U(1)xSU(2), these two were the products of each other: q/|q| exp(q-q*).
Up to this point, I am playing by the book, and have no disagreements.
Now I try to represent SU(3). Sounds like it might require two independent quaternions. Multiplying two randomly assigned quaternions together will not generate a new group, it will generate another member of the group q/|q| exp(q-q*).
Instead of multiplying the two together, I first took the conjugate of one, and then formed the product:
(q/|q| exp(q-q*))* q'/|q'| exp(q'-q'*)
A question I don't know how to answer is how to write this. At first I made a silly mistake, and jotted down U(1)xSU(2)xSU(3). That has 12 elements of its Lie algebra, and I only used 8. In terms of the calculation, I know I calculated q/|q| exp(q-q*) twice for two randomly select quaternions, about 12,000 times for the youtube animation.
> electric charge would be easily violated by some strong force interaction
If true, the proposal would be wrong. Working with the nuts and bolts as I do, I see no reason why the q/|q| - the basic symmetry of EM and thus charge conservation - are in any danger by taking a conjugate and forming a product with another.
> We probably shouldn't see discrete electric charges under such a scenario.
If true, the proposal would be wrong. Again, I see no reason to suspect this will be the case. I can see the U(1) symmetry doing its thing whether inside SU(3) or not.
As I understand it, chromodynamics is such that it is nearly impossible to calculate anything. I doubt I'll ever understand any of that work. This definitely feels like it is a new direction, so there is hope.
I most like the "ontology", a fancy word for "why". There is no reason why these three groups of all the possible ones out there should be relevant to three of the four forces of nature. Nor is there any idea of how to bring in the forth. The animations give a visual explanation: to be able to describe any possible event in a smooth, continuous way from an observer at 0, 0, 0, 0 one needs the symmetries U(1)xSU(2) times the conjugate of another U(1)xSU(2). Make the ball a different size if you want to include gravity. A simple and visual explanation.
I respect your critique that much more needs to be done to make the case persuasive, but at least I can carry my work around on my ipod and show it to friends, and advancement of a sort.
doug
Thanks for this explanation, I think I get your point.
There is a way this proposal uses the word "gauge" in a different way than is usually done. Gauge means to measure. What is normally meant by this is you can add in a gauge field and it won't change a darn thing. EM is a gauge field theory, as is GR.
I want to use gauge to mean how a measurement is made, with the choice being how the spacetime manifold changes (the connection) or how a 4-potential changes or any combination of the two. So I could decide to work in a completely flat spacetime and do all the work with a 4-potential. Or I could decide to work with a boring (constant) 4-potential, and have the manifold bend as need be. I am making a choice on how the measurement is being made. No field is being added in. I can choose to do a mix and match, using a Newtonian potential to get the g_00 part right, and a dynamic metric with the g_11 a bit bigger than one so the light bending around the Sun works with this combo meal.
Talking with folks schooled in GR, I emphasize the case where the 4-potential is there in spirit, not in the calculation. I solve a differential equation, and out pops an exponential metric:
dtau^2 = exp(2(G^(1/2)q - GM)/c^2R) dt^2 - exp(2(G^(1/2)q + 2GM)/c^2R) (dx^2 + dy^2 +dz^2)/c^2
I almost never write the G^(1/2)q part because unified field theory is not a good thing to claim to do. So for an electrically neutral system, it is zero.
Now to your point, this could be approximated using the Taylor series like so:
dt^2 = dt^2 - (dx^2 + dy^2 +dz^2)/c^2 - (2 GM/c^2R + O(2)) dt^2 - (2 GM/c^2R + O(2))(dx^2 + dy^2 +dz^2)/c^2
This has the local Lorentz symmetry, and a first order perturbation. Cool. I've known that for a long time, but this is a different way to think about it. Exponentials are very math friendly, so in my case it probably would not help in a calculation. It does connect to what other people do with gravity proposals.
Thanks,
doug
The short answer: Jesus, those are tough technical questions that I don't have the answer to.
The creation and annihilation operators are in your quantum field theory book, in the section on Lorentz invariant quantization of a 4D wave. I was unable to understand how to write a spin 2 propagator (I gave some papers from Wienberg in the 1960s to a Ph.D. physics friend of mine, and he also did not figure it out).
I do know that the force equations that arise by varying the action with respect to the 4-velocity have like charges attract. I do know that the field equations that arise by varying the action with respect to the 4-potential have like charges attract. I do know the particles travel at the speed of light c if the trace of the field strength tensor is zero.
> Can you identify your rank-2 symmetry generators with the Lorentz group
I don't know why this has to be done.
doug
You are correct that people have tried to make bigger gauge groups. So far that has not worked. I am trying to make things smaller. We know there is EM and its symmetry is U(1). We know there is the weak force and its symmetry is SU(2). We know there is the strong force and its symmetry is SU(3).
With programs, the problems often appear at the interfaces. The tensor product of these three groups is presented as simple. Yet it means that you have to add a special flag onto both the groups U(1) and SU(2). The reason is that they are subgroups of SU(3), so any element of U(1) times SU(3) will be in SU(3). Identical particles don't like labels.
The animations are a visual justification of the 4 known forces in a smaller group, so I am in conflict with current research. Oh well.
doug
Your impression sounds correct to me. The vacuum field equations I use are linear, and NOT a linear approximation of GR. Instead of binding gravity to the second rand field strength tensor T^uv, the coupling is between a 4-potential and the 4-momentum density. The difference between these two is not big. The only difference is that gravity fields are not sources. This is consistent with EM, were EM fields are not sources. Consistency is good.
e m2gem.pdf
The trick here is to be like EM when necessary, and different where necessary. There are no photon-photon interactions (unless things get odd), and likewise, there are no graviton-graviton interactions. That does not prove they theory is renormalizable. I don't have the skills to prove or disprove that point. I am hopeful, because it is basically the manifestly Lorentz invariant quantization of a 4D wave, but with 2 spin fields, a spin 1 for EM, a spin 2 for gravity. In text books, two of the modes of emission for the manifestly Lorentz invariant quantization of a 4D wave have to be made virtual. I put those virtual modes to work.
I am not a professional, so cannot post to Arxiv. I consider this paper a draft since it has not been read and reviewed by anyone with real qualification. http://www.theworld.com/~sweetser/quaternions/ps/
doug
On point of contention is EM tensor_product Weak tensor_product Strong. That leads to a nice, neat, 12 degrees of freedom. That also reflects the history of particle physics, where we learned about EM first, then the weak force in the 1930s, then then eight-fold way in the 1960s. Things is, U(1), SU(2), and U(1)xSU(2) are subgroups of SU(3). I didn't read that, I saw it, and that was confirmed by talking with a math guy from Mathematica at an APS meeting. I think the standard model should more tightly reflect what goes on with the math, not the history of our discoveries in particle physics. I know this makes EM and the weak force feel "too close" to the strong force, but the math side of my brain says too bad, that's the way it goes. This is not a high energy theory. It is an odd assault on the second x of U(1)xSU(2)xSU(3).
:-)
> * I don't understand your concept of inertial mass breaking gauge symmetries.
Great question. The Standard Model is only about EM, the weak and the strong forces. The Higgs is there to bring in inertial mass. And gravity as everyone knows, doesn't get invited to the party. We all know that is wrong. We need gravity right there. The graviton also must be gauge invariant to travel at the speed of light. So do I want it both ways? Of course I do!
Here's how it is done. Use an asymmetric tensor, d^u A^v. This tensor is reducible, so it cannot represent fundamental forces. Cleave it in 2, to an antisymmetric tensor, d^u A^v - d^v A^u, to do the work of EM with a spin 1 field, and a symmetric tensor d^u A^v + d^v A^u to do the work of gravity with a spin 2 field. The antisymmetric tensor will always be gauge invariant, cruising at the speed of light. The symmetric tensor will also be gauge invariant if and only if the trace happens to be zero. This is the house where the graviton lives. The graviton does the work of gravitational mass. When the trace is not zero, that breaks the gauge invariance of the model. The trace of d^u A^v + d^v A^u is a scalar field that does the job the Higgs is suppose to do. The job is done better because now there is a particle explanation of the equivalence principle: tr(d^u A^v) = 0 is for the graviton and gravitational physics, and tr(d^u A^v != 0) is for the scalar field needed for interial mass.
I'll ramble to anyone over a beer. I bought an IPod just to demo these animations
doug
Sorry Charlie, the animations of the Standard Model are up on YouTube, http://youtube.com/watch?v=ExNPiMcVXww
U(1) is a unit circle in the complex plane. SU(2) is a unit quaternion which is easy to animate if you have software for the job (barf out thousands of exp(q-q*), sort by time, drive through POVRay). Electroweak is the product of the first two. The animation of SU(3) tells you what the standard model is about, namely the ability to smoothly describe any event seen by an observer at 0,0,0,0. Gravity is about the sizes of things, so scale the ball to different sizes in a smooth way, and that is the symmetry behind gravity.
It is inertial mass that breaks the symmetry of standard model, not some phony Mexican hat dance around a false god of a vacuum.
doug
Here's my explanation. Quantum mechanics results from the collision of the 2 biggest ideas in physics: calculus and 4D spacetime. The way to handle these mathematically is with quaternions, which most nerds are not aware of. It is like a complex number, but with 4 parts, one for time, 3 for space. They can be added, subtracted, multiplied or divided like a real number.
Take the derivative of a quaternion function. Oops, not going to work. If you recall the definition of a derivative using the limit process, there is a small differential element that goes to zero. Because quaternions do not commute, writing that on the left is different from writing it on the right. There are papers where people work with such a buggy definition.
Steal from L'Hospital's playbook (which was stolen from Bernoulli). Take 2 limit processes, letting the 3-vector go to zero first, then the scalar. Technically that would be a directional derivative along the real axis. This is the world of classical physics because everything is order in time: a comes first, then b, then c. Imagine a movie with 10 frames, and each one has a clear place in a time line.
Now reverse the limit processes. Oops, same problem again. This time the fix is to take the norm of the derivative: that will be the same whether the differential element is on the left or the right. This is as much as can be known. This is the land of quantum mechanics. Instead of 10 frames of an animation played one after another, the 10 frames are possible states the system can be in. You could superimpose the 10 frames, saying this is all the states the system could be in. Doing a measurement picks out one of the 10 frames, nothing more, nothing less.
Get the math right, and the answer is clear.
doug
quaternions.com
quaternions.sf.net
All the folks using Command Line Quaternions over at quaternions.sf.net do 4D work, from, well, the command line. Any other interface is for wimps. The result is an animated gif. 60 downloads already!
Time + 3D = 4D. This is the way to do analytic animations.
doug
Hello:
n tum/standard_model/standard_model.html
The standard model has the symmetries U(1)xSU(2)xSU(3). The one in the middle, SU(2), is a unit quaternion, where a quaternion is like a real or complex number, but has four parts. I have developed the software to visualize quaternions at http://quaternions.sf.net/ using one number for time, three for space. SU(2) can be represented by the quaternion function exp(q-q*). Feed a thousand random quaternions into exp(q-q*), and get POVRay to make a nice animation. Do the same for q/|q| exp(q-q*), and you have a visual representation of the electroweak symmetry. Smash two of these together, and you get the symmetry of the standard model.
Visually, there is a clear message: if you want to smoothly represent all possible events in spacetime as quaternions, the group description must be U(1)xSU(2)xSU(3). You won't read that in a journal because it has to be done with animations.
http://www.theworld.com/~sweetser/quaternions/qua
doug
Hello:
I've posted clips from TheStandUpPhysicist.com, the education arm of my unified gravity and EM field theory using quaternions. I got a nice OK from YouTube, no questions. At Revver, they said thanks, but are you sure you have permission to use that song? It was "Math Prof Rock Star" by Jim's Big Ego, a perfect tune for the most nerdly narrowcast theoretical physics show ever. The song is licenced under the creative commons license attributions, non-comercial, share-alike license. Hoping to make billions off of the shows, I had negociated with JBE's business manager, and had agreed they get a percentage.
I told the revver folks that story. Sorry, that wasn't good enough. They wanted to see either the contract or get an email from JBE. A few emails later, and the clips are up on Revver.com. So I think revver.com is investing time and money in the process of due dilegence on copyrights.
Revver is better, they share the wealth.
doug
TheStandUpPhysicist.com
I'd certainly like to play Texas hold'em against you. Remember, probability was developed to win more often at cards, and to make more money. There is nothing spiritual about it. There is a lot of diversity out there, whether you are a boy, a girl, or intersex (and the odds are not 50-50-50 fortunately). The odds are better to be conceived as a boy baby, but girl babies survive better. Sorry, not enough stats on intersexuals.
I am working to be analytical about animations using quaternions. The brain SUCKS at remembering visual stuff. Instead, the brain is great at shop & compare. That is why artists use easels by the way - it's not just to hold up the canvas, but because visual memory is so bad, but comparison is so good, so the artist's work can quickly be compared to "the real thing".
a tion/Dynamic_graphs/1276.html
an example animation - http://www.theworld.com/~sweetser/quaternions/qem
the project - http://quaternions.sourceforge.net/
Hello Mark:
L agrangian_to_tests.nb.pdf and http://www.theworld.com/~sweetser/quaternions/note books/Lagrangian_to_tests.nb).
> That is a situation [hair standing up due to a wee bit of static electricity] where forces cancel out.
There are two ways to look at what gravity and EM do: as fields or as forces. If one takes the Lagrangian, and does the variation with respect to the potential, one gets the field equations. If one takes the same Lagrangian, and does the variation with respect to the velocity, one gets the force equations. I still remember exactly where I read that in Landau and Lifshitz! So sure, one can view the issue as forces canceling (which really only makes technical sense for my GEM or gravitomagnetism, not GR). One can describe the exact same thing using a field equation approach. The Maxwell equations are logically consistent, so the story generated by looking at forces must be the same as the one for fields.
> It is demonstrably different from this.
Even if I concede this point, it does not address my reply that we know darn well how to characterize a vacuum, whose algebra is the same as Jq - Jm = 0, Jq != 0.
> And when you square that term you don't get the E&M action.
The asymmetric tensor gets contracted, and the result is
L_GEM = - rhom/gamma
-(rhoq -rhom) phi + (Jqx - Jmx) Ax + (Jqy - Jmy) Ay + (Jqz - Jmz) Az
- 1/2 (d^2 phi/dt^2 + d^2 phi/dx^2 + d^2 phi/dy^2 + d^2 phi/dz^2
+ d^2 A_x/dt^2 - d^2 A_x/dx^2 - d^2 A_x/dy^2 - d^2 A_x/dz^2
+ d^2 A_y/dt^2 - d^2 A_y/dx^2 - d^2 A_y/dy^2 - d^2 A_y/dz^2
+ d^2 A_z/dt^2 - d^2 A_z/dx^2 - d^2 A_z/dy^2 - d^2 A_z/dz^2)
Take the derivative of L_GEM with respect to A^mu, and the resulting field equations should have a familiar look to it as far as EM is concerned. This Lagrangian will not look familiar with expectations based on linearized GR or GR itself. The proposal works differently, using only one connection, not the divergence of two connections as happens in the Riemann curvature tensor.
There is nothing wrong with being skeptical. I am of my own work. Starting from the above Lagrange density, I have derived the field equations, solutions to those equations, and put it all in Mathematica (it is not pretty, such is the nuts and bolts nature of the software, but it is available here:
http://www.theworld.com/~sweetser/quaternions/ps/
Any well-trained person will remain skeptical until with good old paper and pencil they calculate for themselves the divergence of the Christoffel of the exponential metric. I would bet you did not do that, so your stance is reasonable.
At the end of the day, I only care about technical issues, they last.
doug
Hello Mark:
:-) Seeing EM requires a little algebra. The source of the GEM unified field proposal is a reducible second rank asymmetric tensor, d^mu A^nu. Any asymmetric tensor can be represented as the sum of a symmetric and an antisymmetric tensor:
Thanks for glancing at the paper.
>For instance, in equation (2), if you make the mass density equal to the charge density, then you get nothing.
I'll presume you find discussion of a vacuum is a sensible thing, where J=0. People study the propagation of waves in a vacuum which can also be done here. You have pointed out that there is another way to have J=0. The same solutions will apply to the case.
Note that the balance of mass and electric charge cannot happen for fundamental particles because mass charge is so far smaller than electric charge. For example, an electron's mass charge is sixteen orders of magnitude smaller than the electric charge of the electron. This is a macroscopic situation you propose looking at. There are classical situations where gravity is balanced by EM: think of hair standing up due to a wee bit of static electricity.
> But with opposite charge, you do get something.
The total charge, electric charge minus mass charge, can be positive or negative, depending almost entirely on what is going on electrically. If J is positive, the particles will repel. If J is negative, they will attract. Electric charge can have 2 signs, but mass charge can only have 1 (the difference is due to the properties of their respective field strength tensors, which I am about to come to...).
> Your action, equation (1), contains neither E&M nor linearized gravity.
I don't want linearize GR, so I am glad you cannot spot it
d^mu A^nu = 1/2 (d^mu A^nu + d^nu A^mu)
+ 1/2 (d^mu A^nu - d^nu A^mu)
The second term is F^mu;nu (and I am sophisticated enough to care about the semicolon, since the theory is about having the choice to work in curved spacetime).
I have done no work with macroscopic media, which is why there is nothing on D or H. Only if I were to establish these equations are sufficient to do gravity and EM would such an investigation be warranted.
>I'd suggest that if this is something you are really interested in, you take some courses and learn the fundamentals before you start putting together a theory.
There is nothing wrong with this suggestion, but I do think it is meant to be a put down. As to fundamentals I know how to work with, that would include what a Lagrange density is, how to generate the field equations by the calculus of variations, finding perturbation solutions to the field equations, and working with Christoffel symbols.
I want to thank you for at least flipping through the paper and citing equations. Those equations do not map to either linearized GR or gravitomagnetism. That does not make them right or wrong, just alien. I suspect that you did not calculate the divergence of the Christoffel of the exponential metric. That sound like a run on sentence full of hard math!
For those in the listening audience that are unfamiliar with what a Christoffel symbol is (a topic that only comes up in GR courses), it involves taking 3 different derivatives of a 4x4 metric tensor, and contracting that with another metric tensor. Scary, even for hard core nerds. Thing is, for the metric I work with is WAY EASY. The metric is static, no time dependence, so that drops one of the three derivatives. The metric is also diagonal, so another one of those derivatives drop out. One is then taking the derivative of an exponential, which is the exponential times the derivative of the exponent. One ends up with charge over distance!
del_u Gamma_v^0u A^v = Del^2 GM/c^2 R
Doing that calculation for the exponential metric, and getting charge/R was one of the most amazing calculations I have done in my life.
Life is odd,
doug
Thanks for the reference. Gravitomagnetism is an example of starting from the field equations, kind of what Einstein did with GR. The more proper thing to do is to start from the action (which is the sum of every type of interaction that can happen inside a box), which is what Hilbert did. In my proposal, there is a antisymmetric tensor in the action that does the work of spin 1 photons, and a symmetric tensor that does the work of spin 2 gravitons.
The field equations are generated by varying the action with respect to the 4-potential. One ends up with a Gauss' law and an Ampere's law. The vector identities for EM, no monopoles and Faraday's law, are still true, but they do not apply to the gravity analogues in my approach.
A significant difference is that I use covariant derivatives (I'm sure the authors who work on gravitomagnetism formally are writing covariant derivatives, but they are not actually putting them to use). A covariant derivative is like the derivatives we all learned, plus a connection term which has derivatives of the metric inside it. Any measurement of change needs to account for both the change in the thing being measured, and a change in the rulers used to measure the changes. So one ends up with something that looks much like Gauss' law, which for a 4-potential looks like:
Del^2 phi = G^(1/2) rho (I have chosen to make my units all like electric charge)
Now the challenge is to find a solution to this differential equation when phi is constant. With normal derivatives, only a vacuum will work. There is a non-trivial solution if the Del^2 is a covariant derivative. One has to look for a metric, calculate the Christoffel symbol for that metric, and then take the divergence of that, and show it equals 4 pi G^(1/2) rho, in other words, to solve this equation:
del_u Gamma_v^0u A^v = G^(1/2) rho
The metric that solves this divergence of the Christoffel is known in the literature as the Rosen metric. What I wrote earlier was the Taylor series expansion of the exponential metric.
Thanks again for the reference. This is not an attempt to simplify GR or to copy directly off of a Maxwell cheat sheet, but it will be hard to convince anyone who doesn't calculate the divergence of the Christoffel of the exponential metric that is the case.
doug
Hello:
The measurement is still in the range of first order parametrized post-Newtonian accuracy. What the Donkey Kong that means is that these are the coefficients to the metric that are being tested:
dtaU^2 = (1 - 2 GM/c^2 R + 2 (GM/c^2 R)^2) dt^2
- (1 + 2 GM/c^2 R) dR^2/c^2
- R^2/c^2 dtheta^2
- R^2/c^2 sin^2 theta dphi^2
It is the 5 integers there (1, -2, +2, -1, -2) that are confirmed by this experiment. That is NOT NEWS, because it is not new. Shapiro got the same results. What would be news is if the experiment got to second order parameterized post Newtonian accuracy. I asked Prof. Clifford Will an expert on experimental tests of GR when where the data hunters going to gather that data. He said he knew of no one even discussing it. The reason is that the data must for 2nd order PPN effects must be a million fold more accurate, so we need data that is 99.99995% accurate.
I care a lot about 2nd order PPN tests, since that is were my proposal to unify gravity and EM using a 4D wave equation differs. GR says the metric should go here:
GR:
dtaU^2 = (1 - 2 GM/c^2 R + 2 (GM/c^2 R)^2 -3/2 (GM/c^2 R)^3) dt^2
- (1 + 2 GM/c^2 R + 3/2 (GM/c^2 R)^2) dR^2/c^2
- R^2/c^2 dtheta^2
- R^2/c^2 sin^2 theta dphi^2
GEM (gravity and EM):
dtaU^2 = (1 - 2 GM/c^2 R + 2 (GM/c^2 R)^2 -4/3 (GM/c^2 R)^3) dt^2
- (1 + 2 GM/c^2 R + 2 (GM/c^2 R)^2) dR^2/c^2
- R^2/c^2 dtheta^2
- R^2/c^2 sin^2 theta dphi^2
At first order PPN accuracy, the coefficients (1, -2, 2, -1, -2) are the same. At second order, they are different. That's the data I need. I'll probably be dead before it shows up.
doug
The point of gravity is not to do work. If the Universe was completely empty, then to measure the distance between a pair of events would require this metric:
dtau^2 = 1^2dt^2 - 1^2/c^2(dx^2 + dy^2 + dz^2)
Why include the 1^2? It is to emphasize that they are constants, the same for everyone, no matter where they are in the Universe. Yet there is nothing else in the Universe to check on it. Add a blob into the Universe, and those constants no longer work. They are very good. Only nerds with telescopes and atomic clocks could tell it was a wee bit different from one.
General relativity has a way to determine the functions that take the place of the 1^2. They start with a connection, which has three derivatives of a metric. Then they work with the Riemann curvature tensor, which is the divergence of two connections. This means now there is a second derivative of a metric. It is the second derivative of a metric that leads to an equation that can be solved. The problem with GR in my opinion is that one has to compare 2 paths, and that breaks the math machinery of quantum mechanics.
I take a different approach. Gravity is about doing nothing. Once there is something else in the Universe, you must do something, but the very least amount of something possible. That has a name: a simple harmonic oscillator, better known as a slinky. The math gets a little scary because it is a 4D slinky, with two modes for EM (the transverse ones) and two modes for gravity. Still, one is trying to do close to nothing, and that is a slinky.
One gets into 4D kinky slinky physics with a 4D wave equation. That itself has two covariant derivatives. If you look at two convariant derivatives acting one after the other, the trained eye can spot a second order derivative of a metric. That leads to a differential equation that can be solved to yield a metric.
The metric from my work ain't the one for GR (the Schwarzschild solution), it is prettier. Beauty always wins in physics, just like in the movies. It has exponentials in the place of the 1^2, and exponential appear again and again in physics. Why? Well when the exponent is super small, as it is for gravitational systems, it basically is 1. Only the first, and sometimes second terms matter. Those terms are identical for the exponential metric and GR. At second order parametrized Post Newtonian accuracy, the metrics are different, so unlike silly string theory, the proposal can be accepted or rejected based on an experiment.
Gravity is the least you can do because there is other crap in the Universe.
doug
The reason several billion dollars are going into the electromagnetic crop circle known as the Large Hadron Collider is to detect the Higgs particle. The standard model of physics predicts all the detected particles we have seen out there. The model can be represented by the symmetry groups U(1) for EM, SU(2) for the weak force, and SU(3) for the strong force. Well, it can do that so long as none of the none forces has a mass. Oops, that's not what the experimentalist tell us.
So what do theoretical physicists do? They try to sneak in something without breaking it. That is what is known as Higgs mechanism, or the false vacuum Mexican hat dance. Instead of saying the vacuum is a vacuum is the home field of zero, it is claimed that vacuum is utterly false, there is a Higgs field everywhere there can be a where, so that fundamental particles can get some mass when they need it (always). This mathematical trick works because it doesn't mess up the symmetry in the Lagrangian (a fancy way of writing all the interactions that can happen in a volume), but does get the vacuum to add the mass back in.
You may have noticed gravity is not in the standard model. Guess what is going to happen when gravity gets in? Gravity will break the nice symmetry, and no Higgs boson will be needed.
The uber-sophisticated will complain (whine) that gravity is done by a spin 2 particle, but the Higgs is all about inertia, so it must be spin 0. This is not a flaw, it is a sign from Einstein, specifically the equivalence principle that gravitational mass (the spin 2 thing) must be cow-tied to inertial pass (the spin 0 thing). There is no way to wrestle a steer to the ground unless those two are expressions of one and the same thing.
The way I do it, because of course I have my own personal unified field theory, is to use a second rank symmetric field strength tensor for gravity and the spin 2 stuff, and then use the trace of that very same tensor for the spin 0 Higgslike stuff.
Blowing sophisticated bubbles out my butt,
doug
My bad, you are correct. Thanks
doug
A little story... When I came up with this idea a few years ago, I showed it to a friend who recently got a Ph.D. in physics. He pointed out that as a vector expression, it was in error, because the V dm/dt term does not have to point in the R_hat direction, and neither does the V c dm/dR. Bummer. Actually, I felt more like drinking (seriously). The correction was to put a new directional vector on the other side, the V_hat. In a galaxy, the mass is moving around in the direction of V_hat. Vector algebra problem solved.
It was also forgotten, until this recent result with the bullet galaxy. This recent result indicates the gravitational source looks separate from the light source. With MOND that plays with the exponent, that is a puzzle. With dark matter, you can put the dark matter where you like, so it is easy. Why there should be any correlation between dark matter and light sources is now a mystery.
With the chain rule, we have two contributions, one that has to do with things speeding up, the other with where stuff is in spacetime. It is possible for these two effects to not align. Like a rocket ship, the center of mass doesn't have to be where the rocket is. Relativistic rocket science is tough!
doug
The relativistic 4-force law is d mV^u/dtau, where V^u has four places, and tau is the spacetime interval, sqrt(t^2 - R^2). Nothing is going fast, so we get classical laws. In the case of gravity, the road from d/dtau goes to d/dt. Simple, and standard enough.
d/dtau is asking about changes with respect to spacetime intervals. We know the changes with respect to time work for our little solar system. What I am suggesting is that a change in spacetime may in the classical limit also be seen as a change in space. That would require c*d/dR to have the same units.
There are limits to what can be done in ASCII, so they appear as derivatives.
There is nothing radical about the V dm/dt, which people sometimes mention does not amount to squat. There is nothing radical about saying the "truer" force law must be a 4-force law. There is nothing wrong with the units in the switch from d/dt to c*d/dR. Don't worry, I do think it is a darn strange thing to do, but the data is forcing us in an odd direction, and at least the math here is far more constrained, as there are NO new factors or mass distributions, just relativistic rocket science.
Here's Newton's law of gravity:
d mV/dt = - G M m/R^2 R_hat
It doesn't work for galaxies, it doesn't work for the big bang, it is broken for almost anything BIG. It also has a tiny bit of error that GR corrects, but that is minor. The problems with this law are HUGE. So we have two schools of thought. One wants to stuff the big M box with dark matter:
d mV/dt = - G (M + Dark_M) m/R^2 R_hat
These folks get to put Dark_M wherever it needs to go to get the answer right. Then there the MOND folks who want to mess with the R:
d mV/dt = - G (M + Dark_M) m/R^2 or if dV/dt is small, d m V/dt = - a_0 sqrt(G M/R^2) m R_hat
where a_0 is a new constant in nature that changes the form of gravity's law if tiny. I got my own proposal. Remember the chain rule from calculus?
d mV/dt = m dV/dt + V dm/dt
That V dm/dt is the stuff of rocket science. We know it is not relevant for stars cause those big star things and galaxies don't change. But we could, just for the fun of it, do a relativistic swap-out, and consider:
d mV/dt = m dV/dt + V dm/dt + V c dm/dR
Force is a change in momentum, which can be seen either as the usual acceleration, the rocket-ship effect, or as where stuff is distributed in space. That sounds like what is going on. So my proposed modification is this one:
d mV/dt = m dV/dt + V dm/dt + V c dm/dR = - G M m/R^2 (R_hat + V_hat)
Too bad I suck at numerical integration or I'd try and see if it could match real data sets. I like it because it uses stuff we know is true (the chain rule) with a fun twist to make an old law point in a new direction.
doug
m^2 c^4 = E^2 - P^2 c^2. We cannot actually say what the signs of any of these are relative to each other since all the terms are squared. You do quote the predominant view that everything is positive. Too bad the simple algebra does not support the case.
I get what 0 Kelvin means - no vibrations. This issue is different. I know that it takes energy to lift something, and I can get energy back by letting it fall. Therefore I have experience with positive energy (the lifting) and negative energy (the falling). If I were to film the lifting, then play it in reverse, it would look like the falling. So a particle that I decided to define as a positive energy system would - if I ran the film backward - would be its antiparticle. This is what happens on a technical level with Feynman diagrams to calculate probability densities.
A zero-energy plane does not make sense to me. I have no experience with them.
doug