Unifying Gravity and EM

[sweetser]

Dirac algebra and quaternions

Hello Carl:

> Since my background is in elementary particles, I'd prefer that you go with the flat space and explain it again from that point of view and would appreciate that.

No problem, so long as it is clear the ability to choose between a metric theory like GR, and a potential theory like Newton's approach is at the core of what is "new" about GEM. The quotes are required, because all I am doing is exploiting a symmetry property of a covariant derivative, which has been around for a while.Quaternions have been discovered and rediscovered many times. The first to find them was Gauss, because Gauss discovered everything. Only much later did they realize this bit of one of his notebooks was an uncredited invention. Hamilton discovered them while trying to come up with a rule for multiplying triplets (can be done, but division does not work). Rodriguez discovered them while trying to do 3D rotations, their one wide use today.

The Pauli algebra is almost the quaternions. For doing calculations in physics, they make things a little easier, because there is an extra factor of i. It is simple to set up the Lorentz group using Pauli matrices. It is not simple to do with real quaternions. De Leo first did it in 1996 or so. Peter Jack was the first person to write the Maxwell equations using only real quaternions, and I repeated the trick independently a year later.

There is a cost to convenience, and it is very subtle. With the Pauli algebra, one can multiply two non-zero numbers and get zero. How many pairs of these are there? About an infinite amount, plus or minus 42. In some ways, these infinite zeroes don't do anything but complicate statistics. It is my belief, without evidence, that these bogus zeroes could be the reason one has to do regularization and renormalization in quantum field theory. This is 100% speculation. I could prove the point by doing a calculation using quaternions exclusively (no gamma matrices), and demonstrating that regularization and renormalization were not needed to make the calculation behave. I am quite certain I could not do such a calculation on my own, only able to act like a adviser for people skilled in the arts of quantum field theory. I have no expectation that this speculation with be confirmed or rejected in my lifetime. It is a favorite speculation of mine though :-)

I am a big fan of time! The reason is that time lies straight down the diagonal of a quaternion. With all of our incredibly tiny relativistic velocities, it says that we change hardly at all in space, that nearly all of the change we experience is through time.

It is one of those funny "size" issues: most people will say that the Dirac algebra is bigger since quaternions fit inside the Dirac algebra, along with all those bogus zeros. I claim that being a division algebra will allow you to do more, and it is the size of what can be done that matters most.

What is the biggest, most important algebra? The one the user thinks they can do the most in :-) In my case that would be the quaternions, in yours, a variation on the Dirac algebra. I actually have a limitation with quaternions, it is the pea under a mountain of mattresses for me. I have no idea how to handle the connection. It is also clear that far too few folks work with quaternions, so to communicate, I had to translate all the work I did initially in quaternions into tensors.For anyone interested in seeing good old technical conflicts, I posted something in slashdot.com about my work, and got in a classic nasty discussion (well, he was the one to toss insults, and I did learn something, that a simple model of mine was too simple. Apres the name calling part, he is back to the banal defense of the status quo).

HTTP://science.slashdot.org/comment...mmentsort=3&mode=thread&pid=15604834#15605147

 

[Don J]

For anyone interested in seeing good old technical conflicts, I posted something in slashdot.com about my work, and got in a classic nasty discussion (well, he was the one to toss insults, and I did learn something, that a simple model of mine was too simple. Apres the name calling part, he is back to the banal defense of the status quo).

HTTP://science.slashdot.org/comment...mmentsort=3&mode=thread&pid=15604834#15605147

A good discussion netheless without too much math formulas...only pur discussion about the subject at hand...glad that you finally try to challenge your theory outside the relatively confortable independant research section of this forum...its hard to play in the major league Isn't it?
Maybe its time to try
http://www.bautforum.com/forumdisplay.php?f=18

Warning: -avoid presenting your theory in the against the mainstream section where the discussion can turn harsh about your ideas -

 

[sweetser]

I consider slashdot and this forum to be the random league. Someone random person comes in with a comment, which I usually have to get them to repeat a few times, and eventually I will see their point. I've learned three things from the first 12 pages of this thread:
1. the field strength tensor indices should both be up, $\partial^{\mu}A^{\nu}$,
2. to get the Lorentz force from the action I need to include the inertial term in the Lagrangian, $-\rho/\gamma$,
3. the GEM action breaks U(1) symmetry for massive particles.

In the slashdot discussion, I learned that my simple argument for a transverse mode of emission is too simple. I have to rely on how the field is quantized to justify the gravity waves are longitudinal or scalar modes of emission.

Harsh? I don't care about that. I make a note of it, and focus on the technical content.

Let me list the initials of the major leaguers I have dealt with, and by that I mean professors who are good enough to have their research funded: AG, SD, and CW. There were so busy, I spent 45 minutes with #1, and ten minutes with the other two, which is not time to present the idea. Again, this is an observation. The pitch to Seth Lloyd is like triple A league. He might get to reading the paper.

I am going to outsource the critique. I know someone who was a physicist in Russia, so I'll try and get feedback from the other side of the globe. I also have a friend in India who is works in at an institute, and will try to work that too.

This forum has helped me in measurable ways, so thanks to all who have participated so far.

doug

 

[CarlB]

Doug,

The factorizations of zero in the Pauli algebra are of the form $$(1\pm \sigma_u)/2$$. To get equivalent factorization in the quaternions requires that we add an imaginary unit to it, as in:
$$(1 + k\sqrt{-1}) \; (1 - k\sqrt{-1}) = 0$$

So is the Pauli algebra equal to a complexified copy of the quaternions?

Carl

 

[sweetser]

bingo, bingo.

Here is my ontological problem with that approach. The quaternion (a, b, 0, 0) behaves with other quaternions of this form EXACTLY like a complex number. So it would be fair to call this a+bi. The quaternion (a, 0, c, 0) also behaves EXACTLY like a complex number. Then there is (a, 0, 0, d), which also could be written a+di, and nothing about how it plays with (e, 0, 0, f) would be different from complex numbers. Now when we call a quaternion a+bi+cj+dk, and have 3 distinct imaginary numbers, it looks like cheating to add in a forth i that behaves like the first one used to. One could claim that as long as we make the rules, it is a fine and very productive thing to do for a mathematician. Thing is, I don't care about mathematicians, I care about Nature, and she is more demanding of silent perfection in how thing work (if they don't work, they die, so that is as demanding as it gets). I understand that there are gaggles of humans that are skilled at manipulating complexified quaternions. It is my belief that Nature does not use that algebra. An odd belief, but quaternions.com demonstrates my practice.

doug

 

[sweetser]

Black holes (not)

Hello:



In a different forum, someone asked me about black holes and this proposal. For politeness sake, if you have any questions related to GEM theory, please do so right here.



The short answer is that I know the math behind very dense gravitational sources is going to be fundamentally different from what appears in general relativity, but I do not understand the details (and the details matter). The differences will be so significant that current efforts to understand black holes will have to be dismissed should GEM theory succeed both on experimental and theoretical grounds.



I have two ways so far of finding physically relevant applications of the GEM
theory. The first is the exponential metric that solves the field equations for a single spherical source with the choice of a constant potential. The second is a potential that in an approximation has a derivative with a $1/R^2$ dependence. The potential gets plugged into a Lorentz force law to get to te same exponential metric expression (odd but true). In these two approaches, there is a point in the derivation where I say that the change in time is tiny compared to the changes in distance, so assume the small time contribution approximation. That is how one gets to the kinds of solutions we see often in Nature: changes in space dominate the near vacuum of a Universe we live in 13.6 billion years apres the big bang, with enough of a nod to changes in time that special relativity is respected.



Now consider the case where the changes in time are as significant as those that happen in space, a condition which may appear for very dense gravitational sources. The potential whose derivative is $1/R^2$ via an application of perturbation theory will no longer be applicable. The potential will have a derivative that is $1/R^3$, and then have a force that had the same inverse cubic dependence on distance. How odd! I have heard it said that an inverse cubic law is unphysical. That is true for a classical law of gravity. The math may give a different story for dense sources.



I recall from classes on differential equations taken decades ago that to say an equation was singular had a precise technical meaning. Nearly all researchers consider the point singularity that appears in general relativity as something worth working on, not a thin ice warning that could open up and drown a large body of work. There is a singularity for my field equations, but it is not a point singularity. Instead the equation blows up lightlike intervals, when $tau^2=0$. That may turn out to be a better deal, because we know there are particles like the

photon and graviton that live on the lightlike surface.



Should the GEM hypothesis get a following, the behavior of singular solutions will be a fun area of study.



doug

 

[A_Wanderer]

Doug Sweetser type-person:

This is not in the nature of a reply, yet now and then a question is worth a thousand words - did Godel and his progeny ever have anything discrete to say about completeness:consistency as related to GUT and so your little GEM?

 

[sweetser]

Logical consistency

Hello Wanderer:

My first introduction to Godel was through the book "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. That inspired me to buy "Godel's Proof" by Nagel and Newman, a book that got to the point in 102 pages instead of 777. I have reread that book more times than any other on my bookshelf, so I have a feel for the technical detail. There is no technical connection whatsoever between GEM (or as far as fields, G, E, B) and Godel, Escher, Bach, other than a happy accident.

I tried a logical consistency argument on Hofstadter himself. Darn, I wish I could find my notes from his talk he gave at Harvard, but I recall it was an overview of physics history. His point may have been - my memory is fuzzy - about how physics has gone for consistency over being radical. Work that goes against consistency loses.

I got all excited by the thesis, and wrote out a question. We have a logically consistent theory for EM, that would be the Maxwell equations. We have a logically consistent theory for gravity, that would be general relativity. These two theories are in conflict with each other. So we have to choose between the two, either the Maxwell equations or general relativity, and if we choose one, the other will become a historical footnote, and is wrong. Given the connection between Maxwell, quantization, and the standard model, I would say the weight of history is for the consistency of Maxwell equations over general relativity. Granted we don't know which one is correct, but given his sense of the history of physics, which theory does he think may be shown to be most consistent in the future.

A little bit about the setting: it was a packed large lecture hall at Harvard because Hofstadter has some media draw what with the Pulitzer Prize for the book most physicists have read and enjoyed. The big names in physics in Boston were in the house. There were plenty of young people, and a few really old ones who you know probably figured out some very important things in physics or math in their prime. I was excited by my question, because it would press Hofstadter on his own thesis to say something edgy: either Maxwell's field equations were flawed, or Einstein's field equations were flawed. Logic is that tough.

After his talk was done, I got to asked my question first. I recall feeling the question was audacious: I was uncomfortable with saying in public there was a choice, and with that choice, one would say Maxwell remains right, GR is wrong, or GR remains right, and the Maxwell equations are wrong. I was that clear in the confrontation.

Hofstadter played defense. He noted he as a historian of science, and was not a researcher. He was unwilling to speculate on any future direction for physics research. As far as I can tell, no one in the audience found my question of interest. I know someone else in the audience who heard the question did not think the choice between Maxwell and GR made sense (he was a string theorist).


Einstein and Godel both worked at the Institute for Advanced Studies at Princeton. They went on walks together, discussing physics. This was late in Einstein's career, so all he was working on was unified field theory and the logical foundations of quantum mechanics. I know they collaborated on a paper together, one on closed loop solutions to the Einstein field equations. I don't know more of the history than that.

GEM is not grand, it is extra ordinary.

doug

 

[CarlB]

The short answer is that I know the math behind very dense gravitational sources is going to be fundamentally different from what appears in general relativity, but I do not understand the details (and the details matter). The differences will be so significant that current efforts to understand black holes will have to be dismissed should GEM theory succeed both on experimental and theoretical grounds.

Doug, I've got my Java applet for gravitation simulation almost done. It now produces correct Newtonian orbits and the vast majority of the user interface works perfectly. I'm working on the equations of motion for the general relativity version, which will assume a flat space coordinate system as done by Lasenby, Gull and Doran as in:
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/grav_gauge.html

And what does this have to do with you? Because the Lasenby Doran and Gull gauge version of gravity works on flat space, it is very natural to compare with the Newtonian version. And I'm guessing that since your theory also works on flat space it will also be easy to simulate. And therefore I've also got a spot in the simulation for the Sweetser version of flat space gravity.

What I need from you, if and when you've got it, is a flat space version of force (i.e. change in velocity per unit time) as a function of position and velocity (i.e. phase space). To get you started, here is the Java code for the Newton theory. (All units are natural.)

private DeltaV Newton(PhaseSpace PS) {
DeltaV DV = new DeltaV();
double R = Math.sqrt(PS.X*PS.X+PS.Y*PS.Y);
R = R*R*R;
DV.DVX = -PS.X/R;
DV.DVY = -PS.Y/R;
return DV;
}

The above routine computes "DV" or change in velocity, from "PS" or position in phase space. The calculation is made in two dimensions (i.e. the z direction is ignored and all orbits stay in the same plane). The change in velocity therefore has two components, (DVX,DVY), and the phase space has four components, (X,Y,Vx,Vy).

For the Newtonian calculation, the force does not depend on the velocity, so there is no use of things like "PS.Vx" or "PS.Vy".

Anyway, the orbit is found by integrating the acceleration numerically. The applet allows you to change the initial condition and see how the particle orbits for the different physics assumptions differ.

I've written it so that the physics is encapsulated in a simple subroutine as shown above so that the Java will be easily modified by someone who doesn't want to learn the ins and outs of object oriented programming of a user interface.

Now when this is done, I'll put the applet, along with its source code, up at my website //http:www.GaugeGravity.com[/URL] , which is intended to promote the Lasenby, Gull and Doran version of GR. I've put my current (test only) version up on the net here:
[PLAIN]http://www.gaugegravity.com/testapplet/SweetGravity.html

The buttons Einstein, Newton and Sweetser don't work, and I need to readjust it so that it runs faster, which you can accomplish (with some loss in numerical precision) by hitting the "faster" button a couple times. The numerical precision isn't adjusted according to how close you are to the gravitating point, so it can get a bit dodgy on near misses. And when you alter the initial parameters (with a carriage return), it stops the simulation and to start it again you have to press "go". I'll make the stop/go button be red / green so it is more obvious what is going on.

By the way, to translate from LGD's version of the Schwartzschild metric back to the usual GR theory, the reason that they work on a flat space (i.e. coordinates are x,y, z and t), is that their ds^2 is not diagonal, but instead includes a drdt term. Thus their solution is not symmetric with respect to time, which I think is a great thing.

Carl

 

[A_Wanderer]

Hello Wanderer:

My first introduction to Godel was through the book "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. That inspired me to buy "Godel's Proof" by Nagel and Newman, a book that got to the point in 102 pages instead of 777.

Hi.

Thud. Seven hundred and 77 pages of a POPular derivation of a technically and imaginitively profound mathematics. You have more patience and tolerance than I could muster in a hinayana trance! Thank heavens you found the Nagel/Newman. Better yet would be to first extract Godel from that framework and wrestle with it unencumbered by focal bias.


What I was suggesting was: Does not Godel's grand reductio establish that any GUT which expression is more complete and and so (inversely?) LESS consistant (in comparison to any theory on an opposing gradient) shall be closer to the truest description/expression/model of the measurable universe (all its forces from the Plancky, pointy ultrino to the BB and any and all strange, possibly stringy, curved, n-dimensional, & etc. in between or amidst) - given that the impetus for GUT from the start IS unity/completeness? Despite, I know a 2500+ year historical record with a rather strong attraction to the beautiful elegance of consistancy.

It is most counterintuitive that inconsistancy then becomes a marker for fundamental veracity/agreement the more complete the GUT (or for that matter the GEM) expression. Yet this surprising in-your-face inconsistancy, oddly, or evenly, may be a consistant balance of inconsistancies when placed next to wave/particle and the where/when of quantum subatomics.

I briefly apologize for being rather off-topic - Sorry!
But sometimes, I can just not resist, and have a weakness for the gravity of my own conceits. And groaners.

I dislike my own speculations. (Clearly NOT theories, nor even hypotheses.) I am very fond of consistancy.


the Big Bang - Everything from nothing! Is this not perfect elegance?


fondly,

Bill Snyder
aka
A_Wanderer

 

[sweetser]

Java applet

Carl:

Wow!

Sorry, just a gut reaction. It will take a few days to think about this, but thought I should put on record my initial reation.

Another reflex reaction...

>A flat space version of force (i.e. change in velocity per unit time) as a function of position and velocity (i.e. phase space).

Spacetime can be treated as flat. Force cannot be treated as m dv/dt. Instead, use the chain rule Luke.
$F = m dv/dt + v c dm/dR = - G M m/R^2$
If you get GR correct, you should end up with an animation that is identical.

Damn, need to get ready for work...

Kudos, kudos,
doug

 

[CarlB]

Kudos, kudos, doug

Yes, the Cambridge geometric algebra guys are awesome.

I've uploaded the next version of the applet. This one draws multiple test bodies, which makes for a more pretty display. I've set the initial conditions so as to give a demonstration of the conservation of angular momentum in the Newtonian potential:

http://www.gaugegravity.com/testapplet/SweetGravity.html

I guess it's possible that I'll add the GR simulation within the next 24 hours, and if so, I'll update this note accordingly.

Carl

 

[Norman Albers]

I am running hard to catch up with you and am almost there. Forgive me for not yet having read this entire disucssion; I shall, but need to start communicating ideas with you. We need to let go of the zero current and charge densities. I have shown minimal, clear construction of electrons and photons by allowing infinitesimal divergence in the field. Now I am studying GR (last chapter in Adler, Bazin, Schiffer) to find the lay of the land. We must allow the vacuum to cook up this inhomogeneous stuff! I have found fundamental problems, in that my photon fields can express a Lagrangian with three different orders of time derivatives, not useful.

 

[sweetser]

Godel and speculations in physics

Hello Bill:

What I was suggesting was: Does not Godel's grand reductio establish that any GUT which expression is more complete and and so (inversely?) LESS consistant (in comparison to any theory on an opposing gradient) shall be closer to the truest description/expression/model of the measurable universe (all its forces from the Plancky, pointy ultrino to the BB and any and all strange, possibly stringy, curved, n-dimensional, & etc. in between or amidst) - given that the impetus for GUT from the start IS unity/completeness?

I am unfamiliar with any way of taking Godel's incompleteness theorem and translating that into either a field equation or an action. As far as I know, the theorem remains in the house of logic.

Did you know that logic can be translated directly into algebra? Scientific American had an article on this a number of years ago. The translation is particularly relevant for fuzzy logic. Here's the cannonical example. There is a card which says on one side: "The statement on the other side of this card is true." OK, call that $x$. On the other side, the card reads: "The statement on the other side of this card is false." That would then be $1-x$. So what is x?
$$x=1-x$$
$$2x=1$$
$$x=1/2$$
Math I can solve :-) The card is full of half truthes. The point of this story is that it may become possible to translate something at the foundations of logic into algebra someday. What would be required is a way to translate numbers into differential equations. I hope that sounds like BS, because I am clueless as to how to do that.

...next to wave/particle and the where/when of quantum subatomics.

I'd have to start a new thread on this forum to treat this right, but my one liner reply is that if you do 4D calculus correctly, the reason causality is different between classical physics and quantum mechanics is clear. Check out the video "Why Quantum Mechanics is Weird" at TheStandUpPhysicist.com.

I dislike my own speculations. (Clearly NOT theories, nor even hypotheses.) I am very fond of consistancy.

My level of discomfort is dictated by how much of it can be translated directly into math. I have wondered about Godel, and am totally uncomfortable with it since zero can be translated into algebra. I am very comfortable saying I have a proposal to achieve unification of gravity and EM because it is so specific, I can feed it directly into Mathematica, a fact that so far has failed to make an impression with professional physicists (a busy lot).

I do have speculations in between. Let me give you the details of one of them. The standard model has the gauge symmetry U(1)xSU(2)xSU(3). There are obvious questions to ask: why three groups, why these in particular? The answer we have is clear: we have no idea. This is the kind of problem that is beyond hard. There is nothing you can do to "work" on it. All you can do is be aware of the clarity of our ignorance about the standard model.

What kind of symmetry characterizes the 4D wave equation at the heart of my unified field proposal? Because my background involved much work with quaternions, one of the things I know is that the group SU(2) are the unit quaternions. How does one make a unit quaterion? Easy, just make sure the diagonal is zero, $A-A^*$. The unit quaternion has three of the four degrees of freedom available to a quaternion. What should the fourth one do? The group U(1) is usually represented as the complex numbers with a norm of one. The group is Abelian, that is to say it commutes with other members of the group. Quaternions are well know as not being Abelian. Under special circumstances, quaternions can behave like an Abelian group. The prime example is if all the quaternions point in the same darn direction. I realized a normalized quaternion would commute with its SU(2) sidekick, like so:
$$\frac{A}{|A|} (A-A^{*}) = (A-A^{*}) \frac{A}{|A|}$$
This is the electroweak symmetry. If I write my GEM field equations like so,
$$J_q - J_m = \square^* \square \frac{A}{|A|} (A-A^*)$$
this has U(1)xSU(2) symmetry! Now the unified field theory contains gravity through the diffeomorphism symmetry, EM through the U(1) symmetry which is not perfect due to massive particles, and the SU(2) symmetry of the weak force which is behind nuclear radiation.

Now we come to the part that I do not understand: SU(3) symmetry. I know its Lie algebra has to have 8 parts to it. The multiplication table has to be different that the standard quaternion multiplication table. It is possible that the $\square^{*} \square$ part of the GEM field equations have what it takes. There are 8 parts, that is easy. What I (and only I) call the Euclidean product, $a^{*} b$, is not assiociative, because $(a b)^{*} c \not= a^{*} (b c)$. The Hamilton product, $a b$ is associative because $(a b) c = a (b c)$. Quaternions under the Euclidean product still are a group: there is an identiy (1, 0, 0, 0), there is always an inverse, and all products are still quaternions. I am not enough of a professional math guy to prove SU(3) connection. The one course I failed, got an outright E, was a Harvard class on group theory and particle physics. As for the E, it is so a person failing at Harvard remains a cut above the rest.

Now both questions about the standard model have the chance of being answered. The symmetry of the GEM unified field equations is Diff(M)xU(1)xSU(2)xSU(3). The four forces of Nature, gravity (Diff(M)), EM (U(1)), the weak force (SU(2)) and the strong force (SU(3)) all fit comfortably in the same home. This is the sort of specific speculation that keeps me crazy.

doug

 

[sweetser]

Studying GR, studying GEM

Hello Norman:

Sean Carroll's lecture notes on General Relativity are freely available on the web. I printed them out, and for each chapter, transcribed them into my own form. That process took two months, and now I think I have a concrete handle on the math behind GR.

This forum is more a call and response venue: a question gets asked, I reply, and we go back and forth a few times. It is not organized in a logical way :-) For that, I recommend either downloading the half hour shows at TheStandUpPhysicist.com, or ordering the DVD (1 sale in the world so far, to a physicist friend). Although each show is only a half hour, it would take more time if you tried to confirm that the math makes sense.

doug

 

[CarlB]

Now both questions about the standard model have the chance of being answered. The symmetry of the GEM unified field equations is Diff(M)xU(1)xSU(2)xSU(3). The four forces of Nature, gravity (Diff(M)), EM (U(1)), the weak force (SU(2)) and the strong force (SU(3)) all fit comfortably in the same home. This is the sort of specific speculation that keeps me crazy.

I think that trying to get SU(3) in there is just one symmetry too far.

You do not have an explanation for why there are exactly three generations of particles. These kinds of things, that is, the representations of these symmetries that happen to correspond to the elementary particles, are at least as important as the symmetries themselves.

But if you assume that the leptons and quarks are composites of three particles, then the SU(3) becomes natural at the same time as you get three generations automtically. And the masses of the leptons become understandable by the Koide relation.

So long as your algebra contains the 3x3 complex matrices, it is of course possible to force SU(3) into it. But this does not imply anything interesting. One could fit any of a large number of symmetries into 3x3 complex matrices. To show a real derivation, you would need to not only pick out SU(3) and all that, but you also need to show why the particular representations that are seen in the elementary particles show up.

As an example, consider the electroweak symmetry. It is not enough to show SU(2) x U(1). What you nee to show is that one ends up with an SU(2) doublet and two SU(2) singlets for each flavor.

If you take a look at the algebra, you will find that this form, a doublet and two singlets, arises naturally. But the same cannot be said of SU(3). SU(3) arises much more naturally as the result of assuming that the elementary particles are composite.

But hey, if you want to put SU(3)xSU(2)xU(1) into complex 4x4 matrices (as is more natural for a Clifford algebra or quaternions, I suspect), then you should read this paper:
http://arxiv.org/abs/math.GM/0307165

Carl

 

[bda]

Hi Doug,

Pardon me for hopping in like this but I thought you could possibly use one little piece of information about the exponential metric. You may or may not know that the exponential metric seems to have been first proposed by Houssein Yilmaz in "H. Yilmaz, New approach to general relativity, Phys. Rev. 111(5), pp. 1417-1426 (1958)". You can do a search on "Yilmaz metric" and a lot of references come up.

In its Euclidean form the exponential metric has been used by Hans Montanus and myself; among other references see for instance: "J.M.C. Montanus, Proper-time formulation of relativistic dynamics, Found. Phys. 31(9), pp. 1357-1400 (2001)" and "J.B. Almeida, Geometric drive of the Universe's expansion, http://www.arxiv.org/abs/physics/0507102 ."
Jose

 

[bda]

But hey, if you want to put SU(3)xSU(2)xU(1) into complex 4x4 matrices (as is more natural for a Clifford algebra or quaternions, I suspect), then you should read this paper:
http://arxiv.org/abs/math.GM/0307165

Carl

As author of that paper may I suggest also http://www.arxiv.org/abs/quant-ph/0606123

Jose

 

[sweetser]

Exponential metric

Hello Jose:

The exponential metric is elegant, in an almost measurable way. The number of strokes of the pen required to write it down is small. If the exponent goes to zero, the terms run to one. Exponentials appear over and over again in critical physics equations. There is a good reason why. The exponential is a small step away from unity that embodies a simple harmonic oscillator.

I am not the best student of the literature, but I do have a copy of the Yilmaz paper. I was able to find 4 papers that had the exponential metric:

N. Rosen. (note: this appears like the first reference)
A bi-metric theory of gravitation.
General Relativity Gravitation, 4(6):435-447, 1973.

H. Yilmaz.
Physical foundations of the new theory of gravitation.
Annals Phys., 101:413-432, 1976.

S. Kaniel and Y. Itin.
Gravity on parallelizable manifolds.
113 B(3):393-400, 1998.

Keith Watt and Charles W. Misner.
Relativistic scalar gravity: A laboratory for numerical relativity.
1999.

The way these four papers generated that metric were not elegant. Misner says he is doing this just so numerical calculations go faster, an argument of convenience over conviction. Rosen tosses in another metric field, and because that can store energy and momentum, it means that for an isolated source, gravity waves can have a dipole mode of emission, which disagrees with experiment. I don't recall the details of the other two, other than I did not get excited by said details. I believe they were variations off of the rank 2 field theory of GR, whereas I am trying to show a rank 1 field theory is a different way to implement a metric theory for gravity via symmetry.

The field equations I have were called beautiful by Feynman (in reference to the EM equations in the Lorentz gauge, not as a unified field theory). The equation that generates the exponential metric is also elegant in its directness - it is the divergence of the connection:
$$\rho_{m}=2\partial_{\mu}\Gamma_{\nu}{}^{\:0\mu}A^{\nu}$$
I remain stuck in 4D where few people do gravity research. Retro man!

doug

 

[sweetser]

Alterations for Newton's classical force law

Hello Carl:

To make a direct connect to the physics literature, we need a different kind of applet. The issue is to generate a velocity profile for a disk galaxy given the mass distribution function as a function of the radius. Let me do this in crude ASCII graphics. We start with a mass/area function that has an exponential decay:

m/area
|.
|.
| .
| ...
-------...
R

From that we calculate the velocity profile:

vel. calculated
| .
| . .
|. .
|.
-------------
R

This has the "Keplarian decline", which is what Newton's theory should generate if the applet is written correctly. Test out GR in the flat spacetime if you like, overlay them, and a difference should not be visable at this resolution. What is seen in Nature is this:

vel. observed
| . . .
| .
|.
|.
-------------
R

When programming, there are so many ways to do things, it helps to be specific. Let's use the mass distribution profile for the galaxy NGC3198. It is a galaxy that is too faint for you or I to ever see with our eyes, no matter what size telescope we used (something like a magnitude of 23). The velocity ramps up to $150,000 m/s$ and stays there. The total mass is $1x10^{40} kg$. The mass per area as a function of the radius is $m/area = 37 Exp(-R'/2.23') solar masses/pc^2$.

The velocity profile is not the only problem with Newton's law. The solution is unstable, so disk galaxies should collapse, but they don't. That's unreasonable, because galaxies last a long time.

Let's consider the forces: it is gravity versus the centrifugal acceleration:
$$m \frac{V(R)^2}{R} - \frac{G m M}{R^2} = \frac{d m V}{d t}$$
To see the proverbial Keplarian decline, the centrifugal forces and gravity are in balance, so solve for V:
$$V=\sqrt{\frac{G M}{R}}$$
If the mass drops off as the square root of 1/R, the velocity can stay constant. The observed light curve instead makes it look like the mass drops off exponentially, much faster than the square root of 1/R.

Algebraically, there are three things that can be done. The first is to "Stuff the Mass box", which goes under the name of the dark matter hypothesis. Folks cannot figure out what dark matter is, but they do know we need more of it than the stuff we know huge amounts about. We cannot see the stuff directly yet despite the extraordinary care astronomers use to analyze light. Since we cannot see it, and don't know what it is made of, folks at computer terminals make up a dark matter distribution that can generate the velocity profile and lead to a stable visible mass distribution. I hope this area of study sounds suspicious to you.

The second approach is "Switch the Equation". It goes under the name of MOND, or Modification Of Newtonian Dynamics. It is claimed that when gravity gets super wimpy, then it becomes a 1/R force law:
$$\frac{V(R)^2}{R} - \frac{G m M}{R^2} = \frac{d m V}{d t} iff \frac{G M}{R^2} > 10^{-10}$$
$$\frac{V(R)^2}{R} - m \sqrt{\frac{a_0 G M}{R^2}} = \frac{d m V}{d t} iff \frac{G M}{R^2} < 10^{-10}, a_0 = 10^{-10}$$
MOND does a good job with real data. That is is strength. The theory is a weakness. Recently someone figured out the Lagrangian required to get this sort of thing to work out, and I heard it apparently is not a pretty site. Suspicious? You should be.

And the third possibility is "The relativistic chain rule for a distributed mass source". That probably is not familiar. Well, the chain rule should be. What does a force do? If you say it is mA, technically you are wrong. Force is a change in momentum, so $F = m\frac{d V}{d t}+ V \frac{d m}{d t}$. The second term is literally the stuff of rocket science. For a spiral galaxy that doesn't change its mass over eons, that term can safely be assumed to contribute nothing.

Technically, my expression for force was also wrong. The relativistic force is $F^{\mu}=\frac{d m V^{\mu}}{d \tau}$. The important thing to focus on is the $d \tau$. That Greek letter is for a change with respect to the spacetime interval, not exclusively the time interval. This means that formally, it could be a change in space that the rocket term describes. The classical force would then be:
$$\frac{V(R)^2}{R} - \frac{G m M}{R^2} = m \frac{d V}{d t}+ V \frac{d m}{d R/c}$$
It is like the rocket force term, but applies to space. There is not a label for it, so I'll call it the rocket-space term, a flip of space rocket because this is a flippy idea. What a rocket-space term says can be described. What does a force do? It is a change in momentum. There is the familiar sort of change in momentum, when something changes its velocity. Well, in the outer reaches of a spiral galaxy, there is ZERO change in velocity, even though there still is a gravitational force changing momentum. That's a real puzzle. The force must be changing something else: where mass is in space. That is exactly the kind of curve I wrote earlier - the amount of mass per area drops exponentially in th outer reaches. The change in momentum as one moves away from the center is seen as the change in mass times a constant velocity. In words, it is an exact match.

This is really a new idea. That is rare in physics. It is worth a try, so see if it is consistent with data from a specific galaxy. I should say I have a way to derive this expression, but it takes longer to do so.

****
So this is what I am thinking about programming-wise. Presume a mass/area function of $m/area = 37 Exp(-R'/2.23') solar masses/pc^2$. Use the Newtonian equation to calculate the velocity for R over a range of say .1 to 30 arcseconds. Plot velocity versus R. I know it is easier to skip the units, but try not to as a check that the max velocity is in fact $150,000 m/s$, and the total mass is $10^{40}$. Newton's theory gets the peak right. Then include the rocket-space term and see what happens to the curve.

doug

 

[bda]

Hi Doug

Hello Carl:

And the third possibility is "The relativistic chain rule for a distributed mass source".
...

This is really a new idea. That is rare in physics. It is worth a try, so see if it is consistent with data from a specific galaxy. I should say I have a way to derive this expression, but it takes longer to do so.

Actually I wrote about this a few years ago with someone who has since passed away. We never agreed that the paper was fit for even placing in arxiv, so it remaind in my hard disk until now.

Note that I had not yet got the exponential metric quite right at that time, so now the paper would have to be revised in that aspect; this should not produce significant alterations to the main conclusions.

Jose

 

[bda]

Doug,

Note that I had not yet got the exponential metric quite right at that time, so now the paper would have to be revised in that aspect; this should not produce significant alterations to the main conclusions.

Jose

I had a quick look at the paper and I now realize there are a few mistakes, none of them serious and all easy to correct. The main difference after correction will consist on doubling the exponent in Eq. (7), but that will have no consequences for the discussion and conclusions.

Jose

 

[sweetser]

Hello Jose:

The exponential metric does not lead obviously to the space-rocket term. There is an error of omission in this express:
$$\frac{V(R)^2}{R} - \frac{G m M}{R^2} = m \frac{d V}{d t}+ V \frac{d m}{d R/c}$$
Force is a vector equation. One must include them. You'll notice that the space-rocket term points along $\vec{V}$, not along $\vec{R}$! The correct vector expression is:
$$\frac{V(R)^2}{R}\hat{R} - \frac{G m M}{R^2}(\hat{R}+\hat{V}) = m \frac{d V}{d t}\hat{R}+ V \frac{d m}{d R/c}\hat{V}$$
The rocket-space term suggests gravity works classically in a new direction. This is ONLY relevant for masses that are distributed over a significant amount of space. It is the passive mass small m whose distribution is changed. Of course the sum of the passive small mass is the active mass M. If you feel it is a bit confusing to have the mass in different part of the same equation, that is the way rocket science works!

doug

 

[CarlB]

$$V = \sqrt{GM/R}$$
If the mass drops off as the square root of 1/R, the velocity can stay constant.

Please forgive a naive amateur, but to get the velocity constant, don't you have to increase the mass proportional to R^2, that is the mass per unit area has to be constant? [Edit: Okay, now I see it. Integral of mass has to be proportional to R, but mass is proportional to area. Nevermind.]

Were I asked to do the simulation you're talking about, I would do it with a large number (maybe 1000) sample points. It would run amazingly slowly in Java. Amazingly slowly. But you could eventually get a result out of it. That's a second stage operation. For now, you still need to tell me what the dv/dt equation is for just a test mass orbiting around a point mass. It turns out that extracting this information from flat space gravity theorists is harder than I expected.

I've now added the logic to throw three types of balls in the air, Newton and two others. I believe I have the correct equations for general relativity, and for the Cambridge geometric algebra relativity, and I should upload the applet later tonight. I've also changed the colors, and added bright white test masses so that you can see them retracing their orbits.

Carl

 

[bda]

Doug

Hello Jose:

The exponential metric does not lead obviously to the space-rocket term.

I agree; no matter which metric one uses the main thing is that velocity is basically determined by

$$\frac{v^2}{r} = -\frac{\mathrm{d}V}{\mathrm{d} r},$$

where $V$ is the gravitational potential; the metric introduces a correction which is significant only for very small r and large M.
You then plug in $V =G M(r)/r$ and apply the the derivation rules
this then results in

$$v^2 = \frac{G }{r}}\left(M - r \frac{\mathrm{d} M} {\mathrm{d} r} \right);$$

it is then obvious that one can get constant velocity.

In my paper I apply this to measured velocity profiles of real galaxies and derive the respective mass distributions; I then compare those with observed light intensity and H1 profiles.

Jose

 

[CarlB]

Doug, I found another author working on combining E&M with gravity, but from a different perspective. I found it quite convincing, and likely compatible with what you're doing, but simpler for a flat space particle guy like me to understand:

For example, see "An Electromagnetic Basis for Gravity" listed on:
http://www.mass-metricgravity.net/

Some papers that you can download from arXiv are:
http://www.arxiv.org/abs/physics/0012059
http://www.arxiv.org/abs/physics/0101033
http://www.arxiv.org/abs/physics/0601013

I just ordered this book which seems to be the most inclusive
reference (the links to the APS preprint server on his website no longer work as APS closed it down so the only free papers I could find on the web are the above three arXiv papers):
http://www.buybooksontheweb.com/peek.asp?ISBN=0-7414-1466-X

The two of you talk quite over my head with respect to gravitation. Anyway, you should send him an email, and see what the differences are between your theory and his. If his theory is the flat space version of yours, then I now have an equation of motion that I can simulate for your version of gravity.

I loved his most recent paper on arXiv, (last one listed above). It explains the apparent acceleration in the Hubble data as being a consequence of an interaction between gravitational potential energy and time. This makes the red shift less than what is apparently observed by just half, which is just enough to eliminate the apparent acceleration and turn the Hubble expansion into a constant velocity again.

Maybe the above paper has something to do with the MOND thingy, but like I said, you can talk with him about it, it's over my head.

Carl

 

[sweetser]

Hello Carl:

I was told by an MIT professor of some standing that I could not claim to have a modern field theory unless I had a Lagrange density. This was very scary at the time because I had the field equations, and had never worked with a Lagrangian before in my life. It was something I had read Feynman talking about, but always thought it was too difficult for me to ever comprehend. Turns out that it is not too tough. If anyone reading this thread thinks the nuts and bolts of a Lagrange density are too difficult, download lectures 1 and 2 from TheStandUpPhysicist.com where I go through the details. If you rewrite the equations I present there yourself, and give yourself some time, the logic is straightforward - it has to be, this is the basic math of how things exchange energy in the Universe.

It was really scary, I didn't think I could find a relevant Lagrange density. I remember how dumb I felt sitting down trying to find one that would generate my field equations. No joking aside, I suck at math. The only way this could work was if it was a bit simpler than EM, which it turned out to be.

The point of this story is that I hold up other research to the same standards that I set for myself. So I did this google search: site:mass-metricgravity.net Lagrangian
That looks specifically at everything Collins has put up for the word Lagrangian. I checked for Lagrange density, action and Hamiltonian too, but found nothing. One needs the Lagrangian to do quantum. So I'll wait on investing effort into understanding his work until he has one.

Nothing like having a well-defined, tough test. It means I can skip over reading most papers. Virtually all of string theory would fail the same test.

doug

 

[CarlB]

The point of this story is that I hold up other research to the same standards that I set for myself. So I did this google search: site:mass-metricgravity.net Lagrangian
That looks specifically at everything Collins has put up for the word Lagrangian.

Tisk. Tisk. He probably won't read your theory because you have no published articles, nothing up on arXiv, and no academic position. If you went to his website, you'd discover that it is only a one page stub and searching it is pointless. He put his unpublished research onto the APS preprint server and that preprint server is no longer available. So there is no way to search the web for the information you're asking for. When I tell you that I see similarities in your work that should be enough to interest you.

He gave me the equations of motion for his theory and it's easy enough to turn that into a Lagrangian. I've asked him by email if he has done this. I would guess that he has, but if not, it's easy enough to do it myself. L = T + V, [edit]uh... well... $$L = T \pm V$$ or something like that.[/edit] and he's quite clear as to what kinetic energy and potential energy is in his theory.

My own point of view is that the equations of motion (or, in QFT, the wave equation) is more fundamental than the Lagrangian. Of course you are aware that QM has been stuck for 30 years. The physicist who gave you that particular piece of advice could very well be a part of the problem rather than the solution.

Oh, and I'm still waiting for you to provide me with equations of motion that work in flat space. I expect that I will have simulations with GR and gauge gravity comparisons with Newton by Monday or Tuesday. The other half of working on all this is that I feel that I am beginning to get a better understanding of gravitation. This is stuff I should have done back in grad school.

Carl

 

[sweetser]

Velocity profile applet

Hello Carl:

He probably won't read your theory because you have no published articles, nothing up on arXiv, and no academic position.

Yup, I know that. And I have a simple standard before I try and publish it: I need someone who understands the Maxwell equations at the Lagrange level, and knows the formalism of general relativity to read my work and see if it is valid. I'll remain in this state until that time. And if I get such a review, and they show me where I have made a mistake, I will point it out here. Note: I have made errors in this thread, but so far, none of them are killers.

If you went to his website, you'd discover that it is only a one page stub and searching it is pointless. He put his unpublished research onto the APS preprint server and that preprint server is no longer available. So there is no way to search the web for the information you're asking for. When I tell you that I see similarities in your work that should be enough to interest you.

If you have a mature proposal, you'd mention the action and the Lagrange density outside more formal presentations.

He gave me the equations of motion for his theory and it's easy enough to turn that into a Lagrangian. I've asked him by email if he has done this. I would guess that he has, but if not, it's easy enough to do it myself. L = T + V and he's quite clear as to what kinetic energy and potential energy is in his theory.

It is in the first paper, equations 65 and 66: T + U = 0. Oops. That's a problem. There is no way to take 0, and do a variation on that to generate the field equations. That is a deadly technical flaw. It also makes quantizing the theory impossible.

My own point of view is that the equations of motion (or, in QFT, the wave equation) is more fundamental than the Lagrangian. Of course you are aware that QM has been stuck for 30 years. The physicist who gave you that particular piece of advice could very well be a part of the problem rather than the solution.

If you have a self-consistent theory, you must be able to go from the Lagrangian to the field equation and to the force equations and to the stress-energy tensor and to the Hamiltonian and through the process of quantization. This is a matching set. It requires a lot of calculus, none of it particularly hard, but the math steps are intimidating. For me, one is not better than another, but having the complete set is necessary before you claim to have a field theory for anything. This is an OLD standard, going back to the nineteenth century. It is cool that the Lagrangian of those times still plays a central role once we got a handle on quantum mechanics in the early part of the twentieth century. The question my approach addresses was first raised in he 1760's when Joseph Priestly realized that electrostatics must be an inverse square law based on a discussion with Ben Franklin. Certainly it was clear to all involved by 1930 that any explanation of gravity had to get along with quantum mechanics, and GR did not. So for me, the time line of these issues is longer.

Oh, and I'm still waiting for you to provide me with equations of motion that work in flat space. I expect that I will have simulations with GR and gauge gravity comparisons with Newton by Monday or Tuesday.

The exponential metric is identical to the Schwarzschild metric to first order PPN accuracy. If programmed correctly, there is no way to tell the difference. Go literally a million-fold more accurate, and the exponential metric has 12% more bending than GR. For a point source, this is not the way to see a difference.

from an earlier post...

Were I asked to do the simulation you're talking about, I would do it with a large number (maybe 1000) sample points. It would run amazingly slowly in Java. Amazingly slowly. But you could eventually get a result out of it. That's a second stage operation.

This slow, repetitive calculation for a thin disk galaxy with an exponentially decaying mass distribution is where it's at. You should have the program generate the velocity profile graph and save it. It turns out one needs elliptical integrals to solve the rotation profile for such a galaxy, and no, I am not good enough to do that (Alar Toomre of MIT was the first to do it in the early 60s, kind of amazing since one would have thought the problem was easy, but it is not).

If one eliminates V hat from this equation:
$$\frac{V(R)^2}{R}\hat{R} - \frac{G m M}{R^2}(\hat{R}+\hat{V}) = m \frac{d V}{d t}\hat{R}+ V \frac{d m}{d R/c}\hat{V}$$
what remains is Newton's law. Now I could go about and figure out how my proposal tweaks this a bit away from plain-Jane Newton, but if I did it right, it should be no different from GR at the resolution of the applet. It is for a system with a distributed mass that GEM theory is noticeably different.

doug

 

[CarlB]

It is in the first paper, equations 65 and 66: T + U = 0. Oops. That's a problem. There is no way to take 0, and do a variation on that to generate the field equations. That is a deadly technical flaw. It also makes quantizing the theory impossible.

I went and looked at the equations 65 and 66. In Newtonian theory, when a particle is at rest at infinity, its potential energy is zero and its kinetic energy is zero. These add to zero. If you examine particles which have dropped in from infinity, the kinetic energy increase and the potential energy decreases so that the sum stays at zero.

This is a well known fact of Newtonian gravitation. This is not the place where you would start with an effort at quantizing a theory. To do that, you need to write the T and U for all possible particle positions and velocity, not the values for a very specfic single trajectory.

The exponential metric is identical to the Schwarzschild metric to first order PPN accuracy. If programmed correctly, there is no way to tell the difference. Go literally a million-fold more accurate, and the exponential metric has 12% more bending than GR. For a point source, this is not the way to see a difference.

What I'm hoping for is that you will write me a force equation for movement of uncharged particles on a flat space metric. The Schwarzschild metric is not flat, so under this requirement, you will not be giving me something that is identical to the Schwarzschild metric.

For example, the Cambridge gauge gravity is built on a flat space metric and sure enough, their equations of motion are different from that of the Schwarzschild. But they give orbits that are exactly identical to that of Schwarzschild. I know this very well from the effort of programming the computer to draw the orbits. The Schwarzschild particles will get stuck on the event horizon while the Cambridge particles go on to the singularity. And for orbits that avoid the event horizon, the two theories will give results that are identical (after correcting for their different assumptions about the relationship between radius and coordinate time). This I will demonstrate by showing particles that collide in one theory, collide also in the other, but with their positions as a function of time being otherwise (i.e. at different radii) different.

Can you give me a link to your flat space metric or anything else I might start with so that I could get an equation of motion? The Collins proposal matches GR to well within the current accuracy of measurements of light bending and Mercury precession, but I know that my simulation will show differences. I'm quite good at this sort of thing. I should have it running in a few days as I've half programmed in the (rather difficult) general relativity and gauge solutions.

In the unlikely event that I am unable to distinguish two different theories with double precision arithmetic, then it is time to talk about some other, more complicated simulation. At first, baby steps only. But I doubt that your theory will differ so little from GR that I cannot distinguish it with the accuracy I'm working at.

Carl

 

[bda]

Hi Doug,

The exponential metric is identical to the Schwarzschild metric to first order PPN accuracy. If programmed correctly, there is no way to tell the difference. Go literally a million-fold more accurate, and the exponential metric has 12% more bending than GR. For a point source, this is not the way to see a difference.

Well, yes and no. The Schwarzschild metric in isotropic coordinates is

$$\label{eq:isotropic} \mathrm{d}\tau^2 = \left(\frac{\displaystyle 1-\frac{M}{2r}}{\displaystyle 1+\frac{M}{2r}}\right)^2 \mathrm{d}t^2 - \left(1+ \frac{M}{2r}\right)^4 \left[ \mathrm{d}r^2 - r^2 \left(\mathrm{d}\theta^2 + \sin^2 \theta \mathrm{d}\varphi^2 \right) \right];$$

and the exponential metric is

$$\mathrm{d}\tau^2 = \mathrm{e}^{-2M/r} \mathrm{d}t^2 - \mathrm{e}^{2M/r} \left[ \mathrm{d}r^2 - r^2 \left(\mathrm{d}\theta^2 + \sin^2 \theta \mathrm{d}\varphi^2 \right) \right].$$

It is true that they are PPN equivalent but this criterion applies only far away from a black hole. If Carl makes the simulation near a black hole horizon the orbits originated by the two metrics will be very different.

Jose

 

[sweetser]

Hello Jose:

For now, my focus is on a huge problem in classical gravitational theory, the rotation profile of thin disk galaxies. There might have a black hole in the center, but the big problem is after the peak velocity is reached. Newtonian theory, as an approximation of GR, gets the peak correct. The exponential metric itself does not correct the problem. It is the rocket-space term that is my hypothesis for a solution to the problem. Gravity still is in play after the peak is reached, but the change in momentum is not due to a change in the velocity of the particles. Instead the change in momentum is due to the change in the amount of mass of the galaxy after the peak velocity is reached. I have to show numerically that both terms are in play.

doug

 

[sweetser]

dv/dt

Hello Carl:

This might do the trick. I have a potential which solves the 4D wave equation. For Newton, a 1/R potential solve $\rho=\nabla^2 \phi$. For the 4D wave equation in this classical problem, we want as solution really, really close to 1/R, but not quite. Hence the use of perturbation theory. It gets a bit more complicated in details to have the potential apply only to gravity, and not gravity and EM (such is the plight of unified field theory). In the interest of full disclosure, the weak gravity, electrically neutral potential is here:
http://theworld.com/~sweetser/quaternions/talks/IAP_3/1119.html
The important thing is the derivative of the potential. That is much simpler. It is:
$$\nabla^{\mu}A^{\nu}=-\frac{\sqrt{G}M}{c^2 \tau^2}\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{array}\right)$$
This derivative of the potential has the correct inverse square dependence. You drop this into Lorentz force law,
$$F^{\mu}=\frac{\partial m U^{\mu}}{\partial \tau}=-\sqrt{G} m U_{\nu}/c \nabla^{\mu}A^{\nu}$$
Contract this and do some simplifications, and one has an expression for the 4-velocity:
$$(\frac{\partial U_0}{\partial \tau}, \frac{\partial \vec{U}}{\partial \tau}) = (e^{-G M/c^2 R}, e^{G M/c^2 R})$$
This is still a relativistic expression. In a low velocity limit, one ends up with:
$$\frac{d v}{d t} = e^{G M/c^2 R}$$
That should be simple enough to program, a baby step as it were.

doug

 

[bda]

Doug

For now, my focus is on a huge problem in classical gravitational theory, the rotation profile of thin disk galaxies. There might have a black hole in the center, but the big problem is after the peak velocity is reached. Newtonian theory, as an approximation of GR, gets the peak correct. The exponential metric itself does not correct the problem. It is the rocket-space term that is my hypothesis for a solution to the problem. Gravity still is in play after the peak is reached, but the change in momentum is not due to a change in the velocity of the particles. Instead the change in momentum is due to the change in the amount of mass of the galaxy after the peak velocity is reached. I have to show numerically that both terms are in play.

Look at my paper attached to post #196; it does just what you want.

Jose

 

[sweetser]

Gravity as a repulsive force going backward in time

Hello:

I went to see "Jim's Big Ego" at the Lizard Lounge in Cambridge, MA. The band wrote "Math Prof. Rock Star", which I subsequently have used as the title song for the "Stand-Up Physicist" community access TV series. The concert was good, skilled musicians making sardonic slashes at society. I hung out afterward to give Jim a DVD with twelve of the TV shows. We chatted about physics (OK, it was mostly me doing various physics riffs), but Jim had a question: instead of gravity being an attractive force going forward in time, why couldn't gravity be a repulsive force like EM going backward in time? I tried to answer the question there and then, and in this post provide a more permanent record of my reply.If we think about gravity using general relativity, the theory put forward by Einstein and Hilbert, the question is poorly formed. In general relativity, gravity is exclusively about geometry. There is no force of gravity to be either attractive going forward in time, nor a repulsive force going backward in time. A force is something that can act at a point in spacetime: you apply a force to my nose with your fist, it breaks. General relativity doesn't work

that way, it is nonlocalizable. This can be seen by looking at the Riemann curvature tensor, which is the difference between two connections (the difference of two derivatives and the difference between two products of these two connections):

$$R^{\rho}{}_{\sigma \mu \nu}=\partial_{\mu} \Gamma^{\rho}{}_{\sigma \nu}-\partial_{\nu} \Gamma^{\rho}{}_{\sigma \mu}+\Gamma^{\rho}{}_{\mu \lambda}\Gamma^{\lambda}{}_{\nu \sigma}-\Gamma^{\rho}{}_{\nu \lambda}\Gamma^{\lambda}{}_{\mu \sigma}$$

The field equations of GR uses the difference of two contractions of this tensor:

$$R_{\mu \nu}-g_{\mu \nu}R = 8 \pi T_{\mu \nu}$$

The field equations can never apply to a point, because they always reference the difference between two paths. One cannot discuss the force or gravity at a point in spacetime, or the energy of gravity at a point in spacetime which is just force times distance, all for this same reason. If force is nonlocalizable, the question makes no sense.

In GEM theory, the question is well-formed. Gravity can be viewed as pure geometry, as a pure force, or a mix of the two consistent with what is seen. I have even found a path through a maze of math where the effect of a force is a dynamic metric of spacetime! Since gravity can be viewed as a force, energy is localize, a big plus in my opinion. Let me write out the force equation:

$$\frac{\partial m v^{\mu}}{\partial \tau}=J_{\nu}\partial^{\mu}A^{\nu}$$

Remember those word problems back in 8th grade? The challenge here is to translate the words of Jim's question directly into the expression above. If time were to go backwards, then the derivative on the left hand side would changes signs. To be a force where like charges repel would require a minus sign on the right hand side, just as goes on with the Lorentz force of electromagnetism. Plug those sign changes in:

$$-\frac{\partial m v^{\mu}}{\partial \tau}=-J_{\nu}\partial^{\mu}A^{\nu}$$

This is the first force equation multiplied by -1. Jim's question concerns transforming one force equation into another. Unfortunately, I don't think the transformation between these two force equations gives us any insight into how the force of gravity works. So Jim's question is not "wrong", it is not informative. It was great that the question was precise enough to map to a specific equation.

So now I am going to speculate on the origin of the question, one possible reason it may have been asked in the first place. Gravity dominates our lives, otherwise we might get up and leave the planet. Gravity can kill from falls or through avalanches. Gravity appears like a very active force shaping our world. It is hard to imagine an active attractive force. Such a force would have to find you and convince you to head their way. A force that shoved everyone in the face, no matter their race, color, or creed, that is easier to imagine. A pushy, repulsive force is easy for the mind to envision. That is the way two electric charges with the same sign work. Neutral particles ignore all the shoving.

Ah, neutrality, that might be the key. Nothing is neutral as far as the gravitational force is concerned. That is the real mystery: how come there is absolutely nothing that can ignore gravity? Photons which have neither electric charge or a mass, the ultimate cream puffs, still bend in both time and space to the curvature of spacetime that is gravity. I don't think it is reasonable to imagine anything going out, finding every photon without exception, and either pushing or pulling on that photon. The cause of the unsubtle force of gravity is subtle. Because there is other stuff in the Universe, where that stuff is is a place you cannot be in spacetime. Other stuff subtracts the possibility of where. So as stuff goes through spacetime, it goes where it can. Since there is less where near stuff, the easiest darn path goes through the possible. There is less possible near stuff, so to balance the path, it will look like the path travels closer to the stuff.

How does Nature keep track of all the stuff? How does she balance the books? In standard physics, I don't think people consider this question. For me it is central. The counting system is based on events in spacetime. Events can be counted. There is the event of this electron living over there now. A moment later, it is still there. I would use a quaternion to keep track of where it is in time and space because a quaternion has a slot for time and three for space. Quaternions come with an accounting system: they can be added together, subtracted, multiplied and divided like a real number, but they have these 4 sub-parts to map to our continued existence (time) in 3D space. Armed with a number and +. -. x, and /, the Universe can do all the math necessary.

Nearly all the math necessary is math very close to one, so really, almost nothing is happening, all the parts are staying near the identity. Yet we do move in time, and a wee bit in space. How is that measured, a distance involving both time and space? Recall how distance always seem to involve squares, the Pythagorian theorem? Same thing is true with quaternions. I am in Waltham at three minutes to seven, and hope to be in Jamaica Plain around 8PM to watch a bit of the Tour de France. The time change is about and hour, the spatial distance, about 10 kilometers. The distance in spacetime is
$$((1 hr)(\frac{60 min}{1 hr})(\frac{60 sec}{1 min}))^2 - ((10 km) (\frac{1000 m}{1 km})(\frac{1 sec}{3x10^8m}))^2=12959999.999999999 sec^2$$
Why all the darn 9's? That is because the 7 kilometers count for almost nothing (the factor of the speed of light is what squashes it to insignificant). There are two take home messages. First, time is money, or at least is the Swiss franc, of spacetime, dominating everything else. Second, changes in space, although tiny, are not nothing, and we have to understand their role.

If there was nothing else in the Universe, the spacetime distance would be:
$$d \tau^2 = 1^2 dt^2 - 1^2 dR^2/c^2$$
There is stuff in the Universe. In GEM theory, I could still use this metric, and then be required to use a 4-potential (a scalar potential theory is good enough to account for changes in time, but not space). Or in GEM theory, the presence of stuff could change those constant 1's, making the first 1 a little smaller, the second 1 a little bigger. The amount of change is stupidly small. Why? Well this Earth is stupidly stable. We are hanging around one.

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I hope no one found the preceding explanation completely satisfying. That is what it feels like to wonder how things really work. There comes a point where what you know runs out. I was riding my bike this evening, some 13,000 miles away from the center of the Earth. I could tell when my bike had to be a mere 5 feet further from that point because it meant I had to climb a hill (0.001 miles). Even with all my math tools, it appears miraculous that the bike knows the difference between a few feet more and a few feet less to the gravitational center. No matter how much you know, there is always another few feet to climb from your center of knowledge. Every once and a while, enjoy the glide down the hill and accept that you live in a Universe whose wonders will aways be more than you can embrace.

doug