How Can Gravity and Electromagnetism Be Unified Through a Rank 1 Field Theory?

[CarlB]

Doug,

I have great sympathy for the way you're doing this. I think that tensors are icky because they make the assumption that there is only one sort of symmetry in an object. I'd much rather leave things in algebraic form.

Quantum physics is now done entirely by enforcing symmetry relations so tensors make a certain amount of sense, but in doing this, it becomes impossible to predict relations between different symmetries. Hence the neverending search for a unifying symmetry. What I'd prefer is a unifying symmetry that just happens to have the symmetries one needs, and this is exactly what you are doing.

Getting back to exp(A^u - A*^u) A^u/|A|. Can you write down what this if A happens to be infinitesimal? That is, for first order in the components of A, what is the above? I'd write down what I think it is, but since you're nearby, my tendency towards laziness is multiplied by my tendency towards bossiness and delegation, and probably squared by my tendency to not want to appear stupid in public (the small residual which has survived my being a physics crank), all these things conspire to induce me to ask you to compute the first order for the thing.

Oh, what the heck. We write

A = a_1 + a_j j + a_k k + a_l l

Then A-A^* = 2(a_jj+a_kk + a_ll)
and the exponential of this is
1 + 2(a_jj + a_kk + a_ll)

I'll leave the rest for you. Please feel free to change notation to something more readable.

[edit]Hmm. Sure looks like when we multiply this by A/|A| we're going to get back the whole algebra. In other words, we will get back four degrees of freedom, as advertised. And the commutation rules seem correct.[/edit]

Carl

 

[sweetser]

U(1)xSU(2)xSU(3) in detail

Hello Carl:

I like this line:

What I'd prefer is a unifying symmetry that just happens to have the symmetries one needs, and this is exactly what you are doing.

This was a happy accident, honest!

Since you want me to comment on notation, I'll modify your A by including a basis for the first component, and shift to ijk which is more common. For the sake of consistency, I will make all the "amplitudes" small a_something, and all the basis vectors e_something, like so:
A=a_0 e_0 + a_i e_i + a_j e_j + a_k e_k
The Taylor's series to first order of the exponent is:
exp(A - A^*) = 1 e_0 + 2 (a_i e_i + a_j e_j + a_k e_k) + O((A-A^*)^2)
Now we are going to multiply this by A/|A|. We know this is going to be "easy" in the sense that we do not have to calculate the cross product, it will always be zero:
((a_j (2 a_k) - a_k (2 a_j)) e_j e_k = 0
It is the cross product that makes a quaternion non-Abelian, so if the cross product is necessarily zero, then this particular quaternion product is Abelian. Lazy man does not have to write out both to prove it.

Multiply out the normalized quaternion times the exponent:
\frac {A}{|A|} exp(A - A^*)
=(a_0 e_0 + a_i e_i + a_j e_j + a_k e_k)(1 e_0 + 2 (a_i e_i + a_j e_j + a_k e_k) + O((A-A*)^2))
=a_0 e_0^2 + 2 a_i^2 e_i^2 + 2 a_j^2 e_j^2 + 2 a_k^2 e_k^2, 4 a_0 a_i e_0 e_i, 4 a_0 a_j e_0 e_j, 4 a_0 a_k e_0 e_k)/|A|
In flat spacetime using Hamilton's rules, it looks like this is a standard quaternion. In spacetime where you choose to account for some effect using curvature, the basis vectors may no longer have a norm of 1. This might be the way for me to avoid using some of the tools of differential geometry (and I will keep that claim vague, because that's its state).

So for me, this shows that the product A/|A| exp(A-A^*) has the electroweak symmetry, U(1)xSU(2). Cool.

Now the road out to SU(3). I am not going to calculate q q'. We know that quaternion multiplication without 0 forms a group, so q q' will be in the same group. We also know that quaternion multiplication is associative, (A B) C = A (B C). A little bit of thought will let you see that the Euclidean product, q* q', is not associative. The different groups end up pointing in different directions, so (A B)^* C != A^* (B C). We can look at that more closely if you choose. The Euclidean product still has an identity, there is always an inverse, but you will need to be more careful with parentheses. The norm of A/|A| exp(A-A^*) is one, the norm of B/|B| exp(B-B^*) is one, so will the norm of A^* B = 1? Of course it will! The conjugate points A in a different direction, but that does not change the norm one bit since you square all the a_n's anyway before adding them together. Since A = A/|A| exp (A-A^*) has 4 degrees of freedom, A* B should have 8. Sounds like a way to represent SU(3) to me.

Have you looked at the animations? Those images build my confidence because I get to use a different part of my brain to "get it", as it were.

doug
http://www.theworld.com/~sweetser/quaternions/quantum/standard_model/standard_model.html

 

[CarlB]

Actually, I was hoping you'd remind me what the multiplication rules are for the e_\chi. You haven't, so now I need to figure them out for myself. Looking in wikipedia:
http://en.wikipedia.org/wiki/Quaternion
I see that I need:
e_i^2 = e_j^2 = e_k^2 = e_ie_je_k = -1
e_ie_j = +e_k and cyclic.
e_je_i = -e_k and cyclic.

In multiplying out A/|A| exp(A-A*) it becomes clear that we need to keep stuff to second order in A. Hmmm.

Let's write A = b + cB where b and c are real numbers, and B is a normalized quaternion. That is B = xe_i + ye_j + ze_k, where (x,y,z) is a unit vector. And I'm ignoring e_0. Then BB = -1.

Compute (b+B)/|b+B| exp(2cB)
= (b+cB)/|b+cB|(1 + 2cB + 4ccBB/2 + 8cccBBB/6 + ...)
= (b+cB)/|b+cB|(1/0! + (2c)B/1! - (2c)^2/2! - (2c)^3B/3! + (2c)^4/4! ...)
= (b+cB)/|b+cB|(cos(2c) + sin(2c)B)
= ( (b cos(2c) -c sin(2c)) + (b sin(2c) +c cos(2c))B )/|b+cB|.

Now I could easily make a mistake computing |b+cB|, perhaps you will take it from here.

Of course what I'm doing here is thinking in Clifford algebra terms. That is, write everything in terms of scalars (i.e. b) and vector (i.e. cB) terms, and then keep stuff grouped according to blade. This sometimes works. That is, it sometimes works in explaining this to somebody in Clifford algebra terms.

(I couldn't stand your notation where a_i is a real number but e_i is a basis vector. This is too similar for my limitations.)

Also, are you planning on going to any conferences this season? I thought I would pretty much sit the year out, perhaps going to the gravity conference in Australia.

 

[jhmar]

Classical and quantum mechanics are both so precisely defined, they can start from exactly the same place.

But QT predicts, Classical physics does not predict. My paper shows how classical theory can predict and in doing so, it shows that QT, while correct; predicts only a fraction of all possible particles.

In the quantum approach, operators are the observables which leads to measurements that are averages.

This process of averaging is carried over into experimental work (see PDG 2004 tables). I show that the rejection of some experiments and the averaging of the remainder are incorrect. All the experimental results are compactions of a single elementary particle; probably the graviton.

What bothered Einstein was why causality for classical physics was different from causality in quantum mechanics.

The difference is caused by Einstein’s inclusion of movement. In order to understand what particles are and why they have there particular quantities it is necessary to examine structure not movement.

When the change in space is smaller than the change in time, information travels at less than the speed of light, and we have classical causality. When the change in space is larger than the change in time, information travels at less than the speed of light, so all we can measure is average values of what happens.

Knowledge travels at the speed of the knowledge carrying particle, usually photons. This varies according to the density of the particles that the photons are passing through (gravitons). Each gravity field is its own time zone and this is the base of all those theories about time slowing down near black holes. Again there is no need for averaging and no disagreement with QT; classical theory is simpler.

About the vacuum...
A vacuum is empty and can do absolutely nothing. Ever.

Newton’s graph of a gravity field without a central mass shows that absolute vacuum does not exist; the (vacuum) zero point has zero dimensions. It is where absolute vacuum would be if it existed.
It is vacuum force emanating from the ZP that controls the structure of matter creating particles that are vacuum fields with mass (vacuum force carrier).

To the folks at CERN betting billions on the Large Hadron Collider to detect the Higgs, I will go on record to say they are going to fail.

I would agree that there is no Higgs particle, but there are certain similarities between a Higgs field and vacuum force carrier that lead me to suggest that they are the same entity.

I have used the maximum attachment allowance to send you some tables from my proposal, only a fraction of table 4 can be included, but this should be sufficient to show the concept.
john

 

[sweetser]

Conferences

Hello Carl:

Of course what I'm doing here is thinking in Clifford algebra terms.

I'll get back to you on this point next week. I have never studied Clifford algebra, oops. I will read up on them, and see if I can say anything sensible.

As far as conferences, I will be wasting some time and money at the April APS meeting. I'm in session "E12. Alternative Theories of Gravity". One guy will talk about plants and gravity. Two people have gamed the system so they can talk for two time slots. At least it is in Florida.

At the end of May, there should be the Eastern Gravity Meeting in NYC at Columbia. They still haven't made an official announcement, but it is in the works.

I may go to SIGGRAPH to show off the quaternion animations. Those folks use quaternions, so I might find an interested audience. That meeting conflicts with the one down under.

doug

 

[sweetser]

Hello John:

There are quite a few statements in your note I find fall into that class "not even wrong." Sorry about that, but I can be precise.

>But QT predicts, Classical physics does not predict.

Newtonian mechanics, the Maxwell equations, even special and general relativity are considered to be classical physics, for the reason I gave (experimentalist can measure a number, the measurements are not quantized). The classical theories are only theories because they make predictions.

> All the experimental results are compactions of a single elementary particle; probably the graviton.

Different particles have different intrinsic spins. The graviton is predicted to have a spin of 2, while electrons have spin 1/2. One of the biggest implications of the difference in spin to characterize particles is the Pauli exclusion principle which applies to thos with half integral spin, but not integral spin. I would expect that should you make clear that there is a single elementary particle, that your proposal to the Independent Research Forum would be rejected on that ground alone. In the GEM proposal, the unified field strength tensor has a spin 1 particle, the photon, to do the work of EM, and a spin 2 graviton to do the work of gravity. Both are massless and travel at the speed c. The also differ because a photon is a transverse wave, while the graviton is a scalar or longitudinal mode of emission.

>The difference is caused by Einstein’s inclusion of movement. In order to understand what particles are and why they have there particular quantities it is necessary to examine structure not movement.

Modern field theory focuses on groups. The standard model was developed in the 1970's, so Einstein's post-mortem opinions don't matter on the subject at hand. Carl and I are working out the details of the groups U(1)xSU(2)xSU(3) which is at the heart of the standard model.

>Knowledge travels at the speed of the knowledge carrying particle, usually photons. This varies according to the density of the particles that the photons are passing through (gravitons). Each gravity field is its own time zone and this is the base of all those theories about time slowing down near black holes. Again there is no need for averaging and no disagreement with QT; classical theory is simpler.

I keep things simple, and work with the vacuum, or isolated currents in a vacuum. All my work so far has been for low energy densities. The math for high energy densities will be different for the GEM proposal, but I do not have the details.

>Newton’s graph of a gravity field without a central mass shows that absolute vacuum does not exist; the (vacuum) zero point has zero dimensions. It is where absolute vacuum would be if it existed. It is vacuum force emanating from the ZP that controls the structure of matter creating particles that are vacuum fields with mass (vacuum force carrier).

My proposal is not Newton's, so this point is not relevant. A vacuum is both well defined and observable: it is a place with an average energy density of zero. There is not such place, but most of the Universe is an excellent approximation of a vacuum, with a hydrogen hanging out in a cubic meter. There are no vacuum fields doing anything because they have no energy to do anything. The logic is simple. The deviation of the average is not zero, and that does not depend on the experimenter, but on a basic property of quantum mechanics, namely that complex numbers are needed.

>I have used the maximum attachment allowance to send you some tables from my proposal, only a fraction of table 4 can be included, but this should be sufficient to show the concept.

I will refrain from commenting further until/if your work is accepted here. As my comments above should indicate, I don't think your work does fit the criteria in the few bits I have seen. I appreciate your sincerity, but I cannot invest more time in work where I see fundamental errors are made. We will have to respectfully disagree with each other probably on most points I've tried to make.

doug

 

[CarlB]

Doug,

What is |A^2=|a_0 e_0 + a_i e_i + a_j e_j + a_k e_k|^2 ?

I'm guessing it has to be a_0^2 \pm (a_i^2 + a_j^2 +a_k^2), but I'm not sure which sign to take. Am I right in supposing that it is a real number? I like to work with the + sign, but I'm guessing you're defining it as the - sign.

I need this to continue the analysis of the U(1)xSU(2) thingy. I'm sure it's obvious, thanks for bothering.

 

[sweetser]

The norm of a quaternion

Hello Carl:

The norm of a quaternion is:
|A|^2=norm(A)=scalar(A^* A)=a_0^2+a_i^2+a_j^2+a_k^2
The norm can equal zero if and only if A=0, otherwise it is positive definite. The rule for the inverse of real, complex, and quaternion numbers can all be written exactly the same way:
A^{-1}= A^*/|A|^2
For the real numbers, the conjugate operation does nothing, but also don't hurt anything. For complex numbers, the imaginary part will flip a sign, then get hit by the norm. Same for quaternions.

So the answer is definitely the + sign.

doug

note added for fun: I have an animation of the norms of a bunch of quaternions. They all sit at the same place in 3D space: 0, 0, 0. What changes is how far in the future they are from now, t=0.

 

[CarlB]

Thanks. Continuing the calculation,

Compute (b+B)/|b+B| exp(2cB)
= (b+cB)/|b+cB|(1 + 2cB + 4ccBB/2 + 8cccBBB/6 + ...)
= (b+cB)/|b+cB|(1/0! + (2c)B/1! - (2c)^2/2! - (2c)^3B/3! + (2c)^4/4! ...)
= (b+cB)/|b+cB|(cos(2c) + sin(2c)B)
= ( (b cos(2c) - c sin(2c)) + (b sin(2c) + c cos(2c))B )/|b+cB|

= ( (b cos(2c) - c sin(2c)) + (b sin(2c) + c cos(2c))B )/sqrt(bb +cc).


In the above, b is the amplitude of the temporal part of the quaternion, c is the amplitude of the spatial part, and B is the direction of the spatial part.

Okay, the spatial SU(2) information is in B and that comes through unchanged. The vector (b,c) codes the relative strength of the time and space parts of the original quaternion. It's divided by its length so it becomes a unit vector in two dimensions. Then it is rotated by the absolute value of the value of 2c:

\frac{1}{|A|}\left(\begin{array}{c}b\\c\end{array}\right) -&gt; <br /> \left(\begin{array}{cc}\cos(2c)&amp;-\sin(2c)\\\sin(2c)&amp;\cos(2c)\end{array}<br /> \right)\;\frac{1}{|A|}\left(\begin{array}{c}b\\c\end{array}\right)

 

[sweetser]

Cool! I'll see if I can confirm this numerially.

 

[jhmar]

My proposal is not Newton's, so this point is not relevant. A vacuum is both well defined and observable: it is a place with an average energy density of zero. There is not such place, but most of the Universe is an excellent approximation of a vacuum, with a hydrogen hanging out in a cubic meter. There are no vacuum fields doing anything because they have no energy to do anything. The logic is simple. The deviation of the average is not zero, and that does not depend on the experimenter, but on a basic property of quantum mechanics, namely that complex numbers are needed.

Einstein improved on Newton's work, he did not prove or claim Newton was wrong. (NASA still uses Newton's formula for gravity). It a simple calculation to show that Newton's original formula gives the force arising from the combination of two vacuum fields obeying the Standard Inverse Square Law that applies to vacuum fields with a central point.
Einstein's and subsequent adjustments to Newton's law are necessary to take into account external factors they do not alter the cause of the original structure.
Are you now saying that Newton and Einstein are wrong?

 

[sweetser]

Hello John:

Newton had a huge volume of work. Newton's theory of gravity is wrong and as an approximate theory is useful. Newton's theory of gravity applies to masses which might be surrounded by a vacuum, but masses have a positive energy density, so there is no longer a vacuum if there is a mass.

General relativity is a rank 2 field theory. My GEM proposal is a rank 1 field theory. If my proposal is correct, then general relativity is wrong and as an approximate theory is useful. My proposal will probably be more useful because the non-interacting field equations are linear.

doug

 

[jhmar]

but masses have a positive energy density, so there is no longer a vacuum if there is a mass.

The flaw in this reply is your assumption that we know what mass is, you are assuming that mass has a positive energy density; I am assuming that the vacuum force carrier is the something that is the cause of density and use bubble chamber experiments to justify that assumption. But that is a debate that will have to wait a decision on my submission. For the present I am content to understand the points on which we disagree; which is why I joined in the debate. Unless you have anything to add I will leave you and CarlB to enjoy your mathematics, thanks for taking the trouble to persevere with my replies.
regards
jhmar

 

[sweetser]

Essays on Gravity

Hello:

I submitted a paper to a contest, the 2007 Essay on Gravitation sponsored by the Gravity Research Foundation. It was set up in the middle of the last century by the businessman who founded Babson College (I'll let you guess his last name). He had all kinds of hopes for anti-gravity devices, and wanted to fund fun/fringe work.

The first year of the contest did not go well: it sounded too wackie. His friend George Rideout suggested that the description of the contest should be toned down and award "provocative" works on gravity. Now this contest is part of the establishment. I recognized 3 names among the winners: Ellis, Smoot, and Wald. There were 33 honorable mentions, so this is a real contest. There are five cash awards, going from $500 to $5000.

http://www.gravityresearchfoundation.org/competition.html

So far I have avoided the full-fledged peer-review journal because I need to work with someone who reads and understands those journals first. This contest sounded a little looser, which is more my style.

The winners will be announced on May 15. Sometime after that, I'll post it here. For now, I will leave the abstract.

Geometry + 4-potential = Unified Field Theory
(same as for my APS talk in mid April)

Gravity is the study of geometry. Light is the study of potentials. A
unified field theory would have to show how geometry and potentials could
share the work of describing gravity and light. There is a long list of
criteria that must be satisfied to have a reasonable hypothesis, from
recreating the Maxwell equations, to passing the classical tests of
gravity, to demonstrating consistency with the equivalence principle, and
working well with quantum mechanics. This essay works through many of the
common objections.

I stayed up to 3 AM trying to polish it, and then skipped out of work because I am too old to bounce back that quickly. Great day out here in Massachusetts. I like the essay. The same guy, George Rideout, is still running it, as he sent me an enthusiastic email reply.

doug

 

[sweetser]

Visual Representation of the Standard Model

Hello:

A week after the national April APS meeting, I'll be up in Orono Maine early on a Saturday morning showing pictures of the standard model for a regional APS meeting. This work was a direct result of writing software to animate quaternions. Here is the abstract I submitted.

(abstract)
Software is used to visualize unit quaternions SU(2) as a 3D
animation. Random quaternions are run through a quaternion exponential
function. The results are sorted by time and placed in a frame of the
animation corresponding to their 3D coordinates. The resulting
animation shows a sphere with an apparent distain for the past. The
visual representation of electro-weak symmetry looks like a complete
sphere with a bias for the past. The animation for U(1)xSU(2)xSU(3) is
the smoothest image of an expanding/contracting sphere that could be
created. Any pattern of events can be represented by this
group. Spheres of slightly different sizes nearby on the manifold
would belong to the group Diff(M) which is at the heart of gravity.
(/abstract)

If you want to look at pictures, click here.

http://www.theworld.com/~sweetser/quaternions/quantum/standard_model/standard_model.html

Later,
doug

 

[sweetser]

Talk on YouTube

Hello:

I will be jetting to Jacksonville in a week to give a talk, "Geometry + 4-potentials = Unified Field Theory". I bought a Mac for Keynote, a presentation software program I had heard good reviews. They were true. I was able to create a simple presentation with nice transitions. I don't expect many people in the room since this is an "alternative" gravity theory session other than the presenters and the moderator and at most two stragglers.

Keynote makes it easy to export an HTML page, and I put that up on quaternions.com. It would be much better if audio could be included with the slides. A little investigation lead to a program called "profcast". That program will record the audio, synced to the clicking of the slides. It does not get the transition animations, but that is minor. I captured the slides and the audio at a good quality, and put it up on youtube:



This is 17 minutes long, and I only will have 12 on next Saturday, so I'll have to be more brief. It was great to record, compress, and upload in a few hours.

As always, I'd appreciate any comments on the content.
doug

 

[rewebster]

Nice presentation--One comment:

at 15:08 or so, you call it a 'hard' sell--I might say something like "it becomes a potentially interesting sell" or something else besides 'hard'----'hard' puts up a barrier for the idea

 

[sweetser]

Good suggestion. I'll change the phrase in the live presentation, see if I can swap it out of the video.

 

[sweetser]

Nope, not the standard model

Hello:

At the APS meeting, I talked to a guy (Oleksandr Pavlyk, don't ask me to pronounce it) from Mathematica about my efforts to visualize the standard model. He pointed out a clear error. The group U(1)xSU(2)xSU(3) has the the Lie algebras u(1), su(2), and su(3), which have degrees of freedom 1, 3, and 8 respectively. Add that up, and the standard model has 12 degrees of freedom in its Lie algebra. What I worked with was two quaternion, which has 8 degrees of freedom.

The fellow also said that SU(3) has U(1) and SU(2) as subgroups, something I was unaware of. Now that I look back on those animations, they are easy enough to spot. At the algebra level, it is clear they belong as subgroups, since they are used to generate SU(3).

I have heard that people who are very sophisticated with the standard model put some sort of caveats on the usual U(1)xSU(2)xSU(3) description. I do not recall what those caveats were.

I have some confidence that the animation I created is SU(3): the norm is one, and it has 8 degrees of freedom. I don't know how to write in group theory the algebra I have done (the conjugate of U(1)xSU(2) times a different element of U(1)xSU(2)). I could just toss in another quaternion, but that strikes me as bogus. These two quaternions smoothly cover any possible event in spacetime, bar none.

doug

 

[sweetser]

The Standard Model Symmetries on YouTube

Hello:

YouTube is now hosting new animations of the symmetries of the standard model. Everyone thinks they know what U(1) looks like (a circle in the complex plane). Even that looks different than I expected once animated. As for SU(2) and SU(3), all I've seen is algebra. I have each animation separately, and a 2' 40" collection of them all.

The groups of the standard model and gravity (the 5 videos below):

The group U(1)
The group SU(2)
The group U(1)xSU(2)
The group SU(3)
The group Diff(M)xSU(3)

As always, comments are welcome. Enjoy, I think this makes the standard model fun to think about.

doug

 

[rewebster]

how did you come up with the angled position of the U(1) in the xyz? is that specific?

 

[sweetser]

Hello:

The plane for U(1) was chosen at random. The software is setup so a specific quaternion can be used to start forming the group, and that one quaternion would determine the angle.

doug

 

[rewebster]

Well, it seemed that the others 'could' be of any orientation due to their symmetry and the U(1) stood out to be chosen or selected to be in that exact orientation for some reason.

 

[sweetser]

Representing U(1) - by itself - is arbitrary with what is effectively a 3D imaginary unit. Complex numbers in contrast have a 1D imaginary unit. When calculating U(1)xSU(2), the generators of U(1) and SU(2) need to point in exactly the same direction. That direction can be arbitrary, but must be shared if the U(1) is to commute with SU(2) as it must to be a faithful representation of an Abelian group.

doug

 

[rewebster]

Representing U(1) - by itself - is arbitrary with what is effectively a 3D imaginary unit. Complex numbers in contrast have a 1D imaginary unit. When calculating U(1)xSU(2), the generators of U(1) and SU(2) need to point in exactly the same direction. That direction can be arbitrary, but must be shared if the U(1) is to commute with SU(2) as it must to be a faithful representation of an Abelian group.

doug

If the U(1) is arbitrary (random?) by itself, would showing it to be more random be more correct? --The one thing, and I don't know how important it would be, is that IF at a arbitrary/random setting it COULD end up laying in the y-z plane and your 'time transversing' line along the x-axis (x-y plane) would have quite a bit different 'read out'---what significance would that be/have?

 

[sweetser]

Hello:

Sorry for the delay in replying. I am not getting the "something has been posted" emails.

The way I view physics, it is a game of describing events in spacetime. Spacetime can be viewed either as a 4D real manifold, or a 1D quaternion manifold. If you choose to work in the quaternion manifold, then choose a quaternion with a norm of 1, and q^n will form the group U(1) because it depends only on one quaternion and is Abelian because quaternions commute with themselves.

In physics, U(1) is where the E and B fields live, the transverse mode of emission of light. One can make the circle appear in only 1 complex plane, say the tz plane. I should be free to do other things in the tx and ty planes. I'll have to think about how to implement that. From my experience, it is the time + space planes that matter, not space x space. Until I create the animation, I don't know. Good question.

doug

 

[CarlB]

Doug,

I've been spending a lot more time messing around with gravity stuff than I usually do, and have come to appreciate something I think you said about your theory -- that it is a vector based theory.

The reason this jars my memory is that my favorite version of GR also seems to be expressed as a vector field, a velocity vector field. This is the gauge gravity. Their stuff matches GR exactly (provided you avoid the weird topological stuff inside of where you can't observe anyway). For a black hole, the gauge gravity stuff says that the natural coordinate system is Painleve.

There is a wonderful paper on the "river models of black holes" that explains how Painleve coordinates describe a Schwarzschild black hole as a velocity vector field:
http://www.arxiv.org/abs/gr-qc/0411060

The velocity vector field is
\vec{v}(r) = -\frac{\vec{r}\sqrt{2GM/r}}{r}

Can you comment on how your theory works as a vector theory? Right now I'm interested in the field produced by a single mass point. I'm still interested in simulating it, and I think I know enough now that I might be able to work out the equations of motion by myself. But I'd like you to check my work.

Carl

 

[MeJennifer]

There is a wonderful paper on the "river models of black holes" that explains how Painleve coordinates describe a Schwarzschild black hole as a velocity vector field:
http://www.arxiv.org/abs/gr-qc/0411060

The Painlevé chart is an excellent description of the Schwarzschild metric!

Flat background, the gravitational force shows up as Galilean not Lorentzian, using Newton's escape velocity. It models a comoving observer of the gravitational pull from infinity.

Also check the Doran chart that models the Kerr metric (a rotating point mass). Again a flat background!

By the way, it seems to me that since the Painlevé and the Doran chart chart resp. a static and stationary spacetime, there is no reason we could not define the orbital equations on a Euclidean 5 space. That should simplify the orbital equations since we should be able to express the effect of the gravitational field completely in terms of a non relativistic translation and a time invariant deformation of the Minkowski spacetime. No?

 

[rewebster]

Hello:

Sorry for the delay in replying. I am not getting the "something has been posted" emails.

The way I view physics, it is a game of describing events in spacetime. Spacetime can be viewed either as a 4D real manifold, or a 1D quaternion manifold. If you choose to work in the quaternion manifold, then choose a quaternion with a norm of 1, and q^n will form the group U(1) because it depends only on one quaternion and is Abelian because quaternions commute with themselves.

In physics, U(1) is where the E and B fields live, the transverse mode of emission of light. One can make the circle appear in only 1 complex plane, say the tz plane. I should be free to do other things in the tx and ty planes. I'll have to think about how to implement that. From my experience, it is the time + space planes that matter, not space x space. Until I create the animation, I don't know. Good question.

doug

The way I see your animation/diagram is/could be that the 'circle' is gimbled (moving at a high rate) on all three axes with time transversing on all three planes at the same time ('-' -> '+') showing/plotting for least separation of the intersection(s) along any of the axes (x, y and/or z--or, to be able to show it in animation, a combination of any two (x,y,z) tranversing across at one of any given/every possible random vector). --that should still show the two points separating and rejoining in the double (overlapping '+' and '-') harmonic/sine (?) 1/2 wave looking pattern. Would that work?

(writing out, now, may appear to be easier than the diagram/animation--hmmm?)

 

[sweetser]

A river runs through it

Hello Carl and MeJennifer:

I have trouble getting excited about any work done on black holes by anyone, good, bad, or ugly.

In the first derivation I did to get to the exponential metric, I struggled to find the right way, no steps skipped. At one point, I thought I had it. Alas, Mathematica did not agree. A bit of retooling, and all the parts fell in place.

At one specific place, I have to make an assumption: the perturbation of the time term was trivial compared to the one for space. That apparently is what needs to be done to generate the exponential metric is consistent with weak field tests.

When a mass is in a very small volume, the assumptions used to derive the exponential metric are no longer valid. That would drastically change the kind of equations that govern motion. What I had done was take a perturbation from a 1/R 4-potential solution, and include a wee bit of a time variation. What if the I look for the non-perturbation solution? I haven't explored it more is because I would get lost. This is way different. There were two different warning signs. First is that the units for the expressions involve had a G (like Newton's law) and a c (like metric solutions in GR) and an h (like anything to do with quantum mechanics). The non-perturbation theory has the units of a quantum gravity theory. The second bigger issue is that the non-perturbation solution is a 1/R^2 4-potential, which results in a 1/R^3 force law.

I am posting here in "Independent Research" forum because I am walking the crank line. Say you have a 1/R^3 force law, and doors close immediately. I would not get the chance to say that the gravity we know - Newton's law that respects special relativity - is a 1/R^2 force law, it is only the quantum gravity theory that the force becomes a 1/R^3. There I things I know because I've done the math with paper and pencil that I don't like to talk about. Selling the exponential metric has been trying enough. A quantum gravity solution that implies that no work done on black holes is correct - there is no chance that will be accepted until derived independently by others.

There turns out to be a few great ironies at work here. Folks who take a quick glance find the 1/R^2 potential with 1/R^3 force, and thus dismiss this approach. The do so without working out the units, since they are professionals using natural units where G = c = h = 1. It was really amazing to see those three amigos in one expression!

doug