Unifying Gravity and EM

[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

 

[sweetser]

Getting lost in terms

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?)

Maybe its too late for me, but I did not understand what your wrote. I see nothing "moving at a high rate". I happened to scale the animation to fill 10 seconds. I could also scale it to fit into a time frame of 10 picoseconds, or a thousand years. Nothing is marked. It is quite the abstraction.

The reason time is in the 3 complex planes is because time is the real axis that is shared by the three imaginary unit vectors (I happen to use i, j, k for convenience, but r, theta, phi or some other combination of spatial unit vectors are possible).

I have see what sine and cosine look like. It was not anything like I expected. I think there is one image up on quaternions.sf.net, but I need to add a few more to the collection.

doug

 

[sweetser]

4-vectors of motion

Can you comment on how your theory works as a vector theory?

This line reminded me of one of the paths to the exponential metric. There is a Lorentz 4-force in my proposal:
$$F^{\mu}=-Jm_{\nu}(\partial^{\mu} A^{\nu}+\partial^{\nu}A^{\mu})$$
I drop in the weak gravity, electrical neutral potential into this, and solve for the 4-velocity. Here is the 4-velocity, weak gravity solution to the 4-force equation:
$$(\frac{d t}{d \tau}, \frac{d R}{d \tau}) = (exp(-G M/c^2 R), exp(G M/c^2 R))$$
Sorry, but I do think those $\tau$'s are necessary.

doug

 

[CarlB]

I don't mean to imply that I think that current representations of what happens in black holes are correct. I always thought the "event horizon" of the Schwarzschild metric was silly, and that's why I wrote the Painleve simulator, to show that one could redo GR and eliminate that stuff. But more than that, even though the Painleve coordinates are an improvement, I doubt that they are correct.

I'm doing black hole simulations for pretty much the same reason Kepler would have done epicycle simulations.

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.

I don't see this as a problem at all. Heck, I don't even see what your point is. You do get the apprximate 1/r potential with the exponential thing don't you? This is approximately valid in the far field isn't it? Of course it is, otherwise you'd never have invested this much time in it.

Any gravitation theory should give equations of motion (and I'm interested in yours even if they are valid only for large r), and those equations of motion can always be written as a power series in r. Unless the theory is straight Newtonian gravity, there has to be forces other than 1/r^2.

In fact, when I write the equations of motion of the Painleve metric as a power series in r, it has some pretty crazy terms. Letting the particle velocity $$\dot{x},\dot{y}$$ be of order 1 (i.e. for simulating light rays), the orders of the forces are:

$$\begin{array}{rcl|r}\ddot{x} &=&-\sqrt{2}\dot{x}(\dot{x}^2+\dot{y}^2)/r^{1.5}&1.5\\&&+1.5\sqrt{2}\dot{x}(x\dot{x}+y\dot{y})^2/r^{3.5}&1.5\\ \hline&&-x/r^3&2.0\\&&+3x(x\dot{x}+y\dot{y})^2/r^5&2.0\\&&-2\dot{y}(x\dot{y}-y\dot{x})/r^3&2.0\\ \hline&&+3\sqrt{2}\dot{x}/r^{2.5}&2.5\\&&+2\sqrt{2}y(x\dot{y}-y\dot{x})/r^{4.5}\;\;\;\;\;&2.5\\ \hline&&+2x/r^4&3.0\\ \hline\end{array}$$

As far as diverging from Newton in the far field, one only worries about terms that have powers of 2 or lower, and that are not multiplied by the particle velocity. (If they are multiplied by particle velocity, then they go to zero as particle velocity is small compared to speed of light.) In the above, you will note that there are lots of such terms, but they all have particle velocity involved.

To look at forces for small velocities, ignore all the terms multiplied by velocity. What's left is:

$$\begin{array}{rcl|r}\ddot{x} &=&-x/r^3&2.0\\ \hline &&+2x/r^4&3.0\end{array}$$

In other words, in Painleve coordinates there is also a 1/r^2 potential leading to a 1/r^3 force (i.e. the $$+2x/r^4$$). But the Newtonian contribution dominates at large r. Schwarzschild is similar, but it only looks nice when you write it in terms of powers of r and (r-2). And in any case, the actual orbits for Schwarzschild and Painleve are identical. All that differs is a redefinition of time, depending on the radius.

By the way, this talk has influenced me to go and make the calculation for the deflection of light from my equations of motion. I'll do it over on my "independent research" thread.

I'm really looking forward to your equations of motion for the exponential solution. It should be possible to see the various tests approximately achieved. For example, the ratio of the bending of light in Newton's and Einstein's theories is only 2x in the limit of small bends. This is something that should be obvious but (stupidly) I did not realize it until I simulated light bending and found that Einstein's light bends far more than twice Newton's prediction close to a black hole. It amazes me that this surprised me.

The simulation is already good enough to pick off small far field effects. And I'm soon going to implement R-K numerical methods (4th order) and expect it to get a lot more accurate soon.

Carl

 

[sweetser]

Hello Carl:

At long last, I may be able to deliver the equations of motion for the GEM proposal for a spherically symmetric, non-spinning, electrically neutral mass. All that needs to be done is to rearrange the 4-velocity in my earlier post.

Here is that 4-velocity again:
$$(\frac{d t}{d \tau}, \frac{d R}{d \tau}) = (Exp(-G M/c^2 R, Exp(G M/c^2 R)$$
Let's eliminate the pesky $\tau$. Multiply both sides by the inverse of $\frac{d t}{d \tau}$ which is $\frac{d \tau}{d t}$ or equivalently, $Exp(G M/c^2 R)$
$$(1, \frac{d R}{d t}) = (1, Exp(2 G M/c^2 R)$$
Looks like an elegant answer to me. Does this satisfy your request?

doug

 

[rewebster]

If I'm understanding your animation of the U(1), it's (yours is) an arbitrary (random) representation as a circle in x,y,z---doesn't the representative circle go through all possibilities 'gimbaling' through the x,y,z co-ordinates through time/conditions and describes a hollow sphere in x,y,z --but as a circle at any given instant/circumstance?---maybe I'm understanding your animation wrong.

 

[sweetser]

Hello:

There is no circle in x, y, z. There are 3 ellipses, one in the t-x plane, one in the t-y plane, and a third in the t-z plane. If there is a circle in one of these three complex planes, then there are lines in the other two.

In the quaternion animation, the line is straight through space. It is true that the line could point in any direction in 3D space.

doug

 

[rewebster]

yes--excuse me,--ellipse(s)

I had stopped the play and didn't get the audio

I guess what I have been looking at is if the animation for U(1) is a fair and/or generalized representation, IF the position of the ellipse(s) is/can be random, or if there was a way to show that it COULD be somehow represented AS arbitrary/random and still maintain an accurate visual depiction through all other possibilities.

 

[sweetser]

I am quite confident the animation is fair because the numbers that are fed in using the rules that generate U(1). Pick out any event, form the product with another event. The result will be another element of U(1). Now take those first two events, but multiply them in reverse. You get the same result because these quaternions commute. The generator of all these quaternions is one quaternion. That is the way the Lie algebra u(1) works (I used a small u).

There is freedom to point U(1) along any unit vector in 3D. The unit vectors do not have to be Cartesian, just part of a set of three unit vectors that span the 3D space. One could represent the group U(1) along a curved line with the choice of spherical coordinates.

doug

 

[rewebster]

There is freedom to point U(1) along any unit vector in 3D. The unit vectors do not have to be Cartesian, just part of a set of three unit vectors that span the 3D space. One could represent the group U(1) along a curved line with the choice of spherical coordinates.

doug

This is what I've been alluding to, or trying to get to. --and after listening (this time) to the audio, there's not this variable mentioned--as a open variable/freedom/arbitrary choice. Is there any way to animate this as a 'freedom' and still maintain the 'point' model that is the result of the time line? It would seem a little more 'correct' as a more representational model or mentioned that this freedom is there.

 

[sweetser]

This is what I've been alluding to, or trying to get to. --and after listening (this time) to the audio, there's not this variable mentioned--as a open variable/freedom/arbitrary choice. Is there any way to animate this as a 'freedom' and still maintain the 'point' model that is the result of the time line? It would seem a little more 'correct' as a more representational model or mentioned that this freedom is there.

I've only got 30 seconds to yak in that audio, so it is not surprising I don't go into all the issues. You were right to feel like U(1) does not fill up a volume of spacetime. Yet it is only one of the four known forces of nature. Have you listened to the YouTube video on the standard model? U(1) is doing the work of EM. SU(2) does the work of the weak force, and fills up much more space. U(1)xSU(2) goes everywhere, but not evenly. The group SU(3) could be covered by all the possible angles in space of the group U(1). This was already known because U(1) is a subgroup of SU(3), so play with enough U(1)'s and SU(3) is done.

 

[CarlB]

$$(1, \frac{d R}{d t}) = (1, Exp(2 G M/c^2 R)$$
Looks like an elegant answer to me.

Eventually I need a 2nd order differential equation like:

$$\frac{d^2R}{dt^2} = f(R, dR/dt).$$

For example, the Newtonian equations of motion are:

$$\frac{d^2 R}{dt^2} = -\frac{2RMG}{|R|^3}$$

so you should probably approach this for large R and small velocity dR/dt (assuming you don't have a MOND effect).

To get to this from the line element I will have to make some assumption about the orbits. In GR, the assumption is that the orbits are geodesics of the line element. I can make that calculation, (by calculus of variations) though it will take some effort.

What I'm saying is that if your orbits follow the geodesics of the line element that I think you are using, i.e.:

$$d\tau^2 = \exp(-GM/c^2R)dt^2 - \exp(+GM/c^2R) (dx^2+dy^2+dz^2)$$

then I can eventually figure it out. This is a simpler line element than GR gets, so it should be simpler to get the orbits out of it. Or did I leave out a factor of two and get some signs wrong in the above?

 

[rewebster]

Have you listened to the YouTube video on the standard model?

sorry---I couldn't get passed the one of you with your foot behind your head

 

[sweetser]

Equations of motion

Hello Carl:

Small correction to the metric you wrote: there needs to be factors of 2 in both the exponents.

I just got a useful book, "Mathematica for theoretical physics: Electrodynamics, quantum mechanics, general relativity, and fractals". They showed how to get the equations of motion for the Schwarzschild metric. Being a good biologist by training, I was able to clone the answer for GEM:

$$\frac{4 G M R'^2 e^{\frac{4 G M}{c^2 R}}}{c^2 R^2}-\frac{4 G M t'^2 e^{-\frac{4 GM}{c^2 R}}}{R^2}+R \phi '^2-2 R'' e^{\frac{4 GM}{c^2 R}} ==0$$

$$\frac{4 GM R' t'}{R^2}+c^2 t''==0$$

$$-2 R R' \phi ' - R^2 \phi ''==0$$

There is a 1-to-1 correspondence between the terms in the Schwarzschild equations of motion and the GEM equations of motion. Corrections appear in only one of the three equations as expected.

Is this form more helpful? It now looks more official.
doug

 

[CarlB]

There is a 1-to-1 correspondence between the terms in the Schwarzschild equations of motion and the GEM equations of motion. Corrections appear in only one of the three equations as expected.

Cool. Would you kindly edit in the difference? Of course these equations of motion are differential equations in s or tau instead of t, but I'm sure I can convert them. And to simulate them efficiently, they will have to be put into Cartesian form.

 

[sweetser]

Equations of motion

Hello Carl:

Here are the equations of motion, for the Schwarzschild metric first (see if it looks Kosher to you), in Cartesian coordinates, presuming we are in the z=0 plane of a system that stays in said plane, so all z's can be ignored.

$$\frac{2 G M(x x'+y y')t'}{\left(x^2+y^2\right)^{3/2}}+c^2\left(1-\frac{2 G M}{c^2\sqrt{x^2+y^2}}\right)t''=0$$

$$-\frac{G M x t'^2\left(1-\frac{2 G M}{c^2\sqrt{x^2+y^2}}\right)^2}{\left(x^2+y^2\right)^{3/2}}+\frac{G M\left(x x'^2-x y'^2+2y x'y'\right)}{c^2\left(x^2+y^2\right)^{3/2}}-x''\left(1-\frac{2 G M}{c^2\sqrt{x^2+y^2}}\right)=0$$

$$-\frac{G M y t'^2\left(1-\frac{2 G M}{c^2\sqrt{x^2+y^2}}\right)^2}{\left(x^2+y^2\right)^{3/2}}+\frac{G M\left(y y'^2-y x'^2+2x x'y'\right)}{c^2\left(x^2+y^2\right)^{3/2}}-y\text{''}\left(1-\frac{2 G M}{c^2\sqrt{x^2+y^2}}\right)=0$$

All the derivatives are with respect to the Lorentz invariant distance $s = \sqrt{x^2 + y^2 - c^2 t^2}$. Looks kind of scary to me.

Proceed anyway. On to the GEM equations of motion in Cartesian coordinates:

$$\frac{2 G M(x x'+y y')t'}{\left(x^2+y^2\right)^{3/2}}+c^2t\text{''}=0$$

$$-\frac{G M x t'^2\text{Exp}\left(\frac{-4 G M}{c^2\sqrt{x^2+y^2}}\right)}{\left(x^2+y^2\right)^{3/2}}+\frac{G M\left(x x'^2-x y'^2+2y x'y'\right)}{c^2\left(x^2+y^2\right)^{3/2}}-x\text{''}=0$$

$$-\frac{G M y t'^2\text{Exp}\left(\frac{-4 G M}{c^2\sqrt{x^2+y^2}}\right)}{\left(x^2+y^2\right)^{3/2}}+\frac{G M\left(y y'^2-y x'^2+2x x'y'\right)}{c^2\left(x^2+y^2\right)^{3/2}}-y\text{''}=0$$

All the t, x, y, t', x', y', x'', y'' terms are in the same place. I was able to collect the exponential into one term (exponential are so convenient).

I can provide the Mathematica notebook if anyone is interested. The result is not this pretty (meaning the result had to be simplified "by hand" and is subject to human error). I did check that the units for all the terms are the same.

doug