Spinning the Interaction Story: Attraction and Repulsion (3 of 3)
By Doug Sweetser | January 2nd 2012 11:58 PM | 13 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

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The more I read on this topic, the less confident I get. If you wish to read a solid technical explanation, I recommend a reply by David Simmons-Duffin sent to me by Henry. It all comes down to Lorentz invariance and unitarity when considering propagators.

In technical tales, if so much as one step is not understood, the path gets lost. I'll point out a few places where I get confused. He writes down the potential between two stationary particles for an arbitrary spin-L force carrier in momentum space:

$\tilde{V}(\vec{k}) = q_1 q_2 P_{\mu 1,...\mu L;\nu 1,...\nu L}(\vec{k}) n^{\mu 1}...n^{\mu L}n^{\nu 1}...n^{\nu L}$

I better just quote the man's description of this beast:

where $P_{\mu 1,...\mu L;\nu 1,...\nu L}(\vec{k})$ is the propagator of the force_carrier, evaluated at a spacelike momentum $\inline k = (0, \vec{k})$ and $\inline n^{\mu}=(1,0,0,0)$ is a unit vector pointing in the time direction.  All these factors of n enter because the spin-L force-carrier couples to a symmetric tensor current, which is given by $\inline q n^{\mu 1}...n^{\mu L} \delta^3(x)$ for a stationary particle. The propagator is traceless in the $\inline \mu$'s and also in the $\inline \nu$'s since it is a two-point function of a traceless symmetric tensor (the force carrier).

Some of that makes sense, some doesn't.

Here is a problem: I don't know how to go from the potential in momentum space to a force in regular spacetime. With the field equations, the switch employed a sign change due to the D'Alembertian (a k2 in momentum space). There is no D'Alembertian in the force equation, so that trick is out, and I don't know the replacement.

While Simmons-Duffin worked out the odd spin case, the even one was skipped. That is too bad for me since it is the case of interest. He does mention the critical role of an odd number of metrics going into the propagator for spin 1. When I calculated the propagator for a tensor field in the previous blog, it too had an odd number of propagators.

He does provide a simple summary table:

Spin even:
q1 q2 > 0 attractive
q1 q2 < 0 repulsive

Spin odd:
q1 q2 < 0 attractive
q1 q2 > 0 repulsive

This looks simple, but upset me the more I thought about it. What was bothering me are the number of charges in the four fundamental forces of Nature.

For gravity, there is only one charge. That is a deep mystery to me. Granted gravity is super weak, but if there were two types of gravity charges roughly balanced in the Universe, there wouldn't be a Universe as we know it. I have a vague memory of a erudite discussion of general relativity where someone said that was the case in GR because only positive values of energy density were plugged in. Should I erase this vague memory, or is there any value in it? So at this point in my physics education, I accept that there is but one sign for the values of mass, but don't know why that is so.

At the other end is the strong force. There is quantum chromodynamics (QCD), with 3 colors and their anti-colors. These can be combined into 8 linearly independent states for the gluons. Due to the phenomena of confinement, we don't get to see any of these colors or gluons. They travel the speed of light all of 10-15m before giving in to the power of confinement. Perhaps the reason there is no "Coulomb-esque's law" for the strong force is there is no simple pair of charges like there is for EM.

The weak force with its non-Abelian SU(2) gauge symmetry looks too complicated to fit into the above model. I don't recall any discussions about the weak force being attractive or repulsive. The weak force is all about decay, decay, decay.

The only fundamental force that would have a q1 q2 product that could be positive or negative looks like EM.

After the exercise of these three posts that relate spin, charge, and attraction/repulsion, I still accept, from authority, that a spin 2 mediating particle is a requirement for a reasonable proposal for gravity. I feel better about the calculation going from spacetime to momentum space, then seeing the values of the polarization states of the transverse spacelike wave.

Doug

Snarky puzzle:
How would a ruling from the United States Supreme Court that mass had to be treated as -m instead of +m effect you personally or professionally?

Google+ hangout: arrange by email if interested.

Next Monday/Tuesday: A New Toy Model for the New Year

Linguist logic and so called mathematical expressions now dominates science. It is difficult to understand by the inventor scientist himself - if he is isolated from his work for a week.

We need to bring back physics to commonsense logic. This we do by understanding Intuition, observer and observed all together.

Vijay Gupta
Proponent - Unary law Space contains Energy
PicoPhysicts

I didn't get much out of this post unfortunately.

Also, if gravity self-couples, then why don't we get confinement for that like the color force? Or at the very least, something different than Coulomb's law?

I agree with Doug. The linked article is simultaneously too indepth (I don't understand how to manipulate those symbols or even know we can write it that way), and too cursory (there clearly are many many unspoken assumptions being made). It's not even correct to say gravity has a scalar charge. What is the "mass charge" of a photon then? It is more proper to say gravity couples to the stress-energy tensor (and therefore it is more clear how light couples to it now).

I don't know what if anything can be taken from that article as is. Can any of the many usual helpful readers point out what exactly are the big assumptions he's making that are confusing me and Doug?

You queried:

Also, if gravity self-couples, then why don't we get confinement for that like the color force? Or at the very least, something different than Coulomb's law?
First, the "confinement for ... the color force" is not just about the "self-coupl[ing]" of the "color force", it also has to do with the very large strength of that "color force" (of its coupling, not just to itself, but to other colored entities).

As an illustration, the SU(2) weak nuclear "force" also "self-couples", yet we see no similar "confinement" in this case either.

On the other hand, when one uses the "spin-2" description for gravity (such as in Doug's previous article, where he is following Feynman's approach which begins exactly this way), wherein gravity does not "self-couple", one actually finds the resulting theory is internally inconsistent!  However, once one works at eliminating the inconsistencies, one finds that the simplest, self-consistent theory is that of full blown General Relativity (GR), which does have gravity "self-coupl[ing]".

In fact, one of the first things one has to start with, in making the "spin-2" theory internally consistent, if I remember correctly, is to add in the stress-energy of the "spin-2" field on the "source" side of the equation:  The beginning of the "self-coupl[ing]".

However, you are correct that this "self-coupl[ing]" does have its consequences, such as "black holes".  (I know that Abhas Mitra makes his claims that "black holes" don't exist—and he is correct that we have no proof that any objects we have seen out in our universe are, in reality, these mathematical solutions found in GR—and I know Sascha Vongehr has an article, Black Holes Demystified, wherein he asserts that "black holes" are nothing stranger that Newtonian "dark stars" where gravity is sufficient that the escape velocity exceeds the speed of light; but "black holes", within GR, do have some rather different and, dare I say "strange" features that are a result of the gravitational "self-coupl[ing]".)

David

Black Holes Demystified, wherein he asserts that "black holes" are nothing stranger that Newtonian "dark stars" where gravity is sufficient that the escape velocity exceeds the speed of light
Let me split a hair here: I did not assert that they are nothing stranger, because they are much stranger than simply dark stars, even though better descriptions do not involve any essential (as opposed to coordinate) singularities. I know you know, I just am a little afraid that you again put me a little too uncomfortably close to this Mitra guy who apparently does not want to understand relativity ("timelike stays always timelike" as if it could otherwise and all that).

Sascha:

I am sorry if I overstated your view expressed within your Black Holes Demystified article.  While the "escape velocity equal to or exceeding the speed of light" aspect is, in a sense, comparable between Newtonian "dark stars" and General Relativistic "black holes", they also have very different characteristics, in terms of what this "escape velocity equal to or exceeding the speed of light" actually means to external observers and what they will observe.

I didn't mean to "put [you] a little too uncomfortably close to this Mitra guy".  I was simply trying to let people know that there are alternative thoughts on such things, and provide them with links if they wished to look further.

David

I don't think either of you should feel all that bad about your difficulty in understanding the linked article by David Simmons-Duffin.  While David is not completely off on what he is expressing, there are some issues that I would have to research to clarify.

On the other hand, while CuriousReader is completely correct about the coupling of gravity to the stress-energy tensor, David's article is taking a shortcut of only looking at the time-time component (given by the nμ vectors that only have the time component as non-zero).  (This becomes rather "important" when going beyond "spin-2", since we know of no physical, symmetric tensors with more than two indices that can serve as "sources".)

However, while David is quite correct that the forms of such interactions are highly restricted by Lorentz invariance (or, even more so, with general invariance of the Lagrangian) and unitarity, I think his assertion that these restrictions (particularly unitarity) determine the sign of the "propagator" may have some trouble:  If an operator is unitary, then so is its negative.

Another important consideration, in my opinion, is that there is no physical significance to the overall sign of the metric:  Whether the metric has signature (+,-,-,-) or (-,+,+,+) can have no physical significance or consequences (at least so far as we have been able to ascertain).

I'll have to do some research to determine what "rules" are actually used to "fix"/determine the signs of propagators (its simply been way too long since I've had to deal with such things).

David

If an operator is unitary, then so is its negative. ... Whether the metric has signature (+,-,-,-) or (-,+,+,+) can have no physical significance ... I'll have to do some research to determine what "rules" are actually used to "fix"/determine the signs of propagators
Yes, I think the other David means how it works out in the sum after you decided for any of the arbitrary conventions throughout. E.g., the overall sign of the metric is arbitrary, but 1 x -1 x -1 x -1 = -1 and -1 x 1 x 1 x 1 = -1 also. He is sure a very long way to go before his explanation is down to the grandma level he desires.

Here is my mostly classical justification for a spin 2 field for gravity.

We are working with fundamental forces that can arise from static fields.

In EM, one has a rank 2 anti-symmetric field strength tensor. Change the order of the indices and the sign flips. We know the EM field is mediated by a spin 1 field and that there are two charges.

The effects of gravity can be explained by a dynamic metric. A metric is a symmetric rank 2 tensor. Changes in a symmetric rank 2 tensor will also be a symmetric rank 2 tensor. Change the order of the indices on the tensor, and no signs change. In a "grandmotherly" way, it sounds like there is one charge and the spin must be more than a spin 0. It is not like EM, ergo the particle mediating the force for a symmetric rank 2 field strength tensor is spin 2.

A current coupling term would need a calculation similar to one done in  the first two blogs of this series. The transverse states would need to be +/-2. That is required for the proposal to be logically consistent.

Doug:

Unfortunately, trying to use the metric (or even its deviation from the Minkowski metric hμν, as done in the "spin-2", linearized gravity approach) simply doesn't create a "logically consistent" "proposal", in any complete sense.

Besides, the General Relativistic (GR) entity that most closely corresponds to the Electromagnetic (EM) vector potential (and to the corresponding SU(2) and SU(3) "potentials") is the Christoffel symbol(s), not the metric.  In fact, the entity corresponding to the EM "rank 2 anti-symmetric field strength tensor" is the GR Riemann Curvature Tensor, that has corresponding ant-symmetries.

As a point of fact, even the coupling of GR Christoffel symbols to (tangent space) vectors is completely analogous to the way the "vector" potentials of U(1), SU(2), and SU(3) couple to their respective "fiber" spaces:  Corresponding "covariant" derivatives.

This actually doesn't make it look very "spin-2" like at all.

So, about the only thing that differentiates the GR entities from the other (Yang-Mills) entities is the pure field portion of the Lagrangian:  The GR entities are able to form a simpler invariant scalar for this purpose.

There have actually been attempts to use the Yang-Mills pure field portion of the Lagrangian for gravity.  However, this yields results that are inconsistent with observations!  (It's too bad, because we "know" how to handle the Yang-Mills like cases, but we are still having trouble with the GR case.*)

David

*  There are those, including string theory, that have tried combinations of various invariant scalars for the GR pure field portion of the Lagrangian.  I have heard that certain combinations actually are indistinguishable from standard General Relativity, at least to the precision of our measurements.  Unfortunately, I am not familiar with all such experimentally consistent approaches.

If anyone can point me to a survey like article on alternate pure field Lagrangians that yield results consistent with experiment, I would be most grateful.

This actually doesn't make it look very "spin-2" like at all.
Thanks for the comment. If a straight-forward approach was available, it would have been done before.
If anyone can point me to a survey like article on alternate pure field Lagrangians that yield results consistent with experiment, I would be most grateful.
Well, I want to play with toys that have this kind of property :-) Will see if I can build one next week. To me, a toy is not worth playing with unless the solutions to the field equations plugged into the force equations get both plain old Newtonian gravity and light bending around the Sun.

Doug:

I think you know that I don't expect you to be able to create such "toys".  ;)

One of the nice features about simply varying the pure field portion of the Lagrangian is that the interactions, and the "force" like characteristics remain unchanged.  In fact, the whole theory remains a "metric theory".  The only thing that possibly changes is the "field equations".  In fact, there is even the possibility that such a change will actually result in an equivalent field equation, in which case, the only thing that changes is the prescription for "quantization".

However, I do know that at least some work has been done along such alternate Lagrangian lines by various researchers in various theoretical contexts.  I have definitely seen articles on some alternate pure field Lagragians (often described as "superstring inspired", or some such).  Unfortunately, since I have not found a good means for searching for such work, I have put out this "feeler" to find out whether anyone else knows of anything along these lines.

Since all the (polynomial) alternatives, up to Yang-Mills form/order can be parameterized with only five constants (Lagrange multipliers, including G and Lambda [the cosmological constant]), I would expect that the various research work that has been done along such lines could be brought together in a single survey like work, somewhat similar to the Parameterized Post-Newtonian (PPN) formalism.  In fact, I am somewhat surprised that I have not seen such parameterized Lagrangians included within PPN works I have read.

Maybe, it has just been too difficult obtaining solutions in such a general case.

David

"Will see if I can build one next week."

I think it is clear we both don't understand the content in the last few posts well enough to reproduce it ourselves, let alone apply it to derive our own conclusions. I think this content is interesting, so can't you focus on that for awhile longer so we can learn more? How about following up on some of David's comments?

Or if you understand the spin stuff better, how about explaining your spin 1 comments on gravitomagnetism? At what point in the series of approximations does everything turn into spin 1? If there was only spin 2 and no spin 1 to start with, how does neglecting the smallest terms in the Lagrangian lead to spin 1 appearing out of nowhere and spin 2 being over-powered or disappearing?

I don't want to become one of the commenters also requesting you to slow down, but could I request maybe focus the posts on learning more? It would be more interesting (and helpful, and useful) for all of us.

"To me, a toy is not worth playing with unless the solutions to the field equations plugged into the force equations get both plain old Newtonian gravity and light bending around the Sun."

Fist a quick comment as this goes with my previous comment on the stress energy tensor. To handle light like GR does, you'll can't treat mass as a scalar like in the mass "q" charge in your article above or as you previously did in your theory ideas (this was actually mentioned by several commenters back when you were designing your own theories, but I'm not sure you understood the objections).

Now back to Newtonian gravity and light.
I've seen comments made about Newtonian gravity only bending light half that of GR, or that the gravitational red-shift in the Pound–Rebka experiment agrees with the potential energy of Newtonian gravity. Even the assumptions in just these comments could lead to an interesting article.

I think I understand these topics well, and can see how misleading these comments could be if taken out of context of the assumptions in which they were stated.

If you take Newtonian gravity as it was first understood and taught in introductory classes (a theory described in flat space and universal time), and add in Maxwell's equations ... you get neither of these effects. Light does not bend, nor red-shift.

So even in this simple example, it is worth-while to dig into to learn what is going on here.
So it ruffles even my feathers when you say stuff like "... I still accept, from authority", when it appears you don't understand what you are "accepting" but are going to try to derive conclusions from it anyway.

I don't understand a lot of the physics contained in the last few quantum field theory discussions. I think we'd both gain a lot if we focused on following up on these. I don't enjoy as much the articles where you decide to propose a theory, and everyone has to guess why it is wrong. Not as useful, or as fun, as an article trying to following up on the physics more.

An amateur theory to replace GR may be sensationalist and get more people responding, but I want physics on "Science 2.0" not sensationalism. Most of the things I like about Science 2.0 are physics, and most of the things I don't like about it can be filed under writing to be provocative. I think this is what turns many people off about Science 2.0.

The spin 1 story in EM is rock solid, everything fits. The spin 2 story for gravity does not work at this time. No matter how many posts on the subject are done, there will be things that don't work.

Relax. A toy it a tool to learn something, not a sensationalist claim. I am motivated by research goals. Without those dreams, I wouldn't invest the time I do on this project.