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    E=mc^2...In Spaaaace
    By Hank Campbell | January 12th 2013 09:11 AM | 76 comments | Print | E-mail | Track Comments
    About Hank

    I'm the founder of Science 2.0® and co-author of "Science Left Behind".

    A wise man once said Darwin had the greatest idea anyone...

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    The only thing that might have saved a sequel like "Ace Ventura, Pet Detective, Jr."(1) was putting it in space. If you put ":In Space" at the end of anything, it gains instant credibility it might otherwise not have. (see The Muppets)(2)

    But some people, like Abraham Lincoln, don't need more credibility, they are already heroic.  So making Lincoln a Vampire Hunter is just showing off. And Albert Einstein doesn't need to go into space - but maybe his iconic formula, E=mc^2, does.

    In Einstein's Theory of Special Relativity, E stands for energy, m stands for mass and c stands for the speed of light (which gets squared - how can you square the speed of light when nothing can go faster than the speed of light? Read on)  Energy and mass have symmetry and are fundamentally the same thing and can even be converted into each other.  This was important in 1945 when it was time to blow up some cities in Japan.

    First, if you are new to physics, that c^2 may seem a little confusing, since you have been taught that stuff cannot exceed the speed of light, not even if the cables are installed by Italian government union laborers.  The c^2 number is instead to show a relationship, not that anyone expects you to go almost 450,000,000,000,000,000 MPH to make fission happen. Kinetic energy is proportional to mass and that is important to understand when energy and mass convert into each other.  Sticking with that Miles Per Hour concept, your car's kinetic energy is proportional to mass, velocity^2, and that determines how long you will skid before you come to a halt if you slam on the brakes. If you double your MPH, the skid distance is not twice as long, it is 4X as long because of that velocity^2. Science media is always agonizing over the perfect metaphor, the One Example To Rule Them All, that makes sense to everyone, so if mine is no good, tell us what makes it clearer.(3)

    Yet maybe Einstein needs to go into space to get the respect of everyone.

    Einstein is the Abraham Lincoln of physics; the only way to make him bigger than he is in the public consciousness would be to have him out there killing vampires. For that reason, everyone in physics would love to debunk him. E=mc^2 came from Einstein's Special Theory of Relativity and it has been validated too many times and in too many ways to count. It is evident in everything from smoke detectors in your house to the sun. We also owe him for making scientists cool.

    Lord Kelvin
    Einstein
    Airplanes can't fly
    Tall people age faster
    Nothing new to learn
    Funnier than J.C. Maxwell
    NOT COOL
    COOL

    He also made theoretical physics cool. He somewhat upended the traditional scientific method where you examine nature and figure out the math; he saw the math and then figured out how to validate it experimentally.  And, credit to his brilliance, physicists did just that. The entire High Energy Physics industry is due to Einstein.

    The problem is that now everyone in theoretical physics thinks they are Einstein and the public thinks that, because Einstein was a theoretical physicist, those must be better and smarter than experimental physicists: they get to think about stuff and get paid for it. I offer, as evidence, that character Sheldon on "The Big Bang Theory" TV show. He has never once worn a Science 2.0 shirt and if you want something done, you are better off making friends with the engineer, but people think he is the smart one regardless.

    Theoretical physicists think so too. Theoretical physics has, in many ways, provided an open season for biannual "Theory of Everything" proponents who know some math and then declare that it is up to the experimental world to prove them right. It's more like Hypothetical or even Speculative Physics than what Einstein did.

    But Einstein takes on all challengers and recently he had an interesting one. University of Arizona physics professor Andrei Lebed made waves hypothetically determining that E=mc2 may not hold up in certain circumstances. He says we might be thinking about mass the wrong way.

    Mass is mass. Going back to the car example again, the mass of the car determines that its inertia, like what would happen to another car if you hit it at some velocity, is the same as its gravitational mass, the weight an object has due to gravity acting on its mass. Right?  Right.

    Introduce modern mathematics which, because it is not science grounded in natural laws but is instead a language, can be used to propose worm holes, time travel and universes where dinosaurs are our galactic overlords. Theoretical physicists come up with new physics math every month, with the disclaimer that 'it has yet to be proved experimentally' but Lebed has something interesting in his claim. Because sometimes the weight of a quantum object has a slightly different reading, if it is because inertial mass and gravitational mass are different - mass is not consistent - then E=mc^2 doesn't work after all.

    Gravity is the result of a curvature in space - the greater the mass of an object, the bigger the dent it makes in that fabric we consider space to make space seem less crazy as a concept. More mass means more gravitational pull and so the curvature of space is what makes gravitational mass different from inertial mass. And then it becomes an arXiv paper and gets mainstream media attention because a theoretical physicist says he can debunk Einstein.

    What do other physicists think?  Like in every other field of science, physics is made up of different personalities so when he presented his maths at  the 13th Marcel Grossmann Meeting on General Relativity a few months ago, some people were open to it and some were derisive. 

    Here is his idea: A hydrogen atom consists of a nucleus, a single proton and a lonely electron orbiting the nucleus. In rare instances, the electron orbiting the nucleus will jump to a higher energy level. Then it settles back down. According to E=mc2, the hydrogen atom's mass will change along with the change in energy level. Lebed says he thinks something different will happen when that atom is moved far away from other objects, like Earth.

    Perhaps, he contends, E=mc^2 does not work the same where Earth is not curving space. The  electron could not jump to higher energy levels where space is flat, because there is no curvature of gravitation.  Closer to Earth, the change in energy levels might happen, due to that curvature of gravitation and we could detect differences in mass.

    "Instead of measuring weight directly, we would detect these energy switching events, which would make themselves known as emitted photons – essentially, light," he said in his statement. You clearly would need a lot of hydrogen atoms to get meaningful results. He proposes sending a mission containing a tank of hydrogen and a sensitive photo detector into space.

    Will it work?  Hey, I have no idea, I am not even a theoretical journalist much less a theoretical physicist. I don't believe anything at all until an experimentalist tells me I should.

    "But my calculations show that beyond a certain probability, there is a very small but real chance the equation breaks down for a gravitational mass," Lebed said.

    Will it mean the end for Einstein?  I am not going to invest in that new warp drive start-up company just yet.

    Citation: Andrei G. Lebed, 'Breakdown of the Equivalence between Passive Gravitational Mass and Energy for a Quantum Body', arXiv:1208.5756

    Notes:

    (1) That's a real movie. Like "Smokey and the Bandit 4" ("Bandit: Bandit Goes Country") I would never have even known it existed, except I needed to find some really stupid sequel ideas. 

    (2) Not the recent Muppets movie, with Jason Segel, that was terrific. The original sequels were pretty bad, though.

    (3) Even Nova can't come up with the perfect metaphor and they get a bajillion dollars in taxpayer money to write stuff.

    Comments

    > Gravity is the result of a curvature in space

    That should clearly be correct to "Gravity is the curvature in spacetime (plus the obsession of Nature with finding local extrema to 4-D paths through spacetime - why local? because otherwise I would surmise P=NP)

    The thought that Lebed's idea could be tested by observing what interstellar or intergalactic hydrogen gas (ionized or not) does (which has been going on since the 50s) immediately occurs. Would it work?

    Halliday
    Yatima:

    You stated:

    The thought that Lebed's idea could be tested by observing what interstellar or intergalactic hydrogen gas (ionized or not) does (which has been going on since the 50s) immediately occurs. Would it work?

    Now, I'm a Theoretical Physicist, not an experimentalist, so I'm not absolutely certain.  However, Lebed's idea depends upon there being no other form of excitation of the hydrogen gas:  No thermal excitation (or, at least, thermal excitations that are sufficiently small that one can actually detect the small "curvature" signal), no radiative excitations, and no perturbations due to electric fields, or such like.

    The fact is, I don't even see his idea of flying a spaceship full of hydrogen and detectors as practical, considering the radiation environment of outer-space.  Yeah, how is the hydrogen going to be sufficiently shielded from such an environment, since we can't even shield astronauts from lethal levels of such radiation.

    David

    Halliday
    Hank:

    I've been looking over Andrei G. Lebed's papers on this subject, on arXiv, this past week.

    I absolutely, wholeheartedly support trying to find systems to test the equivalence of inertial and gravitational mass (any and all of the equivalence principles).  Regardless of how "elegant" such is, or how "elegant" the theory of General Relativity is, such should always be questioned and tested.

    That's simply good science at work.

    In fact, a fortiori, I wholeheartedly support all good attempts to devise tests of that never-never land where Quantum Mechanics and General Relativity cross paths.  After all, how else are we to be guided in reconciling these two great theoretical edifices.

    Unfortunately, I am more than a little disappointed in Lebed's work.

    First, he gets the Post-Newtonian approximation of the metric near a gravitating body wrong:  He expands the spacial terms to second order, but only expands the temporal term to first order!  Even if one were to model the spacetime around the Earth as perfectly spherical, and non-rotating, the temporal term should have a second order term in the gravitational potential, or the spacial term should have no term proportional to the gravitational potential.

    Next, he neglects the spacial dependency of the gravitational potential, in order to treat the spacetime as if the only thing involved was a rescaling of space and time, vs. Special Relativity.

    He then used the Dirac equation in flat spacetime, using these transformed space and time coordinates.  Followed by a non-relativistic Pauli approximation of the Dirac equation, and expanding the transformed space and time coordinates, in order to get a non-relativistic Pauli approximation in terms of the "usual" space and time coordinates.

    However, once again, he neglects any spacial dependency of the gravitational potential.  In fact, he even appears to neglect the gravitational potential term within the temporal portion of the Pauli approximation, even though that is a significantly larger term than the spacial terms.  (A very similar error to his initial error with the Post-Newtonian approximation).

    Of course, this is to say nothing of his emphasis on the lack of commutativity between the electron energy operator and the "gravitational mass operator", he derived using the above methodology, even though the electron energy operator doesn't commute with the gravitational potential that multiplies the "gravitational mass operator" anyway.  (Unless, of course, one neglects, once again, the spacial dependence of the gravitational potential.)

    Unfortunately, with so many mistakes, it is truly difficult to know what would be the case if this had all been done correctly.

    In addition to the aforementioned errors, it also appears that another of Lebed's errors is/was the extent to which he relied upon the erroneous work of others.  This is most starkly illustrated in his oft expressed need to resolve issues of inertial vs. gravitational masses by means of temporal averages (always referencing the same sources), along with his oft expressed "paradox" (and its supposed resolution) for a free photon vs. a photon in a mirrored box.  (This latter problem appears more properly rooted in an all too common misuse of "relativistic mass" vs. actual, inertial mass.)

    Of course, one shouldn't be too harsh in such matters since "gravitational mass" is not so easily identified within General Relativity, since the source term is non-scalar, but a tensorial quantity, and the "interaction" is, likewise, non-scalar.

    In short, I am quite disappointed in this work, and in so many of the sources it relied upon, at least to the extent that he properly used and/or characterized them.

    David

    Hank
    Sure. If this were more solid it would be in a journal and not just on arXiv and in the Proceedings from the meeting. I am always going to want to see how smart people respond to these arguments and his math just made my eyes glaze over a little because, really, math can do anything. I figured the real insight would come after I talked about what he was doing.
    The Stand-Up Physicist
    In my Unified Field Theory (UFT, not UFO) fantasies, I imagine the reason for gravitational mass being precisely equal to inertial mass is justified by an appeal to quantum field theory.  If they had to be different, I would like a reason for the difference.
    Halliday
    Doug:

    Please tell me, even as a "Unified Field Theory (UFT, not UFO) fantas[y]", is "an appeal to quantum field theory" the "reason" why centrifugal "force" is proportional to inertial mass?  What about the Coriolis "force"?  What about the "force" that pushes you back in your seat when your car speeds up, or pushes you forward when you slam on the breaks?  All of these "forces", and all other inertial "forces" are proportional to the inertial mass of the bodies upon which they "act", so that the acceleration of such bodies is always independent of their inertial mass.

    Is "an appeal to quantum field theory" the "reason" for any of that?  Even as a "Unified Field Theory (UFT, not UFO) fantas[y]"?

    Is it even desirable?

    David

    The Stand-Up Physicist
    Have you ever written Newton's second law in an inertial reference frame using quaternion operators?  How about writing a force law in a rotating reference frame?  I did that early on, but I still don't understand all the implications.  I will try to blog about that sometime in February.

    The UFT fantasy was based on an experience I had. One term I thought would lead to a dual metric/potential expression for gravity. A second term led to EM. There was a cancellation between the two terms. None of the terms you listed involve gravity.  It all looks like the inertial mass side of the ledger.
    Halliday
    Doug:

    You are correct that "None of the terms [I] listed involve gravity [in the Newtonian sense]. It all looks like the inertial mass side of the ledger."

    So, by this, am I to conclude that you would not, even as a "UTF" "fantas[y]", think that "the terms [I] listed" would involve "an appeal to quantum field theory" as a "reason" for their existence/nature/character?

    David

    The Stand-Up Physicist
    The fantasy is about explaining the up-to-our-best-measurement equivalence of gravitational and inertial mass. There are quite a number of ways to look at the inertia of inertial mass. A rotating reference frame provides a number of ways examine inertial mass.  As you point out, the acceleration is always independent of the inertial mass.  At this time, my speculation does not provide a reason for that property of inertial systems.
    Halliday
    The point, my friend, is that plain old ordinary Newtonian Mechanics offers quite a good explanation for why inertial "forces" are always proportional to the inertial mass, and, hence, why their accelerations are always independent of (inertial) mass.

    The fact is that there is absolutely nothing in Special Relativity, or Quantum Mechanics that changes that picture.

    On the other hand, as Einstein thought upon the question of how to incorporate gravity into (Special) Relativity, he noticed that gravity, "up-to-our-best-measurement[s]", exhibits this very same behavior:  A "force" that is always proportional to the inertial mass, and, hence, one where its acceleration is always independent of inertial mass.  (See, for instance, Einstein's elevator, and spinning disk thought experiments.)

    The truth is that, according to the theory Einstein came up with (General Relativity), gravity is simply another inertial "force"!  If gravity is a "true force", then so are the other inertial "forces" I mentioned.  On the other hand, if those other "forces" are "fictitious forces" (as they are often referred to within Newtonian Mechanics), then so is gravity.

    So, in this view, gravity is no more, and no less a "force" than any of those other inertial "forces" I mentioned.

    Therefore, in this light, if any of these inertial forces needs some "reason" besides that given by Newtonian Mechanics, and upheld by Relativity (both Special and General), and Quantum Mechanics (even all forms of Field Theory), then so do they all!

    So, now what do you think?

    David

    The Stand-Up Physicist
    The elevator thought experiment does not get the tidal forces correct (details matter). There is no tidal acceleration force on the elevator, while there is one for gravity.

    Richard Epp gave a half hour talk about the rotating thought experiment, available on line.  For a rotating system, there would be a tidal force, but it works opposite of what gravity does.  For a gravitational system, the feet push down the scale, and the tidal force pushes in.  On a spinning system, if the speed is adjusted just so, the feet can push down the same amount, but the tidal force pushes things apart.  I think there may be something to the difference in tidal forces.

    Epp gave a decent description of the precession of the perihelion of Mercury issue, a calculation I blogged about a while ago.  Two things to note.  First, there is an exact solution to the problem, no approximations required.  I have not had the time to digest it, but those interested can take a look.

    The second point was that this was a well known problem for Newtonian gravity.  Einstein solved the problem by coming up with a new theory of gravity, a truly difficult job I myself have failed at.  Yet today we do have all kinds of problems with Newtonian gravity: gravity clusters with things moving too fast, thin disc galaxies spinning too fast, all the galaxies moving too fast now.  That sounds like a similar situation: we have well defined problems between observation and theory.  If one expresses these problems in terms of g, the acceleration of gravity here on Earth, as Milgrom has done, both dark energy and dark matter arrise at way beyond trivial amounts of acceleration (try ten orders of magnitude smaller than 9.8 m/s2). I still dream of a theory that could be dual: one angle it has the weak and strong equivalence principles and makes a distinct math prediction, and an other angle uses potential theory but ends up at the same distinct math prediction. It is the potential path that might stand a chance at building a bridge to quantum mechanics.  You do make a fun point though, that such a bridge must be a little odd since if it was just like any other fundamental force (non-fictional as it were), then the duality to a metric theory would be lost.
    Halliday
    Doug:

    You are being distracted by "red herrings".  The fact that different inertial "forces" can have different characteristics (differences at first and/or second order, for instance) doesn't negate them being inertial "forces".  (Besides, you should check more closely into what stipulations Einstein placed upon the thought experiments.  As an additional point, there are potential matter-energy distributions that produce quite different tidal-like "forces".)

    Additionally, Einstein most definitely did not create General Relativity (GR) with the intent of answering the perihelion of Mercury problem.  It was simply another test for any "new theory of gravity and mechanics".

    As for the "dark matter" or "dark energy" issues:

    First, the best candidate solution to the accelerated expansion of the universe (the so called "dark energy" problem) is Einstein's own "greatest blunder":  The cosmological constant within GR.

    As far as "dark matter" is concerned, it is a mystery.  However, what other potential explanation will be able to include the optical distortions (gravitational lensing) that are observed to be associated with this "missing"/"dark" matter?  On the other hand, I have noticed that all cases where this "missing"/"dark" matter is invoked to answer the question of motions, it is within the context of Newtonian Gravitation and Mechanics.  I have yet to see a fully General Relativistic treatment of such motions.  (Admittedly, this could still run afoul of gravitational lensing observations.  On the other hand, the "corrections" could still be significant, just as in the case of the perihelion of Mercury.)

    The point is that there is nothing in your arguments that negate the present "picture" that "gravity" is simply another inertial "force", and, hence, people that attempt to "build a bridge to quantum mechanics" by attempting to treat "gravity" as a "true force" are almost certainly "barking up the wrong tree".

    David

    The Stand-Up Physicist
    It is tricky to get a handle on this topic.  I find that tidal forces are worthy of consideration.  It bothers me when people claim there is no difference between the accelerating elevator and one on a planet.  Careful folks will throw in the caveat that they meant over a small volume of spacetime (small enough to ignore tidal effects).  It would be interesting to know if someone has figured out an mass-energy distribution that mimics the rotating satellite.  Gravity being universally attractive might not allow a repelling sort of tidal force.  A small fish perhaps, but a fish.

    Sorry if my comment implied the precession of the perihelion of Mercury was a main driver for GR.  It was the clash with Newton's gravity law in potential form:


    Change the mass density, and the scalar potential must change everywhere instantaneously.  That does not play nicely with special relativity.  In Chapter 7 of Misner, Thorne, and Wheeler, they go over an all-star list of physicists who show that if you wish to start with Newton's law, and the only requirement is to get along with special relativity, then you end up at Einstein's field equations.

    People consider that a powerful endorsement of GR.  I am skeptical.  If I want to do spacetime physics, I need to start with spacetime quantities.  Don't start with rho, a simple change in coordinates will make that spill over into a mass current density.  Don't start with a single-valued potential phi, a simple change in coordinates will make it spill over into a 3-vector potential.  Start at the wrong place, end up at the wrong place.  I don't have a logically consistent counter-offer, but there is a reason for my skepticism.

    Certainly everyone in Einstein's time knew that Newtonian gravity could not explain the precession.  People were actively searching for a planet always on the other side of the Sun (such a sneak).  It was a no-see-'em, just like either dark energy or dark matter.

    I agree, the most popular solution for the accelerated expansion of the Universe is the cosmological constant of GR.  Popular solutions have a bad track record in physics.  The source of my skepticism on dark energy was the young researcher who didn't even understand how one might express the accelerated expansion of the Universe in terms of g (9.8m/s2).  At a meeting, he sounded like he thought my question was nonsense, like my units were wrong.  It is not a trivial calculation to do, but Milgrom has done it, and the great acceleration is wimpy, while its scale is impressive.

    I have read a few papers on the rotation profile of thin disk galaxies.  From discussions, I am pretty sure that someone has given GR a test drive with such models, and they confirmed it did not materially change the calculation.  What I find frustrating is in this context, people ignore the accomplishments of MOND.  That theory had but one parameter, an acceleration of 10-10g. When gravity gets that wimpy, then the 1/R2 force law becomes a 1/R force law.  Play your favorite dark matter games, but for some absurd reason, the distribution of dark matter has to follow that pattern for well over a hundred galaxies.  My bet is that Milgrom and MOND are similar to the Balmer and his lines of hydrogen: there was truth in the simple algebra that was only revealed later with Bohr's model of the hydrogen atom.  I think it is a math issue, not missing stuff.  Time will tell.


    I was hoping to communicate a more subtle approach.  Gravity cannot be another force like any other force or it would have been figured out quite some time ago by folks so much brighter than me and sometimes you.  My mumblings to myself are about spacetime "pre-calculus", how should one go about constructing a mathematical field in spacetime after which mathematical analysis has already been all worked out.  The fun thing about this whisper thin idea is that if one could connect the study of change (calculus) with groups, there might be a reason behind the universal properties of Nature.
    Halliday
    Doug:

    You have some good stuff, here.  :)

    You are correct that it is wrong to "claim there is no difference between the accelerating elevator and one on a planet."  The proper way is as you say of "careful folks", such as Einstein.

    No inertial "force" need completely mimic another inertial "force" in order to be considered such.  To try and require such is to chase a "red herring".

    As far as "gravity" is concerned, it should be sufficient to note that there is no definite relationship between the first order "force" and the second order "tidal force" terms, since it all depends upon the distribution of mass-energy-stress.  The central mass distribution is a special case, only.

    You continue with:

    ...  In Chapter 7 of Misner, Thorne, and Wheeler, they go over an all-star list of physicists who show that if you wish to start with Newton's law, and the only requirement is to get along with special relativity, then you end up at Einstein's field equations.

    I certainly wouldn't characterize that chapter of Misner, Thorne, and Wheeler's Gravitation in quite that way, or with anything quite like that "force".  However, I do think it is correct to say "that if you wish to start with Newton's law, and the only requirement is to get along with special relativity, then" the simplest (complete and internally consistent) system "you end up" with is "Einstein's field equations."

    Of course, even if your statement were true, this could never be taken as anything like a "proof" that "Einstein's field equations" are "true" or "correct", since that can only be determined by Nature (not the rag) through observation and experiment.

    You then go on with:

    ...  If I want to do spacetime physics, I need to start with spacetime quantities.  Don't start with rho, a simple change in coordinates will make that spill over into a mass current density.  Don't start with a single-valued potential phi, a simple change in coordinates will make it spill over into a 3-vector potential.  ...

    I would say you are quite correct in this assessment.  However, it does lead me to ask whether you think the way Einstein derived General Relativity (GR) followed something similar to what is found in Chapter 7 of Misner, Thorne, and Wheeler's Gravitation?

    While I know that Einstein had at least a few "false starts" in his pursuit of getting Newton's Universal Theory of Gravitation to be consistent with Special Relativity, and I wouldn't be surprised if he hit upon some things similar to what is found in that chapter, I also know that the most correct way to derive GR is "to start with spacetime quantities."

    Perhaps you should actually go through the other chapters of Gravitation, or some other good text, and see what the actual spacetime underpinning are of GR.

    Next you state:

    I agree, the most popular solution for the accelerated expansion of the Universe is the cosmological constant of GR.  Popular solutions have a bad track record in physics.  ...

    I actually don't know, nor do I care, whether "the most popular solution for the accelerated expansion of the Universe is the cosmological constant of GR."  (Emphasis added.)  What I do know is that the use of the cosmological constant of GR is the simplest solution that introduces the smallest set of "unknowns".  It is also the most likely proposition to "fail", since it has the least flexibility to accommodate any number of possible variations we may observe in any of a number of experiments.

    So, the reality is that the cosmological constant of GR is "the one to beat" as far as any and all "solution[s] for the accelerated expansion of the Universe".

    The only problem, then, is why the cosmological constant of GR is so small.  (Quantum Mechanics [QM] has held, for a long time, that this term should be far closer to infinite.  In fact, QM really has no way for this to be anything besides infinite.  The only way we get anything like a finite quantity is to assume various things about "energy cutoffs", and such, that aren't actually a part of any coherent theory.)  Having it be exactly zero was the only actual "alternative" to "infinite" that seemed to be "acceptable" for so long, before the acceleration was observed.

    Once again, Nature (not the rag) must trump even the most elegant theories.

    By the way, I understand your interest in Milgrom and MOND.  However, this has failed more than a few observational tests.  Once again, Nature (not the rag) must trump even the most elegant or desirable of theories.  (Of course, this is to say nothing of the fact that MOND is quite incompatible with Special Relativity.)

    If you have any references of any attempts "that someone[/anyone] has given GR a test drive with such models [the rotation profile of thin disk galaxies, or clusters, or anything else]" I am most interested, regardless of what they confirm or refute.

    You then end with some musings about "spacetime 'pre-calculus', how should one go about constructing a mathematical field in spacetime", and "connect[ing] the study of change (calculus) with groups".

    Are you not aware of the intimate history of connection between "the study of change (calculus)" and groups (usually infinite, continuous groups)?

    It would seem that there is already much there that your are not yet aware of.

    David

    P.S.  As an aside on the connections between groups and "the study of change (calculus)":  It used to be that groups were used to study the nature of systems of partial differential equations.  Such analysis could have great benefit in constructing solutions to such systems, and otherwise understanding the nature thereof.

    However, modern field theories have essentially turned the relationship upon its head.  There is now the tendency to construct a system of partial differential equations based upon the group properties, rather than the other way around.

    Now, please don't read this as a over strong indictment of modern methods, especially within particle physics.  When one is trying to construct a phenomenological model (like the Standard Model), as a system of partial differential equations, it is quite reasonable to use group properties in organizing the knowledge imparted by experiments, along with other simplifying assumptions, thus leaving some set of parameters that are likewise determined by experiment.

    Halliday
    Doug:

    While I stand by my statement

    No inertial "force" need completely mimic another inertial "force" in order to be considered such.  To try and require such is to chase a "red herring".

    I did happen upon some interesting tidbits pertaining to gravity looking-like the uniform gravitational field, and a "rotating satellite".

    In one text, where the author is actually speaking against the "strong" principle of equivalence, he has the following as a footnote:

    It would seem that a perfectly homogeneous gravitational field (zero tidal force) cannot be distinguished from the pseudo-force field of uniform acceleration.  This is true, but not very relevant:  perfectly homogeneous gravitational fields can only exist under extremely exceptional and unrealistic conditions.  It can be shown that uniform fields are only possible in regions (cavities) completely surrounded by a continuous distribution of mass.

    So, you'll notice that he actually gave away the fact that such a solution actually exists.  He just discounts it, that's all.

    In the case, of a "rotating satellite", we already have such:  The accelerated expansion of the universe in the form of the cosmological constant.  You see, if you go out into "open" space, far enough away from the local group of galaxies, you will have to continually fire rockets just to maintain your relative position to Earth (our galaxy, our galactic group, etc.), just as the outer walls of the "spinning satellite" must continually push you back toward the axis of rotation to keep you within the satellite.

    It's a huge satellite, to be sure, and it is as if it is rotating quite slowly, but it does have an extremely similar characteristic.  In fact, the same author that revealed a solution for a uniform/homogeneous gravitational field also provides the modification of the Schwarzschild black-hole solution to Einstein's equations with the cosmological constant.  Expressing the modification in terms of the "gravitational potential", the potential changes from the usual -GM/r to -GM/r - Λr2/6 (using the author's sign convention).

    You'll notice that the second term is like that of a bowl, with the "force" being either directed inward, toward the center, or outward, depending on the sign of the cosmological constant.

    (You'll also notice that the cosmological constant does not have units of acceleration [or, equivalently, inverse distance or time].  However, the square root of its absolute value does.  [I think I had pointed that out once before.])

    Anyway, I just thought these additional tidbits might be interesting, if not helpful.

    David

    The Stand-Up Physicist
    I think I am ready to "let the tide go out". When I first learned about GR from Edwin F. Taylor, I recalled something along the lines of the tides being the only thing about gravity that was real.  I don't recall the exact phrase, only the general theme.  I also have a dim memory of people saying a not physically realizable mass distribution could have no tidal effect.  As I walk around thinking this issue through, the tides are a positional issue of the underlying factor, namely gravity itself.

    We are both for little flexibility in a proposal.  I will now upgrade my view of the cosmological constant because it has little flexibility.  My dream (and I do emphasize dream to those who wish to whack me on the head with stale bread) is for a new approach that has no flexibility.  Due to its limited flexibility, a non-zero cosmological constant is the best proposal on the table.  For me, it is not good enough.

    As far as MOND is concerned, I started to write up what I knew (limited) and found someone who created a few pages about the MOND versus Dark Matter debate.  He is honest about some of his MOND biases, but is not a complete fan-boy.  If interested in the technical details, spend some time clicking around "The MOND Pages" by Stacy McGaugh. One big picture view is that MOND does well at the galaxy level with rotation profiles and the Tully-Fisher relation while dark matter comes up short.  Go to collections of galaxies or cosmology, and MOND comes up short while dark matter does better.  

    McGaugh states well a feeling I have:
    I guess what I find most unfortunate is the following fallacy:
    MOND is wrong because [insert your reason here], therefore I don't need to think about it.
    Let us accept for the moment that MOND is indeed falsified. Even in this case, it has been quite well established that MOND provides an economical mathematical description of rotation curve data. That must be telling us something. At the very least it is an organizing principle that dark matter obeys, if only in rotating disk galaxies. Consequently,
    The observed MONDian phenomenology requires an explanation in the context of ΛCDM.
    Unfortunately, I see no discussion of this critical point. I have spent an inordinate amount of research time trying to address this very question. I have not published a single thing on it because I have never come up with anything remotely satisfactory. This is a hard problem, but an essential one to crack if ΛCDM is really the correct picture.
    At least there is consensus that this is a hard problem.

    McGaugh has a list of articles far better than mine on the subject, but I did not see anything based on looking at titles to suggest someone has documented their efforts to apply the Einstein field equations to a rotating galaxy and seen no difference to a purely Newtonian approach.  This is a negative result, and a hard one to do at that.  The details of how one does the experiment would matter, but it is doubtful such details would be reported.

    You wrote:
    Are you not aware of the intimate history of connection between "the study of change (calculus)" and groups (usually infinite, continuous groups)?
    There is a huge amount of synergy between calculus and groups.  It comes up in the study of Lie algebras that generate Lie groups that play such a critical role in physics.

    What I was trying to suggest was about mathematical analysis, the proverbial foundations of a rigorous approach to calculus.  I spent a year with Rudin's "Mathematical Analysis".  By page 5, he defines a mathematical field, a critical structure for all that follows.

    On repeated occasions, I wanted things to be mathematical fields.  If you recall, I argued that the hypercomplex numbers were a mathematical field due to the ability to spot those elements that did not have an inverse and just remove them.  I lost that argument because that is an illegal move in the sport of defining a mathematical field.  I also hope to be clever in dealing with zero and the positive integers, but that also did not pan out.  A few times I wrote quaternions were a mathematical field when I should have always written they are a division algebra (something I have known all along, but I blame faulty hardware and the pressure to respond in a timely fashion).  Calculus with quaternions remains a problem because the small divisor will not commute.  Of course I could have just accepted Frobenius' theorem about finite division algebras, but one does learn some things from struggles.

    By page 103, the careful man gets around to the definition of a derivative readers expect to see:

    I believe this was the kind of notation used by Leibniz, so right from the start, this has been good enough.  It is good enough for classical physics.

    Here is where I start speculating.  That definition of a derivative is also used ubiquitously in work in relativistic quantum field theory because it is still true.  But it may not be the richest definition we could use.  Once one needs to account for the sum of all possible histories, the idea of using a 1D field as the starting point needs to be altered.  One cannot say spacetime is just like 4 1D structures because even classical things in spacetime don't commute - see a spinning bicycle wheel as an example.

    A message I read into the standard model is that group theory has to be a player at the foundations.  I don't see a group at the table of the limit definition of a derivative.  If instead I have to account for the past in spacetime being separate from the future of spacetime, meeting only in the neighborhood of now, a structure like that could have links to group theory.

    If there is any truth to this line of speculation, it would require a team of minds that are skilled in mathematical analysis, the standard model, general relativity, and possibly category theory.  Why the last element on the list?  The standard model and general relativity make many testable predictions that are confirmed by experiment.  They are not going to be wrong in the typical way we talk about wrong (it did rain last Tuesday or it didn't).  Instead one would need to show for example that the gauge relationships in the standard model (U(1), SU(2), and SU(3)) were effectively the same as the math structure proposed for doing spacetime calculus with a richer definition than the limit one.  Part of my speculation is that I  don't think there is a way for those groups to be in there in the exact form as one might expect.

    I tried and failed to get a "totally symmetric" kind of calculus for spacetime.  It remains a belief I cannot support that such a creature exists.  This is my mathematical unicorn.  It would be this kind of creature that would be used for, well, any study of change.  Everything has inertia, unless it is the one thing that doesn't, namely light.  

    This is mostly a math speculation that has to connect to a huge amount of physics.  I don't expect progress to be made.  The speculation does make a prediction about future research in physics.  We will continue to get stories of further proof that dark matter exists, that dark energy exists, that inflation must be true - year after year.  Observational data will improve.  Yet nothing comes together.  Same old, same old, with GR not playing nice with quantum mechanics as has been the case since the 1930s.
    Halliday
    Doug:

    You may want to check into such things a little further.  In at least a couple of senses, General Relativity (GR) was actually "not good enough" for Einstein.  He really wanted something that incorporated "everything" with no "flexibility" whatsoever.  I don't know if that will ever be achieved.  However, the fact that once we incorporate all the constants of Quantum Mechanics (QM) with all the constants of GR, all units become "one" is suggestive.

    Now, as for MOND, there is absolutely no reason, whatsoever, to have to provide an "explanation" for "MONDian phenomenology", whether "in the context of ΛCDM" or any other model.  The only thing(s) that ever need any explanation are observed phenomena.  This is independent of whether MOND, or any other model, "fits" any particular class of observations any better than any other model.

    By the way, I can just about guarantee that "apply[ing] the Einstein field equations to a rotating galaxy" will be "seen [to have] difference[s] to a purely Newtonian approach."  I state this due to the differences in the two theories at anything beyond first order:  "a purely Newtonian approach" is but a first order approximation to "apply[ing] the Einstein field equations", regardless of the situation.  (By the way, it is not so much "how one does the experiment", especially since "the experiment" is really just obtaining observations of already existing phenomena.  The issue is in the theoretical predictions that are being compared to those observations.)

    On the subject of groups related to calculus and analysis...  I still think there is much already there that you simply haven't noticed or heard of, yet.  There is already a much richer relationship than you seem to acknowledge, or appear to even imagine.

    Furthermore, even in infinite dimensions, one must still have the one dimensional derivative.  Furthermore, so long as one has vectors, one may reduce any multidimensional derivative in terms of one dimensional derivatives.  Actually, this all comes back down to the nature of manifolds:  As one focuses on increasingly tiny regions of a manifold, one is able to approximate the manifold, to increasing accuracy, as a flat vector space.  (The same thing is what relates Lie algebras to Lie groups:  The groups are the manifolds, while the algebras span the vector space tangent to the manifold.)

    By the way, "the idea of using a 1D field as the starting point needs [not] be altered" even "Once one needs to account for the sum of all possible histories":  One simply has an infinite dimensional space/manifold.  (Besides, the mathematics has been long worked out in at least a couple of different ways.  Admittedly, there is certainly the possibility that there are more elegant methods yet to be elucidated.)

    You say that "A message [you] read into the standard model is that group theory has to be a player at the foundations."  I think you may not be alone in this "reading", however, there is quite another way of looking at the relationship between the system of differential equations of the Standard Model (SM) and the group symmetries.  (See, for instance, my P.S. in my message, above.)

    As for your observation that "[you] don't see a group at the table of the limit definition of a derivative"...  I have been thinking about writing an article on the great similarities between Yang-Mills (group theoretical) field theories, and GR.  I think it is quite illuminating to see what in GR is actually most like the vector potentials in Yang-Mills theories.  (Remember, Maxwell's Electromagnetism is a special case of a Yang-Mills theory.)

    As for the issue with "GR not playing nice with quantum mechanics", which, as you point out, "has been the case since the 1930s", or so...  This is always a most fascinating subject for me.  Of course, as I have pointed out elsewhere, there is actually a "trifecta" that don't quite "play nice" with one-another:  GR, QM, and Statistical Mechanics (or even its more restricted "persona" as Thermodynamics).

    Yes, the latter tends to be forgotten, or assumed to be a "solved problem".  (The problem is that while both GR and QM "play nice" with Special Relativity, Statistical Mechanics has yet to be reconciled with even Special Relativity.)  It may help if we (collectively) broaden our search for "reconciliation" to include all three.  Maybe.

    Anyway, I only bring this up along the lines of there being much more for you to bring within your grasp if you wish to work along these lines.  However, I think the place for you to start may be with gaining a better understanding of the rich relationship between calculus, analysis, and groups/algebras.  There definitely seems to be far more there than you seem to imagine.

    Another possible avenue, as I have mentioned before, may be Clifford analysis.  Maybe.

    David

    The Stand-Up Physicist

    David:

    Perhaps I worded it wrong, but while I think MOND as a proposal has too many flaws to be true, I still accept that it has placed an algebraic noose around any other proposal.  The force of gravity has a 1/R dependence when gravity fields are super wimpy.  So says the data for both the rotation profile and the Tulley-Fisher relation.  If it had only needed 1/R^3, that would be easy to account for with GR.  But a force proportional to 1/R?  GR can do a force proportional to 1/R^2, and higher order correction terms from space and time bending, but 1/R is a magic trick.

    I am a fan of algebraic nooses.  It reminds me of the Balmer equation for the absorption lines of hydrogen.  Odd law Balmer found.  Niels Bohr came up with the creative idea of quantizing angular momentum.  Fit the algebraic noose perfectly.

    There is a h u g e study devoted to the logical foundations of quantum field theory, trying to come up with axioms to make it all work.  And yes, I know next to nothing about the technical details other than they eventually get "stuck" in some sense (otherwise they would be bragging about how great their answer was).

    Lots of other good comments, but I thought I would say something about thermodynamics.  That is a frustrating subject for me.  I feel like I have seen a glimmer of truth, but have not been able to connect it to anything larger.  

    There is the "Arrow of Time" problem, namely that physics laws are identical if time goes forward or goes in reverse.  Stefan Boltzmann himself was aware of the issue: add just a pinch of asymmetry with the second law of thermodynamics, and everything works great.  The question is why the second law gets to be asymmetric.  A fellow named Hew Price has an entire book on the subject.

    In my opinion, it is the question that is wrong.  The better question concerns the "Arrow of Spacetime".  That has a handedness, from the space part of spacetime, not the time part.  Write absolutely every law without exception in a relativistic way, and even if the speeds are dog slow, there will be a trivial amount of handedness.  This is one of the reasons I have stuck so strongly with quaternion expressions for physics equations: a wee bit of handedness is built in.

    The Lorentz group can be used to reverse a time interval.  It is a global sort of transformation, so do reverse the process, use the same member of the Lorentz group.  With quaternions, one can make a local function of spacetime to reverse a time interval.  To reverse the reverse requires a different local function, differing only in the cross product term.  In the classical realm, that would be a stupid tiny difference, but it would be a difference.

    I have not been able to build that observation out into any real thermo.  Frustrating.  So it goes.

    Doug

    Halliday
    Doug:

    If you want to be serious about MOND as a contender vs. General Relativity (GR) I recommend that you take a look at Tensor–vector–scalar gravity (TeVeS).  At least it is relativistic.  What's more, it actually uses a metric (actually two related metrics [gag...]).

    However, from the standpoint of "little flexibility in a proposal" I would say it looses "hands down":  It doesn't just have a finite set of arbitrary parameters; it has an arbitrary function (like having an infinite set of arbitrary parameters [potentially even an uncountably infinite set]), in addition to a couple of arbitrary parameters, and a non-dynamical scalar function over spacetime.

    (Since the non-dynamical scalar function has no derivatives anywhere within the full Lagrangian [density], it actually acts as a Lagrange function, which serves to pull an entire portion [or two] of the Lagrangian [density] out as separate equations.  This is perhaps the worst thing I have ever seen with respect to a Lagrangian formulation of a supposedly "physical" theory.)

    Really, if it weren't for the use of two related metrics, one could consider TeVeS as simply being GR with additional scalar and vector fields, where the scalar and vector fields could be considered a particular "Dark Matter" theory.  (It's actually quite possible that after one expands the so called "physical metric" of TeVeS in terms of the metric used in the Einstein-Hilbert portion of the Lagrangian [density], one would simply have GR with additional fields [from the scalar and vector fields] that actually interact with "ordinary" matter.  So, a "Dark Matter" theory with some "Dark Matter" interactions with "ordinary" matter.)

    David

    The Stand-Up Physicist
    Thanks for the reference.  I had yet to read that MOND, as proposed, does not conserve momentum.  That would have got me to rescind the proposal if it was mine :-)  Making it behave better after the fact ain't pretty.  By comparison, GR makes a graceful transition from the relativistic form to Newon's theory.

    The plight of physics today has some parallels to biology in the 40s.  There was no doubt there was some sort of genetic material out there.  People argued over what the stuff was made of, with protein being a leading candidate.  The nucleic acids look quite boring.  People give Schrödinger the physicist credit for the idea that a boring molecule would be a prime candidate for the genetic storehouse of information.  Work with viruses indicated nucleic acids must do the job all by themselves.  There was also a bit of apparently dull chemistry balancing between the different bases.  In hindsight, it were those bits of biotrivial that were part of the triumph of DNA.  Once known, further mysteries were solved, and continue to this day.


    I view MOND as a clue from a broken theory.  In its classical domain where it competes with Newton's break-the-speed-of-light gravity theory, the force law depends on one parameter, and the force gets a 1/R dependence.  I consider that a clue to a better proposal.  There were lots of "wrong" fish in the hunt for the genetic material, which may include my interpretation of MOND.

    Speaking of algebraic constraints, I have a question about dark energy and the cosmological constant.  Do you view it as a case where gravity makes things repel each other?  On a big picture level, isn't that bad, bad, bad?
    Halliday
    Doug:

    You state:

    I view MOND as a clue from a broken theory.  In its classical domain where it competes with Newton's break-the-speed-of-light gravity theory, the force law depends on one parameter, and the force gets a 1/R dependence.  ...

    (I assume your "the force gets a 1/R dependence" part is referring to the way the "force law" asymptotically approaches a 1/R dependence for very small "acceleration".)

    Actually, MOND depends upon an arbitrary function, η, in addition to the "one parameter" you refer to.

    While it is true that the function, η, is constrained for very large and very small arguments—namely, for |x| ≫ 1, η(x) acts approximately like one (1); while for |x| ≪ 1, η(x) acts approximately like x—it is sufficiently unconstrained (even if we were to require it to be invertible, and smooth) that it is equivalent to an infinite set of parameters (at least a countably infinite set, and quite possibly still uncountably infinite).

    The AQUAL Lagrangian version of MOND depends upon an arbitrary function, f, that is related to η as df(y)/dy = η(sqrt(y)).  Since AQUAL is the non-relativistic limit of TeVeS, I expect that the arbitrary function, F, in TeVeS is similarly related to η and f.

    So, no, even in its classical domain, it depends upon far more than "one parameter"!

    David

    The Stand-Up Physicist
    There are plenty of reasons to not like MOND.  I just read another one:
    A: MOND can be interpreted as either a modification of gravity through a change to the Poisson equation, or as a modification of inertia through a breaking of the equivalence of inertial and gravitational mass.
    Yuk.  It is one of those beliefs I have based on the overall success of general relativity to date along with a good number of experiments that inertial and gravitational mass will be identical no matter how big or small the accelerations happen to be.  MOND supporters have to appeal to the idea that the symmetry can be broken for super tiny accelerations.  

    The function mu is not constrained by observation.  Or by theory.  Pack it with as many parameters as fits the mood, but I don't think it alters the predictions of the proposal.  How one builds mu is both arbitrary and not important to the empirical result.  The same cannot be said for a.  Change that and MOND does not match the data.

    I vaguely recall reading that functions like mu come up in other subjects in physics (was it solid state physics?).  At this cutoff, the physics changes... something along those lines.
    Halliday
    Doug:

    By the way, the Greek letter η is called 'eta', while the Greek letter called 'mu' looks like μ.

    Yes, the theory certainly doesn't constrain the function η much.  So it is free to take on whatever form will best "fit" whatever data is desired.

    Of course, if we could make measurements within the realm of (gravitational) accelerations on the order a0, we could do more direct tests of MOND/AQUAL/TeVeS, though the arbitrariness of η would actually make even that endeavor quite troublesome.  ("Oh, that set of data will fit just fine if η, in that realm, takes the form ...")

    The "trouble" is that we can only do more or less direct tests within our own Solar system, where (gravitational) accelerations are reasonably large, while we have galactic scale observations that are almost strictly within the realm of very small (gravitational) accelerations.

    Of course, we also have cosmological scale measurements (e.g. expansion and acceleration of the Universe itself), that must also be explained.  (TeVeS has, at times, needed the addition of massive [~2eV] neutrinos [a form of "Dark Matter"] in order to fit a wider class of observations.)

    My feeling is that the "success" of MOND/AQUAL/TeVeS is probably telling us far more about the nature of "Dark Matter" than it is about the characteristics of gravity.

    David

    The Stand-Up Physicist
    My feeling is that the "success" of MOND/AQUAL/TeVeS is probably telling us far more about the nature of "Dark Matter" than it is about the characteristics of gravity.
    Fair enough.  My issue with the Dark Matter area of study is that it looks like monkeys typing away at a computer simulation.  At this point, Dark Matter doesn't feel like a physics hypothesis which tend to be exceedingly precise, see any tests of GR or what happens with all that data pouring out of the LHC.  I wish them luck trying to pin down the basics like what dark matter is made of.  I will continue to fish for math omissions, because those will apply to all scales of the Universe.
    Halliday
    Doug:

    Yes, functions like η have shown up in other phenomenological models in history.  In all other cases I'm aware of (such as the heat capacity, and density of states of solids), there did turn out to be a quite natural explanation that "answered" the phenomena, but almost always in such a way that the "true" version of the function turns out to be far more complicated than anyone could have ever guessed.  ;)

    David

    The Stand-Up Physicist
    David:I believe the MOND proposal uses a mu and not an eta, but that is not an important detail.  The fact that the proposal uses such a function is a clear sign that one should keep on the lookout for a "true" version.  Said beast will look quite a bit different than MOND, except under a particular set of conditions.
    Doug
    Halliday
    Doug:

    You state:

    I believe the MOND proposal uses a mu and not an eta, but that is not an important detail.

    Actually, you're quite right!  My bad.  :{

    I don't know how I made that mistake.  :/

    All I can figure is that my dyslexia has a new form I didn't know it had.  ;}

    David

    The Stand-Up Physicist
    Greek dyslexia requires an advanced degree :-)
    Halliday
    Doug:

    You asked:

    Speaking of algebraic constraints, I have a question about dark energy and the cosmological constant. Do you view it as a case where gravity makes things repel each other? On a big picture level, isn't that bad, bad, bad?

    Well, since "gravity" is simply about curvature of spacetime (driven by mass-energy-stress), no, it is most certainly not "a case where gravity makes things repel each other".  Gravity, in this sense, neither "makes things repel" nor attract.

    The cosmological constant (actually, the product of the cosmological constant and the metric of spacetime) can be considered as a form of "energy" of spacetime itself, that, like all energies (including mass and "stresses") affects the shape of spacetime.

    Another way of considering it is like unto Sascha's "model" where space sort of "creates" more space "in between".  Kind of a self "replication".

    Does that help?

    David

    Hopefully David can comment on this, but I thought the very essence of the strong equivalence principle is that gravity is explained fully explained by the metric with no extraneous fields. Hence why the null results of searches for "fifth forces" are supporting tests of the strong equivalence principle.

    Look at Brans-Dicke theory. With certain range of parameters it is consistent with current tests. It is a metric theory of gravity, but involves a "potential" as well if you wish to call it that. Would that fit your initial hopes for a gravity theory? However because of this extra field it does not satisfy the strong equivalence principle (even if it does squeak under any current experimental test of it, as long as parameters are chosen in a certain range).

    So if you want to have a theory of gravity with a potential field (that is quaternion valued or whatever), I think you have already failed to satisfy the strong equivalence principle. Your starting point itself appears invalid, as you are proposing a "fifth force".

    The Stand-Up Physicist
    My speculations are not along the lines of Brans-Dicke theory.  I did have a specific kind of duality proposal at one point in my life.  What is duality?  I like the start of an article on the subject of duality by Sacha.  If you develop a duality principle for work on strings, you would stand a decent chance of winning one of those big money prizes.  What I noticed was if one took one covariant derivative of the Rosen metric with a static 4-potential, and then one normal derivative (because spacetime is oh-so-flat), then one would have a theory that was identical to a flat-as-a-board proposal with 1/R 4-potential.  That is how duality works.  It is not, I have the metric theory, and I will staple on a potential.  Instead it is I have a dynamic metric and a boring potential, or I have a boring metric and a dynamic 4-potential.  That kind of duality could be consistent with the strong equivalence principle.

    I know that the Rosen metric proposal for gravity lost out on the energy wave emissions from binary pulsars.  The mode of emission is like a water balloon, a quadrapole.  Yet the Rosen proposal would have dipole modes of emission.  Oops.  It is quite the challenge to make a proposal so simple that it excludes dipoles.
    "What I noticed was if one took one covariant derivative of the Rosen metric with a static 4-potential, and then one normal derivative (because spacetime is oh-so-flat), then one would have a theory that was identical to a flat-as-a-board proposal with 1/R 4-potential."

    I was curious what you were referring to and I looked through your articles to find what this claimed duality is. A search showed a decent place to start is here:
    http://www.science20.com/standup_physicist/shoot_it_head-80394
    You don't define a duality and then derive something with it. Instead your "derivations" are just a sequence of new statements/assumptions. To show the extent of your error, the theory you originally thought was "dual" to the Rosen metric for a spherically symmetric body and experimentally indistinguishable from GR, that same theory you later found out actually had a repulsive static force between masses. Even after changing the theory to fix that sign error, the theory in the static case showed NO deviations from Newtonian gravity (and in the more general case was not rotationally symmetric), so it was very far from being GR-like and definitely not "dual" to the Rosen metric. So even if you can't understand the root or details of your errors, it should be extremely apparent that your "duality" you thought you saw was nothing of the sort, as the two give different predictions. So even if you don't return and learn from that error, at least acknowledge it is an error and stop carrying it around as 'inspiration' of some sort.

    Skimming through some of your other articles it seems this is a common problem. Instead of suggesting an idea and exploring the consequences of it, you evolve the idea in the middle of "deriving" consequences of it, making incorrect conclusions in the process. This is like "cargo-cult" science. Don't manipulate things towards where you want them to go. Instead explore an idea, learn from the results/consequences, THEN choose a new idea to explore. Make sense?

    Getting back to the point which you seemed to miss:
    Introducing a gravity theory as a 4-potential in flat-spacetime will _already_ violate the strong equivalence principle.

    If one of your requirements on a new gravity theory (as you stated above) include the strong equivalence principle, then you need a new starting point. Learning from these big-picture ideas instead of just throwing things into Mathematica can save you lots of time on your quest.

    The Stand-Up Physicist
    While skimming, I hope you noticed that I rescinded the GEM Unification proposal.  The proposal was not as bad as some claimed in the threads, but I have no interest in substantiating anything with regards to the rescinded proposal.

    You can choose to be completely unimpressed that the covariant derivative of the terms in the  exponential metric looks like the kind of potential Poisson would have recognized.  That is similar to the kind of reaction others at this site have to squaring an event in spacetime quaternion and noticing the first term happens to be invariant under a Lorentz transformation.  As far as I can tell, readers here are bored with the observation that the four derivative of a four potential in the quaternion form - essentially all the way a 4-potential can possibly change in spacetime - turns out the a way of writing the electric and magnetic fields.  Those concrete observations I can check with Mathematica are worth my time.

    Since you appear to admire David's contributions, try to note his style.  While there has been moments of tension, in general there is a respectful discourse, even in this very thread.   There is no cargo-cult here as that would involve a group of people.  Please refrain from advising me on how I should work, and I will do likewise (other than I suppose this comment itself).

    I remain a fan of Mathematica.  It is unbiased, coming in with no big-picture ideas.  I do realize its many limitations.
    I know you like the picture you put on your lunchbox, but as pointed out multiple times, the Rosen metric is not "dual" or "identical to a flat-as-a-board proposal with 1/R 4-potential". To claim duality, both viewpoints of approaching a problem must be equivalent when calculating a solution. You do not have this, as geodesics on the Rosen metric never matched predictions for trajectories from your 4-potential gravity theories. Please stop claiming this duality.

    The Stand-Up Physicist
    To claim duality, one would need a theory for gravity.  I don't have theories plural or singular.  This is a speculative thread.  I am sorry if you interpreted my comments to mean that I think the GEM Unification Lagrangian was somehow back to life in a valid form.

    Oddly enough, you did bring up what I to this day consider a most powerful suggestion, the graphic in the lunchbox.  Here it is



    General relativity is on the left.  Newton's scalar theory is on the right.  Newton's scalar potential is most excellent.  It does fail certain tests, getting light bending around the Sun only half right for example, so we need to do better.  The current leader is the metric theory of general relativity.  My rescinded theory doesn't work (listing its flaws is banal).  Do I believe that someone might make a potential theory that does work?  I think it would require real mathematically ingenuity to do it and I doubt I have the skills for the task.  This graphic still gives me hope that such a duality may exist.  It could well turn out that one needs to accomplish many of same things as GR, but using a modified math engine.  There is no telling what the future brings.
    Halliday
    William Stanley:

    I hope you are aware that there are at least three levels of equivalence principle.  Yes, the strongest does indicate that the effects of gravity are "fully explained by [a] metric with no extraneous fields", with the additional stipulation that the only coupling is the minimal coupling through the "covariant" derivative.

    Of course, if Andrei G. Lebed's proposition had been done correctly while suggesting a deviation, this need not violate some of the weaker forms of the equivalence principle.

    David

    I didn't realize about the additional minimal coupling stipulation.

    Is that why some say f(R) gravity can violate the strong equivalence principle? I've never understood that, since the field equations still just involve the metric coupled to the stress energy tensor, but can involve higher order derivatives than GR.

    BTW, I really enjoy your style of writing. When you finish your spacetime introduction articles, I'd love to see you venture into discussing gravity theories. Even just a good description of some precise definition of "minimal coupling" would be interesting, as trying to discuss the bizarreness of "self-force" felt by orbitting charged particles in GR leads to all sorts of confusion (for me at least). I'm also curious if there is an intuitive way to see how some tensor quantities cancel at certain dimensions, such as why Gauss-Bonnet gravity is the same as GR in 4-d spacetimes. Or it would be neat to discuss f(R) gravity, and in what sense it can be viewed as GR with a scalar field. Or what extensions like Lovelock gravity give us in higher dimensions (is there a purpose besides just looking at a more "general" GR in higher dimensions?).

    Anyway, please keep writing.
    Thank you.

    Halliday
    William Stanley:

    The "minimal coupling stipulation" comes in when one states the strong equivalence principle in term of a "metric with no extraneous fields".

    On the other hand, one may also specify the strong equivalence principle in a manner like that used by Misner, Thorne, and Wheeler in Gravitation:  "in any and every local Lorentz frame, anywhere and anytime in the universe, all the (nongravitational) laws of physics must take on their familiar special-relativistic forms.  Equivalently:  there is no way, by experiments confined to infinitesimally small regions of spacetime, to distinguish one local Lorentz frame in one region of spacetime from any other local Lorentz frame in the same or any other region."

    Later, they continue with:  "The laws of physics, written in abstract geometric form, differ in no way whatsoever between curved spacetime and flat spacetime; this is guaranteed by, and in fact is a mere rewording of, the equivalence principle."  Then they state:  "The laws of physics, written in component form, change on passage from flat spacetime to curved spacetime by a mere replacement of all commas [partial derivatives; flat-spacetime gradient] by semicolons [covariant derivative; curved-spacetime gradient] (no change at all physically or geometrically; change due only to switch in reference frame from Lorentz to non-Lorentz!).  This statement, like the nonchanging of abstract geometric laws, is nothing but a rephrased version of the equivalence principle."

    (Warning:  There is a caveat, namely "factor ordering problems in the equivalence principle", but that issue remains the same with the "minimal coupling stipulation".)

    Contrasting this to another author that chose to steer clear of the strong equivalence principle, but, instead, chose to focus on a weaker form of "uniqueness of free fall" (what Misner, Thorne, and Wheeler also refer to as the weak equivalence principle), and the principle of "general invariance".

    As that author puts it:  "Unfortunately, although the principle of general invariance places considerable restrictions on the possible interactions of gravitation and matter, it does not determine these interactions uniquely.  Thus, Eq. (19) [Fμν = 4πjν] remains entirely consistent with general invariance if we add an extra term (FμνR) to the left side.  To rule out such extra terms, it is customary to appeal to the principle of minimal coupling.  This principle asserts that the equations of motion of matter in the presence of gravitation are to be obtained from those that hold in the absence of gravitation (equations of special relativity) by replacing ημν by gμν and ordinary derivatives by covariant derivatives; no other changes are to be made.  Since in local geodesic coordinates the covariant derivative reduces to the ordinary derivative, we can also express this principle as follows:  in local geodesic coordinates the equations of motion are those of special relativity."  (Emphasis in original.)

    Upon close inspection, one finds that these prescriptions are equivalent.

    David

    Wait, I'm confused now.

    I'm probably misunderstanding, but it feels to me like in response to my question on minimal-coupling in light of considering the self-force charges feel in curved spacetime, that your response is that GR+EM is not minimally coupled.

    "Equivalently: there is no way, by experiments confined to infinitesimally small regions of spacetime, to distinguish one local Lorentz frame in one region of spacetime from any other local Lorentz frame in the same or any other region."

    As I referred to before, the self-force felt by a charge in curved spacetimes confuses this topic for me. Wouldn't a simple counter-example to the above be that a charged particle and a neutral particle set in motion inertially can follow the same path in flat spacetime, but the charged particle will follow a different path in curved spacetime. This is because the vector potential of EM is not "minimally coupled" to the metric as defined in those quotes, as the coupling involves the Ricci tensor. This experiment can be done as "local" as you want, since you only need to measure the proper-acceleration of a point particle at a single point in spacetime.

    So I may be misunderstanding, but it sounds like you are saying "empty" GR with no other interactions (ie. only contributions to the stress energy tensor are in dust solutions) satisfies the strong equivalence principle, but GR + EM violates the strong equivalence principle.

    ...hmmm, wait a second...

    I just realized that due to gravitational waves, even a neutral particle with a mass could feel a self-force. Does this count as non-minimal coupling even in "empty" GR? For example let's consider the local Lorentz, rest frame of a neutral particle that interacts only via gravity with the environment. In a flat spacetime the particle will not feel a proper acceleration. In curved spacetime, it can. For example a particle orbitting a large mass. And yes, this self-force is "real" in that it gives a non-zero proper acceleration. If that is non-intuitive consider the following: In Newtonian gravity, there is no gravitational radiation, and the path of the particle in this experiment would be independent of the mass since the gravitational "acceleration" is independent of the mass. In GR though there is gravitational radiation, and it is not proportional to the mass, therefore the path taken would depend on the mass (indeed, orbital decay depends non-linearly on the mass) and therefore cannot be encoded entirely by geodesics according to the metric. The orbital decay thus shows a non-zero proper force due to back-reaction of the gravitational acceleration. So there appears to be non-minimal coupling even in "empty" GR.

    Is there some caveat of the "definitions" that I'm missing?
    If we use the definitions in your quotes which refer to the equations of motion, it appears they fail.

    I'm thinking that it may make more sense to define minimal coupling not in the equations of motion, but in the action. Here the coupling between the vector potential of EM and the metric appears as minimal as I think one could make it. Similarly in GR, in the action the coupling of the stress energy tensor to the metric looks as minimal as it could be to me.

    Does that sound like a fair change to make the definitions more useful?

    Halliday
    Wait a minute, William Stanley.  You are conflating many separate issues.

    First and foremost, my response had nothing whatsoever to do with your "question on minimal-coupling in light of considering the self-force charges feel in curved spacetime".

    I was trying to help you see where "the additional minimal coupling stipulation" comes from and "a good description of some precise definition of 'minimal coupling'".

    As for issues such as "the bizarreness of 'self-force' felt by orbitting charged particles in GR", that was far beyond my intent.

    However, while on the subject, it is true that the coupling of the metric to "the vector potential of EM ... involves the Ricci tensor", it is also true that 1) the vector potential has no bearing upon the motions of classical charged particles, and 2) the involvement of the Ricci tensor in the vector potential coupling is completely covered by the "factor ordering problems in the equivalence principle" caveat that I mentioned.  (As far as both texts I mentioned are concerned, this is completely within the "minimal coupling" criteria.)

    Now, when it comes to issues of "self-force/self-energy" with regards to charged particles, this is far from a problem unique to General Relativity (GR), curved spacetime, etc.  This has been a longstanding issue with Electromagnetism (EM) at least as long as we have had Maxwell's Equations.

    Furthermore, one must be very careful when trying to describe, let alone set up an experiment involving charged particles, especially if they are to be unneutralized by opposite charges (like a single electron or a single proton, rather than a Hydrogen atom [although, in such cases, we get into additional potential issues involving quantum particles]).

    You then continue with:

    I just realized that due to gravitational waves, even a neutral particle with a mass could feel a self-force. Does this count as non-minimal coupling even in "empty" GR?  ...

    First, if the particle is able to be appreciably influenced by its own distortions of spacetime, then it is not—by definition—a test particle, and this has absolutely nothing to due with any "non-minimal coupling" of any kind.  Having particles couple with fields that couple back to particles is simply a reality of any field theory.  It can either be negligable, in a given situation, or it must be dealt with (often within a numerical/computational model).

    However, after all this, the following statements you have made are troubling:

    ...  For example let's consider the local Lorentz, rest frame of a neutral particle that interacts only via gravity with the environment. In a flat spacetime the particle will not feel a proper acceleration. In curved spacetime, it can. For example a particle orbitting a large mass.  ...

    Yes, troubling, not for what they seem to imply about reality, but troubling in the light they seem to shed upon misconceptions you appear to harbor.

    So, in order to clarify, does a tiny, uncharged, non-magnetic ball bearing in orbit around a large, uncharged, non-magnetic mass, isolated in deep space, "feel a proper acceleration", if we are able to neglect gravitational radiation (after all, are we not assuming a sufficiently large mass differential that the ball bearing can be considered a test particle?)?

    David

    P.S.  I tend to work primarily with the action, so I tend to, operationally, consider the "coupling" from that standpoint.  However, this is not about me.  I was simply trying to share "a good description of some precise definition of 'minimal coupling'".

    Thank you very much for taking the time to give a detailed reply. It is appreciated.

    Starting with what feels like the most pressing issue:
    "does a tiny, uncharged, non-magnetic ball bearing in orbit around a large, uncharged, non-magnetic mass, isolated in deep space, "feel a proper acceleration", if we are able to neglect gravitational radiation"

    No of course not. I tried to be clear that the proper acceleration was from the back-reaction to the gravitational radiation. I reread and see I made a typo at one point:
    "The orbital decay thus shows a non-zero proper force due to back-reaction of the gravitational [radiation]"
    Regardless of where the confusion came from, let's move on. I'm not some crank claiming a particle moving in an "curved" path in coordinate system sense necessarily means that there is a non-zero proper acceleration. I understand the difference between "coordinate acceleration" and proper acceleration. A free-falling test particle following a geodesic feels no proper-force. The end.

    Now, while I'm not a crank, I do have a lot to learn (obviously), so please interpret my education short comings as merely my curiousity trying to extend beyond my current understanding. I do not mean to promote a weird theory or anything, I'm just trying to learn.

    "First, if the particle is able to be appreciably influenced by its own distortions of spacetime, then it is not—by definition—a test particle, and this has absolutely nothing to due with any "non-minimal coupling" of any kind."

    So another condition of tests of the strong equivalence principle is that the experiments can only involve test particles? It also feels strange to say to ignore a coupling because if we drive one term towards zero the coupling vanishes. There's something larger here I'm not understanding.

    "Having particles couple with fields that couple back to particles is simply a reality of any field theory."

    Yeah, I guess that is usually true. Why doesn't it happen in Newtonian gravity though?
    Or if we want to stick to 'metric' theories of gravity, would Newton-Cartan count?
    Intuitively I surmise it is because there is no gravitational radiation. But still the metric and particle still influence each other, mathematically what prevents this from feeding back? What cancellation makes the self coupling go away?

    "1) the vector potential has no bearing upon the motions of classical charged particles"

    Well, it does affect the motion of the charged particles. But I agree that there is a gauge freedom, such that there is a class of potentials which have equivalent "bearing" upon the motions of classical charged particles. Are you saying that this coupling to R doesn't affect the motion of the classical charged particles? I don't think that is true, so I'm probably misunderstanding your point here. (Or revelling another error in my intuition.)

    "2) the involvement of the Ricci tensor in the vector potential coupling is completely covered by the "factor ordering problems in the equivalence principle" caveat that I mentioned."

    I admit I saw that, but didn't know what you meant. I should have asked. Can you tell me more?

    P.S. I've been searching around for more gems by you in the discussion threads. Holy cow! There is a lot of good stuff tucked away. If you just copy pasted some of your more involved discussions, it could make for some really interesting articles. I'm glad I finally decided to post something. I'm learning much more from this and reading the articles on this site currently. It's a shame others won't find these discussions easily unless purposely googling for them.

    Halliday
    William Stanley:

    You clarified:

    ...  I'm not some crank claiming a particle moving in an "curved" path in coordinate system sense necessarily means that there is a non-zero proper acceleration. I understand the difference between "coordinate acceleration" and proper acceleration. A free-falling test particle following a geodesic feels no proper-force. The end.

    I'm very glad you did not take offense at my perception and clarifying question.  I did try to express the fact that I was simply trying to clarify a possible perception, with terms like "seem" and "appear", so I'm glad you took it that way.

    You continue with:

    Now, while I'm not a crank, I do have a lot to learn (obviously), so please interpret my education short comings as merely my curiousity trying to extend beyond my current understanding. I do not mean to promote a weird theory or anything, I'm just trying to learn.

    I try to consider even those that come off as far more "crankish" or "crackpotish" than you ever did as people that simply do not, yet, understand.  So, no problem whatsoever.  :)

    Now, unfortunately, I'll have to get to the rest of your message later, hopefully this evening.  I need to be heading off to work.  (I had more time earlier this week because I was home, sick.)

    David

    Halliday
    William Stanley:

    I will probably not get to all this tonight, so please allow me to focus on what seems to be most pressing, as far as I can see.

    "1) the vector potential has no bearing upon the motions of classical charged particles"

    Well, it does affect the motion of the charged particles. But I agree that there is a gauge freedom, such that there is a class of potentials which have equivalent "bearing" upon the motions of classical charged particles. Are you saying that this coupling to R doesn't affect the motion of the classical charged particles? I don't think that is true, so I'm probably misunderstanding your point here. (Or revelling another error in my intuition.)

    Of course, I should not have said a flat out "no bearing", since the Electromagnetic field, which is what does directly influence the motions of classical particles, does depend upon the derivatives of the (4-)vector potential.  The point I was trying to make, without using too many words, was that the entire system—charges, to Electromagnetic field, back to charges ("particles couple with fields that couple back to particles")—can be completely formulated, characterized, and solved without ever having to use the (4-)vector potential.

    The (4-)vector potential, classically, is nothing but an auxiliary equation than can be used to simplify the problem, but is otherwise quite unnecessary.

    So, let's do an exercise:  Can you write the equation of motion of a charged particle within an electromagnetic field, using either the four (4) vector potential, or the electromagnetic field tensor?  (You may use HTML to format it, here, or you may write it in something TeX like.)  The four dimensional tensorial notation, even if you use component notation, is far more compact than trying to write this in the usual 3D notation.  You may write it for the flat spacetime of Special Relativity (SR), or the curved spacetime of General Relativity (GR).

    Then there's the following part:

    "2) the involvement of the Ricci tensor in the vector potential coupling is completely covered by the "factor ordering problems in the equivalence principle" caveat that I mentioned."

    I admit I saw that, but didn't know what you meant. I should have asked. Can you tell me more?

    Well, let's start this one with a second exercise:  Please write whatever equation involving a coupling between the (4-)vector potential and the Ricci tensor, within EM-GR, that you think is troubling to you.  (I'm reasonably certain I know which one it is, and if I'm right, this is one of the equations Misner, Thorne, and Wheeler used as an example of "factor ordering problems in the equivalence principle".)

    Well, that's all the time I have, for now.  I look forward to your work on the exercises.

    David

    E=mc^2 has long had problems associated with it. For instance, when photons, gluons, or neutrinos are studied, Einstein's equation must be modified for relativistic effects and becomes:

    E^2 = (pc)^2 + (mc^2)^2

    Where p is the allowable momentum of the massless particles. Remember if E=mc^2 then a massless particle could have NO energy. Whoops, nice save Lorentz!

    The widely accepted perspective in modern physics is that the luminiferous aether does not exist and has been completely out of favor and poo-pooed for quite some time now. The aether is what Maxwell and his other contemporaries reasoned was the necessary medium for which light travels. Like a wave of water the momentum in any wave comes from the medium from which it travels. The wave of water, like a wave of light, has no mass nor momentum, in of itself.

    Interesting that in many areas of physics there are now people beginning to bring back the aether concept. They are, however, not calling it aether, instead they have cleverly developed new terms such as dark matter and dark energy. I suppose they must save face and secure exclusive access to the Nobel prizes they're drooling for.

    Hank, the self-depreciating attitude that you are not smart enough to understand the basic principles of physics is humorous and I suppose disarming but it is completely untrue. It is however the necessary attitude for any charlatan or snake-oil salesman pitch for cash, power, and peer recognition.

    Associating Sheldon with modern science is hilarious. Sheldon sees himself as the most rational person in his world. He even envisions himself a new evolutionary branch of human. If anyone were to identify the most irrational person on that show it would undoubtedly have to be Sheldon. To quote Sheldon: "Its funny because its true."

    Hank
    Thank you for validating my PhD in Theoretical Phys Ed.  :)
    E^2 = (pc)^2 + (mc^2)^2 isn't a modification of E = mc^2. It's simply the entire relativistic energy equation, whereas E = mc^2 is that E^2 = (pc)^2 + (mc^2)^2 reduces to as momentum approaches zero. There is no problem.

    When the momentum of what approaches zero, ah yes, massless particles.

    This is a self-contradictory concept. Like many concepts introduced in introductory physics courses, critical thinking becomes overwhelmed by the necessity of assimilating everything that is thrown at you.

    Spend more than a few seconds actually contemplating what you are being sold and you will fall behind your academic rivals. Therefore the typical, especially the undergraduate, student will never actually apply critical thinking to anything they encounter.

    Welcome to the Machine...

    Attaching a link to a Pink Floyd video which isn't even available doesn't prove that you've cracked the case wide open, Sheerluck. Scientific truths are demonstrated truths, and the merit of modern physics is -very- well demonstrated. Got an iPhone? Then you're playing with stuff you philosophically deride every time you touch that gorilla glass.

    rholley
    Sheldon sees himself as the most rational person in his world
    I don’t watch the Big Bang Theory, but from what you say, the following seems appropriate, an extract from Chapter 2 (the Maniac) of Orthodoxy, by G.K.Chesterton

    “If you argue with a madman, it is extremely probable that you will get the worst of it; for in many ways his mind moves all the quicker for not being delayed by the things that go with good judgment. He is not hampered by a sense of humour or by charity, or by the dumb certainties of experience. He is the more logical for losing certain sane affections. Indeed, the common phrase for insanity is in this respect a misleading one. The madman is not the man who has lost his reason. The madman is the man who has lost everything except his reason.

    The madman's explanation of a thing is always complete, and often in a purely rational sense satisfactory. Or, to speak more strictly, the insane explanation, if not conclusive, is at least unanswerable; this may be observed specially in the two or three commonest kinds of madness. If a man says (for instance) that men have a conspiracy against him, you cannot dispute it except by saying that all the men deny that they are conspirators; which is exactly what conspirators would do. His explanation covers the facts as much as yours. Or if a man says that he is the rightful King of England, it is no complete answer to say that the existing authorities call him mad; for if he were King of England that might be the wisest thing for the existing authorities to do. Or if a man says that he is Jesus Christ, it is no answer to tell him that the world denies his divinity; for the world denied Christ's.

    Nevertheless he is wrong. But if we attempt to trace his error in exact terms, we shall not find it quite so easy as we had supposed. . .  ”
    Robert H. Olley / Quondam Physics Department / University of Reading / England
    Interesting, I might just have to look deeper into this material. The prejudicial use of of the term "madman'" makes me hesitate in his reasoning. One must constantly be asking the question: is the sage the "madman" or are the priests the ones who are insane? From the priests perspective it is the sage who is always the "madman" but the sages motivation stems from his viewing the priests as "madmen."

    It is always the easier road to travel by going along with the status quo, especially if popularity and politics is the modus operandi. However there is always the danger in group think of mass hypnosis, which always results in mob rule. With group think suppression of the truth is nearly universal which results in the corruption of the underlying organizational system.

    "Republics decline into democracies and democracies degenerate into despotisms"
    -- Aristotle

    Halliday
    Photodady:

    The only "problems" that E=mc2 "has long had" "associated with it" has been people misunderstanding what it actually means.  All too often, people see the "E" in E=mc2, and the "E" in the energy momentum four vector, or its square (pseudo-)norm E2 - (pc)2 = (mc2)2 (the more proper way of writing your "relativistic effects" equation), and think they are simply "the same".  Then they rationalize their mistaken understanding by modifying the meaning of E=mc2, with such means as "relativistic mass", or saying the "equation must be modified for relativistic effects" (even though the original equations already include "relativistic effects").

    You see, you are absolutely correct that "if E=mc^2 then a massless particle could have NO energy", in the sence of the "E" in that equation. However, this is absolutely no "Whoops" that needs any "nice save", by Lorentz or anyone else.

    PB tried to correct your error.  Unfortunately, he referred to "momentum approaches zero" (emphasis added).

    It's not when "momentum approaches zero", but when total momentum is zero, exactly.

    If a person reads Einstein's original paper (even translated into English, via a competent translation), with an adequate degree of reading comprehension, they will find that Einstein was strictly referring to a system with its center of mass at rest with respect to the reference frame in which he derives his iconic equation (E=mc2).

    So, the question is, how does "a system with its center of mass at rest with respect to [a] reference frame" manifest itself?

    A reference frame that is at rest with respect to the center of mass of a system is a reference frame in which the total momentum of the system is exactly zero (p = 0)!  Nothing more.  Nothing less.

    In this case, as PB already pointed out, the square (pseudo-)norm of the energy momentum four vector, E2 - (pc)2 = (mc2)2, quite naturally reduces to Einstein's iconic equation (E=mc2).

    Nothing else is necessary.  No conflicts need correcting.

    Now, as for a system consisting of only a single massless particle:  There doesn't actually exist any "reference frame that is at rest with respect to the center of mass of a system" consisting of a single massless particle.

    However, if one wishes to take an infinite limit, one does, indeed, find that the energy, E, in such a "reference frame" is just as zero as is the mass of the massless particle!  (You may want to consider Einstein's own thought experiment of "riding" an electromagnetic wave, and asking, what would one observe?)

    I'm sorry, but it is you that has yet to fully apply critical thinking to such issues.

    David

    "Strictly speaking, of course, the second of Einsteins relativity postulates prevents a Lorentz transformation to the rest system of light, since light moves a c relative to all inertial frames. Consequently, the term rest mass has no operational meaning for light."

    Modern Physics, Fourth Edition, Tipler - Llewellyn

    If you can't bedazzle them with brilliance, baffle them with bullshit.

    Let me clarify the allowable momentum of a massless particle.

    From E^2 = (pc)^2 + (mc^2)^2

    Where in the relativistic equation p = ymu , where y (gamma) is the Lorentz portion and m is mass and u is the velocity.

    Now y (gamma) = 1/( ( 1 - u^2/c^2)^(1/2))

    When considering a photon (light) u = c and Einstein's equation then breaks down, thus the necessity for replacing the relativistic p with the "allowable momentum of a massless particle." This is also the reason that the concept of rest mass has no meaning in this context.

    The equation for the energy of a massless particle then becomes:

    E = pc

    Which IS modifying the equation to fit the data, in other words fixing the theory to fit the data!!!!!!!!!

    Hank
    This is not correct. We know space can travel faster than the speed of light, it doesn't mean that you can go faster that light and Einstein is wrong, it means things without mass are not bound by the speed of light.

    Every theory matches data, that is why it is a theory. Until then, it is a hypothesis.
    I certainly don't have a problem with traveling faster than the speed of light. In my example I was speaking of a photon (light) which, if I'm not mistaken, travels at the speed of light. It is here where Einstein's equation has a discontinuity when u = c. Unless I'm not aware of a new mathematical method defining a zero denominator.

    Additionally what is the meaning of the imaginary component of the resultant energy in the equation when u > c ?

    I'm being facetious here, I'm using the modern theories to point out the inconsistencies. I'm well aware that I won't be taken seriously by any established "modern scientist" who derives their living from the hypothetical theories themselves, its simple self preservation.

    Actually the speed of any wave is a property of the medium in which it travels, if you were to enter somewhere in the universe where the aether didn't exist there would be no barrier to traveling at any speed.

    However, quantum physics is now telling us that space ain't so empty, the vernacular is stupendous. Call it quantum foam, zero point energy, quantum randomness, quantum noise, dark matter, dark energy, etc. etc. etc. etc. When it all comes down to it they are all just tip-toeing around the obvious and are simply describing the luminiferous aether.

    "Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality."
    -- Nikola Tesla

    Halliday
    Photodady:

    You stated the following proviso:

    ...  Unless I'm not aware of a new mathematical method defining a zero denominator. 

    While it is far from new, are you not aware that we can handle many cases with a "zero denominator", provided the numerator likewise approaches zero in a related manner?

    David

    The value of p for massless particles is an experimentally derived value, it is NOT mathematically derived. The limits of P as u -> c and m -> 0 has no meaning (in the real world) since for a photon m=0 and u=c, always. The equation is said to "remain defined" because of the experimental evidence NOT because of the math involved. (quote from text previously cited)

    As for the quantum theory derived p , p = hc/lambda, where h is plank's constant and lambda is the wavelength of the particle. Plank's constant is yet another value that is experimentally (read NOT mathematically) derived.

    Really now, I'm getting bored here, yawn...

    Halliday
    Photodady:

    Your reply is a complete non sequitur.

    Did you not notice the context of the message to which you were replying?  Or did you simply choose to ignore such "trifles"?

    David

    P.S.  If you need additional context, just compare and contrast what I said about "a related manner" in my message to which you were supposedly replying, vs. what I said about "when there is no definite relationship between the factors" in my message below.

    Hank wrote in this article:
    "The problem is that now everyone in theoretical physics thinks they are Einstein and the public thinks that, because Einstein was a theoretical physicist, those must be better and smarter than experimental physicists: they get to think about stuff and get paid for it. I offer, as evidence, that character Sheldon on "The Big Bang Theory" TV show. He has never once worn a Science 2.0 shirt and if you want something done, you are better off making friends with the engineer, but people think he is the smart one regardless."

    Are you, by chance, related to Sheldon Cooper?

    Halliday
    Photodady:

    Another complete non sequitur.

    Even further, you compound your mistakes by asking me whether I am "related to" a fictional character.

    I was actually hoping for better from you.

    David

    OK genius, educate me.

    Show me the solution for the energy of two different photons, one with a wavelength of 650 nm and another at 430 nm. Show me how your solution is somehow simultaneously convergent to these two separate cases without using quantum theories or any other experimentally derived values.

    Use ONLY the simultaneous multivariate limits that you described for solving the Einstein's energy equation and show all of the work.

    Halliday
    Photodady:

    This is still not within the proper context.  However, I will tell you that I have already expressed when a "solution is somehow simultaneously convergent" (above), and when it is not (below).

    If you will also read other of my messages, even just within this forum (under this article), you will also see that I am not a Theoretical Physicist that fits the characterization Hank expressed in his article.

    Now, if you actually want to discuss the proper relationship between experiment/observation and (mathematical) theory, then feel free to post such a message within the appropriate context of one of the threads in this forum, or start a new thread, below.

    David

    By the way the solution to this problem was EXPERIMENTALLY solved by Max Plank. He found the constant of proportionality of the measured energy of a photon times its wavelength was proportional to the speed of light. He labeled this constant h.

    E = hc/Lambda

    Halliday
    Actually, Photodady, you have grossly mischaracterized what Max Plank did (you apparently need to look up Black-body Radiation).  However, you are correct that his constant, h, was obtained by way of a parameterized curve fit to experimental data (where the principle parameter to be fit was his constant, h, of course).

    The "wonderful" thing is the way his little curve fitting parameter keeps "showing up" within quantum theories.  ;)

    David

    I can concede that my history and timeline were a little sketchy. In fact it was Einstein himself who used the portion of Planck's Law to solve the shortcomings in his equation. In fact it wasn't until Einstein published his theory that this portion of Planck's Law became widely accepted. I also read that Planck's Law weighed heavily in the conception of Einstein's second postulate.

    I think that you may be referring to Planck's Law in its entirety, whereas the portion that Einstein used E = hf or E = hc/Lambda was derived from Planck's work with electronic oscillators and the electromagnetic radiation emitted by them, specifically at n integer harmonic multiples of the fundamental resonant frequency.

    Until I see actual results I can not concede that Einstein's relativistic energy equation can be solved for massless particles. This is why the text that I cited stated that rest mass has no meaning in this context. it is rest mass that is used in Einstein's equation.

    As explained before, we can use E = m c^2 for a system with no net momentum.
    There is no special case for photons; this works just fine for photons as well.

    Consider a system of two potons moving in opposite directions, with each photon having some energy E_0.
    What is the net momentum of this system? zero. So we can use E = m c^2 here. It doesn't matter that photons are involved.

    What is the total energy of this system? E = 2 E_0
    So what is the rest mass of this system? It is 2 E_0 / c^2

    Halliday
    AnonymousPhotodady (aka Photodady):

    Unfortunately, it appears your "history and timeline" are still "a little sketchy".  Einstein did not use "the portion of Planck's Law to solve" any "shortcomings" in Special Relativity.  However, Einstein did use it in his paper on the photoelectric effect.  (Perhaps that was the connection you were thinking of when you stated that "it wasn't until Einstein published his theory that this portion of Planck's Law became widely accepted.")

    Einstein's second postulate of Special Relativity (the constancy of the speed of light [in terms of various reference frames]) had nothing whatever to due with any portion of Plank's Law.  Instead, it was most heavily influenced by Maxwell's equations.  (As Einstein put it, as I recall, he believed Maxwell more than he did Newton.)

    You are absolutely correct that "it is rest mass that is used in Einstein's equation[s]", both his iconic E = mc2, as well as his more general squared (pseudo-)norm of the energy-momentum four (4) vector, E2 - (pc)2 = (mc2)2.  (Actually, arguably, "it is rest mass that is used in" all of Einstein's relativistic equations that involve mass.)

    While, as expressed within "the text that [you] cited", and as I have already expressed in one of my early messages to you, it is not proper to "transform" to the "rest frame of a photon", one can still perform an "improper" limit to such, within appropriate contexts.

    However, with even more "force of truth", one may use invariants (so they yield the same answer regardless of reference frame), such as the squared (pseudo-)norm of the energy-momentum four (4) vector, E2 - (pc)2 = (mc2)2, to obtain the invariant rest mass of any system, including that of a single photon.

    David

    P.S.  Another area where you show that your "history and timeline" are still "a little sketchy" is in your assertion that "E = hf or E = hc/Lambda was derived from Planck's work with electronic oscillators and the electromagnetic radiation emitted by them, specifically at n integer harmonic multiples of the fundamental resonant frequency."

    I think you need to look more carefully into this.  There is absolutely no way for Planck, or anyone, to obtain Planck's constant, h, "from Planck's [or anyone's] work with electronic oscillators and the electromagnetic radiation emitted by them", unless and until one works with single (or few) photon systems.  The fact that there are "integer harmonic multiples of the fundamental resonant frequency" is a general characteristic of many, many oscillating systems.

    Gerhard Adam
    Oops ... I guess I must have deleted that non-response.  Sorry.  :)
    Mundus vult decipi
    Halliday
    Photodady:

    from E2 - (pc)2 = (mc2)2, if m = 0, then E2 = (pc)2.  It's as simple as that.  No limits necessary.  No infinite values to contend with.  No "bad"/"improper" transformations.  None of that.

    Now, as for your use of p = γmu:

    This equation is principally used in order to retain a similar look to Newtonian mechanics.  It is certainly not a fundamental definition, and is only ever to be used in the cases when there is a non-zero (rest) mass.  (A more proper equation, though, again, only ever to be used for non-zero mass, is p = mU, where U is the proper velocity, and both are four vectors, of course.)

    When trying to apply such an expression when m goes to zero, while u goes to the speed of light will always result in an indeterminate value (as is the case anytime one approaches anything like 0/0, or infinity times zero, when there is no definite relationship between the factors).

    So, the only equation that "breaks down" is your misuse of the expression for three-momentum.

    No "modifying [any] equation to fit [any] data".

    Just your misuse of one little equation.

    David

    LOL

    Halliday
    |D
    Halliday
    Photodady:

    Short of your last line, this is all completely consistent with what I stated.

    Now, if you are "baffled", I'm sorry, and I can help, if you are willing.  There is no "bull...".  All is eminently consistent, though the way Special Relativity is all too often communicated and/or taught could be greatly improved, I'm sure.

    David

    Quentin Rowe

    Ah, the 'ol C^2. The squares show up often in physics, and I've often pondered the significance.

    Back in 2006, I rode a BMW R100S from LA, up thu Washington State, and all the way across to Long Island, New York. Have a guess at what famous piece of real-estate I visited there, Hank! ;-)

    Anyways, after having spent a bit too long dithering on the winding roads, I got to spending more time on the freeways, and what wonderful roads they were. I spent a great deal of time traveling at night along uncongested lanes at 80-100 mph. Damn the deers, they would have to learn to give way!

    Less concerned about my carbon footprint, which was already deeply imprinted by flying over the Pacific ocean, and more concerned about the hole in my wallet from burning too much gas, I invoked the law of frictional losses as one speeds thru the air and rolls over concrete at high speed. Knowing that it followed similar V^2 laws to momentum (not to mention energetic deer impacts), I took great comfort in realising all that gas I was saving by not going 10mph faster, and therefore way up into the steep part of the losses curve. I know - delusional, yes, but it's all relative after all! ;-)

    I'd like to call you out on this:

    I don't believe anything at all until an experimentalist tells me I should.

    I'm not sure if this applies to you, but what most people really mean by this is that if something is unknown, they will believe the default consensus until proven otherwise. Those that choose to believe the non-consensus until proven otherwise get a harder time.

    Hank
    Sure, I was being a little facetious. An experimentalist can't reproduce human evolution, for example, yet I accept it. However, when it comes to physicists on a math fetish, it's the lack of any consensus that bugs me. Doing some math and declaring it is now up to experimentalists to prove it is why theoretical physicists today can be insufferable. There are too darn many of them and the quality drops a lot. 

    Because math can do just about anything - not being bound by the laws of reality is nice that way - you'll be better off being skeptical than saying something might be right because one guy wrote an arXic paper claiming it could be mathematically possible. Obviously, some crackpots go to the opposite extreme and claim they can never believe anything until an irrational metric is reached - motorcycles should not be allowed until they are 'proven safe', for example.
    Halliday
    Doug:

    [Since it's getting rather cramped up there, I'm replying down here.]

    You stated:

    ...  I will continue to fish for math omissions, because those will apply to all scales of the Universe.

    In at least one sense, I tend to agree.  This is why I think someone needs to solve the galaxy rotation "problem" using as close to full General Relativity (GR) as reasonably possible, in order to see what modifications result.

    The "trouble" is that GR is quite non-linear, so this is difficult (though one may be able to use some of the newer computational GR methods to simulate this).  However, by this same token, GR, unlike Newtonian mechanics with gravity (or MOND), allows for the distortion of spacetime to "feed back" upon itself to produce additional effects.  In essence, gravity "generates" gravity (black holes are sort of the ultimate expression of this "feedback").

    In essence, the asymptotic approach of MOND to a 1/R "force" law (from a 1/R2 Newtonian force law at close range) is a statement that the (effective) "force" is even longer range than Newtonian gravity.  Perhaps this is a "hint" of "self generation".

    Maybe.  Maybe not.

    Another difference between GR and Newtonian like systems (including MOND) is that in Newtonian like systems angular momentum cannot affect the gravitational "force", yet angular momentum most certainly affects the distortions of spacetime in GR, resulting in very different "forces" not predicted by Newtonian like systems.

    Is the angular momentum (per mass) of a spinning galaxy enough to have an appreciable effect through GR?  I'm not certain, off hand.  Since the system is (essentially) only interacting via "gravity", one cannot fully answer this without allowing the full system (the galaxy, with all its "particles" [stars, etc.], and spacetime itself ["gravity"]) to come to equilibrium:  The galaxy (and its motions) affecting spacetime, and spacetime affecting the motions of the galaxy.

    Could just GR be the answer?  I find it unlikely, but we will not know without checking more closely, I believe.

    David