APS Talk: Electromagnetism in a gravity field for the quaternion gravity proposal

Transcript

this is a talk I gave at the fall ApS meeting at the Massachusetts College of Liberal Arts titled electromagnetism in a gravity field for the quaternion gravity proposal now when I was discussing this with the person who was moderating this section she didn't know what a quaternion was so I said the first thing to do is use the blackboard and explain that word if you are really not familiar with it so what is a quaternion well actually I'm going to start this with some physics that night Einstein figured out in 1905 he said this is a really important invariant quantity this is known as an interval and for inertial observers they will all agree upon that Einstein was not a math guy he didn't know what Matthew was doing he was a physics and logic guy and he figured out all the wonderful logical consequences that happen because of this relationship particularly that that minus sign that causes all kinds of fun things and it took a couple of years before his math teacher said hmm you really are dealing with a rotation in space-time and this is a contraction where what you have is a metric tensor and you are dealing with a colored pea a contraction of two four vectors and it was again Einstein who realized that you know in front of these things over here are really constants and in order to do gravity these guys have to be functions of something and a functions basically of M over R and so what you end up doing is doing calculus on this and this leads to the Einstein field equations the need to vary this in such a way that it can account for how gravity works now when I look at this I say wow that's super simple that looks like it's the square of something well the square of what well the old square the only thing it could be you must have a DT and a dr and put a little and you must square that thing now if this was an imaginary number so this was only one symbol long we'd know what the answer is you go DT squared minus D R squared Oh what we want to DT D are alright but this thing is really three things so it is in fact the mathematics of quaternions that says that's really a three thing that's really a three thing and this is a perfectly fine thing to do so this is a quaternion squared and if you know what a complex number is you've got this is there anything else well yeah there is there is a plus a cross-product so if you had you know we P cross P with but since those pointing in exactly the same direction that equals zero so you know this already it's just complex numbers where instead of having an eye you get an i J okay sort of thing and that's it and the rules are exactly the same plus your cross-product so that's well and good but I really think if you think about this in terms of information theory to sound really super fancy okay what you end up what you're what you care about is this one value over here right to get that in this case you need ten values here because this is a symmetric tensor and a symmetric 4x4 tensor and so they're ten independent values and then you've got here for for this guy so you use 14 kind of values just to get one and that seems kind of crazy I'm here with this quaternion squared you've got four all right and you end up with this one okay and then three over here or you end up with four so that seems really what's more balanced in terms of information theory I scarred two before I end with four instead of starting with 14 and then hanging up with what with 1 and what I did in the fall of 2015 who as I said gee it was Einstein who said you know if this is an invariant then I end up with special relativity and that's a hugely wonderfully rich area of study of the natural world well I've got these three over here first of all what are these called means I go bigots in your lexicon how could that this be super important to special relativity and this thing doesn't even have a name and that's one of the things that figured out I call it space times time because it's space-time Sam and the question I asked at that time was well what happens if two observers agree on that value what sort of physics do I have if they agree upon this and that was the starting step towards my investigation of quaternion gravity and I hope and we'll see how far I can go with it all right and with that completed I went and gave my talk like so alright we know that light bends both in theory and in practice Einstein gave us the thought experiment of what a photon would look like to an accelerating elevator and in that case it would look like the photon was falling and due to the equivalence principle which said that you really can't tell the difference except for some title effects between a an elevator and being gravitationally attracted that argues quite forcefully that light must Bend due to gravity but that's in theory and in practice we have no so seen during eclipses a little bit of light bending I should say that those experiments were really kind of touch and go for a while it really took some experiments done in my home state of Massachusetts to to use radio telescopes where you could do it any old time as the brightness of the Sun didn't drown out that sort of signal and and particularly when you looked at this start of the approach as the that absolutely matched the kind of predictions that came out of general relativity so we have more than enough data with you know also light reflecting off of planets and stuff and coming back and timing that but one of the more fun predictions is this thing called gravitational lensing where you have a very distance quasar and right in the path is another galaxy and so then the light we get deforms this ring around that galaxy and there are a few cases where we can see it see it and that's called gravitational lensing all of these just make so solid the idea that light is bent by gravity and there's just no question but what about the fields the fields of light would be of the electric and magnetic fields well those don't change at all in a gravity field the reason is that here is the the what we call the field strength tensor for electromagnetism and I've used the covariant derivative and then we we have to make some assumptions here about what that means in this case I'm using the assumptions that show up in general relativity that it's metric compatible and torsion free so we use this Christoffel symbol of the second kind I think and we write that all out and because the Christoffel symbol is symmetric with the MU and the new up there then those kind of white wipeouts so people always write down the normal derivative because this is formally called actually let's let's wait on that um so the math here as I say says as directly as possible that the E and the B field do not make do not change so if you had a box that had had an e in B field and you measured those fields and then you took that box and you put it right outside of a black hole the E and B fields measure the same strength now somebody over dinner pointed out that what I'm talking about is an exterior derivative exterior derivatives appear in other places in physics for example the Li algebras u 1 and u 2 that can be used to generate the continuously groups su 1 and su 2 which are part of the standard model now while I'm saying the E and the B fields do not change their the energy of those fields does change because the energy of the field is B squared plus e squared and you must contract that with the metric tensor and a metric tensor here would be very different from a metric tensor right next to a black hole so it's a little bit subtle but could keep up and keep in mind that difference but now isn't there really a conflict between like bending and the E and the B fields not changing at all I mean when I think about the e field I usually think about it in terms of the potential and it's not the potential this is the derivatives of the potential you hate take a time derivative of the three potential a or you take a vector operator like Dell and acted on a scalar operator tential fee so you've got time and space kind of going there and if you think about the Schwartz shield metric and how that the time term gets a little smaller but the space one gets a little larger you can see how those two effects could cancel then that's actually reasonable all right but what about magnetism well magnetism is a spatial operator del and a spatial kind of three vector potential so that's space on space now with space changes and both of them change then be a better change to me that's make some sense all right but you're gonna have to figure out what the metric is to do any e/m this is what's called this it's a it's part of its background structure that you must supply that somehow e/m can't figure out on its own and one way to do this would be to use a photon and you'd use the photon and due to that gravitational redshift you'd be able to figure out what what what metric to use but that's really kind of bizarre right a photon is simply an excitation of the the quantum expression of an electromagnetic field and you're using it to figure out the metric tensor which you can't figure out by just looking at the fields themselves to me that general relativity every time I kind of dived into it further has become more and more elegant and to find something that doesn't quite make sense that's that in itself is pretty cool so there's another reason to maybe not be comfortable with this situation in general relativity because the e field I think of as being proportional to Quantum's of electric charge and if you have three of them up high in a gravity field and three lower down in a gravity field well it sounds almost reasonable that it wouldn't change three is three but a magnetic field is those charges in motion and thinks in motion they are always affected by gravity so why shouldn't be B all right so now we're going to get into my proposal my proposal uses these quaternions and in particular the squares of a quaternion and we have these things called equivalence classes for both special and relativity and put my attorney and gravity proposal so what we do is we say you've got two observers they're looking at the same event P and they make you their measurements and they're not the same but they square those measurements and they compare the real parts and if the real parts are exactly equal then we can say something about the two observers we can say that one observer is moving at a constant velocity relative to the other observer and if we use the space x time quantity we can figure out exactly how one observer is moving relative to the other all right so what happens on the other side what happens if their imaginary parts their space times x value are exactly the same hmm well that is my quaternion gravity proposal so tools observers are equivalent if the imaginary parts of the square are using using quaternions are identical so the only difference between those two observers is that one is farther away from a gravitational source than the other now they can't also be moving they must be not moving at all and can you combine these two and make an even harder kind of statement yes but I've kind of avoided that I want to get the easy stuff first see if it is reasonable see if it has a chance of being right okay now for gravity problems you know what we expect to do is first solve a differential equation and then use algebra to eliminate constants and match the data this actually happens with the Einstein field equations say I've got a solution that's not enough you have to show things like are you sure that when the mass goes to zero your solution goes to flat space-time that's one of the constraints another constraint is that you to be able to pick out Newton's law of gravity it's still it's still useful particularly for eliminating all the constants now in special relativity there are no field equations instead you've got this invariance that you must respect well that's an algebra problem and you have to make sure that you conserve that that quantity is the same for these two inertial observers and so you just solved the algebra problem to fit the bit data well the same thing is going to happen for my quaternion gravity proposal there are no field equations hmm so it is just an algebra problem needed where you need to batch the data so I thought I would provide a little graphic to show what these these things are here's a classical physics special relativity and quit turning gravity so classical physics oh well that should be all straight lines and it is because time and space are completely independent of each other pretty cool then in 1905 Einstein came up with this spacetime diagram where the zero isn't black I like to say and Newtonian physics is just no D R equals zero or DT equals zero here we're thinking about the squares of DT squared minus dr squared being equals zero the interval and then we have these hyperbolas of constant intervals now with quaternions that's a real value if the real value is positive you have a time like situation if the real value is negative then you have a space like separation and then I have this graph that nobody talks about that I am claiming is all about gravity and it basically is the new classical physics where you actually respect to speed of light or you could look at it as as the light-cone rotated by 45 degrees either one is a valid way to look at that graph and then I ask you as the viewer to say I wonder what physics that would be because we know special relativity is usually important shouldn't this one also be just because it's an m3 imaginary numbers well so what go ahead and think about that for a while all right so what are the algebraic constraints on a problem involving gravity that you'd have to be like consistent with well you'd have to be in a situation where you could say you know if M goes to 0 or R goes to infinity this looks like flat space-time nothing's happening you'd also need to be a function of M over R because you want to be able to pick out Newton's solution from long ago and you'd want to be consistent with all weak field gravitational tests that actually pins down about five of the coefficients in the Taylor series three for the time part and two for the spatial part and you need to be I have their space times time value constant because that is my hypothesis that I'm exploring and another one I I put in there is that it should be our harmonic solution because if you think about the earth going around the Sun four billion times once a year as a pretty consistent kind of oscillating system so with those things in mind one sort of proposal that would work is if it was an exponential factor in front of the DT and the dr and then we square it up and we get a term in there that looks a lot like the interval you get from the Rosen metric of general relativity and that metric is is definitely consistent with all week field tests so far and in my case there's nothing against this being time-dependent you know if if you were had a system a binary system that somehow was undergoing changes well in you know the the the distance R was a function of time there's there's no problem with putting that in there and and making it work out it was yes it would change in time but it would still be still fit fit all five of these constraints and I point that out because of course there have been gravity waves we're saying hey you got to be able to allow time dependence and I think it's pretty obvious there should be time dependence and this and therefore there should be this possibility of wave-like things all right so what does the quaternion gravity proposal say about what happens with electric fields and magnetic fields and so the time derivative gets a minus exponential this space part gets a positive exponential and that means boom no change in the e field so we're not disagreeing with general relativity yet but for the B field you get a positive and a positive and those don't cancel so you have this sort of effect on the B field now that actually is the subtle issue it means that the result is not going to transform like a tensor okay so this is fundamentally different from general relativity where in order to transform like a tensor the B field must not change and that clear difference leads to a clear type of experiment and that is there is this vector called the pointing vector which is e cross B and just measure it at different heights now if it's the same then well gr remains true and if it is different well that actually would indicate that the the quaternion gravity proposal is right now I have emailed to people who had papers with a pointing vector in their title and I said hey dude do you know anybody has done this and they are like no I mean a different thing which is fine but what I like is how clean this is about a difference between my proposal and general relativity and I suspect it's really really difficult definitely like one of these things where you have to measure your pointing vector to you know eight or ten digits if you just move your apparatus by you know ten meters or something like that and it won't be easy because that's the way tests of general relativity always are but as I say it would be very kind of clean situation at least if if it turned out in favor of quaternion gravity but you can also see why it probably hasn't been done before because this is kind of a null test it's like measure it and see it doesn't change if we were to measure it and see a change it probably would have been done but it hasn't all right so so for dealing with special relativity and gravity these are now both problems in algebra not in field theory and this is really great for gravity in that there's no field there's no graviton and there's no quantum gravity of course it's a problem for me because they're people whose jobs it is to figure out a quantum gravity field theory in fact I saw somebody had a there's a job posted for in philosophy the philosophy of quantum gravity and I'm here to say I don't think there is any of that mmm well the math is going to be kind of simpler in a way but actually in another way it's not as simple because it's like everything is piled up on top of each other and it's it's very dense and gets scary like am i spelunking the right way or am I kind of lost and the problem is that tensors are everywhere everyone uses them for every problem and anytime you see a little Greek letter up or down and I'm saying no there should be no Greek letters in physics that would involve a and just almost mind-boggling rewrite of how we do physics and that's not too popular an idea but it's where a man all right so if you want to see slides of this I use this bitly shortly URL shortening service V P stands for visual physics another big thing in mind VP - DM for the slides if you want to see this a site that's just devoted to teaching about my quaternion gravity proposal it's V P - Q G and my main site quaternions comm can you believe I own it is it V P - Q alright thank you for listening