Space-time Baseball using Quaternions

Transcript

[Music] hello I'm Doug Switzer and this is a talk I gave titled space time baseball using quaternions that's a picture of Minkowski who was Einsteins math professor now he was not impressed with Einstein's math skills cuz Einstein actually didn't go to a lot of the classes but his friend did and he was impressed however with special relativity I mean he says here from now onwards space by itself and time by itself will recede completely to become mere shadows and only a type of the union of the two will stand independently on its own and given this guy's a full fledged math professor that's really kind of radical language so we're going to think about baseball we got bleacher Bob and he's watching Newtonian baseball where we treat time as a parameter now that's a good thing because we're going to take the derivative with respect to time in order to get the velocity and the second derivative with respect to time to get the acceleration and time here is actually kind of boring as a parameter and in fact there is a really small collection of people who actually think time doesn't exist Julian Barbour being the current champion of that view now I think that's definitely wrong because if you really parameterize time you're actually destroying information specifically time itself see because when you're collecting the data to make your parameterised expression you're actually writing down the time and the location the time in the location again the time and location again and then forming your function but if you don't have the time you don't have the parameterization of the distance but isn't this kind of boring cause now we take the time derivative in that one and the second time derivative and ends up being zero hmm well I think that's actually a way to define that you're you're in this classical regime because we can think of other people watching exactly the same baseball game we could consider neutrino tony who has sitting on top of a neutrino which is flying by at you know 99.9 percent the speed of light and he gets to watch relativistic baseball because if he was at home plate when that ball was hit he'd be like six million miles away by the time it landed in right field so he actually has to take the derivative with respect to the interval DT D tau and that is not boring at all that's the gamma that shows up all over the place and special relativity and I actually prefer this way of doing basic physics because it works both for special relativity and it works for you know Bleacher Bob because lychee Bob's just going to say well that first term is 1 and the second derivative term is zero all right but let's take a closer look at this ball in flight through space-time so it covers one unit of space and I'm going to claim it's one unit of time and what changes though is our language so space - time well we have to think about home plate now going to right field future cyclically so home plate now again I'm going to right field future again alright and if I draw a graph of that I've got that middle sheet of paper as labeled as now and then the sheet in the further away is the future and the ball goes from home plate now into right play right field future so what we can start to think about is we can start to think about let's see ah reflections in a mirror because those will be space reflections as you can see we take we go from home plate now not only to right field future but left field future and that takes twice the amount of space because we've got left field and right field it takes exactly the same amount of time because we're going from now into the future and when we draw it in a graph it looks like uh you know little seagull or something like that now this isn't really physical in the sense that we start with one ball and we end up with two but we know how I did the trick I just used a mirror now we can think about what a reflection around time would be okay so the ball starts at home plate now goes to right field future oh and there what's it doing going back like that okay I've got cyclic boundary condition so it's going from right field passed to home plate now to right field future cyclic so right field passed to home plate now to right field future and again this isn't so physical because balls just don't jump up from right field and make it to home plate we are only using one unit of space there just one thing of art but we are using twice the amount of time because we go from the past to now to the future but what would happen if we did both go ahead and try and take a guess because I was kind of surprised by this all right if we do both a reflection in time and a reflection in space what we do is we go from left field passed to home plate now to right field future so we use twice the amount of distance we go from left field to right field we use twice the amount of time because we go from the past to now to the future and of course it's not physical because the ball jumps up on left-field and makes it to home plate now before continuing on the proper way to right-field future but I'd still find this kind of amusing because it looks like it's like continuing a path by doing both of these symmetry operations now we're going to ask a question about the differences between these two types of reflections that we have and I think the most important conclusion is they look different yeah a mirror reflection you can always see a pair of balls moving and the reflection in time you've got to remember oh yeah it did that and now it's doing that in the opposite way because we draw in space and yet we animate in time now I think about this in terms of number theory and I see space as having an X Y Z as those being three imaginary numbers and then time being a real number and it turns out there's a type of number called a quaternion and it's on Wikipedia and it's got its own website quaternions comm which I own but the important thing to say about times relationship to space I think is that space is orthogonal to time and yet it's intimately linked mathematically because they're all part of the same number and what is different here is that when people talk about things being orthogonal they almost always mean the same thing they're talking about a right angle you can move up and down you can move left or right but when you move up and down you don't do anything with the left and right when you move left and right you do nothing with you up and down but there's something kind of arbitrary about well what's up and down versus what's left and right whereas here you know if drawing in space is a completely separate thing from animating those sheets of paper we could go through them super fast or super slow and in fact that's what the difference is between bleacher bob and and the neutrino tony neutrino tony was running away so fast he actually goes through them kind of slowly the animations and that's why he thinks it takes such a long distance and actually a pretty long time but what can those two agree about what can bleacher Bob and neutrino Tony actually agree upon well in a formal sense is going to be the interval now there they've got a difference in time a difference in X difference in Y difference in Z and yes we could talk about whether they're going to be differential amounts or whether they're discrete amounts but to me that's a technical detail it's important technical detail but what we do with this is we square it and when we do that then we get x squared minus the distances squared so x squared plus y squared plus Z squared and they're going to agree about about those numbers and because I'm using a quaternion I'm also going to get these two DT DX 2 D DT dy 2 DT DZ which actually doesn't have a name which strikes me as profoundly strange because well it's not that complicated so simple things by now should all have names and yet I call it space times time it doesn't appear to play a role in physics and yet for these two it could clearly play a role they're going to agree to this Lorentz invariant interval and those other three the space times x value are going to be Lorentz variant they're going to change because those two are going at different speeds and how much they change will dictate the difference in speed between Bleacher Bob and neutrino Tony so you could put it to use although as I say it's not done at this time but if you think about the numbers they go in there okay these dimensionless numbers for their home run are like crazy different for space and for time because I really want to treat it as a number whole numbers number numbers don't we have units so I'm using what are called natural units and that means I've got 400 feet and that turns out to be like 400 nano seconds I wasn't aware of this kind of approximate conversion it's kind of simple but then you have to divide that by the plunk time a super small number which makes this a super large number four point seven times ten to the 36th and then you square it and well of course it gets bigger five point five times ten to the seventy-three but that ten seconds well that actually gets even larger okay so it's like two times ten to the 44th which when you square it Wow four times 10 to the 88th and there are 15 orders of magnitude different between those numbers and like nature knows how to handle this but don't ask an experimental as to deal with 15 orders of magnitude different between the two so that's that's that's too bad for us but nature apparently knows how to deal with this perfectly every single time so I'm going to think about a different person to compare with Bleacher Bob I'm going to talk about radio tower Tim who's a way up high in a radio tower is there anything they can agree to because up in that tower gravity is going to be a little bit weaker and so time is going to go a little faster and the ruler that Tim has is a little bit bigger so he'll measure distances is being a little smaller well in general relativity you go through a lot of work to start out with a Lagrangian city you derive your field equations if things are super simple you can you can get a solution to your field equations and it turns out that though the changes in time approximately cancel out the changes in distance now it's not perfect but my new proposal for how gravity works says well actually you don't need to grungy in the field equations the field equation solutions instead it's as gravity's all about this symmetry this cancellation of the changes in time with the changes in space being exact and I have not flashed it flashed it all out and at the level that you know professionals would accept but at least I can show you I think this is what it's all about is that this kind of this type of symmetry of the space times time for two different observers in non observe non inertial observers in a gravity field and I gave the talk wearing this very t-shirt okay so why this teacher well because it has space time or quaternion math on it that I think covers like most of physics if you're in the situation where you're saying hey time space don't do anything with each other they're fifteen or seven whatever orders of magnitude different from each other well that's classical physics and all the lines are straight the lines in black are always zero and the lines in gray are always unity now sometimes a real sometimes they're imaginary you have to pay attention to know which ones which okay and then there is fast physics special relativity and that's where we're thinking about a quaternion thing squared and when you square it the zero is actually the light-cone that's where DT squared minus dr squared equals zero and then if it's equal to plus one well then you get the lines that are in yell if it's minus one then you get the lines that are in the kind of purple section now the final two graphs down there below those are my own and so far I haven't convinced anybody they're of interest but the the one in green for gravity you could see that coming arising two different ways one is take that classical graph and say hey one should not cross zero okay and when you do that you get hyperboles or you say hey what if I take the fast physics guy and are rotated by 45 degrees well then instead of agreeing on a real value of +1 and a real value of minus 1 you say hey those people are really agreeing to an imaginary value of I or an imaginary value of minus I was like well that's that's I think that's a really nice way to connect all those those graphs but so you hold it there's one in pink what's what's on with the pink one oh the pink ones the strangest one I'm gonna have even less success selling that concept than the gravity one but you can see I was following the pattern here of saying hey what's a simple pattern for zero well the simplest pattern of all is just a dot right dead center and then you draw a circle around it one unit away and this is when I'm always taking a conjugate of itself so this is like DQ star times DQ and I'm really am treating all of these planes that has complex-valued planes and a circle in a complex valued plane is the symmetry group u1 which if you look it up on the internet you'll say is the gauge symmetry of electromagnetism you know Maxwell's approach to how light works and that charge conservation is all about that symmetry and I got it by simply drawing a circle oh that's kind of simple now if I think of that D are not like together as a social unit but it's three different ones then that would have the symmetry of SU 2 which is the symmetry of the strong force no the weak force the weak force yeah and then people always say well what about the strong force the strong force is su 3 it's got 3 squared minus one-eighth independent elements that each have a normal one and I have not seen any connection formally between the something called the quaternion group q8 and the group su 3 I think there are different animals but then I wonder whether the quaternion group q8 could do the most important jobs of that hour that need to be done by su 3 essentially covering all the gluons and I can only offer a numerology argument that maybe maybe quaternions are sufficient to do it big but of course I I'm pretty sure I'm never going to be able to formalize that you know and and submit it to any journal because it's just too technically daunting for a fellow of my limited skill set but I will say I saw it I mean I saw these three different symmetry groups when I started working with norms and that's kind of neat because a huge question that remains to this day is why you won su 2 and su 3 I mean yes those work to explain the great collection of particles we use in the standard model but it doesn't explain why it had to be those three and if you say well you actually need to use quaternions and circles and spheres and the whole thing itself might be able to put a limit on it who knows we'll see time will tell but it certainly has been fun to think about space time baseball using quaternions thank you very much [Music]