Classical to Quantum
Wild Egg Maths
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The underlying geometry for classical Special Relativity is affine geometry, which is principally the study of parallelism. We review basic constructions in this subject, some of which go back to Renaissance artists, and allow us to create coordinate grids that align with linear algebra. We also review the cross ratio both of points on a line and lines through a point.
In the two dimensional geometry of timespace corresponding to events on a one dimensional line, the role of signal propagation is crucial, but the nature of the signal not so much: while light is usually used, sound works just as well to develop the Lorentz transformations as we have seen in previous videos. From an affine point of view it is the two signal directions in the two dimensional timespace corresponding to right and left moving signals in the underlying one dimensional space that create the additional structure on which SR is built. We encode that by associating to such a set up an action of the red complex numbers on timespace.
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hello thanks for joining me on this adventure we're looking at special relativity and today we're going to talk about the role of geometry in special
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relativity I've told you about the algebraic side of things which is all about red complex numbers today the
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geometry side rests on an understanding of apine geometry so it turns out this
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is the crucial geometry underlying special relativity and it's unfortunately these days a not very well
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studied subject even though it's very rich it together with projective geometry is one of the two pillars of
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modern geometry and it is very closely connected with linear algebra so in fact
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in some ways it's perhaps the easiest geometry for modern students of linear algebra to understand it also is the
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crucial geometry for the algebraic calculus those of you taken my algebraic calculus course will know or have met
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quite a lot of apine geometry so today I want to introduce this to you in a very quick way I'm not going to go over um
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detailed definitions I'm going to do a lot of drawing of pictures so you'll get a an overview of the subject but it's
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not at all comprehensive and perhaps at some future point I'd love to actually put together a course in apine Geometry
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okay so we're talking about a one-dimensional space and we have observers and they have Clocks by
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bouncing some signals back and forth maybe with with mirrors and they use this uh signal uh to to find out you
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know bouncing a signal off something and coming back then from the initial and
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final times they can make some assessment to where and when some event occurred so this allows each of them to
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come up with sort of time and space coordinates of events but very crucially
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these coordinates are going to be different for different observers and so one of the main points is how do we
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relate or understand the relationship between these different observations or
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coordinates that different observers are going to come up with and it turns out that the relevant geometry for setting
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all this up in a nice mathematical way is out of two-dimensional Aline space so
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this time space which is really recorded here with time and space values this
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naturally forms an apine space and that's sort of the simplest um most natural mathematical setup in which to
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uh frame this discussion so we're going to talk about Aline geometry today and I'm also going to talk about the role of
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the signal lines which is a very crucial ingredient and importantly I'm not going to insist that these signals are are
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light or or sound or anything else in particular it's actually quite a general theory that works in a wide variety of
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situations okay so aine geometry in the plane the main objects are points and
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lines so we'll assume that we have a straight edge but importantly it's not a ruler so the markings on this particular
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thing are Irrelevant for us and the main constru struction that we're allowed is that of parallel lines so that's what we
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can do in aine geometry and that's pretty well the main thing okay so given a line we can create parallel lines so
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here is a parallel line okay you can think about one of those draftsman's tables that has a mechanical device that
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allows one to easily move a line to an parallel line in various positions okay
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so here are two parallel lines and let's choose another Direction okay and some
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other parallel to that okay so these two families of two
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parallel lines give us uh the fundamental object in the subject which
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is a parallelogram so that little um piece there is a parallelogram and one
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of the first orders of business is to be able to extend this parallelogram to a a pattern s to replicate it and the way we
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do that is say same way that Renaissance artists uh did we draw a diagonal say something like this okay and then
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through one of these other intersections say this one here we draw a parallel to the one that we currently have okay we
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could also do it on this side here there's another parallel okay and now what you see what we've
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been able to do is sort of U create the the start of a
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new family of of sort of equal parallel so if we take this intersection here and
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take a line parallel to this one here through
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there then we're getting a copy of the parallelogram that we started with and if we do the same thing on this side we
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get another copy of it okay and we can uh you know carry on for example down here we could take this
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thing here and uh draw parallel like this and we could then get this sort of
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pattern that'll allows us to replicate uh parallelograms okay so that's a an
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important sort of key starting point and there's a various consequences of that
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so in in particular if we want to uh duplicate or triplicate a segment so for
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example if we're uh let's say uh going from here uh let's say to here okay then
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if we just do the construction that I've just shown you then that allows us a way of essentially taking this segment and
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sort of duplicating it because we obtaining this point here and we consider this segment and this segment to be equal in the sense of vectors not
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in the sense of distances because there are no distances in aine Geometry to start with okay but in terms of
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displacements those are those are equal so we have this notion of translation and we actually have also an implicit
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notion of area here because what we're really doing here is sort of subdividing the plane into what we can consider to
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be sort of equal parall pars so this parallelogram is equal to this parallelogram in some fundamental way
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and so we can say that the area of this thing equals the area of this thing so the area of this bigger one here for example will be four times the area of
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this one here and so that's a a starting point to how we could get a kind of a
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notion of area or relative areas in apine Geometry something else that we can do
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is let's say we have U three segments like this here and U we can use those
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three segments to subdivide uh something else so for example suppose that we were interested
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in subdividing let's say uh this segment here into three equal pieces well I
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could do that use a different color uh in the following way so I'm
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going to draw this line here okay and then I'm going to draw parallel
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to that one lines through these two points okay
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and what I've effectively done then is I've created points over here and here so that I've effectively subdivided
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that segment into three uh what we could say equal pieces in the sense that each
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of these segments are equal as vectors they have the same displacements and if I had started with not just three
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segments here but seven segments and done the same kind of thing I could divide that uh by seven so now we have a
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way of actually introducing rational coordinates in some sense to this story
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okay because we could start say with some particular Point say U say this one here okay and
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we could uh say start with the two lines that we started with maybe let's call this one say uh say t just for think
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argument and say this one say x okay so now we have two coordinates and um and we could denote
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that the original parallelogram sort of specified some unit displacements in these two directions so
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if we wanted to we could coordinati we could say this is zero and then this is one and this is two and and in this direction this is this is one and this
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will be two Etc okay and with this device of dividing stuff we can then
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actually um create coordinates not just for integral points like this point here
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then could be represented by the pair T and X which in this case would be one in
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this direction and two in this direction so it' be represented by um oh I said the other way around
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sorry I said one in this direction two in this direction so 1 comma 2 all right
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so that way we can assign coordinates to uh positions and we can do that not just
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with integral uh numbers but also with General IR rational numbers by using this subdivision uh trick so this way we
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can coordinati the entire plane but rather AR arbitrarily by starting with you know
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with one point and starting with sort of two particular directions that I sort of randomly chose and actually starting
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with the corresponding parallelogram uh that um that we used to start the whole game
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off right we started with this parallelogram here okay but so once we have a point chosen
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uh then we get and and and with with these axes then we get an identification
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of of this apine plane um with what you usually think of in linear algebra as
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being a a vector space a two-dimensional Vector space okay and uh and that's
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quite reasonable so once we have a a point chosen then it makes sense actually to talk about vectors so we
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could you know talk about not just this point but we could talk about this Vector okay that Vector we could maybe
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represent it with something a little bit different because it's a displacement so that would be the round bracket one to
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so that would be like a vector if we had another Vector let's say this one here what would that uh yeah that one there
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so that would be um so one in this direction minus one so that would be the vector u 1 minus one okay and we can add
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those vectors let's see if we can do that so add those vectors means drawing a parallelogram so that would be roughly
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about there so this would be uh the vector which is the sum of these two which would be the vector 2 two um 2 one
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how does that sound two in that direction One in that direction okay where again all the
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coordinates are given in terms of parallelisms not in terms of perpendiculars so if I
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actually Drew lines correspondingly so that would be
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corresponding T equals 2 line and there we see um that point really is 2 comma
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1 okay so what can we do in apine Geometry where there there's lots of U
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beautiful uh facts but uh the the main thing here is that just by using
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parallelograms okay we can coordinati things but clearly clearly there's some
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choices involved here we could have started with some different point and we could have started with some different parallelogram somewhere else so we want
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to distinguish and this is such an important point for for physicists approaching the subject we want to
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distinguish between those Concepts that can be enunciated in a way which is
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independent of the actual coordinate system chosen okay so statements
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involving the X and the T coordinates in this particular frame are not as
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preferable as statements which are independent of the coordinate system so we might use a coordinate
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system to prove things but we want our state statements to be uh above the actual choice of uh of a
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system okay so that's um a very brief uh
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sort of view of of the framework of of apine geometry we have a two-dimensional plane with the idea of a straight edge
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and the idea of parallelism uh with our um with uh parallel
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lines okay now let's talk about cross ratios and the associated idea of a harmonic range so cross ratio is an
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important kind of invariant sort of a really fundamental invariant both in projective geometry and in apine
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Geometry but maybe it's a little bit simpler to understand in the apine geometry context because we already have
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this understanding of vectors and the um relative sizes of
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vectors if they are in the same direction so suppose we have a line here and we have points A B C D and in this
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case case I've tried to make them equally spaced so we've constructed these four equally spaced points so we
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can say that in the language of vectors we can see that this displacement here between A and B is equal to the
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displacement between b and c equal to the displacement between c and d and we can write that using say familiar Vector
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notation say if we use that for the a is like the the this displacement from A to
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B then that's the same as B C equals same as um c d we can also write things
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like a c that one there from here to here is twice a because we can naturally
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add vectors in the way that you're you're used to in linear algebra by putting the head of one next to the tail
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of the of the other okay so um and if you look at an equation like this we
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could rewrite this in in as a ratio if we wanted we could say that you know a c
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C to to a you know the the proportion between these two vectors is the same as
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2: 1 that's writing it in proportion language or if we wanted to use the more
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familiar language perhaps of fractions we could perhaps also write it we could allow ourselves to write that the ratio
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of AC to a in this case is equal to 2: 1 so in other words it's two okay so we
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have some kind of possible multiplicative algebra of of vectors but crucially
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vectors that are only in the same direction so if we have some other point over here say e and we consider uh the
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vector from a to e this Vector a e okay
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then that vector and these vectors are fundamentally incommensurable we cannot
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say anything about the relative size OAS or the relative
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aspects of these of these um these vectors in different directions that's
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typically a metrical thing that happens only when you impose a metrical structure on the apine plane as we're
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going to ultimately do in special relativity but just with apine geometry we have to restrict ourselves to
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comparing vectors as long as they're in the same direction of course um we could also have ones going in the negative
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direction also that's fine okay so now having this notion of
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uh of uh vectors and their possible ratios let's talk about the the notion then of a
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cross ratio so suppose that we have four points again on a line let's call them a b c and d again but they're not uh
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equally spaced out now so now I'm going to Define what we might call R of a
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b um say c d okay so this is going to be
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a a number which is uh going to be a ratio
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of of ratios okay so what we're going to do is we're going to take the ratio of
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a to a D in the sense that we've just talked
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about here because they're they're cinear so AC to a d and we're going to divide that by the
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ratio b c to b d
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so uh in the example let me uh just approximately say what what this thing
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might be okay so we're talking about the ratio between this Vector AC and the
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vector a d okay so in this example uh maybe that's like if that's four
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maybe that's five so AC is maybe U if that that if that's four then maybe
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this one is uh is five and the total is is nine so maybe 4 over 9 and BC to BD
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uh so BC to BD 1 2 3 four maybe four of them one two
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two one two three let's say three of them okay so uh that might be in the ratio of um BC to BD maybe 1 to four
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something like that okay and so um then you take the quotient of
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this fraction over that fraction so this thing inverts uh so we get 16 over over
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9 so that would be the cross ratio of these four
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points now What's um remarkable about
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this is that it has a a kind of an invariant property in in a very
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beautiful way namely if I choose some other point for example over
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here okay and I draw some lines from that other point two of the Four Points
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in question there's one of them there's another
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one there's another one there's the fourth one okay so I've
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drawn four lines from this sort of perspective point out here let's call it e and then I'm able to choose any other
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line that meets these four transversally that doesn't okay I can maybe do this
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one here it will meet those four lines in four new points this
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one this one this one and this one
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and so this would be say a prime this one would be B Prime this one would be C
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Prime Prime and this will be D Prime okay let's see if we can uh guess
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what the cross ratio uh is in in in this for the primes okay so if we looked at
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the same corresponding thing maybe I'll try to squeeze it up here R of a prime B
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Prime Colon c Prime D Prime uh what are
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we going to get so it's a Prime C Prime to a prime D Prime so 1
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2 3 4 let's say four okay so it's 1 to four perhaps it be like
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that and um B BC Prime B Prime C Prime to B Prime D
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Prime so we have to take this and compare it to this so 1 2 3 4 five six
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seven maybe let's say six okay so one to six let's say 1 to
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six okay and if we do the ratio there that would be 6 over4 is the same as 3
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over2 is that close to uh uh this thing well it's um not that far off right 3
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over2 16 over9 okay not not dramatically
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far off considering that I didn't do it very accurately but the the statement is that if we did this accurately and did
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this all very carefully then we would find that the cross ratio of the primes is exactly the same as the cross ratio
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of the um of the original points and so that's then a property then of the U of
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the of the four lines emanating from E so we can associate that number that
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cross ratio with actually the four lines emanating from E so we have a kind of a dual notion here we can talk about the
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cross ratio of four points on a line or we can talk about the cross ratio of
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four lines through a point and they have this nice uh property that if you take a
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transversal you go from Cross ratios of points to cross ratios of lines and vice
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versa okay so this is a a a really key Central beautiful thing going back to
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Greek times I think to papasa probably uh which is a a core mathematical property of apine
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geometry and also very plays a very big role in projective geometry too so let's
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return to our onedimensional space and our Observer a with a little clock
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ticking back and forth and the Observer a is looking at some distant object C
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and perhaps there's an event that happens there at some point and a observes that in terms of a signal going
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from a leaving at some time bouncing back and coming back and being registered coming back and from those
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initial and final times a is going to deduce where when uh this event e
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occurred now a is going to want to create a a time space diagram like this
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that we can conveniently make okay with an axis T representing times and uh X
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representing space and the first order of business is to clarify what A's clock
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ticking sounds like that's represented by a sequence of equally spaced points
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and now we meaning equally spaced in the sense of a find geometry okay and these
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could be also in negative times as fine and then this light signal that's being
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emitted here and then is returning that uh corresponds in A's clock to to
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some lines which are emanating either going to the left and up or to the right
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and up depending whether we're talking about a left or right uh oriented signal
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so we can represent the thing here for example by maybe at this time here okay
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this time here uh a is sending out that um that signal okay represented like
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this and maybe at some point there uh then it it comes back and maybe it uh
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bounces back like this okay so then a would say aha this is where I'm going to
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position the event e and uh then deduce um where it is from this
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now the other ingredient of course is the spatial Direction which is naturally associated with the mirror
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so that we can at least on the diagram create some equally spaced points in
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this direction also Etc that uh will then signify equal
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space displacements for a so with this initial and final reading then in this
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case here for example example if this is min -1 0 1 2 3
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then a will want to assign a time value for the event which is halfway between
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these two points the average of three and minus one which is is one okay so
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that one will be the the time coordinate and uh in terms of position
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well the total the total um displacement in time is 4 3 - -1 is 4 and we divide that by
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two it's two so a will reckon that the
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signal has gone two units over and then come back so two two so uh the position
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over here is two in the negative Direction so two bar okay so that's how we've seen before that a would use these
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time coordinates to come up with the time space coordinates for some event e
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now when we look at this situation here the the kind of crucial things really are um the these lines that are Ting
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that are representing the signals the propagation of information perhaps light in in a traditional setting or with our
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bats sound but notice that I'm not concerned that these things go up necessarily at
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at U you know at 45 degrees this is an Aline situation so it means that uh we
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could change the axes we could use axes which are oriented seemingly in different directions
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and still um the the essential aspects of the vector um story here we're just
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using vector addition and Vector arithmetic uh that's still going to be holding so what that means is that we
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have to identify what the essential properties of these um signal lines are
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okay and this is in the direction of replacing Einstein's second postulate so
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we don't have to say that the speed of light is this or that or the speed of sound is this or that what we really
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rather want to insist is that the signals are propagating in a way that's
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consistent with the apine geometry in the sense that the signals
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representing um Transmission in the same direction are always parallel okay so if
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we have some other uh transmission at some say later
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time maybe a time of one then we should want that that's represented by a line
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in our plane which is parallel to this one and if at some point it's then
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bounced back again it should do so in a way which is parallel to uh this one
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here okay so that's the crucial property that we have two specific
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directions that represent left and right going signals
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but we will want that all Observers agree on the on on the the position of
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those or the the directions of those in our time space so what we effectively want to do
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is we want to perhaps we have a so here's our abstract time space perhaps
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we have some designated point that represents say Time Zero for for a then
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what we want is to have these two directions let me actually draw them in red instead
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these two directions represented by some designated lines through this
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designated kind of origin Point okay
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so okay so if we have such a thing so we have our apine plane here but we've augmented it
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in two important ways first of all by introducing a special Point okay uh
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special point and also two lines and let's call them signal line one and
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Signal line two and it's important that uh we distinguish between the first one
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and the second one uh they play different
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roles so what I'm um what I'm getting at here is that the essential geometry of
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special relativity is really encoded in the data that consists of starting with
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an apine plane unadorned without any coordinate system just an apine plane
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together with a choice of a point which you can think of as being a you know a
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place where one particular Observer says okay this is Time Zero for me okay and
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two lines through that point which represent uh the lines where um our
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signals are are going in these two different directions in the corresponding space uh
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story and so this this enlarged system is
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really equivalent to having a metrical structure okay this is a very interesting point that once you have
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this data this is enough additional data okay that basically what you're doing is
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you're um putting on here a a loren um a
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loren geometry but I want to explain that in in in a different way by talking
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about uh the connection with the red complex numbers I want to explain to you
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why once you've done this the red complex numbers
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naturally make an entrance so red complex numbers then let me remind you those are things of the
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form 3 + 2 J where J is some symbol whose square is
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one not minus one okay
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that's a red complex number and we've seen that they have uh
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algebraic properties that are very much like the regular blue comp numbers and
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it's a rich story just as rich in fact as the blue ones but now I want to show
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you why these red complex numbers are automatically figuring when
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you um when you have such a configuration so when we say 3 + 2J
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we're really talking about 3 * 1 + 2 * J there's really two basis elements one
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and J so in in the complex number world you may remember that I prefer to put
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one up here and and J here okay when we were drawing red complex numbers so we
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did something like this we put one there and a j there so that uh three + 2 J
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would be three in this direction and two in this direction perhaps up there 3 + 2
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J okay so now I want to explain why the red complex numbers act on this apine
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plane once you have this additional structure so this C subr
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acts on this two-dimensional time space
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uh situation and and how is that well in order to explain that I need to have two
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linear Transformations one corresponding to one and one corresponding to J the
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one corresponding to one is is sort of obvious one is just the identity so when you multiply by one uh
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everything stays exactly where it is we're thinking about this as being zero
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okay that's like the origin of this two-dimensional space now thought of as a vector
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space and so it all comes down to the operator Associated to J so J what does
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J do that's the crucial thing so here is what J does so suppose that you have a
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point well let's put a first of all a point let's choose a point a here okay
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so there's the point a so I want to tell you what J does to a and it's all in
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terms of parallels so I'm going to
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take the line two and draw a parallel to that line through a I will do it in red
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okay so this is a parallel to a through there
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okay and I'm going to see where it meets line one there and I'm going to
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reflect a in this point so if this point say is uh e then I'm going to choose the
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point over here let's called it B so that the
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vector a e is the same as the vector e b
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it's one situations where I'm just replicating a displacement from here to
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here is the same as from here to here that's the analog in this setting of sort of reflecting in this thing
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here so this thing here I'm going to Define that to be J of
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a and you can see that if you do it twice okay so if you do J of B well it's
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the same thing through b we draw that same line and we reflect it again through e we're going to get back to a
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so yes indeed this transformation which is kind of a reflection in the line one
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but in the direction of the line two uh this is indeed a transformation whose
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square is the identity and it's these two things here that
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combine to um to allow you to multiply by any red
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complex number so if we wanted to for example uh multiply by 3 + 2 J let's see
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if we could do that um if here's the origin and here is a and I just multiply
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by three well it's going to be one 2 3 up there somewhere so that's three and
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If I multiply by uh by J I'm going to get over here okay if I multiply by 2J
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I'm going to get over here okay so there is a one
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one two three sorry there's there's three times a perhaps and uh and this
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one was 2 * B so if I add this one and and
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this one here then perhaps up here that's the result of taking uh 3 + 2
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J sort of acting on
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a so what we get is that the apine
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plane inherits an action of the red complex
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numbers in this um very simple way uh it's really all having to do with j and
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when J is really in some sense these are like igen vectors for J so J has an igen
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value of one in this direction and an igen value of minus one in that direction it's diagonalizable in in in
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these two directions okay so it's actually very simp simple story but once you have this action of the red complex
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numbers on this apine space augmented with these two signal matrices then the
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lence Transformations uh immediately become just red complex number
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multiplications so um with this setup the apine geometry coming together with
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the red complex numbers the whole story the whole Logistics of Laurent transformation becomes essentially just
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a trivial in terms of just we're just multiplying by Red complex numbers that's all what we're doing okay I'm
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going to explain that to you in my next video