Classical to Quantum
Wild Egg Maths
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We look at the Lorentz transformations in the simple case of one dimensional physics, using not light as the preferred signal, but rather leaving that open, so as to include sound and other suitable transmission mediums. We look at the essential mathematical structure of the associated 2D timespace, prominently with the distinguished signal directions establishing a rudimentary metrical structure coming from affine geometry and the notion of cross ratio.
This allows us to reframe our earlier derivation of the Lorentz transformations for sound into just simple commutative multiplications in the algebra of red complex numbers, thus in this situation by passing the need even for matrix linear algebra.
A big thank you to all my Patreon supporters!
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hello everyone thanks for joining me on this adventure we're looking at special relativity in particular the Lorent
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Transformations and I want to show you that essentially these are just red
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multiplications in the sense of complex numbers but the red complex numbers and this is coming from our
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understanding of one-dimensional Relativity in terms of bats sending signals back and forth
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using sound not light so we're stepping back from the postulate of uh Einstein
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that the speed of light is constant in all reference frames so it turns out that we can substitute sound very
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happily and we get a lot of the consequences in fact most of the consequences of special relativity but
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it turns out that sound is actually not really crucial I mean we can do this with any kind of signal as long as the
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signals propagate in a certain Pleasant fashion with respect to the apine nature
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of the time space system that we're we're in so we have say bats A and B
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they both have little clocks that allow them to measure time and then with these
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clocks they are able to send signals and have them return and judge the time and
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positions of U external events and we're interested in the relationship between those time space coordinates uh that two
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observers have now to make this a little bit uh simpler we're going to make a a
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simplifying assumption that their clocks are in some sense sort of coordinated in the following way so
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first of all they are moving with respect to each other okay and we'll assume that at some point they actually
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coincide in their position they're at the same place physically as one goes by the other and we'll say that they are
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homones okay these are homon observers in this one-dimensional world if they
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are coincident when both of them have time equals Z so they've arranged that
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exactly when they cross pass that's when they're going to coordinate their clocks
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to set that to be time equals zero okay so that's a simplifying assumption it's not uh really necessary and one can sort
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of shift things uh to more General situations but this is a really good starting point which clarifies the
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algebra and simplifies it now in this one-dimensional world an observer like a will think about events in this world as
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occupying essentially a two-dimensional space because an event is associated in
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their minds with both a time and a position that forms a two-dimensional situation and in this two-dimensional
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situation what actually intrinsically is there well there is first of all the the
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point that coincides with the time zero where a is okay so a at some point his
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clock takes zero if there's a second Observer homo chronis with a then their
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clock will also tick uh zero at the same point in time space but let's just stick
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with a so if a is there then there's that sort of distinguished point in this time space and from this point there's
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also two uh sort of canonical lines okay and these are the lines that represent
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the propagation of signals both to the right and to the left from wherever a is
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so if at Time Zero a sends out two signals one to the left one to the right
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then these signals are going to propagate out and they're going to create in this time space frame these
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two uh lines and that's really somehow the essential mathematical starting
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point of this entire discussion is this fixed Point here and these two uh lines
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going through it in fact if you removed the The Observer a altogether then that point
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there would diminish in importance it wouldn't be there but still the directions of the propagations of these
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uh these signals would still be an important part of the of the geometry and in fact it's really these two
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directions in time space which correspond to signals moving right and
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left which imbue the two-dimensional World here with the the geometry
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required to make this happen so now let's also consider what happens when our Observer a has a a mirror at some
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fixed distance from a very close and is using that to establish a rudimentary
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clock by sending a signal back and forth that's represented in this time space
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with this parallel line representing the world line of the mirror and the
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bouncing back and forth of the signals is represented by this ladder of red um
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line segments which are always parallel to either S1 or S2 okay and uh and
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that's really the source then of the clock for uh for a and we'll assume that
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things have been oriented so that uh the labeling is such that uh if if there's a
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tick of zero then when that sound comes back let's say that's tick of two so
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halfway there is what a will consider to be the tick of uh one of his clock and
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okay and so we have 1 2 3 4 5 equally spaced in that direction and minus one - 2us 3 or one bar two bar three bar
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equally spaced in in the past now associated with this there's a second
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important Direction and that's corresponding to those events which are
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from A's point of view happening at Time Zero okay so that's establishing a
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notion of simultaneity from A's point of view so there are certain events in the
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time space World which have a distinguished property that they're happening at Time Zero as far as a is
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concerned and what does that mean well that means that say if we send a
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signal at some time say at -2 that that signal will return at time two so the
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start and end times are sort of symmetrical with respect to zero then the event of bouncing is deemed to be at
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Time Zero because remember that's how we create the time coordinate we take the average of the initial and final
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emission uh times which this light signal is sent and come back so in this
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diagram here that would mean that there's another line that we have to draw so let's draw that so for example
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if we um if we extended this uh I should do it in red but let me just do it here like this okay and like this okay so
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this point here is going to be another point which is deemed to have time zero
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for a and that means that this axis then formed this way is going to be important
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uh for a that's going to be the x axis or the spatial axis as far as he or she
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is concerned so what happens then is that there's four lines in this diagram
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actually okay there's not just three there's four lines there's the original lines S1 and S2 that represent our
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signals there is the world line of our Observer a and there's also the
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implied xaxis for that Observer namely those events which are deemed to have time
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coordinate zero in their frame so it turns out that this
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construction that I've um I've done uh really amounts to the following that if
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we think about these two vectors here say this maybe I'll do it in red I think about this Vector in the S1
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Direction and this Vector in the S2 Direction okay then these vectors here
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and these vectors here are the sum of those two vectors s say V1 plus vs2 and
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also the difference V1 minus V2 or V2 minus V1 and that has the implication
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that these four lines are what's called a harmonic range this connects with the
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notion of cross ratio that I talked about in a previous video so these four
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lines here are going to have a very special geometrical relation so this is aine geometry manifesting
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itself here okay these four pencils form a harmonic range in the sense that their
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cross ratio t x t and x and S1 S2 that's
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equal to minus1 another way of saying that in Geometry is that t S1 X S2 so
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this one this one this one and this one form a harmonic
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pencil and I remind you that that uh can also be interpreted in terms of uh the
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points obtained by any transversal so if we take any transversal and see where these four lines meet this transversal
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then those four points are going to form what's called a harmonic
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range harmonic range for points harmonic pencil for lines and uh in terms of
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those points this is often described in in a nice way it's saying that say this
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point here divides this segment internally in the same ratio as this one
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divides it externally so maybe this is in the ratio of one to two let me put it make one to two maybe like like this so
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that might be 1 to two Okay and then that would mean that this to this is
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also in the ratio of 1 to two but the sign is reverse because we're talking about displacements and the direction
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matters Okay so I want to emphasize here that uh necessarily we are touching base
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with a very fundamental aspect of apine geometry built in to the very
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fundamental nature of special relativity World lines and
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Associated spatial directions okay so now I'm going to introduce another idea
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which is very important for us which is a particular identification that we we can end up
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making between events in this one-dimensional world
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points in this two-dimensional time space and actually observers of a
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particular kind so here's an observer a and that
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Observer has a clock so when we talk about observers we typically mean you
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know bats that have set up a little clock so they have a notion of time over
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here at some point something happens that's an event so an event is
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associated with a spatial position and a time position as far as
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any one Observer is concerned as long as they have some means of using the signals to uh to get
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information so we've already seen that we're identifying points in time space
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for example there is a point in time space let's call it uh a it's a point in
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time space that this corresponds with events in this
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one-dimensional space now what we see here in this time space is a distinguished Point O which
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let's assume is the time equals z event for the Observer
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a in question that means that the world line of this
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Observer a can be identified with a line through our time space that passes
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through this
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o and this event we can now associate it
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further with actually an observer in fact more particularly with the event where that observer's clock is
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ticking one okay so here's the world line of a
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we're assuming that a has registered their clock so that this is zero as far
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as their clock is concerned points on here represent time positions for a okay so
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that's the time axis for a now what we're going to do is we're
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going to think about that particular point on this wh line as representing the tick one of A's
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clock so we're setting up a clock so that this is Tick zero this is Tick one
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and then subsequent ticks are just created in an apine Way by creating the apine lad equally spaced points either
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in this direction or in this direction if we
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like so this is a an interesting thing now that points in our event
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space have now a further interpretation we are now identifying them or trying to
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identify them with observers which are homones with this
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fixed Point O so there World lines pass through this fixed point and the point
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in question is ticking of the clock
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one so there's a a few things to say about this so if things are oriented in
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the usual way that this represents a line representing light going to the
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right this represents line going to the to the left so we're sort of having a notion of time going up then you might
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well ask well what about about a point down here does that correspond to the ticking of uh a clock for some Observer
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which corresponds which is at time zero here so we're going to be rather
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open-ended about this so so the possible interpretation or extension of this uh
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to the entire time space we'll we'll be agnostic about that currently we have a
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reasonably standard familiarity if we're sort of in this um this upper cone okay
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but in principle we might be able to make some um discussion about uh more
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General things as well but typically I'll start at least by drawing our pictures like this now there's the
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second thing to note is that there's an important auxiliary Point Associated to
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this point a and that's obtained by uh essentially acting by J remember video
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so back I explained how the red complex numbers naturally act on this apine
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space with a distinguished point and two directions like this so the way that
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that works is that we create a line parallel to S2 through our point so parallel maybe
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like this okay and then we're going to go an
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equal amount in the opposite side so I'm going to measure it here which is a little bit
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cheating I to to make an equal segment I should actually draw parallelograms but
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I'm going to dispense with that and just roughly uh do it okay so there's
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I think this is an Aline construction of reflecting this point in this line parallel to this line here so we get
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this new Point here and let's call it U say a Tilda so this is is equal to
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that in the sense of this Vector being the sense of this Vector not in the sense of the distances being
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equal what this then amounts to is a multiplication by J so this another way
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of thinking about this is this is J * a and this is a way of obtaining our
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second line the spatial line that
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U corresponds to the space axis for the Observer a okay and gives us then a
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natural unit also in this spatial Direction so we get all of this for free
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just because of the pleasant apine geometry which we're assuming underlies
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this time space situation okay so now we come to the
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heart of today's lecture which is this beautiful diagram and some really
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interesting algebra that goes with it that brings out the role of the red
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complex numbers very clearly and what we're doing here is essentially getting
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at the Lorent transformations in a very Elementary way so it's a busy diagram so
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here's how we should start to think about this so the first order of business is a particular point in time
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space and two lines representing the world lines of
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signals maybe it's light maybe it's sound maybe it's carrier pigeons okay
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maybe it's guys paddling up a river in kayaks it's a very general situation so
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there's S1 and there's S2 and now we have a
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point in time space and that point we're going to associate it to an observer
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which is hocon with that point there so its World line is this one here this is
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the world line of Observer a and we're assuming that a has set up a clock in in
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the usual fashion for which this point represents the ticking of one so here is 1 2 3 Etc minus one minus 2 and then in
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the way that I've just described we can get at the point a Tilda so we take this
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line here this line here we're going to take a line parallel to this one through a okay and we're going to go an equal
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displacement from here to here to get this point a Tilda once we have a Tilda
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we can draw the axis the spatial axis through a TIA so now we have two
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axis the time axis for a and the space axis for a and we have natural
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coordinates on each of them and this gives then a frame some coordinates on
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the entire time space by drawing as we do in aine
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Geometry lines which are parallel to the two base lines and uh then end up being
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being sort of equally spaced and then allow us to read off coordinates pretty easily so this is our first coordinate
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this is our second coordinate so for example this point here will'll have coordinates well we go up two in the T
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Direction and we go over one in the X Direction so we get a little parallelogram there so this is the point
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2 comma 1 so this is a way of
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coordinatization that represents Observer A's clock ticking
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one okay good now what we want to do is we want to introduce a second Observer a
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prime and it's also a homon Observer with a that means this observer's world
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line will also pass through that particular point and we're choosing this world
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line and the Observer is a prime and that is encoded by a point on the world
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line which is that point there so that point there represents the ticking of a Prime's clock one and then there's
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equally spaced thing so there would be two there would be minus one so there's the world line of a prime and just as
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before we can find the the the space axis for a prime by
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reflecting we take a line parallel here just as we did before there's S1 we reflect an equal amount in there and we
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get this point a prime Tilda and then the line from our origin to a prime
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Tilda gives us the x axis for a prime so now the time axis
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and this space axis for a prime give us a new set of coordinates so we
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coordinati the space by using lines which are parallel to these ones parallel to these ones okay and so for
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example uh this point point up here would be of coordinate one in the T
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Direction and of coordinate one in the X Direction these two vectors add up to
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that so that would have coordinates one one in the a prime
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system now what's the relationship between these two observers that can be
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concisely encoded by describing the relation of the point a prime
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in terms of the point a so from the A's frame point of view so if we go back to
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the A's coordinates axes here and here then that point there has coordinates
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what so now we're in the a system we're trying to figure out the coordinates well 2 1 this is the point 2 1 and
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generally we use in in general we'll say that the coordinates of a prime in the a
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frame are in general s and r s is the time coordinate R is the space
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coordinate and in terms of um red
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multiplication this is the red complex number s +
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RJ time a because I remind you that the red complex numbers act on
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points in the following way so if we want to act on that point by say 2+ J we
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have to multiply by two in this direction and we have to multiply by J
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in which which rotates us down there okay so we have to take the sum of two times this and that thing there and that
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gives us that point there now in this particular
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example s is 2 and R is 1 so this becomes just two plus
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J time a so if we've expressed the relationship between these two points in
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time space in terms of the red complex number that's required to multiply a by
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to get a prime conversely the coordinates of a in
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the a prime frame are given by this s/ s^ 2 - R 2 comma - r/ s^ 2 - R
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2 and in this case if s is 2 and R is 1
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then S2 - R 2 is three so we get 2/3 here minus 1/3 J so what we're doing is
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we're looking at it symmetrically okay the other way around if we started with
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the a prime frame then we would have to multiply by 2/3 - 1/3 J in order to get
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a so let's see what that means so if we start being in the a prime frame there's
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a basis Vector there's another basis Vector in order to describe this point here what we have to do is we have to go
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2/3 in the a prime Direction so that would be like there okay and we have to go minus 1/3 in the a prime tilted
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Direction so there's a prime 1/3 would be this direction uh one minus 1/3 be over here
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so we have to go up 2/3 in this direction and then minus 1/3 back there and indeed that's plausible that looks
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like the coordinates of a in the a prime
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frame in terms of red complex numbers
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this is just the inverse of
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this how do you invert a complex number how do you find S + iir inverse in
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ordinary complex number arithmetic s + I what you do is you take the complex conjugate that's s minus
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I and you have to divide by the quadrants of the vector namely the
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product of the vector with its complex conjugate which in ordinary complex
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numbers would be s² + r 2 here it's exactly the same story we take the
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complex conjugate of this thing which is s minus r or in terms of these things s minus RJ and then we have to divide by
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the quadrants of this red complex number which is S2 minus r 2 because we're in
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the red situation so that's exactly where this is coming from it's
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just the red complex number inverse of that thing and it's 2/3 minus 1/3 j a
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prime okay so now I want to illustrate where we're in a great position here if you've understood this now you can
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understand the lorence transformations in this dramatically simplified way okay
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so the Lorent transformation allows us to go from one frame for a third event
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to the other frame back and forth so let's choose a third event say this one
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right here e it's another point in time
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space in the a coordinates mind that's that line and that line how do we get to
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there well we should go -1 in our first axis and over one in the space axis so
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this thing here has coordinates minus1 comma 1 which corresponds to the
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red complex number minus1 + J so we can write that e = -1 + J *
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a how about in the a prime Direction
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well here's the a prime T axis here's the a prime x axis we want to express
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this thing as a combination of that vector and that Vector okay we want to get down to here so we should go uh you
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know minus this Vector that would send us down there and then plus this Vector that would get to there so this is -1 *
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this Vector plus 1 * this Vector so it corresponds to the red comp number - 1 +
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J coincidentally and it's just a coincidence these two happen to be the
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same okay but now I want to show you something really interesting
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suppose for the sake of argument that we hadn't a a picture for the a
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prime frame that we didn't have this this geometrical picture but we just knew
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this but we also know that a prime in terms of a is 2 + J *
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a so now we could make this calculation in a different way we could just replace
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this a actually with this expression here 2/3
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1/3 J okay so imagine what would happen if we put 2/3 minus 1/3 J there
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so then we have to multiply this red complex number with this one let's do that how do you multiply these two
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things well it's -1 * 2/3 so that's - 2/3 and then we have to add the product of this * this that's 1 * - A3 so it's -
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A3 so we have - 2/3 - A3 that's min-1 and what about the J component
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well it's this coefficient time this so it's -1 * -3 that's + 1/3 plus 1 * 2/3
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that's plus 2/3 so 1/3 plus 2/3 that's one so we get a one there so we could have used the red
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complex multiplication here to get this thing if we didn't know it let's do
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another example let's choose this point another point
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in time space f so in the a coordinates we go up three and we go over one in the
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spatial Direction so this has coordinates 3 comma minus1 it corresponds to the red complex number 3
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J * a okay so maybe we should try to do it now
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in the a prime Direction but okay that's a little bit uh a little bit more complicated so why don't we do this
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algebra instead okay so we'll do exactly the same as we did before we'll just take this expression here and replace it
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replace the a with 2/3 - 1/3 J * a prime and
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now over here I'm going to do the calculation I'm going to multiply this red complex
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number by this red complex number see what we get so the 3 * 2/3 that's two
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then we have to add the product of this coefficient and this coefficient so there's two minuses make a plus and there's a one * 1/3 so that's 1/3 so we
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have 2 + 1/3 2 + 1/3 is uh
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7/3 so um we're getting
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7/3 what about the J coefficient well there's a 3 * - A3 so that's
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min-1 and there's a -1 * 2/3 that's -
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2/3 so we have -1 - 2/3 uh which is - 5/3 I guess so we
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should get - 5/3 time
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J do a prime what this is telling us is that the coordinates of of this point F in
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the a prime frame are going to be 7/3 that's a little bit more than two
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there's one there's two there's a little bit more than two up here and minus 5/3
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okay there's um one there's -1 - 5/3 going to be out here okay so we
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have to go up here and then we have to go back in this direction - 5/3 that
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looks about right
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so the Lorent Transformations here instead of having to memorize complicated matrices and Gamma
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factors we're just doing red complex number multiplication which I remind you
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is just like the complex number multiplication that you learned back in high school except it's a little bit
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easier because there's no minus sign it's all plus signs right it's all plus
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signs this times this plus this times this and the other way around the next component this times this plus this
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times this so let's summarize what we've learned today if a and a prime are Homo
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chronous observers so their clocks are simultaneously registering zero when
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they meet and if the coordinates of the event
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a prime clock ticks one the coordinates of that in the a frame are s comma R so that's the time
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at which a thinks this event takes place and R is the position that a thinks this
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event took place at then for any event e the time and space coordinates of that
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event e in the a frame so unprimed are related to the time space coordinates in
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the a prime frame same things with primes by multiplying this vector by the
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2x2 matrix s r RS and what we're realizing is that this
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is really just a a red multiplication that is in
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the world of red complex numbers where a red complex number is thought of as a
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pair T comma X that if we multiply these two complex
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numbers here we're getting this one here because how do you get this first thing
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it's this times this plus this times this and that's the same thing that we're getting with the matrix
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multiplication this times this plus this times this it's exactly the same and the second coordinate it's the T here time
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the r plus the x * the S that's the same thing we're getting over here the T * the r plus the x * the S so this this is
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just an encoding of the Lorent transformations in the language of red complex
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numbers and inversely if we want to calculate the prime coordinates in terms
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of the unprimed coordinates then we have to multiply by this Matrix but this Matrix is just
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representing the inverse of this red complex
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number that's how you do the inverses in the red complex number world as we've already seen so this thing here is just
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encoded here and you can think about this thing here as being nothing but just a an elementary rearranging this
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equation where we're taking this thing here and multiplying by its inverse bringing on the other side there so over
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here we've removed ourselves from the non-commutative world of matrix
35:50
multiplication and just framed everything in the commutative world of
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red complex numbers number algebra so this is such a happy situation that we're going to think more
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now about the sort of mathematical structure and in the next video I'm going to explain how we can use some
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sort of conceptual ma familiar mathematical Notions particularly that of a module to
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uh to clarify what's going on here and to say in a very nice way what we're really doing with this um this
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interaction between the red complex numbers and the time space of events uh
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in our onedimensional world