Classical to Quantum
Classical to Quantum | The Lorentz Transformation via sound | Wild Egg Maths
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The Lorentz transformation is at the heart of Special Relativity (SR). This allows us to transform from the Time / Space coordinates of an event from one inertial observer to another inertial observer moving in uniform motion with respect to each other. Here we give a dramatically simplified derivation of the key formula, in the novel context of using sound, not light, to transmit information. Along the way we emphasize the importance of affine geometry for a deeper understanding of SR, including how the cross ratio and harmonic pencils connect with world lines in timespace diagrams, and also cast serious doubt on the validity and indeed meaning of the modern form of Einstein's 2nd Postulate: that the "speed of light" is a universal constant. [Sorry, but it is not, as any Klingon will readily explain to you.] The mathematical prerequisites here are only elementary linear algebra, some careful reasoning, and attention to the precise use of language. We are also able to avoid the usual formulations requiring square roots in the gamma factor: this derivation is purely algebraic. Merry Xmas 2023 and a bit thanks to all my Patreon supporters and Members of the Wild Egg Maths channel!
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hello everyone thanks for joining me on this adventure we're learning about special relativity and today we're going
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to talk about the lorence Transformations which are at the very heart of special relativity but our
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point of view is going to be quite different from Einstein's deration also different from
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the prior work of Fitzgerald lorence po coming essentially from the Michaelson
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morally experiments and trying to understand the nature of light in the context of Maxwell's equations we are
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thinking about a much simpler situation where we just have a one-dimensional
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space and we're only using one sense and that's the sense of hearing so we're
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using sound to communicate information so we're thinking of having a bat and
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the bat is sending a sound signal and then listening to when it comes back and from the initial and the final times of
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sending that signal it deduces something about where and when whatever was was being heard
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actually occurs so this means that say if we have two bats they each have their
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own way of sort of
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coordinatization now in order to make this happen we have to start with each of the bats having an
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internal clock so a has some mirror Arrangement that I've discussed say a little mirror and it bounces back and
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forth some sound creating an elementary sort of sound clock and a prime has some
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other similar kind of clock Arrangement but crucially we're not going to require
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anything about the commensurability of these two clocks they are independent
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clocks and we haven't been able to to compare them yet so what we want to do is we want to um describe the
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relationship between the measurements that bat a gets for an event and the
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measurements that bat say a prime gets for that same event and that
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relationship mathematically is the the content of the Lorent
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Transformations crucially we're going to along the way talk a little bit about the important role of apine geometry in
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setting up special relativity that's a really important topic that we'll have to talk more about and we'll also talk
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about Einstein's second postulate about the speed of light being a universal
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constant I'm going to shed some doubt on that so I don't really believe that's a
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proper statement and we're going to steer clear of that in our derivation of Lorent Transformations as opposed to
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Einstein Einstein used used that crucially in his setting up of the theory so let me remind you that the
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situation is say with bat a here's bat a and events are described by this
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two-dimensional diagram that represents events that are
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occurring somewhere in this one-dimensional space so bat a sets up
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axes a Time axis here space axis here so that an event like this represented by a
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DOT would be something that corresponds to at time two something happening at
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position one bar or minus one so in A's coordinate system this dot represents
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something happening at minus one on his space scale at time two and here this is just a
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representation of his clock this is his world line and here is the world line of
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his mirror and these lines here represent the bouncing back and forth of the sound signal okay so what's
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interesting and important here is that when the sound is going to the right it's represented by something of this
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slope here and that's always the same direction no matter where we are and similarly when light is going to the
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left it's represented by lines that have this constant slope and the way that a measures things
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say to measure the coordinates of that event that a sends out a signal and
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receives it so in this case um sending out a signal from this point here in in
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this direction here so we we call this a T2 because it refers to uh going in the
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left Direction and then it bounces back there and S bounces off whatever it is and then it comes back and that's
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represented here and it comes back at t1al 3 and then there's these rules that
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we talked about last time that uh the time is the average of these two
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readings so in this case that would be two that's why there's a two there and the x coordinate is minus T1 + T2 so
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that' be - 3 + 1 that's 2 ID 2 that's min-1 and so that's why the the the x
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coordinate is minus one here correspondingly is a similar time space
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diagram for bat a prime which has a different mirror the represents the
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bouncing back of sound signals between bat a prime and its mirror so that's what establishes this rudimentary clock
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along the time axis for a prime that's the T Prime axis
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here and um and the XP Prime axis
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is
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coordinatization for example here's another Point representing An Occurrence in the time space for a prime and this
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event can be registered if a prime sends a signal from this point so there's a
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light signal emerging at time minus 2 the light signal is is going to the
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right so it's represented by this motion here until it gets to this point here and then it reflects back that's
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corresponding to this line here and so it's received at T2 Prime equal 5 and if
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we use the same laws 1 12 of the average um 5 + - 3 that's um 2id by two that's
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one so that's why there's a coordinate one there and uh minus T1 that's + 3 + 5
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is 8id 2 is 4 so that's why the x coordinate is four okay so those are the
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individual ways that these bats are going to register time and position
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coordinates for events by recording When sound signals were emitted and when they
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were received taking into consideration which direction they were going okay so now our job is to try to
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relate these ones here but that depends on the relative motions of a and a prime
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so it might be for example that as far as a is concerned let's just look at it from his point of view
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bat a prime she is moving perhaps to the left and perhaps it's the case that at
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time T equals z for a bat a prime is at
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this position here and at time two she is over here
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so that's represented by these two dots that would mean that her world
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line as far as a is concerned would be this line
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here through these two points okay so that would be the world line of bat a
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prime as measured by a okay so U then our job is to try to
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compare how this bat a prim's view of the world relates to bat
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A's view of the world those are the Lorent transformation that we now want to derive now usually
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physicists think that the underlying geometry for special relativity is so-called lorenzian geometry or Einstein
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manowski geometry and that's true to a certain extent but there is an underlying geometry which at some level
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is really more fundamental and it's really at this underlying geometry level that you can understand things in a
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better way and that's the more Elementary geometry of apine geometry this is a
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unfortunately under appreciated geometry it's a sister
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geometry to projective geometry so apine geometry and projective geometry are somehow the pillars of of modern
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geometry and then metrical geometry like ukian geometry or lorenzian geometry are
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things that when Builds on top of these geometries typically on top of apine geometry but apine geometry very simply
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is a geometry let's say of the plane where the the main notion you have points and lines but the main notion is
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that of parallelism so it's what you get when you sort of consistently think about the consequences of having a a
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consistent notion of parallelism and uh that that uh figures
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in in the following way so if we look go back to one of the bats say here's the T axis for that bat and here's xaxis we do
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not have to have these things perpendicular as we started with asbat because after all when a was looking at
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bat a prime the the world line was was slanted okay so it's good to have a
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framework where we're um flexible about the direction of the axes okay they do
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not have to be perpendicular and that's really meaning that we're working in apine Geometry which has the underlying
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group um of Transformations being that of linear algebra namely the the general
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linear group of U of say 2 by two matrices or linear
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Transformations along with translations so that's a more flexible geometry so let me remind you that if
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this is the world line of a bat and here's the the world line say of her mirror that um the way we we set things
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up is in terms of light okay so we bouncing light back and forth okay and let's say light going in this direction
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this direction is represented by that direction in our uh time space and and
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light going in this direction is represented by by this then what you see here is that this little parallelogram
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that's naturally formed here okay is um is one where the the axes themselves are
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forming the the boundary of this parallelogram and the light directions are forming the
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diagonals now that's a very tight and interesting situation which I I sort of
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blow up and and put it over here so these two sides here call them u and v
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in black and then we have these two diagonal vectors maybe if we call this one U and this one V then this one will
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be U plus v and this one here is like U minus V or or U minus V what did I say oh
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actually it should be minus U plus v sorry let me adjust that minus U plus v
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so minus U down this way and plus v okay so these four um vectors or four
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directions emanating from this point form what's called in projective
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geometry or in apine Geometry a harmonic pencil and that's a a very interesting
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configuration uh usually defined in terms of what's called a cross ratio is a fundamental sort of invariant in aine
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Geometry arguably it is the fundamental invariant okay uh cross ratio being
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minus one we're going to talk more about that but I just want to um alert you to
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this strong connection between this sort of fundamental uh worldline diagram and
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this cross ratio and this notion of harmonic pencil so in particular if you know three of these directions then the
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fourth is sort of determined all right like if you know uh these two lines here
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and you know one of these lines then the parallelogram is determined and then the other diagonal is also determined so
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another way of thinking about that is if you think of the light directions as being determined okay then a choice of
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direction for a Time axis will specify a direction for the space axis and
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conversely okay so that's pretty interesting so this is like the fundamental parallelogram whose
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diagonals are light lines and so it's a natural an interesting connection with
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apine geometry here okay and now I have to say something about Einstein's second
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postulate in its modern form so that's not in the form that Einstein originally
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expressed it in his 1905 paper that form actually I have no problems with but the
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modern form is a much stronger form and it says that the speed of light is a
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universal constant so no matter what inertial frame you're in okay whether
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you're in a human spaceship going like this or a klling on spaceship going at Warp 7 if you measure the speed of light
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you're always going to get the same answer which is this standard constant C in me/
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second and rather contentiously I I claim that physicists
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really should have serious reservations about this statement there's serious logical problems with this and there are
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reasonably Elementary logical problems just having to do sort of the philosophical setup about the meanings
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of the words involved in fact to some extent
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relativity is already uh fighting against this posate right this posate is
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in some deep way actually contrary a little bit to the spirit of Relativity
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and let me just give you a brief orientation about that so yes we may on our rocket ship have our
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meters our meters meter sticks yeah and uh you know and then some clocks with
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our seconds defined in terms of cesium atom transitions Etc
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but the Klingons almost surely don't have such a system okay they're going
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warp seven and they're not stopping so it's not as if we can get them to stop and they they dock and and we we go and
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we compare their rulers and our rulers and their clocks and arock they're not doing that they're just constant speed
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that's what an inertial reference frame is okay it's a just constant speed so we
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can only make measurements about their rulers and their clocks we can make
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deductions about them but we have no absolute way and that's the problem that
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when the Klingons um measure speed of light they do experiments there they're surely not going to get this answer
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because they're not measuring in meters per second so it is an unwarranted assumption to assume that other systems
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are all using meters and seconds and that they are the same meters and seconds whatever that
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means okay you're not really allowed to to make that assumption because first of all
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uh when you accelerate meter sticks vastly want to get to Andromeda we're
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going to have to accelerate this thing hugely okay and that's very likely going to affect its physical characteristics
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acceleration and then later on deceleration to what extent is that the
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same uh rule or once we've accelerated to to warp eight or whatever same with
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our clocks okay but there's this question of you know how do we compare even if they
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did some measurement in in terms of their units how do we compare our measurements to their units that's a
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serious problem and in fact there's no real uh way of doing that it it's very
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different from you know you may think well we're able to to measure you know relationship between units if this is a
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meter stick and this is a you know a foot okay then we're able to you know compare uh you know these unit systems
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and then say oh this one is three times this one or whatever but that's a situation where you have unit systems in
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the same overall system that's not at all what we're talking about when we talking about
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Klingons going at warp 8 and us going in the other direction with our our units
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okay so I'm going to talk more about this but I just want to alert you to this um sort of heresy okay I don't
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believe I don't believe this postulate and we are going to step back from using this postulate which in my view is
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either wrong or meaningless depending on sort of the meaning of the words however there is still some notion in that
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direction that we have to put our finger on okay so there's something that replaces it but it can be better stated
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in terms of apine geometry rather than in terms of any kind of measurements of
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speed and the way to do that is in in this direction so I'm not saying this is
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the my final answer to this but it's something in this direction that a better posture would be the light or
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indeed sound or whatever you're talking about waterways moving in the same direction have parallel trajectories in
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the time space so we're in a one-dimensional space situation and here this represents let's say sound or light
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if you like um you know moving so it's representing uh the movement of a photon
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say in this direction okay and this line here represents the movement of another
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Photon okay but they're separated okay we separated
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and a somewhat sort of analogous statement to this one is to say ah these
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two world lines it's not as if they have the same slope okay we're not able to say that because slope is not is not an
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intrinsic quantity in the apine world but we are able to say that they're
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parallel that is an intrinsic and fundamental quantity in the apine world
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so even if we change our coordinate systems with a linear transformation
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okay you know the scale of our units is going to change but the parallelism of
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these two lines is not going to change okay so if light's going in this
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direction we have different particles going in this direction say photons then their world lines are all going to be parallel and conversely if they're going
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in in this direction say represented by these World lines they are also going to be parallel so that's a way of rescuing
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the SE can postulate making it um moving it away from a speed measurement which
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you can't do really not in in an invariant way to a a parallelism um
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measurement and that's really what we're going to be needing in this analysis to derive the Lorent Transformations and uh
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another ingredient that we'll need is just a little bit of linear algebra so let me just remind you about how to
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invert uh some matrices how to put equations in Matrix form so here is the
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time axis here's the mirror we're bouncing light back and forth and note
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that you know we don't um need to specify any particular slopes we're quite happy just to use arbitrary
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directions arbitrary directions for our for our light directions one in the One
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Direction which is representing going right and another in the two direction which is representing going left so
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these as long as all of these guys are parallel and and all of these guys are parallel then we're good okay so in this
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system here then the clock that's sort of generated by this back and forth bouncing has sort of like ticks there
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actually sort of like tick tock tick tock tick tock Etc okay and okay so then here's some event like a dot in the time
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space diagram it's got a t coordinate it's got a spatial coordinate I haven't
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shown the um the x-axis and so let me remind you that T subb and X subb in
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terms of these times T1 and T2 where we're sending out a sound
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signal and then receiving it after it's bounced off that that event there te is
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1 12 T1 + T2 and XE is 12 minus T1 + T2
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and we can state that in a vector form by making a little Vector T sub x sub
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making a little Vector T1 T2 and then rewriting the equations as te = 12 T1 +
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12 T2 and x e is minus 12 T1 + 12
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T2 and then if we want to invert this and express T1 T2 in terms of
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te and XE then we just have to invert this Matrix okay and I remind you to invert a
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matrix a 2X two matrix it's pretty simple you interchange those guys and you multiply these ones by minus one
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okay and you have to divide by the determinant the determinant here is one4
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minus minus 1/4 so it's a half so you have to divide by that half is the same as multiplying by two so the effect is
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that the fractions disappear in you just get 1 1 - one 1 okay so that's a a very
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nice um way of saying that T1 is just te minus XE and T2 is just t p plus XE okay
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so we have the ingredients now let's derive the Lorent Transformations so we're going to think
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about bat a as being stationary so we're going to describe everything in Bat a
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coordinates so prominently here's the time axis for a and here's the space
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axis for a here are the various coordinates there's a clock and I haven't shown you the mirror but here
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would be the mirror and here are the two directions corresponding to sound uh
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going to the right and going to the left so if we wanted to we could augment this
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by having a a sound U clock like as we've seen before and now let's suppose
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that a prime is another bat and she is moving let's say to the left with regard
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to a so as far as a is concerned a prime she's coming towards him a time T equals
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z for him a prime coincides with his origin so that represented there by that
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dot there and then this is the wh line of a prime with regard with respect to a
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and these dots here are the um places
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where the the ticking of a Prime's clock is going so I haven't shown you a mirror but there's some corresponding mirror
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for which this is Z this is 1 this is 2 this is minus one so those are the times
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on uh a primes clock this is the T Prime axis I haven't shown you the X Prime
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axis because we don't need that and now let's consider an event let's say t sub
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be x e represented by a DOT okay so in A's coordinates uh the coordinates of
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that are easily read aable it's uh you know roughly 2 and A2 and and three and
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a half or something like that but how does a prime measure or record where and
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when this event occurred well what she does is she sends a sound signal so from
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here a sound signal to the left so represented by this line here which is
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parallel to this line here and then once it gets there uh it
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goes in a direction which which is parallel to this one here representing the sound
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returning to a prime and gets back there and a prime measures these times say T2
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Prime and T1 Prime and now we want to understand what
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a prime register is for the values here but the other piece of information that
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we need to pin things down here is the position of this world line as far as a
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is concerned and that's all encoded by the position of just this one dot here
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if we know this dot here and we know that a prime is at the origin at Time
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Zero then that determines that line and it also determines a Prime's clock so
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everything is determined by that point there and so that point has coordinates in A's framework let's call those
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coordinates s comma R so s is is the time coordinate
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of that which in this case is two and R is the x coordinate of that point which
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is minus1 or one bar so two comma 1 bar that is
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Sr then these subsequent ticks of a
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Prime's clock are just obtained by taking Sr and
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multiplying those coordinates by some some integer by one by two by 3 or by
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minus one min-2 Etc so let's look at um say the the
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light emanating from here okay so this position here okay what is that well
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we're assuming this is T1 Prime in a Prime's setup
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so the coordinates of this point here as far as a is concerned it's going to be
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T1 Prime Times this point so we have to take Sr and we have to multiply by T1
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Prime T1 Prime s T1 Prime R and then to get to te XE this is down here we're
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going Su multiple of this Vector here this Vector here okay in A's framework
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is the vector 1 one because we're going one one over and one over go from here to here okay so that's
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representing uh this line segment from this point we're going some multiple of
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this vector and we're getting to this point here similarly from here this point here
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which has time coordinate T sub 2 Prime as far as a prime is concerned its
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coordinate with respect to a is just Sr time T2 Prime so it's T2 Prime s T2
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Prime R and from there we're going in this direction well that's represented by this Vector here that's the vector 1
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minus one in A's framework so we're going some multiple in this direction
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from this point and we're ending up again at T sub be XB so let's say that multiple is beta so this point plus beta
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* the vector 1 - 1 = te
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XE okay so we have two equations we have two equations and we
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can now um and they're both Vector equations so each one of actually has two components so we have enough
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information now to take these two equations and to figure out what's really going on okay so here are the
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equations that we've just arrived so here's te XE that's the point in question representing some event and
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these are the coordinates of that event in A's frame all of these things here are from A's point of view so here's
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that first point on the world line the time axis for a prime and Alpha * 1 one
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was this and then here's that second point on that same world line plus beta 1us one okay so from this one here this
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first equation there's really two equations there's T1 Prime s + Alpha =
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te and T1 Prime r + Alpha equals XE so
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in these two equations if we take the difference this one minus this one then the alphas which we don't really care
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about disappear and we get T1 Prime * s - r = t e - x e okay that's good we can
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now solve for T1 Prime it's t sub - x e over s-
29:36
R similarly over here we write down the two equations T2 Prime s + betaal T sub
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e and the second one T2 Prime R minus beta = x sub
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e in this case here we're going to add the equations to get rid of the beta so we're going to get T2 Prime * s + r r
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equal T sub plus x sub solving for T2 Prime we get that it's equal to T sub
30:04
plus x sub over s + r what does this do so from A's point of
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view if we know the coordinates of some event in A's point of view then a can figure out what these times T1 Prime and
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T2 Prime are that a prime is going to have to use to send out sound signals to
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register the event in question okay and once we know
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those initial and final times then the same equations that we
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had before allow us to calculate the time and space coordinates with respect
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to a prime for that same event okay so we get te Prime is 12 T1 prime plus T2
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Prime so we just have to add this and this now the denominators are a little
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bit different so we're going to multiply this top one by S subr and we're going to multiply this bottom one by s- R top
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and bottom then that we'll have a common denominator of s2 - R 2 and then when you think about it you multiply this one
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by s+ R so there's an S + r * T subb and over here multiply this by s- R so
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there's an S minus r * T subb and when you add those things
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okay the the r pieces are going to cancel because they have different signs
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so we have an st and another St and we're dividing by two so we just get
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S and similarly when we look at the XE uh things the same argument when you
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multiply this thing by S Plus R this one by S minus r and you take the the sum
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then you're going to get minus r x sub e please check that write it out just to
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make sure and similarly x sub e Prime is 12 minus T1 prime plus T2 Prime so now
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we have to take minus this thing plus this thing we're still going to use the
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same common denominator s2 - r s so we still have to multiply this by S + r and
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this one by S minus r but now we're taking minus this one plus this one so we have different
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cancellations so now for the te the the S pieces cancel and we're just left with
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the the the the minus r because we're taking minuses so it'll be a minus r
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here and a minus r there then we divide by two so we get minus RTE and for the x e the r is cancel and we get plus
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sxe so this is essentially these are the Lorent
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Transformations okay these are the lorence transformations
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that do what they take us
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from the T and x coordinates for an
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event e according to a and give us the T and x coordinates of that same event
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according to a prime and the only ingredient that's involved other than
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the te and the XE from a are these these two coordinates s comma R those are the
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coordinates of a Prime's clock ticking one so a prime remember was at the
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origin at time zero and then at time tal 1 it was exactly at the point s Comm R
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in A's framework so that's the ingredient in here so let's quickly summarize what we
33:51
just discovered really important okay there's a foundational kind of uh
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formula in in special relativity in fact in in physics so if the a prime clock
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has t0 Prime at 0 0 and T1 Prime at Sr
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and this is all with respect to A's framework then for some event e t e
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Prime x e Prime in other words the coordinate of that event for bat a prime is related to the coordinate of that
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same event for a by this equation we take this vector and we multiply it by
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this Matrix here so that's just stating in Matrix form the equations I gave you
34:37
before so it's s * te - R * XC all over
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s2 - R 2 and the next one minus r t + S
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XE all over s2 - r
34:52
2 and then we can invert this because this is a linear algebra situation so if we want te XE in terms of T Prime x e
35:01
Prime then we have to invert this Matrix so if you take that common denominator
35:06
of 1 / s2 - R 2 out momentarily then what's left is s - Rus RS and if you
35:14
take the inverse of that Matrix the S is interchange and the two minus RS change
35:19
sign so they become plus RS you get exactly this Matrix but you have to divide by the determinant of that the
35:25
determinant of this thing it's s2 - R 2 okay um
35:33
so if you take that that factor let's say like the factor was one was one over five you take the factor one over five
35:40
out so now you have 1 over five time a and you taking the inverse well that's going to be five times the inverse of a
35:47
all right so when you take the inverse of this you're going to get s2 - R 2 * the inverse of that s- r - RS Matrix and
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then the determinant that appears in the bottom of that formula will cancel with
35:59
the determinant that's uh out front and you're just going to get Sr RS this is incredibly beautiful I mean
36:06
this is uh you know mind-bogglingly beautiful okay and it's so simple okay
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and especially this one here okay that to go from time space coordinates uh a prime
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to time space coordinates of a we just multiply this little vector by this simple Matrix that only depends on that
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position s comma R uh where the bat a
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prime clock ticks one as far as a is
36:42
concerned note that um in this formula uh there's no square roots there's no
36:47
gamma Factor you in traditional physics courses you there's a gamma which is 1/
36:52
< TK 1 - b2 c2 um so we are happily
36:57
finessing square roots we like to avoid square roots of course uh when we can we
37:03
tend to think that um formulas not involving square roots or in fact any functions that you can't explicitly
37:10
compute are naturally going to be more fundamental more likely to be fundamental likely to be really at the
37:16
heart of what's really going on and this is very much this really very much ticks that box so I think it's really useful
37:23
to see this in action with an example so here's an example I do hope that uh you
37:29
go through it really carefully and try to visualize what's going on so this is actually the same picture that I showed
37:34
you before so we have bat a we thinking of bat a as stationary or we're putting ourselves in Bat A's position that
37:41
that's really a better way of saying it we don't have to make any a statement about stationary and not stationary
37:47
because that's actually meaningless that's what Galileo taught us right so when we use that language we really
37:52
putting it in quotes it's meaningless we're just putting ourselves in A's position as far as A's concerned they
37:59
are stationary so there's A's time axis and here is A's space axis as before so
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we haven't shown the mirror here and the sound clock but uh these are the ticks
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of A's uh clock and here are the directions of sound represented by the
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diagonals of that little um parallelogram that you get there which is actually a square visually to us okay
38:23
and we're assuming now that if we think about this is being spaced for a second so bat a is sitting here so bat a prime
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is coming like this okay and a Time T equals z according to a a prime is here
38:37
and then she carries on like this and at time two she is over here okay so she's
38:43
going like this she's over there and this is when her clock ticks
38:49
one and so we have an event let's choose that same event here and event that has
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a um a T1 Prime which is two on on her clock and a t T1 Prime which is two on
39:04
her clock and T2 Prime which is minus one on her clock so these are the Whirl
39:09
lines of the light rays that has to be emitted from here bouncing off there there so it's she's she's moving like
39:15
this and and when she's over here light is is emitted and the light goes over here and bounces off and then then comes
39:21
back and meets her uh there okay and then she makes some deductions from that
39:26
okay so the T1 Prime is two there it is there the t2 Prime is one bar or minus
39:33
one there it is there okay so her estimations of the time and space
39:40
coordinate of this event are given by the usual formula 1/2 T1 prime plus T2
39:45
Prime so the average is 1 over two that's a half and x e Prime is 12 minus
39:52
T1 prime plus T2 Prime so that's - 3/2 okay so as far as she's concerned she
39:59
registers that the time and the space position of that event are at 1/2 and
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minus three Hales now I can explain that a little bit um in in the following way so uh it
40:13
turns out that her um her space axis is like this it's still such
40:19
that that if you take a a parallelogram you make a parallelogram uh with this vector and this Vector here like this
40:25
right that these directions are going to be uh the diagonal directions that's
40:30
that's what's determining the space directions once you know that the the sound World
40:36
lines okay so we have these um these things and and what this is saying is
40:42
that to um get to here you go um so in terms of sort of a multiple of this
40:48
vector and a multiple of this Vector you go 1/2 this Vector okay and then minus
40:54
three Hales of the corresponding Vector down here so - 3 - one - 3 something like this so you going up 1/2 there and
41:01
then over there like that right so this represents the combination 1/2 in this direction and minus three Hales in that
41:10
direction right what are the coordinates of this thing as far as bat a is
41:16
concerned well here's the equation T sub e x sub equals the T sub Prime x e Prime
41:26
okay that one there it's just exactly that times this Matrix what is this Matrix this is the Matrix obtained from
41:33
S and R so the crucial thing is the coordinates of that point that's S and R that's
41:39
2 minus one so that Matrix is 2 - 1 -1 2
41:45
okay so now we do the multiplication 2 * this um uh minus this so that's uh
41:51
that's 2 - - 3es that's 5 Hales and this multiplication minus a half uh- 6 is -
41:59
7- 7 so Al together this 5 over 2 that's like there okay that makes sense and - 7
42:06
over 2 1 2 3 so down here like that pretty
42:12
close that's how you go from uh the prime coordinates to the unprimed
42:18
coordinates and and to go backwards if we wanted to get this thing in terms of
42:23
this thing well we'd have to multiply by the inverse of this Matrix which is the the usual thing uh those things are interchanged those minus ones become
42:30
positive ones and we have to divide by the determinant which is three okay so
42:36
that's Len transformations in action and it's really useful to visualize you know
42:43
think about what's happening one-dimensionally okay you just have these bats this bat's stationary this bat's moving like this and there's some
42:49
event that occurs somewhere and both of these bats are trying to figure out when and where this event occurs and the only
42:57
way they can do that is by sending out these sound signals and listening and and and and noting when it went and when
43:03
it came back and making this little calculation and the lence Transformations are just Elementary
43:10
linear algebra first year linear algebra that relate these two little
43:16
computations it's a beautiful thing and it's very simple ultimately
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