Classical to Quantum
Classical to Quantum | Relativistic velocity addition (simplified) | Wild Egg Maths
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In 1905 Einstein introduced his Special Theory of Relativity (SR), with many unintuitive consequences. One of these is that when we consider "compositions" of relative motions, then the relative velocities are related in a different way than in Newtonian mechanics. In this video we use our simplified framework for SR to see that this law is just a simple consequence of algebraic relations between quantities associated to complex numbers. But we do not use the ordinary theory of complex numbers; rather the relativistic version that we call red complex numbers. This is an algebraic arena which is valuable for physicists understanding not just of SR, but also quantum mechanics. This video is part of the Classical to Quantum playlist for Members of the channel. Please consider joining and accessing dozens of specially created videos on a wide range of intermediate to advanced topics. The full playlist is at 
• Introduction to QM and the two slit e...
Transcript
0:00
hello everyone thanks for joining me on
0:02
this adventure from classical to Quantum
0:04
and right now we're talking about
0:06
special relativity in
0:09
particular relativistic velocity
0:11
addition in a simplified fashion which
0:15
connects very closely to just simple
0:20
algebraic manipulations essentially with
0:22
complex numbers although they are the
0:24
red complex numbers not the traditional
0:27
blue
0:28
ones okay so to motivate the story we're
0:31
in a onedimensional space and we'll
0:32
actually consider the same situation
0:33
that we looked at last time okay where
0:36
we have an observer a which is
0:38
stationary from our external point of
0:40
view just to simplify things we have
0:42
another Observer B which is moving with
0:45
velocity V equals 1/2 according to
0:48
a so A and B are what we call autochron
0:53
observers they happen to meet at time T
0:56
equals z okay where which will be there
0:59
because a stationary and there's a third
1:02
Observer uh last time it was a particle
1:04
but this time it's just an observer
1:06
called C who's moving in this Direction
1:08
with velocity minus one4 also with
1:10
respect to a and it's also autoc
1:13
chronous so it will also meet A and B
1:16
exactly there at time T equals z for a
1:20
so these autoc chronous observers each
1:22
one of them establishes sort of
1:24
canonical way of of
1:28
coordinatization
1:31
clock ticking one that's the point C and
1:34
then a conjugate Point C star which
1:37
depends on the uh the signals that we're
1:40
that we're using either light signals or
1:42
sound signals it doesn't really matter
1:44
which okay
1:46
so in a time space the situation could
1:49
be regarded this way if we want to put a
1:52
stationary so it has a
1:54
Vertical World line for us then here are
1:58
the relative IC signal lines okay that
2:02
represent the world lines of sound or
2:06
light emanating to the right or to the
2:08
left in this onedimensional space we
2:11
have time axis we have an x- axis and
2:12
this is with respect to a and the other
2:15
coordinate that for a that's relevant is
2:17
this one here there well all right good
2:21
now in terms of coordinates okay the
2:26
coordinate of a with respect to a is
2:27
obviously one the coordinate of B with
2:30
respect to a is 1 + 12
2:34
J B's World line is here B is
2:37
represented by this point this point has
2:39
coordinates 1 and 1/2 so it's
2:41
represented by the red complex number 1
2:44
12
2:46
J C sub a the coordinate of C with
2:49
respect to a so here is C's World line
2:52
representing this Observer which is
2:55
moving to the left so the war line
2:58
sloping this direction passing through
2:59
the Point 1 - 1/4 so its coordinate is 1
3:04
-4
3:06
J and now what's the velocity how do we
3:09
introduce the Velocity in terms of these
3:11
coordinates well the
3:13
velocity of B with respect to a so we're
3:16
using the same kind of notation with the
3:19
subscript here is with respect to that
3:21
Observer velocity of B with respect to a
3:24
is equal to how do you get that well you
3:26
look at B sub a it has has a t
3:30
coordinate it has an x coordinate those
3:32
represent the the displacements in time
3:36
and space as we go from the the origin
3:38
which is the point 0 0 in A's coordinate
3:41
to the point B which is
3:43
one2 so we have to go up one and across
3:47
1/2 so the delta T is 1 the Delta X is
3:51
1/2 so velocity is Delta X over delta T
3:55
right so we're just going to do exactly
3:57
that so the 1/2 is the Delta X the one
4:00
is the delta T so we'll just take the
4:02
ratio2 / 1 which is a
4:05
half which agrees with what we
4:08
originally say that the velocity of B is
4:10
indeed 1/2 and the velocity of C with
4:13
respect to a well we look at this
4:15
complex number we take the J coefficient
4:19
put that in the numerator that's minus a
4:20
quarter that's the Delta X as we go from
4:24
here to here Delta X changes by minus A4
4:28
and we divide by the delta T T which is
4:30
1 to get minus 1/4 which again agrees
4:32
with our our setup so this is how we're
4:34
going to introduce velocities in this
4:37
situation just in the same way we always
4:40
introduce velocities a Delta X over a
4:42
delta T in whatever coordinate system
4:44
you're working with so a natural
4:47
question then is well what about the
4:49
relationship between B and C if B is
4:52
looking at C what is the velocity of C
4:56
with respect to
4:58
B now you might say well B is going in
5:01
this Direction with speed of a half C is
5:03
going in this direction the speed of a
5:05
quarter you know classically you would
5:08
add those two velocities and say oh the
5:10
speed of C with respect to B should be
5:12
minus
5:14
3/4 but we'll see that that's not the
5:16
way the relativistic world works okay so
5:19
our aim is to find V subb of C in other
5:22
words the velocity of C with respect to
5:24
b or reciprocally V Sub C of B the
5:27
velocity of B with respect to C so what
5:29
we already know we know that b sub a is
5:32
this so a subb will be the inverse of
5:34
that in the world of red complex numbers
5:36
so we take the inverse of that we've
5:38
already seen that's 4/3 minus 2/3 J so
5:41
from that we can deduce that the
5:43
velocity of a with respect to B is this
5:46
quantity that we look at we take
5:48
the X component minus 2/3 divide by the
5:53
T component that's 4/3 we get minus
5:58
1/2 okay okay that's not
6:02
surprising and a sub c is C sub a
6:07
inverse so we have to take the inverse
6:08
of this so we complex conjugate up there
6:12
and the quadrants 1^ 2 - 1/4 s we get 15
6:17
over 16 in the denominator we take the
6:19
reciprocal we get 16 over 15 * 1 and 16
6:23
over 15 * 1/4 is 4 over 15 *
6:28
J
6:31
all right so the velocity of a with
6:35
respect to C because this is a sub c so
6:38
we take 4 15 / by
6:43
1516 that's the same as 4 over 16 or
6:46
one4 so we just make the observations
6:49
that V subb of a is minus V sub a of B
6:53
and similarly V Sub C of a is minus V
6:56
sub a of C so now what we're interested
6:59
in say to calculate something like this
7:00
is to find C subb and we know how to do
7:03
that because of this basic
7:04
multiplicative relationship between the
7:07
coordinates of the various observers
7:10
with respect to each other C subb is
7:12
equal to C sub a * a sub B and it's easy
7:15
sort to remember because it's almost a
7:17
cancellation of the A's okay it's almost
7:20
like what's called Einstein notation um
7:24
in in physics we have a lower index and
7:26
a and an upper index you have to sum
7:28
over them they the kind of disappear so
7:30
what's left is just a a c subb okay so C
7:33
sub a we've just found uh we actually
7:35
know that that's 1 - one4 J and the a
7:38
sub B that's what we found up here
7:40
that's 4/3 - 2/3 J so we make that
7:42
multiplication and we get 3 minus J and
7:45
that tells us that V subb of C
7:48
is the X coefficient which is minus1
7:52
ided by the T coefficient which is three
7:54
has to get - 2/3 and we note that this
7:57
is not Theus 3/4 that we might have
8:01
suspected naively if we were just doing
8:03
sort of classical
8:05
mechanics adding velocities in the kind
8:08
of obvious way so let's be a little bit
8:10
more formal just to set things up if a
8:13
and b are autoc chronous
8:15
observers with B sub a equal little a
8:19
plus little B J then the velocity of B
8:23
with respect to a is by definition so V
8:25
Suba of B it's
8:28
the coordinate
8:30
x / the coordinate T So B A and of
8:34
course this is undefined if a equals z
8:37
and we're not going to always specify
8:40
that that's just obvious if we have a
8:41
denominator in our our formula the
8:44
formula is just uh you know put on a
8:45
shelf it's it's not valid okay so we
8:47
don't worry about
8:49
that okay so an initial Lema let's just
8:52
check this observation that we made
8:53
before that V subb of a is minus V Suba
8:57
of B in general so suppose B sub a is a
9:01
plus BJ as before so that the velocity
9:03
here as we've defined is B over a well
9:05
then a subb is the inverse of this in
9:07
the red complex number so we take the
9:09
complex conjugate a minus B J and we
9:11
divide by the quadrants which is a^ 2
9:13
minus b^ 2 so we get in terms of a t
9:16
coordinate a over a 2 - b^ 2 minus B
9:20
over A2 - b^ s there's the x coordinate
9:24
times J okay so how do we calculate the
9:27
velocity of a with respect to B we take
9:30
this x coordinate which is minus B over
9:34
A2 minus b2 and we divide by the T
9:37
coordinate which is a over a 2 - b^ 2
9:40
and the a^ S - B s's cancel so we just
9:43
have minus B over a which is the same as
9:46
minus minus V sub a of B and here is the
9:51
main result of our lecture a formula
9:53
found essentially by Einstein and it's
9:56
the relativistic velocity addition
9:58
formula says if a B and C are autoc
10:01
chronous observers so they all
10:02
conveniently are passing through the
10:04
same point at time T equals z and they
10:08
set up their coordinates in the uh in
10:11
the standard way using um The Tick of
10:15
their clock at one and the corresponding
10:18
conjugate Point okay so if a b c are
10:21
autoc chronous observers then the
10:24
velocity of C with respect to a is
10:28
velocity of B with respect to to a plus
10:29
the velocity of C with respect to B
10:32
divided by 1 plus the product of these
10:35
two things V sub a b v sub BC so if
10:38
there was no denominator here it was
10:40
just a one then we would get the sort of
10:42
classical gas as to how velocities add
10:46
okay so it's just a little bit of
10:47
algebra suppose B sub a is a plus BJ and
10:51
C subb is C plus d j so that means that
10:54
c sub a which is C subb time B sub a as
10:58
it always is is a plus b
11:02
j that's the B sub a times C plus DJ
11:06
that's the C sub B and when we multiply
11:08
those in the world of red complex
11:10
numbers we get AC plus b d because of
11:12
course j2 equal 1 and we get a d plus b
11:16
c j okay so this quantity here that
11:19
we're talking about V sub a B plus v sub
11:22
b c over 1 plus v sub a b v sub B C
11:25
equals what well the V sub a b we know
11:28
what that is that's B over a the V sub b
11:33
c we know what that is it's D over C so
11:37
we have to add those and in the
11:40
denominator we have to put one plus the
11:41
product
11:42
b/ a * D over C and now we simplify this
11:47
in the obvious way we multiply through
11:49
top and bottom by a * C when we do that
11:53
we get a B * C up here plus an a * D
11:57
alog together in the numerator and in
11:59
the denominator we just get a C+ b d and
12:03
that's exactly V sub a of C because here
12:07
is C sub a okay there it is there so
12:11
this expression is the X component which
12:14
is the coefficient of j a d plus BC
12:17
that's appearing here in the numerator
12:19
over the T component which is AC plus BD
12:22
that's appearing in the denominator
12:26
there and uh yeah so in principle this
12:29
could be very familiar to you because
12:31
you could have learned the same theorem
12:33
in ordinary complex number story okay
12:36
where the the story is it really
12:39
discussing slopes you would you would
12:40
interpret it in terms of slopes you know
12:43
how do how do slopes of of complex
12:45
numbers relate when they're when you
12:48
multiply them okay and it's maybe
12:50
instructive to just go through the same
12:52
exact story with ordinary complex
12:54
numbers and just a minus sign crucially
12:56
that appears okay but it's algebraically
13:00
very simple and in fact sort of elegant
13:02
and you sort of see that I mean is it
13:06
physics or is it mathematics that we're
13:08
seeing here that's an interesting
13:10
question I mean is is this some some
13:11
deep property of of of the physical
13:14
world or is it just an elementary
13:17
property of mathematics which just
13:19
happens to be manifesting itself in the
13:21
physical
13:22
world that's an interesting question
13:25
that um we'll be coming back over and
13:27
over again on our our journey uh towards
13:29
quantum mechanics so this little
13:31
expression here is quite tidy and it
13:33
deserves perhaps its own name so we're
13:36
going to call it V sub one with this
13:39
sort of U plus sign V sub2 and we're
13:43
going to call this the hyperbola sum
13:45
it's like a sum but it's not an ordinary
13:47
sum it's a hyperbola
13:49
sum that's sort of analogous to what in
13:52
other places I call Circle sum okay so
13:55
let's do some examples
13:58
1/2 plus 1/2 1/2 hyperbola sum 1/2 is
14:03
well you have to take the sum ordinary
14:05
sum on the the top that's um2 + 1/2
14:08
that's one and in the denominator you
14:11
have 1 plus the product so 1 plus a
14:13
quarter which is alog together 5
14:15
quarters you take the reciprocal you get
14:16
four fifths how about one hyperbola sum
14:19
one well we get 1 plus 1 on the top and
14:21
we get 1 + 1 in the bottom so we get one
14:26
how about 1/2 hyperbola sum to now you
14:30
might say wait a minute Norman this
14:32
corresponds to something that's going
14:34
faster than our
14:37
signal whether it be light or um or
14:40
sound yeah so doesn't bother us does it
14:44
there's nothing in the theory that we've
14:46
developed so far that restricts us to
14:48
just what's usually called the upper
14:50
light cone like just mathematics
14:53
mathematics works everywhere now you
14:55
might say well the physics doesn't well
14:56
that's that's a question right that's
14:58
not something that's so easily
14:59
established you can claim that but
15:02
actually to to check whether it's true
15:04
or not is a far different story okay
15:06
physicists have not checked whether
15:08
that's true okay so the idea that
15:10
nothing can go faster than the speed of
15:13
light is not something that we are going
15:14
to incorporate just as a matter of
15:16
principle okay it's something that may
15:18
be true and may not be true and part of
15:22
what we're interested in is you know
15:23
investigating things in such a a good
15:25
way that maybe the truth or otherwise of
15:28
such a statement becomes more clear but
15:30
initially at least we're going to be
15:32
open-minded so this is a totally
15:34
legitimate thing to take a hyperbola sum
15:36
of so if we add 1/2 plus two in the
15:39
ordinary way we get U two and a half
15:41
which or five halves if we take one plus
15:44
the product the product is
15:46
one uh so we get 1+ 1 is two so we have
15:49
five halves over two so we get five
15:53
quarter
15:55
another speed you might say that's
15:57
bigger than one how about 1/3 hyperbola
16:00
sum - 1/5 well there it is there 1/3 -
16:03
1/5 over 1 - 115 that's 17th and say
16:07
three hyperbola sum minus three well
16:10
that's obviously zero because you get
16:11
zero in the
16:14
numerator okay an important theoretical
16:17
uh thing about this thing it's probably
16:19
important enough I should put this in a
16:21
box okay it's like an associative law
16:24
for this operation of hyperbola sum okay
16:27
that V1 hyperbola sum V2 hyperbola sum
16:31
V3 equals V1 hyperbola sum V2 hyperbola
16:34
sum
16:35
V3 okay and so you do this exercise and
16:38
show in fact that what you end up
16:40
getting is some kind of variant of this
16:42
where we have V1 plus V2 plus V3 because
16:45
there's three of them and then we have a
16:47
cubic term V1 V2 V3 and in the
16:50
denominator we have a 1 plus well
16:52
there's more than one quadratic term
16:54
there's V1 V2 plus V2 V3 plus V1 V3
16:59
that's quite elegant so the way you
17:02
might want to think about this is
17:03
there's a constant term here there's
17:05
some linear term here which is very
17:08
symmetrical Elementary and symmetrical
17:11
there's a quadratic term here which is
17:13
also sort of Elementry and symmetrical
17:16
and there's a cubic term up
17:19
here so maybe that's a clue to this
17:22
other one generalize this so what would
17:24
happen if we had four velocities and we
17:27
took hyperbola sums of all four
17:29
V1 V2 V3 and V4 what would we get so
17:33
maybe I haven't made it explicit um but
17:36
I've I've said it in directly in a
17:37
number of times that if we consider one
17:40
of these autoc chronous observers whose
17:43
coordinate system is set up the way I've
17:45
I've suggested that we we start with the
17:48
the red signal lines whatever they are
17:51
okay and we have some point representing
17:54
some Observer on the world line then
17:56
we're going to take its conjugate with
17:58
respect to to these two things and
17:59
that's essentially a harmonic conjugate
18:03
and that's essentially some kind of
18:05
reflection in U one of them with respect
18:08
to the other one so with that kind of
18:10
coordinate system then if you ask what's
18:12
the velocity of the the signals
18:14
themselves the answers are always either
18:16
minus one or plus one depending on which
18:18
one you ask okay so that's a um thing
18:21
that automatically happens for all of
18:23
these observers all of these observers
18:24
are going to say if you you ask what's
18:26
the velocity of this thing going this
18:27
way they'll say oh it's minus one and
18:29
what's the velocity of the signal going
18:30
in that direction it's always plus one
18:33
so having said that let's look at some
18:34
inequalities that are quite interesting
18:37
so the first one is if we take two
18:40
numbers A and B and they're between zero
18:42
and one and strictly less than one but
18:45
they're positive okay then the claim is
18:48
that their hyperbola sum will also be in
18:51
exactly the same range it will be
18:53
somewhere between zero possibly zero up
18:56
to one but not being equal to one
18:59
okay so proof If a and b are between Zer
19:03
and one in this fashion then a plus b
19:07
over 1 plus a the hyperbola sum this
19:09
thing here is obviously bigger than
19:10
equal to zero because well A and B are
19:13
bigger than equal to zero and this thing
19:15
here is also bigger than or equal to
19:16
zero so it's uh it's obviously positive
19:19
and what about the other inequality how
19:21
do we know that this thing is less than
19:23
one well this inequality is equivalent
19:25
to the following because if a and b are
19:28
positive then this quantity in the
19:30
denominator is positive so this
19:32
inequality can be restated by
19:34
multiplying through by the positive
19:37
denominator here giving us a plus b is
19:39
less than 1 + a so these two things are
19:42
equivalent under
19:45
this condition now this thing here then
19:48
can be Rewritten by bringing everything
19:50
to the right hand side so we get zero is
19:52
less than well 1 + a B minus a minus B
19:55
and that's just what you get when you
19:57
expand 1 - A * 1 - B so now if a and b
20:04
are less than one and both positive well
20:06
each of these numbers here okay is going
20:10
to be uh strictly
20:13
positive strictly
20:16
positive and so their product is going
20:18
to be strictly positive and these things
20:20
are all in equivalent so this being true
20:23
means that this thing's true means that
20:25
this part of inequality is true okay so
20:27
this has an important
20:29
uh consequence so this is going to end
20:32
up being important for us right
20:34
physically what this means is that you
20:36
know if if one of these observers is
20:37
like a train okay so a is stationary and
20:41
B is like a train okay and then on this
20:43
train B shoots uh a bullet you know
20:46
forward okay so usually as we've seen
20:49
the velocities add that's the way we
20:51
would think about things but in this
20:53
context the velocities don't add they
20:55
they have this hyperbolic sum okay but
20:57
the point is that
20:59
if the train and the bullet are both
21:01
going slower than our signal speed then
21:05
the
21:06
resulting the resulting combination of
21:09
these two things will also be slower
21:11
than the signal speed and that's true
21:14
not just when the signal speed is light
21:17
it's also true when the signal is
21:22
sound it's also true when the signal is
21:25
sound so that same phenomena is
21:28
happening it's just the mathematics of
21:30
it and what happens when we consider
21:32
because we've said well maybe we can
21:34
consider things moving faster than our
21:36
signal speed why not mathematically it's
21:39
it's possible so let's just uh do a
21:41
little computation here what would be
21:43
two hyperbola sum three so this would be
21:48
sum of a train going faster than our
21:50
signal and the bullets going even more
21:53
faster than the signal and what's the
21:54
combination of of these two things okay
21:57
well uh we use the same formula right
21:59
it's 2 + 3 over 1 plus the product
22:02
product is
22:04
six so that's five over
22:10
7 that's less than
22:12
one that's less than
22:15
one that's a kind of curious thing okay
22:18
so we might have to ponder uh the
22:21
significance of that but more generally
22:23
show that if you have numbers A and B
22:25
which are bigger than one then also the
22:28
hyperbola sum of them is positive but
22:31
less than one so generalizing this so
22:34
we're coming into contact now with some
22:37
interesting sort of aspects of special
22:39
relativity and I want to emphasize to
22:41
you that this is not uh not a feature of
22:45
light predominantly okay it's that's not
22:47
really the the thing it's it's just
22:49
having some signal propagation that has
22:52
certain properties uh particularly this
22:54
sort of parallel um postulate property
22:57
which is really
22:59
um you know in the context of an apine
23:01
geometry for our time space okay but in
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particular uh a lot of things that we
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maybe formerly thought were really
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special things having to do with the
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their amazing special role of light in
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special relativity maybe that's not
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really the case maybe these phenomena
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are much wider and and more um more
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General kinds of things that happen in
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uh perhaps even very slow moving
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situations
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is