Classical to Quantum

Classical to Quantum | Special Relativity without equations (Galilean) | Wild Egg maths

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Here is a very elementary exposition of the basic framework for one dimensional Special Relativity, where Galileo's Invariance Principle takes centre stage, and Einstein's other Postulates are dispensed with. In particular we do not assume that "nothing can go faster than the speed of light", and that "the speed of light is the same in all inertial reference frames".

In fact we argue that "inertial reference frames" are an unnecessary fiction, not corresponding to the reality that we find ourselves in --- here on planet earth, trying to make sense of the cosmos. It is more modest and sensible to adopt a position that observers are just localized, with no preordained cosmic reference frame that can immediately be invoked to compare measurements at widely separated places or times. This is in keeping with Galileo's Principle.

Further we do not restrict to using only light as our means of propagating symbols. This possibility was also recognized by Einstein, and we take up the challenge of framing the discussion using sound instead, which has conceptual advantages in that its much slower speed is something we can more easily appreciate, and that bats use effectively in navigation and catching prey with echo location. So we have some reasonably familiar intuitions that can help us understand how measurements of events can be systematized and then compared between two observers.

We will see that this approach allows us to seamlessly connect with the algebra and associated geometry of the red and green complex numbers that we discussed in previous videos.

This talk is meant to be very elementary: no equations will be derived or even posited, but I will give you enough insight to derive the Lorentz transformations that connect the time / space measurements of two observers moving at uniform speeds with respect to each other. It is important to realize that the time and space measurements here are defined, and not assumed to have an a priori meaning.

This video is part of the Classical to Quantum series at Wild Egg maths, and the entire series, along with several others, is available to Members of the channel at

• Classical to Quantum

Your support is very much appreciated!

If you are keen on learning more about the details of this approach to Special Relativity, please check out my 2014 video called Bats, Echolocation and a Newtonian view to Special Relativity at

• Bats, echolocation, and a Newtonian v...

However I will be going over the content of that video, and considerably more, also in this series.

0:01

Hello everyone thanks for joining me on this adventure we're looking at special relativity

0:06

and today a very important lecture I want to try to explain a very Elementary approach to thinking

0:11

about special relativity that doesn't involve any equations but I give you enough structural

0:17

intuitive understanding of what's going on so that you can go off on your own and just with high

0:23

school maybe first year math derive essentially the equations for special relativity all right so

0:30

we're going to simplify things dramatically by stripping away all unnecessary and complicated

0:37

things and just thinking about physics in a very elementary way just a one-dimensional space devoid

0:45

of everything pretty well except for just a one or two observers and maybe a few things

0:51

to observe and we're going to step away from the conundrums of the meanings of fundamental

0:59

terms terms like mass and force and energy and even things like time and and and space so we're

1:08

going to do that by adopting a very simple kind of story where there's only sound okay so this is

1:15

like a one-dimensional tunnel in which we have bats and the bats are just listening to things

1:23

they they emit little clicks the clicks bounce off something and then they receive the click and from

1:28

that information alone they make some deduction as to what was out there so that's all what we're

1:33

going to have so we not adopting the idea of an inertial frame importantly okay so that's a very

1:41

crucial idea in Newtonian mechanics the idea that there's a rigid system of um of of coordinates

1:50

for space and also for time which is also adopted by Einstein to a great extent right this an idea

1:56

of an inertial frame some ongoing sequence of of equally spaced rods let's say with clocks attached

2:04

okay I don't think that's really representative of the way we are in our universe we're really

2:11

more like Point observers here on planet Earth and the idea of some massive inertial frame extending

2:17

Beyond uh you know our solar system or our galaxy is fanciful to me and it imposes already

2:24

assumptions on the story that we don't really need so we're going to stick with just the these

2:30

Point observers okay that's a a really important point and we're going to let go of Einstein's uh

2:39

dependence on on light okay Einstein actually did recognize that there were Alternatives to using

2:47

light as a framework in fact let me quickly quote here this is the meaning of special relativity a

2:55

book he wrote in 1922 um and so he says "At some point the theory of relativity is often criticized

3:01

for giving without justification a central theoretical role to the propagation of light

3:06

in that it founds the concept of time upon the law of propagation of light the situation however

3:12

is somewhat as follows in order to give physical significance to the concept of time processes of

3:18

some kind are required which enable relations to be established between different places it

3:24

is immaterial what kind of processes one chooses for such a definition of time it is advantageous

3:31

however for the theory to choose only those processes concerning which we know something

3:35

certain this holds for the propagation of light in vacuo in a higher degree than for any other

3:41

process which can be considered thanks to the investigations of Maxwell and Lorentz". So he's

3:46

at least opening the door here that we could be talking about special relativity in the context

3:51

of other propagation methods and that's what we're going to be doing here in this lecture

3:58

that has uh several advantages in that it makes it more obvious that that this additional postulate

4:05

of Einstein or sometimes it's maybe considered a deduction that nothing can go faster than the

4:10

speed of light is really not logically justified okay that's not logically Justified it is not an

4:16

intrinsic part of special relativity even though it's often presented as such and in fact the even

4:24

the second postulate which is that you know the speed of light is constant in any reference for

4:30

in any inertial reference frame that also is questionable actually okay because well the

4:36

meaning of that is first of all in question what do we mean by the velocity of anything

4:40

in the Galilean system velocity doesn't have any meaning and even if you do ascribe some

4:45

local meaning well you still have the problem of comparing relative units in systems or inertial

4:52

frames which are moving at speeds with respect to each other so that postulate also to me doesn't

4:59

really play an essential role in the story and that's going to be the same here and what about

5:05

Einstein's first postulate well that's really just Galileo that's really just Galileo's invariance

5:10

principle so um okay the reality is that this the equations of special relativity I really

5:18

follow really just from from Galileo with some s of simple-minded arguments that I'm going to show

5:24

you today um having to do with one-dimensional physics all right so let's start with bat A:

5:29

here's a bat A and the bat A is interested in making some observations and deductions about

5:35

the world so maybe there's some particles over here or maybe there's a second bat B over here

5:40

which the bat A is interested in observing so the bat A does not have a prior notion of a clock

5:47

so the first order of business is as Einstein suggests to establish a clock and a rudimentary

5:53

way of doing this is to use a mirror which can reflect the sound okay so let's say that a has

6:00

a fixed mirror at some close location to him and he's clicking click sound comes back and then as

6:08

soon as the sound comes back he makes another click and so the sound bounces back and forth

6:12

establishing a rudimentary kind of clock so then to ascribe some numbers to this situation the the

6:20

bat could say all right let's say if we start at Time Zero then the receiving click that the next

6:27

one that we get let's call that time t equal 2 if we do that then we can posit that at time tal

6:34

1 it got to the mirror now actually we don't know you know when it actually got to the to the mirror

6:41

because we're not at the mirror we're over here but nevertheless the bat can Define that the time

6:47

at which the reflection took place at the mirror was at Time t equals 1 and that's consistent

6:53

with say defining this unit between the bat and the mirror to be the unit of of of length or or

6:59

better displacement uh in the spatial Direction okay so the mirror and the bouncing of um sound

7:07

back and forth from the bat to the mirror that little setup is sufficient to do introduce the

7:12

idea of time indeed to Define time observationally okay and also to to establish at least a unit of

7:23

displacement in in this uh in this one-dimensional direction so that's that's already a pretty good

7:28

start so now what what's going to happen if we want to observe some maybe a mosquito buzzing

7:36

around over there somewhere or maybe it's a ball bearing you know moving towards another ball

7:40

bearing there's going to be a collision here soon and we want to to make some observations having to

7:44

do with some Dynamics you know but before we do that we should have a way of figuring out

7:49

you know where and when things are occurring so what's the bat going to do the bat is going to

7:54

send out sound waves say send out a signal at T1 to the right so let's agree that at T1 the signal

8:03

is emitted to the right it bounces off whatever and then comes back say a T2 moving to the left

8:10

coming back okay so that's all that the bat has is these two these two measurements T1 and T2

8:17

when the signal to the right was sent off and when it comes back what can the bat deduce from

8:23

that well in parallel to what it did over here it can first of all deduce or ascribe a Time Value to

8:33

the to the event of the reflection of the sound of the object namely the average of T1 and T2;

8:43

T1 + T2 over 2 and that's exactly the argument that Einstein used in his 1905 paper he used it in

8:51

the context of calibrating clocks over here with the clock over here but in fact we can use that

8:58

to dispense with the clocks over there we don't need the clocks over there right if we have this

9:03

very clear way of finding time values for distant events we only need the time value here the clock

9:11

over here that's all what we really need okay so already this idea of an inertial frame having

9:17

clocks uh you know scattered all around it it's Overkill you don't need those clocks you just need

9:22

one clock because you are able to send signals and receive them okay what about the position of the

9:31

event where the sound hits the mosquito or ball bearing and comes back where did that happened

9:38

well we started at time T1 the signal came back at time T2 that's a total time of T2 minus T1 in

9:50

analogy with what we did over here remember we sent a signal at time t equals 0 and received

9:55

that time t equals 2 the difference is 2 minus 0 and we divided that by two to get the position to

10:03

be equals one well we'll do exactly that we'll say that okay this um this thing is going to uh

10:12

Define a position coordinate say x of (T2 minus T1) / 2 if you like that's a definition that's

10:22

the definition of the x coordinate as far as bat a is concerned of an event of ball bearing being

10:32

in a certain place and that place is (T2 minus T1) / 2 so to summarize okay the heart of the matter

10:42

this is the heart of the whole story is that if we have a signal going to the right at T1 coming

10:50

back at T2 then the time coordinate of the event that's being registered is the average T1 plus t

10:59

T2 / 2 and the x coordinate position is (minus T1 plus T2) / 2 and I write it that way so that it

11:11

immediately syncs with what I told you in previous lecture on green complex numbers it's exactly the

11:20

transformation between the 1K basis vectors in green geometry and the uh n1 n2 basis vectors

11:29

that we were talking about that I said was sort of central okay so it's exactly the same story so now

11:35

we have an unambiguous way of going from T1 and T2 to T and X great so now what can bat A do well bat

11:45

A can talk about the velocity now it's velocity with respect to him okay so for example he could

11:52

look at this particle C1 and make two different measurements at different times and record time

12:00

and x coordinates for this thing at two different places and then could compute in the regular

12:06

way what the velocity of that thing is he could also apply the same thing say to bat B and bat B

12:16

perhaps is moving with respect to bat A okay bat B uh maybe is moving in this direction maybe so from

12:25

bat A's point of view perhaps bat B is moving in this direction and so then bat A can you know know

12:33

where bat B is at at at a certain amount of time let's say that bat B is moving uniformly a could

12:39

decide oh you're moving uniformly with constant velocity okay so bat A is now in possession of

12:47

having the tools to for example investigate some Physics some Dynamics so for example he could look

12:53

at these two particles C1 and C2 and could figure out what their initial velocities are they seem

12:59

to be approaching each other and now they're going to now they Collide and then they come back apart

13:04

but how they actually do that suggests that maybe one of them is bigger than the other in some sense

13:11

okay so just these time position measurements are going to allow bat a to make some deductions about

13:22

about collisions and maybe make some deductions depending on on his his theory of relative masses

13:30

of say these two particles okay and now let's turn from bat A to bat B here is bat B with a similar

13:42

setup bat B also has a mirror and has exactly the same idea of using sound bouncing off between it

13:54

and its mirror to establish a clock, and then bat B can make exactly the same computations

14:02

that bat A is making and bat B can look at bat A and notice that oh bat A is moving in that

14:11

direction okay so as far as bat B is concerned bat A is not stationary it's moving in that direction

14:17

and bat B could tell bat A how fast it's moving in the same way that bat A could formerly have

14:24

told bat B how fast she is moving but now now the interesting story happens we have two different

14:33

ways of describing some third independent event say you know what's happening to this particle

14:39

C1 when these two things Collide where do they Collide when do they Collide well A

14:50

and B are going to have different answers to that question but we can figure out the

14:57

relationship between the time and position coordinates of events of a and the time and

15:05

position coordinates of B when we do that I claim that we get the Lorentz Transformations

15:12

the Lorentz Transformations are nothing but an elementary first year university it's probably

15:20

actually high school stuff working out what are the relationships between the

15:26

time space coordinates of bat A and the time space coordinates of bat B of some

15:31

event what about those fancy things of special relativity like you know length

15:40

contraction time dilation the uh relativistic additional velocities those are all immediate

15:49

consequences of the Lorentz transformation and they apply in this situation just as much

15:54

as they do in the Einstein situation with light it's ultimately a Galilean phenomena

16:01

which is a consequence of the fact that we are assuming that these two these two

16:07

bats have a symmetrical position neither of them has a prior or privileged position with

16:13

respect to the world in which they are in they are both equally valid ways of looking at the

16:19

world and then when we get to Dynamics when we start talking about interactions of particles

16:29

momenta energies masses forces perhaps then again the same kind of reasoning leads us to

16:42

relativistic Notions of what usually called four momenta vectors and ultimately relations

16:49

like E equals mc2 so I claim that E=mc2 is not something that is just uh a related to light:

17:01

it's a phenomena that's going to happen in this very general situation so there's an E=mc² here

17:06

but the c in this case it's which is the speed of sound is is one at least it's one for for this

17:12

bat so I've given you enough information now that if you are um you know reasonably proficient um

17:23

undergraduate you can sit down and you can work this all out really all what you need

17:28

is first year maybe a little bit of first year linear algebra and if you've watched my uh red

17:33

and green complex number stories well the thing will be uh dramatically simplified because the

17:39

equations are almost right there okay um so it's it's not very hard dealing with the Dynamics is

17:46

a little bit trickier but uh but if you think about that um without prior um conceptions you

17:52

very well may be able to discover all of that stuff yourself so this is a really uh great

17:58

opportunity to to come to grips with essential physics like you know we're really getting at the

18:04

heart of of Elementary physics here without prior suppositions letting go of Einstein's

18:12

postulates except for the really important one Galileo's invariance principle Galileo

18:19

is really the hero here! Einstein was also very important but ultimately for special relativity

18:27

the critical understanding was already contained in Galileo and it's just that people didn't take

18:37

that seriously enough because of the dramatic success of Newtonian physics which is pretty

18:45

obviously at odds with the Galilean principle but it's so successful that people said okay well

18:55

let's let's forget about the Galilean principle okay and Einstein was the first person who had the

19:00

the Insight maybe Poincare and Lorentz also to a certain extent uh these these guys had the the the

19:07

insight to realize that it was really the Galilean principle which was at the heart of the matter.