Classical to Quantum

Classical to Quantum | Einstein's Simultaneity (1905 paper Section 1.1) | Wild Egg Maths

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We have a close look at Einstein's ground breaking 1905 paper: "On the Electrodynamics of Moving Bodies" in detail in this series. In this video we read through Part I: Kinematical Part, the first section 1. Definition of Simultaneity, to try to get some initial understanding of the important new ideas that Einstein is here introducing.

We will be shortly advocating a different interpretation of Special Relativity, which in my view is more mathematically clear and elegant. It also touches base with earlier thinking of Poincare. But before we get to that, we must first try to understand Einstein's original thinking in this direction.

This is the fourth video of the series "Classical to Quantum" available to Members of the Wild Egg Maths channel and also to Patreon supporters. Here is the Playlist in the Wild Egg Maths channel for this series:

• Classical to Quantum

In this series, which is part of the Explore Research Level Maths program, we investigate how the relations between classical and quantum physics manifest themselves in pure mathematics.

0:01

hello everybody thanks for joining me on

0:03

this adventure where we are looking at

0:06

physics ultimately we're aiming towards

0:10

quantum mechanics

0:11

but it's important for us to get our

0:13

heads around some important issues

0:16

having to do with special relativity

0:18

introduced of course by Albert Einstein

0:20

in 1905 and I'm interested in looking

0:25

very carefully at this foundational

0:27

paper of Einstein I mean really

0:30

carefully because it gives us a chance

0:33

not only to connect with this incredibly

0:36

original and important work

0:39

but also it's a chance for us to

0:43

tackle a lot of foundational issues in

0:47

physics which Einstein is sort of laying

0:50

before us he's saying look we may have

0:52

to think a little bit more originally we

0:55

may have to think in a new way about

0:57

some

0:59

very familiar Concepts which are

1:03

open to let's say interpretation or

1:07

alternate definition

1:09

so in my last video we talked about this

1:12

introduction here

1:14

and today we're going to look at

1:16

section one the kinematical part and

1:20

this subsection one definition of

1:23

simultaneity so it's not that long

1:26

you'll see you know just quickly zip

1:27

over here

1:29

it's only really two pages so we're

1:31

gonna just read this two-page thing and

1:34

I'm

1:36

claiming that there's a lot of really

1:38

important issues here and in my next

1:40

video I'm going to talk a lot about

1:42

those important issues and try to

1:44

enunciate them for you but today what I

1:47

want to do rather is to read it in a

1:50

very slow and careful fashion to give

1:53

you a chance to think about

1:56

what it means to you okay I want you to

1:59

react to this I want you to ask yourself

2:01

okay am I understanding what Einstein is

2:04

saying here

2:06

um I convinced

2:09

of his explanations or his derivations

2:12

or his Concepts

2:15

okay so I don't want to be judgmental in

2:17

this video in my next video I am going

2:19

to be looking at this and this will be

2:21

the stepping stone for a whole new way

2:24

of thinking ultimately about really

2:26

essential core aspects of modern physics

2:29

from a mathematical point of view

2:31

because I'm a mathematician not a

2:32

physicist okay so shall we start

2:36

definition of simultaneity

2:39

let us take a system of coordinates in

2:42

which the equations of Newtonian

2:44

mechanics hold good

2:46

there's a little

2:48

footnote here I.E to the first

2:50

approximation

2:53

in order to render our presentation more

2:56

precise and to distinguish this system

2:58

of coordinates verbally from others

3:00

which will be introduced Hereafter we

3:02

call it the stationary system

3:05

if a material point is at rest

3:07

relatively to this system of coordinates

3:10

its position can be defined relatively

3:13

there too by the employment of rigid

3:16

standards of measurement and the methods

3:19

of euclidean geometry and can be

3:21

expressed in Cartesian coordinates

3:24

if we wish to describe the motion of a

3:27

material point we give the values of its

3:30

coordinates as functions of the time

3:33

now we must bear carefully in mind that

3:36

a mathematical description of this kind

3:38

has no physical meaning unless we are

3:41

quite clear as to what we understand by

3:44

time

3:46

we have to take into account that all

3:48

our judgments in which time plays a part

3:51

are always judgments of simultaneous

3:54

events

3:55

if for instance I say that train arrives

3:58

here at seven o'clock I mean something

4:01

like this

4:02

the pointing of the small hand of my

4:04

watch to seven and the arrival of the

4:07

train are simultaneous events

4:11

just another footnote here and uh the

4:14

footnote says we shall not here discuss

4:15

the inexactitude which lurks in the

4:18

concept of simultaneity of two events at

4:20

approximately the same place which can

4:22

only be removed by an abstraction

4:31

it might appear possible to overcome all

4:33

the difficulties attending the

4:34

definition of time by substituting the

4:37

position of the small hand of my watch

4:39

for time and in fact such a definition

4:42

is satisfactory when we are concerned

4:44

with defining a Time exclusively for the

4:47

place where the watch is located

4:50

but it is no longer satisfactory when we

4:53

have to connect in Time series of events

4:57

occurring at different places

5:00

or what comes to the same thing to

5:02

evaluate the times of events occurring

5:05

at places remote from The Watch

5:08

we might of course content ourselves

5:10

with time values determined by an

5:12

observer stationed together with the

5:15

watch at the origin of the coordinates

5:17

and coordinating the corresponding

5:19

positions of the hands with light

5:22

signals given out by every event to be

5:26

timed and reaching him through empty

5:28

space

5:30

but this coordination has a disadvantage

5:33

that it is not independent of the

5:35

standpoint of the Observer with the

5:37

watch or clock as we know from

5:40

experience

5:45

we arrive at a much more practical

5:47

determination along the following line

5:50

of thought

5:52

if at the point a of space there is a

5:55

clock

5:56

an observer at a can determine the time

5:59

values of events in the immediate

6:01

proximity of a by finding the positions

6:03

of the hands which are simultaneous with

6:06

these events

6:07

if there is at the point B of space

6:10

another clock in all respects resembling

6:13

the one at a it is possible for an

6:15

observer at B to determine the time

6:18

values of events in the immediate

6:19

neighborhood of B

6:21

but it is not possible without further

6:24

assumption to compare in respect of time

6:26

an event at a with an event at B

6:31

we have so far defined only an a time

6:34

and a b time we have not defined a

6:38

common time for a and b

6:41

for the latter cannot be defined at all

6:43

unless we establish by definition

6:46

that the time time and quotes required

6:51

by light to travel from A to B equals

6:54

the time it requires to travel from B to

6:58

a

7:00

let array of light start at the a Time T

7:04

sub a from a towards B

7:07

letter at the B time T sub B be

7:11

reflected at B in the direction of a and

7:14

arrive again at a at the a Time T sub a

7:19

prime

7:22

in accordance with definition the two

7:24

clocks synchronize if

7:28

T B minus t a

7:32

equals

7:33

ta Prime

7:35

minus TB

7:38

so here we have our first equation of

7:41

the paper

7:42

and let me just make a little side note

7:45

okay that this definition of

7:48

synchronization that he's introducing

7:50

could also be stated in another way

7:53

just by rearranging that we could

7:55

restate it as saying that t sub B the

7:59

time coordinate

8:01

on the B side equals the average of T

8:05

sub a and t sub a prime in other words T

8:08

sub a plus T sub a prime over two that

8:12

would be completely equivalent

8:14

mathematical way of stating this

8:16

synchronization

8:18

so he's defining what it means for these

8:21

two clocks one at a one at B to be

8:24

synchronized

8:26

now we assume that this definition of

8:29

synchronism is free from contradictions

8:32

and possible for any number of points

8:35

and that the following relations are

8:37

universally valid

8:40

so he's now assuming some things

8:43

that if the clock at B synchronizes with

8:45

the clock at a then the clock at a

8:48

synchronizes with the clock at B

8:52

the mathematical point of view is the

8:53

kind of a symmetry

8:55

of this of this relation between you

8:58

know

9:00

synchronization of a with B

9:03

second if the clock at a synchronizes

9:05

with the clock at B and also with the

9:07

clock at C

9:09

then the clocks at B and C also

9:12

synchronize with each other

9:16

then he says so thus with the help of

9:18

certain imaginary physical experiments

9:21

we have settled what is to be understood

9:24

by synchronous stationary clocks located

9:28

at different places and have evidently

9:30

obtained a definition of simultaneous or

9:33

synchronous and of time

9:36

the time of an event is that which is

9:40

given simultaneously

9:42

with the event by a stationary clock

9:44

located at the place of the event

9:47

this clock being synchronous and indeed

9:50

and indeed synchronous for all time

9:52

determinations with a specified

9:54

stationary clock

9:57

in agreement with experience we further

10:01

assume the quantity

10:03

T sub a b over T Prime minus t a so this

10:09

this difference between the the value at

10:13

a of the emission of the light beam and

10:17

it's receiving being received at a so

10:20

that's T sub a and that's T sub a prime

10:22

we take two a b what does it mean by a b

10:26

he means

10:27

the distance between a and b

10:31

okay so if we

10:34

Take 2 times a b over t a prime a minus

10:38

t a

10:40

call that c

10:42

in agreement with experience we further

10:43

assume that this quantity C to be a

10:46

universal constant the velocity of light

10:49

in empty space

10:51

it is essential to have time defined by

10:54

means of stationary clocks in the

10:56

stationary system and the time now

10:58

defined being appropriate to the

10:59

stationary system we call it the time of

11:02

the stationary system

11:07

okay so this is very important small

11:09

section

11:10

where he talks about what it means to

11:13

synchronize clocks

11:15

and thereby give us

11:20

a notion of how to compare time at one

11:24

place in space with time at another

11:26

place in space assuming we have these

11:29

synchronized clocks and then further

11:31

he's sort of briefly touches upon this

11:35

quantity C which involves a ratio of a

11:39

distance over time roughly

11:42

um

11:44

even though actually to be honest right

11:46

he hasn't quite defined what he means by

11:49

the distance a b so we can see at the

11:52

bottom here on the relativity of length

11:53

of time so perhaps he will clarify that

11:55

in this next section

11:58

and he's also introducing

12:01

this crucial assumption that this

12:04

constant C is a universal constant for

12:06

velocity of light in empty space

12:09

okay so there's much to be said about

12:11

this and it's so important to have a

12:14

detailed

12:15

careful discussion of the issues that

12:19

are being raised by this by this small

12:22

section of just two pages

12:25

and in my view almost all courses in

12:29

physics do not pay sufficient attention

12:33

to exactly this stage of setting up

12:37

special relativity where we we roll up

12:40

our sleeves and say okay let's examine

12:42

carefully what exactly we are talking

12:43

about

12:45

so in that spirit I now invite you even

12:47

if you're a professional physicist even

12:49

if you're a Nobel prize-winning

12:51

physicist

12:52

I invite you to think very clearly about

12:55

this section and ask yourself

12:59

does this

13:00

make sense

13:02

does it work is it clear

13:05

do we understand what Einstein is trying

13:08

to say here

13:12

what is it that he's actually trying to

13:15

say what

13:16

what is actually going on here

13:19

I think this is an important discussion

13:20

and we're going to

13:24

do exactly that in my next video but I

13:27

do hope that you're going to spend some

13:28

significant time thinking about it okay

13:30

thinking about exactly the issues raised

13:33

in this crucial section of this most

13:36

important paper

13:38

in modern physics