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General Relativity and Quantum Cosmology

History and Philosophy of Physics

Research talk

Researchers

Standard

general relativity

00:00:00

[Music]

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I want to give an introduction to the

00:00:24

way Einstein discovered

00:00:27

generativity let me mention that there

00:00:30

series of books the last one of them by

00:00:33

joggen Ren and his collaborators here M

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Jansen uh which is from an

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authoritative historians of science why

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I will speak as a scientist to explain

00:00:49

the way I see Einstein discover

00:00:51

generativity so

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first let me remind you of gravitation

00:00:57

before

00:00:58

Einstein so uh it started essentially

00:01:02

with Galileo in 1634 we discovered the

00:01:07

law of freeall all bodies fall with an

00:01:10

constant acceleration at the surface of

00:01:12

the Earth small G independently of the

00:01:15

mass and composition so 10 kg fall like

00:01:18

1 kilog but 1 kilog of lead fall like 1

00:01:22

kilog of anything else uh then Newton

00:01:27

came along then General the law of

00:01:30

Dynamics with the acceleration saying

00:01:33

that um when there is a force acting on

00:01:36

a on a body uh there exists a constant

00:01:40

which measures the inertia of the body

00:01:43

the inertial mass Mi and then this mass

00:01:49

inertial mass times the acceleration is

00:01:51

equal to the force but the gravitational

00:01:53

force is special in the sense that not

00:01:55

only it is in 1 / r s but that it is

00:01:59

proportional to the same mass which

00:02:02

appears as inertia so like Newton said

00:02:07

the gravitational force should be in

00:02:09

principle proportional to some he did

00:02:11

not use this gravitational mass and in

00:02:14

principle the acceleration of falling

00:02:16

bodies should contain the ratio of

00:02:18

gravitational Mass the weight to the

00:02:21

inertial mass but the fact that all

00:02:23

bodies fall in the same way this

00:02:25

universality of freeall means that mass

00:02:29

is proportional to weight we will see to

00:02:32

with accuracy now after Newton

00:02:35

laas then transform the gravitational

00:02:38

force in the gradient of a scalar the

00:02:41

gravitational potential fi when it is

00:02:43

negative or U when it is positive which

00:02:46

is the sum of the 1 / R times the mass

00:02:48

of various bodies G being Newtons

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constant and P wrote down the field

00:02:53

equation that satisfies the that the

00:02:57

scalar potential F the gravitational

00:02:59

potential satisfies just to have this in

00:03:02

mind because that was starting point

00:03:05

from Einstein so the pral equation is

00:03:08

this you have a laas operator acting on

00:03:12

F equal the mass density which means

00:03:15

that gravity propagates instantaneously

00:03:17

at the time although even Newton did not

00:03:22

believe that gravity could propagate

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really instantaneously the first one to

00:03:26

put quantitative limit on the speed of

00:03:30

propagation of gravity was laas laas

00:03:33

said okay if gravity propagates to

00:03:35

finite velocity then you expect to

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lowest order that the gravitational

00:03:40

force uh acting on body a from body B

00:03:43

will not be directed towards the

00:03:45

position of body B now but at a position

00:03:48

that body B had had at the time where he

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sent some wave and then he used this to

00:03:53

put limits on the on the velocity of

00:03:56

propagation of gravity and found that

00:03:59

propag

00:04:00

velocity propagation of gravity had to

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be 7 million times larger than the

00:04:05

velocity of light which was a very

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strong limit separately from this uh

00:04:12

when Galileo and Newton found that all

00:04:15

bodies fall in the same way it was very

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surprising to them and Newton was the

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first one to try to make precise

00:04:22

experiment to check whether it was

00:04:25

really true and to what level he found

00:04:27

he did pendulum experiments at the

00:04:30

one part in a thousand level

00:04:33

laas using Celestial mechanics of the

00:04:35

solar system could put limits on the

00:04:38

ratio of gravitational to inertial mass

00:04:40

at the 10us 7 Level and at the end of

00:04:43

the 19th century the baron lauron

00:04:47

fos got limits which were better than

00:04:51

10us 8 reaching 10 to Theus 9 and

00:04:55

Einstein was aware of this uh after

00:04:59

Newton people asked the question was the

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one / R square law of Newton exact in

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the solar system and um it seemed to

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explain everything we knew in gravity in

00:05:11

the solar system except that in 1845 ler

00:05:15

actually found that the planet Which is

00:05:17

closest to the Sun uh Mercury actually

00:05:23

uh I mean is following an ellipse and

00:05:26

the semi major axis of this ellipse is

00:05:29

is moving due to the perturbation of all

00:05:32

the planets this is a large effects of

00:05:35

the order of 500 seconds of AR per

00:05:37

Century uh but there was a residual that

00:05:40

he could not explain of the order of 38

00:05:43

segs of AR per Century that nobody

00:05:46

explained for a long time uh the first

00:05:49

attempts to explain this was due to

00:05:53

Laurence who had a a kind of

00:05:56

relativistic model of gravity in the

00:05:58

sense that Lauren said that maybe

00:06:01

positive and electric charges in Maxwell

00:06:04

Theory are not exactly opposite to each

00:06:06

other which means there is a residual

00:06:09

one / R square force due to

00:06:11

electromagnetic

00:06:13

interactions but he use max equations

00:06:16

and which

00:06:18

contain v Square / C Square FX I mean

00:06:21

one C Square FX to conclude that this

00:06:25

implied on the motion of mercury that

00:06:27

there was an advance of the per alion

00:06:30

but he got something which was not which

00:06:33

was substantially smaller than 38

00:06:36

seconds of Arc Point car was the first

00:06:39

one to introduce the idea that maybe you

00:06:43

need to describe U SpaceTime although he

00:06:47

was using only A nulean Spacetime with

00:06:50

X4 equal ICT so but they could be action

00:06:54

at a distance lawence in variant Force

00:06:57

slow but he had no way of selecting a

00:07:00

special force law so he wrote General

00:07:01

Force law and in 1906 in lectures he

00:07:06

gave at the subone in Paris lectures

00:07:08

that have been published only in 1956 so

00:07:11

nobody knew about this he uh he computed

00:07:14

some effects on the motion of mercury

00:07:17

and um and he did not get 38 seconds of

00:07:20

AR he could have noticed that he was

00:07:22

getting you know 1/4 or one third of it

00:07:25

something not unrelated but he did not

00:07:30

uh so anyway all these attempts before

00:07:33

Einstein uh did not explain uh the

00:07:37

motion of mercury although the motion of

00:07:39

mercury was not like a big Scandal there

00:07:42

was not everybody saying yes Newtons

00:07:46

must be wrong because Mercury is wrong

00:07:48

and many people would say okay it's a

00:07:49

very small effect among many other

00:07:51

things are you sure you got all your

00:07:53

data right you know it was not the what

00:07:56

we would call today a five Sigma

00:07:58

discrepan in in physics anyway in

00:08:03

1905 after previous work evidently by

00:08:06

lawence by prank I won't enter into this

00:08:10

uh Einstein U introduces what we call

00:08:13

today special relativity because he was

00:08:16

the first one not to derive those

00:08:19

transformation laws between the

00:08:20

measurement of time and space between

00:08:22

two relatively moving frames which had

00:08:25

been written down by Lawrence before and

00:08:28

by point

00:08:30

uh but because he was the first one to

00:08:32

say that uh the variable which

00:08:36

represents this what Lawrence called a

00:08:38

fictitious mathematical time T Prime in

00:08:41

the moving frame was not a mathematical

00:08:45

varable but was as Einstein put it time

00:08:49

pure and simple which means really the

00:08:51

time you live if you are there okay and

00:08:54

the time measured by clocks and in this

00:08:56

paper of 1905 it was the first one to

00:08:59

say that if you have two clocks A and B

00:09:02

and they are uh they are synchronized

00:09:05

and they are made in the same way and

00:09:07

then one of them goes traveling with

00:09:09

some velocity and then goes back to its

00:09:13

broader clock when you compare the two

00:09:15

readings of the clock they differ it was

00:09:18

the first one to say that time really is

00:09:20

affected by motion by this quantity that

00:09:23

this is the proper time you have to

00:09:25

integrate this on the polygonal line as

00:09:27

it says and so uh this really changed uh

00:09:32

physics and although at the time the

00:09:35

only one to really understand that this

00:09:36

was revolutionary was Max Blan actually

00:09:40

Max PL had to tell all his students

00:09:42

including summerell saying read this

00:09:45

paper of Einstein so they read the paper

00:09:47

okay we saw equation but we know these

00:09:49

equations nothing new okay you did not

00:09:52

understand go back and understand what

00:09:54

is new and revolutionary and finally

00:09:57

they understood or at least they tolds

00:10:00

PL they did understand uh anyway a few

00:10:03

months later in September 1905 as an

00:10:07

addendum to his paper

00:10:10

Einstein uh writes this paper does the

00:10:14

inertia which means the inertial mass of

00:10:17

a body depend upon its energy content he

00:10:22

makes um a thought experiment where a

00:10:24

body is losing energy in the form of

00:10:27

radiation elatic radiation and then

00:10:30

argues that the inertial mass of this

00:10:32

body after having lost some energy to

00:10:35

radiation is smaller than before by the

00:10:38

quantity of the Lost energy divided by

00:10:40

c² okay and this will play a very

00:10:44

important role in the story of gravity

00:10:46

because as we said before inertial mass

00:10:49

is not only inertia but will be also the

00:10:51

source of gravity the gravitational mass

00:10:54

and then this formula is saying that the

00:10:56

source of gravity must be energy G

00:11:02

okay now uh from 1907 to 19 from 1905 to

00:11:09

1907 um many people developed not so

00:11:12

many actually but special activity was

00:11:15

developed in in several directions and

00:11:18

then the Editor in Chief of this book YB

00:11:22

there Journal

00:11:24

radioactivity electronic ask uh Einstein

00:11:29

to write a review paper on the most

00:11:31

important developments of spal

00:11:33

Relativity first re explaining what is

00:11:35

the basic of special relativity and the

00:11:38

development it was not called special

00:11:40

relativity at the time it was called

00:11:42

just

00:11:43

relativity uh so Einstein wrote um it

00:11:48

took him time he was still working at

00:11:50

the uh uh patent office in burn at the

00:11:54

time and as he said and this looks

00:11:57

believable one day while he was in his

00:12:01

office um or maybe at home he he read

00:12:04

the newspaper of the city of BN the city

00:12:06

of B is particular because it has many

00:12:09

roofs like that uh he read that a worker

00:12:14

who was doing something on the on the

00:12:17

roofs uh

00:12:19

fell and then uh so many people read

00:12:22

this and there nobody concluded anything

00:12:24

about physics but then he imagine this

00:12:27

worker falling with its tools tools of

00:12:30

its trades and then he realized suddenly

00:12:33

that if you fall with your objects with

00:12:36

you as all the bodies all the objects

00:12:38

fall exactly with the same acceleration

00:12:41

it is as if everybody was weightlessness

00:12:44

you know that you the gravity had

00:12:47

disappeared because the thing fall

00:12:49

exactly at the the same rate and then

00:12:52

you realize that probably this meant

00:12:54

that one needed to generalize the

00:12:57

principle of relativity principle of

00:12:59

Relativity was limited to changes of

00:13:02

frame between uniformly uh moving frames

00:13:06

with Rel constant velocities in constant

00:13:08

directions okay and he added while he

00:13:11

was writing this review paper he thought

00:13:13

about it and added the section in his

00:13:15

review paper called accelerated

00:13:18

reference system and gravitational field

00:13:21

in which he expresses in simple words

00:13:25

this idea that initially called the

00:13:27

hypothesis of equivalence and later the

00:13:30

principle of equivalence which has

00:13:32

various version but essentially the IDE

00:13:35

is the

00:13:36

following

00:13:37

um here you have people doing

00:13:41

experiments uh in a body uh in a frame

00:13:46

uh on let's say on the earth submitted

00:13:48

to the gravitational field of the earth

00:13:50

and then bodies fall with a constant

00:13:52

acceleration and things like that and

00:13:55

here you say you are far away in space

00:13:58

and you are in absence of real

00:14:00

gravitational field but you impress an

00:14:04

acceleration by some like in a rocket uh

00:14:09

if the gravitational field is towards

00:14:10

the the the bottom here it has to be

00:14:12

towards the top here and then he said

00:14:15

that once you are in this accelerated

00:14:18

frame moving with this constant

00:14:19

acceleration what was called at the time

00:14:22

inertial forces okay where as real and

00:14:25

we are completely mimicking what could

00:14:28

be a rad gravitational field and he used

00:14:31

this as a tool to get new information

00:14:35

about uh gravit real gravitational field

00:14:38

in particular he deduced immediately two

00:14:41

physical

00:14:42

consequences uh one uh which is that if

00:14:47

um if you are in an accelerated frame

00:14:49

and you compare two clocks because the

00:14:52

two clocks move but move in an

00:14:54

accelerated way there will be a variable

00:14:57

uh Doppler red shift between the two and

00:15:01

then he computed that this Doppler shift

00:15:03

just between moving clocks with some

00:15:05

varying

00:15:06

velocity uh if it is equivalent to

00:15:09

something in a gravitational field it

00:15:11

predicts that the proper time of clocks

00:15:13

must slow down when they are in a

00:15:16

gravitational potential F by this

00:15:18

formula approximate formula or that the

00:15:22

the red shift of like spectral lines has

00:15:26

to be red shifted uh by this where f is

00:15:29

negative so that if you see the red

00:15:31

shift of something in a gravitational

00:15:33

field you should see more red than uh

00:15:36

blue but he also uh had this following

00:15:40

thought experiment that if you are um in

00:15:43

an accelerated frame a ray of light

00:15:46

entering by a window uh because this

00:15:49

thing is going up the ray of light will

00:15:51

look to be bent with respect to this

00:15:54

accelerated frame and then he said

00:15:55

immediately ah but then it means that a

00:15:58

rare of light in a gravitational field

00:16:00

has to be bent because as it

00:16:03

computed effectively the velocity of

00:16:06

light in a gravitational field is slowed

00:16:09

down velocity of light is not the

00:16:11

constant velocity of light in flat space

00:16:13

time but is smaller by the quantity fi

00:16:17

the gravitational potential which is

00:16:19

negative f is minus gm/ r uh and then

00:16:24

that was in

00:16:25

1907 and uh he thought about this for

00:16:29

several years he said I did not know

00:16:31

what to do next but he kept thinking uh

00:16:35

how to make progress towards

00:16:37

understanding Gravity by using this

00:16:39

equivalence between accelerated frames

00:16:42

mimicking a real gravitational field and

00:16:44

a real gravitational field and in 1911

00:16:47

what he understood that if the velocity

00:16:50

of light is really slowed down near

00:16:52

massive bodies then it implies as it

00:16:56

could deduce from the high r or we as we

00:16:59

say in French

00:17:01

principle uh that uh that light will be

00:17:06

deflected by the sun and it could

00:17:09

compute the light deflection by the sun

00:17:12

if a light Ray is grazing the Sun and

00:17:14

then he got this Formula 2 gm/ c² R

00:17:17

where R is the radius of the sun which

00:17:19

gives

00:17:20

0.83 seconds of Arc and then he pushed

00:17:24

astronomers to measure this effect and

00:17:27

luckily for him first the only

00:17:29

astronomer who was keen on doing that

00:17:31

was FR Lish and fr Lish planned to do

00:17:36

this during a solar eclipse in the

00:17:39

summer of

00:17:40

1914 he went somewhere in the East and

00:17:44

because of the war outbreak he could not

00:17:46

do the

00:17:47

experiment um which is good because if

00:17:50

he had done the experiment successfully

00:17:52

would not have found this result but

00:17:54

twice this result because the second

00:17:56

prediction of Einstein in 1950 15 was

00:17:59

twice dis value so it's good that no

00:18:03

experiment disproved him wrong before

00:18:05

that he could change

00:18:07

his his

00:18:09

prediction anyway uh so uh so in 1907

00:18:17

Einstein was in bar uh it took actually

00:18:20

years for people to give him a better

00:18:22

position than being a patent uh office

00:18:25

person although although actually he had

00:18:27

a better salary there than what he got

00:18:29

later when he went to zurk as an

00:18:31

assistant professor so first he got an

00:18:33

assistant professorship position in zurk

00:18:36

then the First full professorship

00:18:38

position he was offered was in Prague so

00:18:41

in 1911 1912 he went to prag and

00:18:45

continue thinking about gravity and in

00:18:48

particular so uh link to what he slowly

00:18:53

uh he convinced himself that what he had

00:18:56

said uh that the velocity of light was

00:19:00

modified by the gravitational field

00:19:02

meant that you need to modify the

00:19:05

SpaceTime metric I should say that you

00:19:09

know when minkovski in

00:19:13

1908 was the first one to understand

00:19:15

really what point car had done and um

00:19:19

and made it more evident to people uh at

00:19:23

the end without citing Point car in his

00:19:26

1908 uh lecture the he died so just

00:19:30

after this

00:19:32

um so this metric of SpaceTime was

00:19:36

introduced by MOSI and P and Einstein

00:19:40

understood little that little by little

00:19:43

that the fact that um the equivalence

00:19:47

principle is suggesting that the

00:19:49

space-time metric near masses where

00:19:52

which change the gravitational potential

00:19:55

should be of this form the same form as

00:19:58

in flat space time except that only the

00:20:01

component of time

00:20:03

g0000 is is different from unity and is

00:20:07

modified by the the gravitational

00:20:10

potential and and and this fact which

00:20:13

was both very useful was also a fatal

00:20:16

Prejudice that blocked him for years as

00:20:19

we will see uh because it is not quite

00:20:22

true uh but he reinforced his conviction

00:20:26

that this was the case by the driving

00:20:28

what is today for unknown reasons to me

00:20:32

the rindler metric because rindler is a

00:20:35

scientist who is alive but this metric

00:20:37

had been first derived by Einstein in

00:20:40

1912 and was well known uh at the time

00:20:44

uh so you derived the first exact uh

00:20:49

transformation of acceleration a way to

00:20:53

have a a nonlinear transformation

00:20:56

between a rest frame and an accelerated

00:21:00

frame such that the SpaceTime metric in

00:21:03

the accelerated frame is time

00:21:05

independent and Einstein does this by

00:21:08

successive approximation and then he

00:21:11

gets that it has to have this form where

00:21:14

C is exactly linear in in a space

00:21:17

variable which is in the direction of

00:21:19

the gravitational field of the

00:21:20

acceleration which means that c is this

00:21:23

linear function of X or in terms of the

00:21:26

gravitational field can written this way

00:21:29

and then he used this as a tool he said

00:21:31

ah so according to my equivalence

00:21:34

principle this is an exact gravitational

00:21:37

field now where C is linear in X

00:21:39

therefore I can write an exact field

00:21:41

equation that c satisfies the laas

00:21:44

equation because it's linear in X okay

00:21:47

and then and then he immediately

00:21:49

postulated that probably the pon

00:21:51

equation when there is a matter density

00:21:53

will be something like this that it is

00:21:56

laas of C Etc

00:21:58

and then uh which where C is square root

00:22:02

of g0 0 you know is the square root of a

00:22:04

component and why he was in Prague uh

00:22:08

but then he he went on he said okay so

00:22:10

let me limit my consideration to static

00:22:13

gravitational

00:22:14

fields uh there is only one component of

00:22:17

the SpaceTime metric which is important

00:22:19

c as a function of X and he he started

00:22:23

exploring that this should satisfy a

00:22:26

generalization of personal equation

00:22:28

because as a physicist he immediately

00:22:30

had the idea that you know eal mc² is

00:22:34

saying that the source of gravity mass

00:22:37

inertial mass equal gravitational Mass

00:22:39

has to be energy but energy cannot be

00:22:42

Just Energy of the rest mass of particle

00:22:46

the gravitational field has energy in

00:22:48

physics any field in space time has some

00:22:51

energy and therefore the source of

00:22:53

gravity should be the sum of the kind of

00:22:56

matter energy and the energy of the

00:22:58

gravitational field something nonlinear

00:23:00

in the gravitational field and then he

00:23:02

started playing by saying okay so we

00:23:04

need nonlinear equations where in the

00:23:07

right hand side of pesson type equation

00:23:09

there will be also gradient of the

00:23:11

gravitational field Square okay uh I

00:23:15

must say that at the time

00:23:19

uh two scientists Abraham and norstrom

00:23:24

picked up on the investigation of

00:23:26

Einstein saying maybe there is

00:23:28

interesting things to do with gravity

00:23:31

but they use a Pur they did not follow

00:23:33

Einstein with the idea that we must

00:23:36

generalize relativity to arbitrary

00:23:38

moving frames they said no no let's stay

00:23:41

in special relativity if if gravity is

00:23:46

only a modification of the velocity of

00:23:48

light it means I have a scal of field

00:23:51

and if I have a scal of field the

00:23:52

simplest field equation I can write is

00:23:55

dalom of the scalar field is equal to

00:23:58

some Source A Spacetime scaler which

00:24:02

they took as what was called at the time

00:24:04

the LA scaler the trace of the stress

00:24:07

energy tensor of matter tunu is in

00:24:10

special

00:24:11

activity uh a

00:24:13

tensor with symmetric tensor with uh 10

00:24:19

components uh the z00 components measure

00:24:22

the energy density the time space

00:24:25

components measure the Flux Of energy or

00:24:29

density of momentum and the space space

00:24:32

components measure the stresses okay so

00:24:35

it is an object with 10 components which

00:24:37

generalizes the notion of energy density

00:24:40

and which embodies the conservation law

00:24:43

of energy uh so but Einstein convinced

00:24:48

himself that you could not have a purely

00:24:50

scalar theory of gravity for several

00:24:53

reasons some wrong reasons actually uh

00:24:56

because when you follow Einstein you

00:24:58

find that he he has both a very good

00:25:01

intuition to go to the good thing but he

00:25:03

he makes mistakes calculational mistakes

00:25:06

conceptual mistakes although he corrects

00:25:08

them all the time U and uh but anyway he

00:25:13

had a very good reason for thinking that

00:25:16

you needed more than just the time time

00:25:19

component uh because separately from the

00:25:23

he he took again the this basic ID of

00:25:26

the equivalent principle saying okay any

00:25:29

accelerated reference frame should mimic

00:25:32

some gravitational field so an evident

00:25:34

one is to go in an accelerated frame but

00:25:38

another one is to go on a roundabout if

00:25:41

you go on something which rotate he he

00:25:44

said ah but this is a gravitational

00:25:47

field and therefore of a special type of

00:25:50

a different type and therefore it should

00:25:52

also satisfy the field equation it is an

00:25:55

example of an interesting gravitation

00:25:58

field and in particular what he

00:26:00

discovered he if you take uh a rotating

00:26:05

disc actually in special relativity you

00:26:08

cannot take a solid disc and make it

00:26:11

rotate but what you can do is you take a

00:26:14

solid disc you melt it so that it

00:26:18

becomes liquid then you can make it

00:26:21

rotate you you wait until it rotates

00:26:24

uniformly with the velocity of rotation

00:26:27

and then you let it uh condense again as

00:26:30

a solid material uh because when you do

00:26:33

that at the end you have a solid body

00:26:36

which rotate uh if you do a simpl minded

00:26:40

transformation between the coordinates

00:26:42

of the rotating system and non- rotating

00:26:44

system you find that the SpaceTime

00:26:46

metric is given by this formula this

00:26:49

formula is just the minkovsky flat space

00:26:51

time but written in coordinates that go

00:26:54

along with the rotation of the disk and

00:26:57

then what uh Einstein understood is that

00:27:00

if you are on a rotating disk and you

00:27:02

measure the ratio between the

00:27:05

circumference and the radius of a

00:27:08

circle you find something which differs

00:27:11

from 2 pi actually because of the soal

00:27:15

laen contraction you find that the ratio

00:27:17

of the circumference to the radius is 2

00:27:19

Pi / square root of 1 minus Omega squ R

00:27:23

sare c² anyway you find that the space

00:27:26

geometry

00:27:28

is curve now it is not the flat geometry

00:27:32

simply because you are on rotating and

00:27:34

for him it was a very intuitive reason

00:27:36

to think no no I need General uh G reman

00:27:42

type uh curve geometries okay and he

00:27:45

convinced himself while he was in Prague

00:27:47

that indeed he needed to have a theory

00:27:50

of gravity which was not only uh you

00:27:53

know the C Square in the time time

00:27:56

component but contained a full gunu a

00:27:59

full SpaceTime metric with a symmetric

00:28:02

tensor with 10 components and signature

00:28:05

minus plus plus plus and in particular

00:28:09

he understood by using his equivalence

00:28:12

principle that if a general

00:28:14

gravitational field is described by a

00:28:16

metric of this type then a test particle

00:28:20

moving in this external gravitational

00:28:22

field will follow a geodesic that is to

00:28:25

say you have to extremize the integral

00:28:28

of the proper length along the world

00:28:30

line of a particle and this defines the

00:28:33

effect of a gravitational field on test

00:28:36

particles so you see that between 1907

00:28:39

and 1912 Einstein little by little

00:28:41

gathered many tools towards what we call

00:28:45

today General activity and in 19 uh in

00:28:49

the summer of 1912 if I remember

00:28:52

correctly uh Einstein now got a full

00:28:55

professorship position in in Zo

00:28:58

so he went back from Prague to zurk and

00:29:02

he knew that in zurk there was his old

00:29:04

friend Marcel grman actually Marcel

00:29:07

grman was close friend because when they

00:29:10

were together at the poly technicum when

00:29:12

they were making their studies and

00:29:14

Einstein often would not go to follow

00:29:17

the courses uh would study books in the

00:29:20

library or whatever but Marcel Gman was

00:29:22

going to all courses taking notes and

00:29:25

before the exams Einstein could pass the

00:29:27

exams because he read he read all the

00:29:29

notes of Marcel Gman Marcel Gman also

00:29:32

was the one to help Einstein to get the

00:29:35

position uh through his father anyway

00:29:38

they were close friends and M goosman

00:29:41

was a mathematician working in in

00:29:44

Geometry although geometry I think meant

00:29:48

uh con I mean did not mean reman

00:29:52

geometry was like synthetic geometry so

00:29:55

actually Marcel Gman did not know the

00:29:56

liter

00:29:58

on reman Spaces but he could study it

00:30:01

and understand it anyway when Einstein

00:30:04

came back from Prague he had a very

00:30:06

clear idea about what he wanted from

00:30:09

what was needed from a new theory of

00:30:12

gravity and uh and I have actually the

00:30:16

only information in the abstract I gave

00:30:18

for these talks is that the strategy of

00:30:21

Einstein was

00:30:22

multiped which means was based on

00:30:25

several different pillars and he wanted

00:30:28

all those things to be satisfied

00:30:31

together uh as requirements and and this

00:30:34

was what prevented him from finding

00:30:36

generativity immediately in a sense

00:30:38

because he wanted too many things to be

00:30:41

true at the same time and he could not

00:30:43

reconcile them immediately so what he

00:30:45

wanted first is to generalize the

00:30:48

principle of Relativity to a very

00:30:51

general class of transformation of

00:30:53

between accelerated frames he had in

00:30:56

mind you know linear acceleration and

00:30:58

constant rotation but then you could

00:31:00

think ah then I can do more complicated

00:31:03

things maybe what it's not totally clear

00:31:06

but something as general as you can he

00:31:09

had the idea of the equivalence

00:31:10

principle which is that any of these

00:31:14

fictitious gravitational fields coming

00:31:16

from Flat space time are really real

00:31:19

gravitational fields and give you

00:31:20

indication about the field

00:31:23

equations uh he had uh what we have

00:31:27

before that in the case of weak static

00:31:29

gravitational fields he had convinced

00:31:31

himself over years that in the case of

00:31:35

weak static gravitational fields the

00:31:37

only component of the scalar field of

00:31:39

the gravitational field would be in the

00:31:42

g00 and as we will see this is wrong

00:31:45

actually but this blocked him okay

00:31:48

because he imposed this and with the

00:31:50

person equation he had in mind the

00:31:52

geodesic principle which is

00:31:56

still totally correct that the general

00:31:59

test particle follows this he had in

00:32:02

mind because by combining equal mc² you

00:32:06

know and Mi inertial Mass equal m

00:32:09

gravitational M gravitational that the

00:32:12

source of gravity should be the energy

00:32:15

density and in special relativity the

00:32:18

energy density is described by this 10

00:32:20

component stress energy tensor which

00:32:24

satisfy in special activity the

00:32:26

Divergence equal to zero and immediately

00:32:29

Einstein in 1912 understood that if you

00:32:32

had an matter in a gravitational field

00:32:35

the law of conservation of gravity would

00:32:38

be the that the covariant

00:32:40

derivative uh of tmu was zero and then

00:32:46

having in mind pron equation he had in

00:32:49

mind that the field equations of gravity

00:32:52

should have as Source the 10 components

00:32:54

teu as the gravitational field and also

00:32:57

10 components Jim you knew it was

00:32:59

natural to say that there should be some

00:33:01

tensor probably linear in the second

00:33:04

derivative of G but probably with

00:33:07

quadratic nonlinear terms in in the

00:33:09

first derivative because he already knew

00:33:12

that the gravity the stress energy T of

00:33:14

gravity has to be a source also so of

00:33:18

The Connection by the time so Mar Marcel

00:33:22

Gman no leita invented the connection

00:33:25

only in 1970 so there was no connection

00:33:29

there were Christopher Sim yeah there

00:33:30

was there was the paper so what he

00:33:33

studied under the influence of Marcel

00:33:36

grman was um first the paper of reman in

00:33:41

which but it is impossible to understand

00:33:42

really what reman did where there was

00:33:44

the reman Christopher tensor with four

00:33:47

indices and then there was a very long

00:33:50

paper review paper written in

00:33:52

1900 written in French by Richie and

00:33:56

leevi

00:33:57

which introduces by the way we don't

00:33:59

need the connection it had the rich

00:34:01

connection coefficients you know so it

00:34:04

had in frames everything was in carton

00:34:06

frames before Carton and everything was

00:34:09

what we call today the connection in a

00:34:11

frame but leita not invented yet the

00:34:15

connection as a separate concept okay

00:34:17

this was called calcu differential

00:34:20

absolute absolute differential calculus

00:34:22

okay to have tensor derivatives and like

00:34:25

that so he studied that

00:34:29

and and then what happened so first yes

00:34:32

first as I said Einstein I understood

00:34:36

that what had to be so yes because I

00:34:39

said here it's a coant derivative so now

00:34:41

what it meant for Einstein explicitly

00:34:44

using Richie and leevi chivita and uh

00:34:48

yes is this so first historians are

00:34:51

lucky in the sense that Einstein kept a

00:34:54

notebook in Zurich from 1912 to 1914 for

00:34:58

his course in in relativity and in his

00:35:00

notebook it contains all the calculation

00:35:03

he did to uh leading to around

00:35:07

generativity okay and this notebook has

00:35:09

been preserved and it's fascinating to

00:35:12

look at it and you see Einstein doing

00:35:14

calculations stopping in the middle

00:35:16

making errors stopping saying okay there

00:35:18

is a problem then trying something else

00:35:20

for pages and pages so we know the way

00:35:23

like for instance on the first pages of

00:35:26

this notebook he writes the explicit

00:35:29

form of the what the conservation law of

00:35:33

uh of the 1030 in mixed components and

00:35:37

in in densitized version which means in

00:35:42

modern notation ordinary derivative of

00:35:45

the T muu with one index down and square

00:35:47

root of the determinant minus 1 half of

00:35:51

the gradient of the metric times T muu

00:35:54

and and this is from The Notebook so you

00:35:57

see that he hesitated about the one half

00:36:00

the sign and thing like that but he got

00:36:01

it right okay but but then this this was

00:36:07

a tool for him to to find the field

00:36:10

equations uh except that uh there was a

00:36:14

Prejudice also here because he he got

00:36:16

this form because it was quite simple

00:36:20

and in this form so this form is saying

00:36:23

that in a sense the Divergence of

00:36:25

something Direct linked to the matter so

00:36:28

it's like the conservation of energy and

00:36:31

momentum of the matter is not zero but

00:36:34

the reason is that there are forces

00:36:36

acting on the matter gravitational

00:36:38

forces so evidently you should not

00:36:40

conserve momentum if there is a force

00:36:42

acting physicists know this and for

00:36:45

instance Einstein knew that if you have

00:36:49

electromagnetism and if you have charged

00:36:51

matter the the law of conservation of

00:36:54

the energy momentum of the matter

00:36:56

contain contains a force term and this

00:36:59

force is the force of electromagnetism

00:37:03

acting on the current and we know

00:37:06

Maxwell equation which says that the

00:37:08

current is the source of f muu the

00:37:11

electromagnetic tensor F muu being the

00:37:13

first derivative the curl of the four

00:37:16

potential so Einstein had in mind that

00:37:19

this equation is similar to this and

00:37:22

that this term is the force uh due to

00:37:26

the gravitational field and this force

00:37:29

is as always in physics the product

00:37:32

of the gradient of a field times the

00:37:36

source of this field okay uh I should

00:37:40

have deleted this let me explain this as

00:37:43

this will be important on a simpler

00:37:46

example not

00:37:47

electromagnetism but uh but this is

00:37:49

something that Anin followed very

00:37:52

precisely let's imagine that we have a a

00:37:55

scalar field uh coupled to some Source

00:37:59

Sigma so the scale of field satisfy

00:38:02

dalom the F equal Sigma and now uh the

00:38:07

stress energy tensor is not conserved

00:38:10

because there is a force acting a scale

00:38:12

of force acting on the source and this

00:38:14

scale of force is the product of the

00:38:16

density times the gradient of I which is

00:38:20

the force okay the force per unit Mass

00:38:23

let's say but now if you replace

00:38:27

by box of five you have that the

00:38:30

conservation law of energy is this plus

00:38:33

this term and now you do what we call

00:38:36

today

00:38:37

ibps integration by

00:38:40

parts Einstein use in this notebook

00:38:43

every on every page ibps integration by

00:38:47

parts to convert a force into the

00:38:50

gradient of some stress energy tensor so

00:38:54

the calculation is very easy you take

00:38:56

this

00:38:57

and you try to write it this way as the

00:38:59

gradient of something and you can do it

00:39:03

if you define the stress energy tensor

00:39:05

due to the energy of the gravitational

00:39:07

field in this form that every physicist

00:39:11

know D muf D muf minus 12 of d square

00:39:16

muu where muu is the flat space time

00:39:19

metric and you take the gradient of this

00:39:22

you find that uh actually uh this uh

00:39:26

goes back to this form okay and then now

00:39:29

at this stage you have derived by ibps

00:39:32

that the sum of the stress energy tensor

00:39:36

and energy of matter plus energy

00:39:39

contained in the field is conserved

00:39:41

which is conservation law of the total

00:39:44

energy of the system okay and this is

00:39:47

this principle what you call the energy

00:39:50

principle that Einstein said okay I need

00:39:52

to apply this to gravity I need to find

00:39:56

field equations such that I can convert

00:40:00

the term accompanying this in in the

00:40:04

form of the Divergence of a stress

00:40:06

energy tensor of gravity and but the the

00:40:10

the problem he had is he had used the

00:40:12

the conservation of tunu in this form

00:40:15

which looks very simple because you have

00:40:16

only DDA gunu so it looks exactly

00:40:20

similar to this you know he said this is

00:40:22

the source of gravity T this is the

00:40:25

gravitational force

00:40:26

and now I integrate by Parts until I get

00:40:30

something uh meaningful okay until I get

00:40:34

an equation which should be of this type

00:40:38

where essentially box of G muu should be

00:40:41

tunu and the reason why he really wanted

00:40:44

box of G minu in some approximation

00:40:46

first he wanted to have on the right

00:40:48

hand side the sum of the energy of the

00:40:51

matter and the energy of the

00:40:53

gravitational field and he also want

00:40:56

wanted box of GMU to be in lowest

00:40:58

approximation directly related to IMU

00:41:01

because he knew that the stress energy

00:41:05

tensor for the simplest matter around us

00:41:08

ordinary matter in this room uh made of

00:41:12

material which moves with very low

00:41:15

velocities is something which is

00:41:17

essentially containing in the z00

00:41:20

component with very small terms in the

00:41:23

space I mean time space component and

00:41:26

even smaller terms in the space space

00:41:29

components so he knew that U was

00:41:31

essentially t00 and he had impos that g

00:41:35

mun was also essentially g0000 the time

00:41:38

time component so everything looked okay

00:41:40

we did an equation of this type by the

00:41:43

way this ID so Einstein for years really

00:41:47

used as guide one of the guiding

00:41:49

principle the fact that the gravity the

00:41:52

energy of the gravitational field should

00:41:54

be the nonlinear Source in the

00:41:56

gravitational field equation and this

00:41:58

idea has been

00:42:00

reacted by fan in the 60s in his

00:42:04

lectures on gravity and transform into

00:42:08

uniqueness theorems by several authors

00:42:11

not only Stanley deser and Bob wal and

00:42:16

collaborator where they proved that if

00:42:18

you start from a massless pin 2 you

00:42:22

start from the massless pin 2 field

00:42:23

equations which are like box of H min

00:42:26

plus other terms equal Z and then you

00:42:28

say I want on the right hand side to add

00:42:31

the energy of the gravitational field H

00:42:33

muu itself and so that it bootstrap

00:42:36

itself in a self-consistent manner it

00:42:38

has been shown that there is a unique

00:42:40

solution which is Einstein's uh field

00:42:43

equations okay in vacuum so part of the

00:42:46

requirement of Einstein was definitely a

00:42:49

modern

00:42:50

requirement uh and was true uh but he

00:42:53

use it together with other part so when

00:42:55

you continue looking at this very

00:42:58

beautiful Zur

00:42:59

notebook at a later page you start

00:43:02

seeing so this are the So-Cal

00:43:05

Christopher symbols so in the coordinate

00:43:08

system this is the leevi Chita

00:43:10

connection not introduced by Le I mean

00:43:13

introduced later by leita but

00:43:16

Christopher had introduced the

00:43:17

Christopher symbols already earlier in

00:43:20

1880 something uh and then this object

00:43:25

with four indices ikm is the reman

00:43:28

tensor the curvature tensor with four

00:43:31

indices which is explicitly written here

00:43:35

and here you see that it's called Uh

00:43:38

Grossman so he alludes to the fact that

00:43:40

this is Marcel grman who told him that

00:43:43

this was the important four index tensor

00:43:46

for a a manal kite okay of a

00:43:50

manifold then it takes so at the

00:43:54

beginning Einstein uses a notation where

00:43:58

all indices are put

00:44:00

down uh because at the time it was not

00:44:04

usual to say there are contravariant or

00:44:06

covariant indices and then you needed to

00:44:09

know when a tensor add indices down or

00:44:12

up and he used the notation where when a

00:44:16

tensor like G munu has inded down first

00:44:19

it's called G munu and the indices are

00:44:21

called munu when it is put in

00:44:24

contravariant form it's no longer called

00:44:26

G but gamma and the indices are called

00:44:29

with Latin indices so gamma KL means G

00:44:33

muunu up okay in this

00:44:35

formula uh and at the beginning you know

00:44:38

at the beginning of the notebook and

00:44:40

during most of the notebook he put

00:44:42

summation indices although he

00:44:44

understands little by little that you

00:44:46

don't need to write Sigma so he invented

00:44:48

the summation convention at the end but

00:44:51

not in the notebook and and this thing

00:44:54

which is the contraction of the uh

00:44:56

remant tensor with two indices on the

00:44:59

two antisymmetric pairs is called the

00:45:01

Richie tensor uh not here but uh but he

00:45:05

considered the rich answer because uh

00:45:09

grman and him had noticed that if you

00:45:12

wanted an object which was linear in

00:45:14

second derivative with two indices

00:45:16

symmetric and the source could be U Rich

00:45:19

was the most evident thing so the first

00:45:21

thing that Einstein tries is to say okay

00:45:24

let's take the rich tensor has

00:45:27

gravitational tensor and then but then

00:45:30

the rich tensor has many components even

00:45:33

when you look at its linearized limit

00:45:36

and then there Einstein invents

00:45:39

coordinate conditions he says okay I'm

00:45:42

going uh it's not clear whether okay the

00:45:46

notion of being generally coant and

00:45:49

fixing coordinates you know was not

00:45:51

clear at the time but still he does it

00:45:53

essentially and actually he says uh

00:45:57

using coordinate such that Delta F equal

00:46:00

Z so these are what we call today I mean

00:46:04

Sergio would call them wave coordinates

00:46:07

usually they are called harmonic in my

00:46:09

time they were called harmonic

00:46:10

coordinates at the time they were called

00:46:12

isothermal coordinates so Einstein knew

00:46:15

about that he knew that you could impose

00:46:17

exactly this condition you know the

00:46:19

condition is written here this is gamma

00:46:22

mu up time gamma mu down Lambda equal Z

00:46:27

which means that you have the two minus

00:46:30

there is a one half here and then one of

00:46:33

this okay so he gives a condition and

00:46:35

Einstein here does a computation and

00:46:38

finds that after this the the the

00:46:42

leading terms with two derivatives in

00:46:44

the richet answer contains only G muu

00:46:48

second derivative of G Alpha Beta down

00:46:51

okay which is exactly the form he wanted

00:46:54

so at this stage AG he nearly found what

00:46:59

he wanted so uh so he proves so in this

00:47:04

notebook early on he he reduces the

00:47:08

Richie tensor by harmonic conditions and

00:47:11

by linearizing I mean forgetting

00:47:14

nonlinear terms because he knew that

00:47:16

they were square and they were small

00:47:17

okay uh then he gets that the Richie is

00:47:20

minus one2 of dalom acting on Jim yunu

00:47:24

he looks for and therefore says Ah as I

00:47:27

want an equation of this type where I

00:47:30

have only the g0 component dominating on

00:47:33

both sides evidently this equation is

00:47:36

what I

00:47:38

need and but so at this stage he nearly

00:47:42

found generativity in 1912 okay but the

00:47:46

problem is that he was convinced by all

00:47:49

the things for years he had worked that

00:47:51

for an accelerated frame only the g0000

00:47:54

component was to to be there in the

00:47:56

Newtonian limit uh so it is okay with

00:48:00

this so this is okay it's compatible but

00:48:03

then in knew that t muu satisfies a

00:48:05

conservation law which says that the

00:48:09

Divergence of T muu modulo non terms is

00:48:12

zero but the ammonic gauge condition is

00:48:16

not that the Divergence of G minu is

00:48:18

zero but that the Divergence of G minu

00:48:20

is 1 half of the gradient of the trace

00:48:24

of G and therefore at this stage there

00:48:26

is incompatibility everything is

00:48:29

satisfied among all these requirements

00:48:31

except one that the conservation load

00:48:33

does not work Einstein modifies his

00:48:37

equations and then says okay I can add

00:48:39

the minus one2 of tunu and then the

00:48:42

conservation law is satisfied these are

00:48:44

the Einstein linearized equation so it

00:48:47

was a surprise to historians that

00:48:49

already in 1912 Einstein had written the

00:48:52

Einstein linearized equations except

00:48:54

that he rejected them immediately

00:48:57

because he said yeah but then it means

00:48:59

that the space component is as a source

00:49:02

now as important as the time time

00:49:04

component and this is against everything

00:49:06

I have done for years so he got very

00:49:10

close but then he said there is

00:49:12

something which does not work the

00:49:15

Newtonian limit as he understood it so

00:49:18

for months with his friend Marcel Gman

00:49:21

they wrote a paper so he had they got

00:49:25

close okay they they got the richet

00:49:27

answer they they nearly got the good

00:49:29

field equation but then they said no it

00:49:31

does not work because of all the

00:49:34

requirements and therefore they said

00:49:36

okay we need to find other field

00:49:39

equations and they wrote a review paper

00:49:42

after working a lot on it they they

00:49:45

convinced themselves that they had found

00:49:48

uh the best equations they could find

00:49:50

alas it was not Richie based okay it was

00:49:53

of this form this Theta means T muu

00:49:56

within this is up because this is like

00:50:00

you know G muu up was called gamma so T

00:50:03

muu if thises up was called Theta muu so

00:50:06

gamma muu is the field equation on on

00:50:09

the right hand side uh in knew the

00:50:12

geodesic principle so in this paper

00:50:14

which is called enor paper which means

00:50:17

outline so they knew in writing this

00:50:20

paper that maybe this was not the final

00:50:22

Theory but this is the best they could

00:50:24

find that's satisfy all the requirements

00:50:28

and and then how did they so they know

00:50:30

the left hand side timu how do you find

00:50:34

what you need okay so they use ibps

00:50:37

integration by parts they said okay now

00:50:41

one uses the conservation law before you

00:50:43

know there was the conservation law of

00:50:46

tunu which contain a force term this

00:50:48

Force term at the gradient of G muu

00:50:51

times T muu but T muu is equal to this

00:50:55

function of G gamma so you want to

00:50:58

transform

00:50:59

this in where you here you say that to

00:51:03

Leading order the the second order the

00:51:06

symbol AS mathematician would say in

00:51:09

Einstein's equations Einstein Grossman

00:51:11

equation would be the block acting on on

00:51:14

G muu up okay you say this has to be the

00:51:17

symbol of the field equation and now I

00:51:19

want this time this by integrating by

00:51:22

parts to be able to Define for me the T

00:51:25

of the gravitational field to check that

00:51:28

this Theory conserved energy globally

00:51:32

okay and so at the end they get this

00:51:36

equation so they say and they claim that

00:51:39

this is unique so in the paper they say

00:51:42

yes now there could be several things

00:51:44

but no no the unique thing is this okay

00:51:48

and the unique thing are these field

00:51:50

equations where essentially so this

00:51:53

object means so gamma alphab beta means

00:51:56

G muu with indices up so this is g muu

00:52:00

with indices up this is the dalom

00:52:04

operator the Beltrami operator acting on

00:52:07

Junu okay so this equation is Beltrami

00:52:09

of jimu acting on jimu up

00:52:13

equals uh equals the stress energy

00:52:16

tensor of matter plus something uh which

00:52:21

Einstein calls the stress energy tensor

00:52:23

of gravity because he proves

00:52:25

that the sum of the stress energy tensor

00:52:28

of matter and gravity satisfies the

00:52:29

conservation law so it could satisfy you

00:52:32

know all his requirements now the conser

00:52:35

energy principle everything except that

00:52:37

he was not generally coant okay so if

00:52:40

coordinate sorry still harmonic ordinate

00:52:43

no here you don't know you are you you

00:52:46

don't use it's not the harmonic

00:52:48

restriction of the rich Tor this failed

00:52:52

although he added the minus one2 it

00:52:54

still failed

00:52:55

so he started from scratch again and

00:52:58

said I want an operator where the symbol

00:53:01

is the belamy operator Define by jimu

00:53:04

acting on jimu as the only symbol

00:53:07

without indeed as if I was in Armon

00:53:09

coordinate but it's not the armonic

00:53:11

Restriction because he did not work with

00:53:13

the conservation of timu he had excluded

00:53:16

this okay here he has the good Newtonian

00:53:19

limit as he believed he needed it and

00:53:22

conservation law so he satisfied all

00:53:25

what he wanted except that he had to

00:53:27

write a term like that which was not

00:53:29

generally coant which was not obtained

00:53:31

by a coordinate restriction from richy

00:53:33

and he knew that he he felt bad you know

00:53:36

he said I would have liked this and okay

00:53:40

so he tried to convince the reader

00:53:42

that's the best and there are good

00:53:43

reasons for this uh in particular so uh

00:53:49

it is not obtained from generally coant

00:53:51

field equations um they say it is unique

00:53:54

but they realize little by little that

00:53:56

it's far from being unique what they

00:53:59

what they did uh what is the cence group

00:54:02

because the idea of Einstein was he

00:54:04

wanted something which maybe was not

00:54:06

generally coent but could be true in

00:54:08

accelerated frames like rotating frames

00:54:11

and linearly accelerated frames and he

00:54:14

is going to find they do not satisfy

00:54:16

this later okay so that's why he will

00:54:20

then with his friend mikelangelo besso

00:54:23

he computed the advance of the perion of

00:54:25

mercury also because at this time he

00:54:28

understood this was a Criterion and

00:54:31

actually in their paper they they got

00:54:34

the factor 100 wrong uh but then they

00:54:37

realized okay they corrected the factor

00:54:40

100 and after correcting it they

00:54:42

concluded that the advance was only 18

00:54:45

seconds of AR per Century instead of 40

00:54:49

okay something like this at the time uh

00:54:53

then for years U it tried to argue that

00:54:56

the rotating dis metric which he liked

00:54:59

was a solution of the field equation and

00:55:02

pages and pages of calculation of

00:55:04

Einstein he red do the calculation in

00:55:06

coent form in sometimes he find they do

00:55:09

satisfy other times the time they don't

00:55:12

satisfy he ask some friend please I have

00:55:14

done a calculation five times I don't

00:55:17

know where is my mistake uh at the end

00:55:20

he convinced himself they do not satisfy

00:55:23

okay but you know he is making many

00:55:26

sometimes he's doing exact calculation

00:55:28

very difficult perfectly okay sometimes

00:55:30

he's doing

00:55:32

mistakes

00:55:33

um and uh after the publication of this

00:55:37

paper the pupil of um Hilbert called

00:55:42

bernes uh who was a logician I think

00:55:45

told him that actually there is a good

00:55:48

tool for knowing conservation laws in

00:55:51

physics is to use a variational

00:55:54

principle because at the time let me

00:55:57

remind you that the So-Cal I mean not

00:55:59

the so-call NS theorem and work is from

00:56:03

1918 but before this it was an exercise

00:56:07

for actually was not so interested in

00:56:10

this it was a great mathematician in

00:56:12

other parts of RS yes and but people

00:56:16

Felix Klein and others knew that they I

00:56:19

mean since lrange everybody knew that if

00:56:22

you had a variational derivation if you

00:56:24

had a l and the lon was invariant under

00:56:27

some symmetry group or some

00:56:29

transformation then you had

00:56:31

corresponding conservation laws okay for

00:56:34

the stress energy tensor of the matter

00:56:37

and ber told this to Einstein and then

00:56:40

Einstein does a lot of variational

00:56:42

computations in 1914 I mean he does not

00:56:45

like theorems and deres that indeed

00:56:48

these equations can be obtained from a

00:56:50

variational principle and as a

00:56:52

consequence you can rederive the team

00:56:55

the conserve stress energy tensor by

00:56:57

lrange n type methods quite fast okay uh

00:57:02

yes so he did that in the following

00:57:06

what so yes so the lran

00:57:09

essentially was uh you take the the

00:57:13

gradient you take uh G mu new derivative

00:57:18

uh Lambda and you take its Square

00:57:22

putting you know in thises up and down

00:57:24

with square root of G and this is the

00:57:27

lonan so the lonan is just the

00:57:30

generalization of grad F square if you

00:57:32

have a scalar field Lon is sare root of

00:57:35

g g mu D mu D mu you do the same thing

00:57:39

where fi is G Alpha Beta and G Alpha

00:57:42

Beta with the correct sign okay and the

00:57:45

correct Factor this was finally so it's

00:57:47

a very simple Lon actually but he

00:57:50

discovered it only as a

00:57:52

afterthought uh okay

00:57:55

now in uh but all the time Einstein

00:58:00

regretted said okay General covariance

00:58:03

the richet answer was really something

00:58:05

that we wanted at the beginning ah yes I

00:58:08

should have mentioned that in the

00:58:10

meantime he um I should have put this in

00:58:14

the slides he invented the so-called

00:58:16

whole argument uh which is a subtle

00:58:19

argument against the idea of having

00:58:22

field equations that are invariant under

00:58:24

all coordinate Transformations because

00:58:27

he said the following let me uh maybe

00:58:31

write

00:58:32

formula that the whole argument is that

00:58:36

if you

00:58:37

have field equation for

00:58:41

gimu and they are

00:58:43

determined uh by maybe kosi data at some

00:58:48

time tal Z and by the presence of matter

00:58:53

here but that uh so matter may be here

00:58:58

and influence the propagation of the

00:59:00

gravitational field but here you have a

00:59:02

hole without matter okay and now if you

00:59:05

have fi equations which are invariant

00:59:07

under any change of coordinate systems

00:59:10

you can make a change of coordinate

00:59:12

system which changes only inside here

00:59:16

and and therefore you will have two

00:59:19

solutions of your field equations that

00:59:22

are identical everywhere except that

00:59:24

they differ here okay and evidently in

00:59:28

modern language we say okay you did a

00:59:30

def morphism it's the same solution in a

00:59:33

different coordinate system but if you

00:59:35

wanted determinism in some sense that

00:59:39

you know the matter and the initial

00:59:41

condition determine uniquely the

00:59:43

solution indeed you have infinitely many

00:59:45

solutions with the same kosi data and

00:59:48

this uh he considered as an argument

00:59:51

saying okay uh although he would have

00:59:54

liked generally coant equation he said

00:59:56

no no because of this I don't want

00:59:58

finally generally coant things and in

01:00:01

modern days you would say no it's an

01:00:03

error but even Hilbert at the time did

01:00:05

not know nobody knew even in mathematics

01:00:08

you know what it meant only leevi

01:00:10

chivita uh uh was probably more subtle

01:00:14

and new but not directly on the whole

01:00:16

argument so anyway all these arguments

01:00:18

of Einstein went in the direction of

01:00:21

this but in the summer of

01:00:23

1915 he was invited to give a set of

01:00:27

lectures many lectures actually uh

01:00:29

during several weeks in gingan and uh

01:00:33

and then Einstein at the beginning was

01:00:35

very happy because you know um at the

01:00:38

time yes I should have said this uh

01:00:42

after

01:00:43

zurk uh the people in Berlin which means

01:00:47

essentially Max Blan made a lot of

01:00:49

effort to attract Einstein to Berlin uh

01:00:53

saying that Einstein clearly was a

01:00:56

rising star more than a rising star and

01:00:59

he wanted the best possible physicist in

01:01:02

Berlin and this was at the time Einstein

01:01:06

had made uh most the most important

01:01:08

contribution to Quantum Mechanics

01:01:10

actually and special relativity and

01:01:12

plank considered that the most important

01:01:14

problems of physics were quantum

01:01:16

mechanics and they were actually at the

01:01:18

time so actually PL was not so happy

01:01:21

when Einstein in Berlin started doing

01:01:24

only gravity because this

01:01:26

work okay up to 19 in 1914 Einstein goes

01:01:30

to Berlin and he continues working on

01:01:33

gravity so so plank was not so happy

01:01:36

that Einstein was not continuing his

01:01:39

genius work on quantum mechanics but

01:01:41

losing time on you know a force which

01:01:44

nobody cares about because the gravity

01:01:46

is a very small thing this relativistic

01:01:48

effects you know a few seconds of Arc

01:01:51

per Century who cares uh why was

01:01:54

convinced that there was a future to

01:01:56

gravity okay anyway um so at the time so

01:02:02

so the colleagues of

01:02:04

Einstein no big colleagues of Einstein

01:02:07

like plank were interested in what

01:02:09

Einstein was doing only Lawrence Arin

01:02:12

Fest and some more minor people like

01:02:15

Abraham and and

01:02:17

norstrom uh so when Hilbert invited him

01:02:20

to uh because Einstein had said yeah

01:02:23

gravity REM

01:02:24

curve space times okay a part of gravity

01:02:27

sober said ah something interesting to

01:02:29

learn so he asked Einstein to give a set

01:02:32

of lectures and Hilbert asked many

01:02:35

questions and to understand all the

01:02:37

details and the arguments the whole

01:02:39

everything and at the beginning in his

01:02:41

letters Einstein said yes I'm very happy

01:02:43

I have colleagues that are really

01:02:45

strongly interested in this little by

01:02:47

little he had done saying that but maybe

01:02:49

Hilbert is going to try to scoop me and

01:02:52

because he realized that some of the

01:02:54

claims he had made were not true that

01:02:57

and then he said okay so I need to think

01:02:59

again about the thing and so after the

01:03:02

summer of

01:03:03

1915 in the in the fall uh Einstein

01:03:07

found a way uh to solve all this

01:03:10

Problems by getting rid of the two

01:03:13

prejudices that had blocked him actually

01:03:16

and these two prejudices were first a

01:03:19

technical thing but which technically

01:03:22

changed the game remember that Einstein

01:03:25

was writing the energy principle how the

01:03:27

energy of matter is influenced by the

01:03:29

gravitational force in this form where

01:03:32

what played the role of a force like

01:03:35

grad ofi was this but then he said yeah

01:03:39

but I can write it this way in terms of

01:03:42

the connection the Christopher symbol

01:03:45

because you know this thing is made of

01:03:47

one half of three three derivative of G

01:03:50

but two of them are cancelled because T

01:03:52

is symmetric so finally there is only

01:03:54

one term left but but this gives the

01:03:57

idea that this should be the

01:03:58

gravitational force and also although it

01:04:02

took time it is when he did in a few

01:04:05

weeks the calculation of the periastron

01:04:07

precession that he realized that finally

01:04:10

it's not a big defect if there are space

01:04:13

components of G okay uh that with the

01:04:16

Newtonian limit it's not important but

01:04:18

it was the last thing that he

01:04:20

understood so in each week so in

01:04:24

November he was now member of the pran

01:04:27

Academia of Sciences which meant he

01:04:30

could uh present a paper at the Seance

01:04:33

of the academy and see it published

01:04:36

immediately because you present a paper

01:04:37

it's considered published and then he

01:04:40

would send immediately actually not

01:04:42

Xerox copies because there existed no

01:04:44

Xerox copies but proofs of his uh to be

01:04:47

publish paper to some colleagues okay to

01:04:51

tell them ah I have done this so in the

01:04:54

first paper in November 4 he goes back

01:04:58

to the remant tenser then he goes back

01:05:02

to the trace of the remant answerer

01:05:04

which we call today the richy answer

01:05:06

this is G is the richy answerer okay he

01:05:09

decomposes it in two parts one that you

01:05:13

call R that it would be the notation for

01:05:16

Richie but this R is not Richie is part

01:05:18

of Richie because he understands that

01:05:22

the other part okay if you

01:05:24

if you restrict so here he understood he

01:05:27

could use nearly General covariant

01:05:30

except with the Restriction that the

01:05:32

coordinate transformation as a unit

01:05:34

Jacobian which means uni modular

01:05:37

coordinate transformation transformation

01:05:39

that you you can vary a lot but you keep

01:05:41

the determinant of the Matrix of

01:05:44

transformation to one okay which is a

01:05:46

small restriction but this small

01:05:48

restriction uh kills actually half of

01:05:51

the richy tensor so it allowed to

01:05:53

simplify very much the richet answer

01:05:55

because the rich answer uh now looks

01:05:58

like this the richet answer you know

01:06:01

usually has four terms you could say

01:06:03

okay two or four who cares but when you

01:06:05

are a physicist and you do explicit

01:06:07

calculation and you have two and one of

01:06:10

them is a Divergence so it's immediately

01:06:12

what you want it's good you can

01:06:14

integrate by Parts without problem and

01:06:16

then you have this extra term okay and

01:06:19

the conservation law of T is written

01:06:21

with gamma so by combining this and this

01:06:24

and also understanding that there exist

01:06:27

a lran derivation actually it could get

01:06:31

that these equations which are nearly

01:06:34

which is so at this stage the equations

01:06:36

are r muu equal Capa T muu where R is

01:06:39

not the full Richie but the part of

01:06:41

Richie which is invariant under

01:06:44

unimodular transformation but you are we

01:06:46

are getting close Okay November 4 and he

01:06:49

proves that he has conservation of the

01:06:50

sum of energy of matter and the

01:06:53

gravitational field

01:06:54

he does it variation and he does a

01:06:57

variational thing

01:07:00

yes okay but then he realized in an

01:07:04

adendum that actually because he did not

01:07:07

like the idea that this system was

01:07:09

limited not General cence but coant only

01:07:12

to something unimodular but still square

01:07:15

root of G was not could not be put to

01:07:17

one so he said ah if I assume that the

01:07:21

fundamental theory of matter is based on

01:07:24

massless particles as we would say today

01:07:26

like electromagnetism you know that in

01:07:28

electromagnetism the trace of the stress

01:07:30

energy tensor is zero because of

01:07:32

conformal invariance so if you have only

01:07:35

massless particles in the sense of spin

01:07:37

smaller than two you have the trace

01:07:39

equal zero so Einstein said ah let's

01:07:41

assume the fundamental Trace is zero

01:07:44

although maybe the gravitational field

01:07:46

has a nonzero Trace which explains why

01:07:49

we don't see Zero trace for dust for

01:07:51

instance then he can use the full Rich

01:07:54

equal tunu so at this stage we have Rich

01:07:56

equal

01:07:58

tunu with now a coordinate condition

01:08:01

that you must work in coordinates where

01:08:03

square root of G is one which is just a

01:08:05

coordinate restriction okay then the

01:08:08

week after he computes the solution of

01:08:12

these field equations Rich equal

01:08:16

tunu uh in vacuum outside the sun

01:08:19

because now he wants to say do I explain

01:08:22

the perion Mercury Peron advance of

01:08:25

mercury so you need to solve this with

01:08:27

zero on the right hand side so that you

01:08:29

don't see if the right hand side is you

01:08:30

know t or whatever so you have to solve

01:08:33

these equations you have to solve them

01:08:35

to second order it's a non-trivial

01:08:37

calculation with a coordinate condition

01:08:39

of determinant one he can do this you

01:08:42

have then to integrate the geodesic

01:08:44

equations of motion you know he does

01:08:46

this in very quickly very quickly

01:08:50

because actually he had worked with

01:08:52

mikelangelo so in 1913 in a paper that

01:08:56

was never published so he had practice

01:08:59

to do all those computation in a

01:09:00

different Theory but he knew how to do

01:09:02

that including contour integrals alak

01:09:04

Koshi to compute everything uh there

01:09:07

were no Loop integrals but still at the

01:09:09

time this was quite technically

01:09:11

challenging to do that and then at the

01:09:14

end the calculation yields for the

01:09:16

planet Mercury a Peron advance of 43

01:09:19

seconds of AR which is still the value

01:09:21

today you know from gr and he says

01:09:24

why the astronomers say it's 45 plus

01:09:26

orus 5 so there as he said he had half

01:09:29

pal palpitations because he got

01:09:32

something which was nearly completely

01:09:35

generally coant and which precisely

01:09:37

explained the periastron Mercury advance

01:09:41

and the last chapter so this was

01:09:44

November 11 or November 25 now he can do

01:09:48

the full game nearly the full game in

01:09:51

the sense that he says okay this is

01:09:53

Richie and in the right hand side I need

01:09:56

to add the minus one2 G mu which you

01:10:00

remember he had obtained in 1912 but at

01:10:03

the time this was incompatible with the

01:10:05

idea of the Newtonian limit where

01:10:07

everything would be in

01:10:08

g0000 but now we understood from his

01:10:11

calculation of the per that it's not

01:10:14

important for the Newtonian limit what

01:10:16

is G and the fact that the space part of

01:10:19

the metric is not flat actually changes

01:10:21

the periastron exactly in the good

01:10:23

quantity that's why before there was

01:10:26

only 18 seconds of AR you get now 43

01:10:29

because you have more components so at

01:10:30

this stage is convinced to be right he

01:10:33

has the conservation law he can write

01:10:35

explicitly the tensor conserve energy

01:10:38

tensor of the gravitational field all

01:10:40

this calculation are used in unimodular

01:10:44

coordinates but at this stage this is

01:10:46

just a technical restriction because he

01:10:48

says this is a fully covariant thing so

01:10:52

uh as you know uh but I won't discuss uh

01:10:56

this because this is essentially the end

01:11:00

uh in the weeks after the paper on the

01:11:04

perion Mercury by the way in the paper

01:11:07

in the previous paper there is a

01:11:09

footnote uh saying that he already

01:11:11

understood this there is in the footnote

01:11:13

it's clear he has understood that he's

01:11:15

not limited to the calculation so that

01:11:18

he got actually the full Einstein

01:11:20

equation the week after but then in

01:11:22

between Hilbert had thought a lot about

01:11:26

things and then wrote a

01:11:28

paper uh that's for years when I was a

01:11:32

student people said ah poor Einstein had

01:11:35

been scooped because there is a paper of

01:11:37

hbert the week before Einstein wrote his

01:11:40

equation with the minus one2 where

01:11:42

Hilbert said that what did hbert say

01:11:46

hbert said uh we need an action

01:11:50

principle for gravity and the action for

01:11:53

gravity T is something integral D forx

01:11:56

sare root of G some

01:11:59

K where he says K is some curvature

01:12:03

invariant does not write it but

01:12:06

implicitly maybe the rich scaler you

01:12:09

know uh and then if you read the

01:12:13

published version of the paper then you

01:12:15

see essentially Einstein equations

01:12:17

because this is called the Einstein

01:12:19

ilbert action okay and from it if you do

01:12:22

a variation derivative a lrange

01:12:25

derivative you get the Einstein

01:12:28

equations except that Einstein was

01:12:30

Furious because what happened is uh in

01:12:33

the paper as Hilbert submitted it to the

01:12:37

gingan academy it did not contain thein

01:12:40

equation explicitly he contained wrong

01:12:42

statement saying you cannot be generally

01:12:44

C that is to say he had understood too

01:12:47

much what Einstein said and he had

01:12:49

imposed some condition still it contains

01:12:53

new new things okay it's not a

01:12:54

negligible paper but what Einstein did

01:12:57

not like is the idea that uh

01:13:01

hbert was

01:13:04

nosty okay to nosty is a

01:13:09

neologism that Einstein invented which

01:13:12

means making ours that is to say uh

01:13:17

essentially plager rizing that is to say

01:13:19

take a result of somebody else and make

01:13:22

it look as if you invented everything

01:13:24

yourself so Einstein was Furious because

01:13:28

in the original version of the paper of

01:13:30

Hilbert Hebert did not even site

01:13:33

Einstein by name

01:13:34

nowhere which is a bit hard because he

01:13:37

had learn everything and then on the

01:13:40

proofs he changed the proofs to add

01:13:42

things that were in the last paper of

01:13:44

Einstein You know which were not in his

01:13:46

original proof so really hbert wanted to

01:13:49

leave you know a statue of saying okay I

01:13:52

have done everything mathematic Ians are

01:13:54

better than physicist and mathematical

01:13:56

methods okay so although this paper is

01:13:59

important contains important thing it

01:14:00

did not contain okay what hbert tried to

01:14:03

make it contain after the fact because

01:14:05

people historians have found the proofs

01:14:08

of hbert where he change on the post

01:14:10

what he changed okay so they they know

01:14:13

that indeed ilbert the publish paper had

01:14:16

all sorts of Correction yes made after

01:14:18

exactly made after Einstein you know to

01:14:20

make it look better okay he suppressed

01:14:22

the parts that Einstein okay so that's

01:14:25

why for for years people we not we not

01:14:28

understanding why Einstein was Furious

01:14:30

you said why but it was Furious not

01:14:33

because it was cooped but because it was

01:14:35

really there was a way of anyway to use

01:14:39

yes so uh but after this Hilbert

01:14:43

recognized saying okay we are we are

01:14:47

superior beings we should not let these

01:14:50

small things interfere with our friend

01:14:53

ship and he he he did everything to for

01:14:57

Einstein to get a big prize uh adding uh

01:15:02

big prize b b prize I think adding that

01:15:07

every uh kid in the streets of gingan

01:15:10

knows more geometry than Einstein but he

01:15:13

was the one which is a kind of compl but

01:15:15

he was the one to to disc discover

01:15:17

General activity so it's a mixed

01:15:20

complement uh but anyway after that they

01:15:23

talked again and and the modern view is

01:15:26

that definitely generativity was

01:15:28

constructed from the beginning to the

01:15:30

end by Einstein with help okay by

01:15:34

colleagues by grman by other people

01:15:36

learning things but that he he kept the

01:15:40

the line of thinking and that uh he he

01:15:43

lost uh time because he wanted too many

01:15:48

things to be true at the same time and

01:15:50

he could not prove because for instance

01:15:52

let me just mention

01:15:53

one the explicit calculation that he

01:15:56

does in this paper Okay including you

01:15:59

know the T so the tunu is defined by

01:16:03

this in terms of the gamma so if you

01:16:05

want to prove for instance in vacuum

01:16:07

that the Divergence of this T is zero

01:16:10

this is an explicit

01:16:12

calculation uh modulo these field

01:16:15

equations which is quite complicated and

01:16:17

which is strictly equivalent to checking

01:16:19

the bian contractor identities by hand

01:16:22

okay because at the time nobody knew

01:16:25

that biun identities existed they were

01:16:28

rediscovered after this okay people say

01:16:30

ah by the way uh bian had written

01:16:33

something if you contract them you have

01:16:35

something useful okay Einstein got them

01:16:38

by hard calculations in his notebook

01:16:41

working with components and using

01:16:43

coordinate conditions until he got all

01:16:47

the general bian identities were known

01:16:49

before or yes bian I forgot I read once

01:16:52

the paper in 1880 something like this

01:16:55

there is a paper of ban where there are

01:16:57

the with four indices and five I mean

01:17:00

you take Divergence they were all they

01:17:02

were all known but they were forgotten I

01:17:04

think in the review paper of Richie and

01:17:06

leevi chivita they never appear one

01:17:08

should check this okay so anyway this

01:17:11

was the end uh so thank you for your

01:17:14

[Applause]

01:17:20

attention so who was the first reverse

01:17:23

the trace sorry who who sorry was the

01:17:28

first to show to reverse the trace of

01:17:31

this equation so Einstein okay he did it

01:17:34

yes he did it here he did it in

01:17:37

1912 and uh I

01:17:40

mean um a controversy was uh even in the

01:17:48

when when ilbert wrote this okay uh

01:17:53

somebody has cut a little part of the

01:17:56

remaining proofs so that some people

01:17:57

said maybe he wrote explicitly

01:17:59

Einstein's equation because L derivative

01:18:02

of this gives AR muunu minus one half of

01:18:05

RG muu but the consensus is that

01:18:08

actually hbert was not able to do this

01:18:11

simple variational computation Hilbert

01:18:14

admire Einstein for having done the

01:18:16

perial perion computation which I'm not

01:18:19

sure even the young people here could do

01:18:22

here in in two hours you know uh

01:18:27

correctly um so at the end no no so in

01:18:31

spite of the historical hiccups uh it's

01:18:35

considered that Einstein really from the

01:18:37

beginning to the end found every element

01:18:40

of Einstein's theory except except that

01:18:44

after this at this stage um he had the

01:18:48

variational principle for these

01:18:50

equations with a restriction to

01:18:53

unimodular coordinates later he wrote an

01:18:56

article with a l variational

01:18:59

derivative understanding that if you

01:19:01

have an action which contains which is

01:19:04

linear in second derivative you can

01:19:06

integrate by parts and have something

01:19:08

quadratic in first derivatives and

01:19:10

writing explicitly that the lron is is

01:19:14

this plus another term which is called

01:19:17

the Einstein form of the action and

01:19:20

using uh not

01:19:24

andand lrange type identities to get the

01:19:27

conservation law directly from the

01:19:29

action

01:19:31

okay are there uh other urgent

01:19:35

questions you you didn't mention the

01:19:37

letters between Einstein and H

01:19:42

they but you mean after this they even

01:19:46

before I mean which were a bit strange

01:19:48

letters because they were not telling

01:19:50

what they were doing at the time they is

01:19:53

taking some distance between what they

01:19:55

were doing and what they were writing to

01:19:57

the other which was

01:19:59

kind what exist also and and this you

01:20:02

will find documented in in this book and

01:20:05

others is that Einstein was uh I mean

01:20:10

realized because indeed of the way the

01:20:13

fishy way maybe Hilbert was acting that

01:20:16

he had better write his own friends like

01:20:19

Aon Fest lawence who were following what

01:20:21

he was doing by expl in what he was

01:20:24

doing at the same time so that they

01:20:27

would know that he was getting these

01:20:29

results in at time T yes so indeed there

01:20:33

was the correspondence is

01:20:38

delicate okay other

01:20:40

remarks no so thank you again for your

01:20:44

[Applause]

01:20:48

[Music]

01:20:51

attention

Description:

Einstein's path to the discovery of General Relativity, from 1907 to November 1915, will be described. A particular emphasis will be given to the multi-pronged character of Einstein's strategy. Thibault Damour (IHES)

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