about the paradoxe of the twin

[Zefram Cochrane]

Partie 1

Bonjour,

http://forums.futura-sciences.com/as...ml#post4496843

Je me dois d'être honnête avec vous, j'ai affreusement pompé ma démo du livre "The principle of relativity' qui regroupe plusieurs articles sur le sujet. Je vais donc la développer en reproduisant l'article en question. Comme il est écrit en Anglais vous n'aurez aucun mal à faire la part des commentaires que je vais parsemer en route.

ON THE INFLUENCE OF GRAVITATION ON THE PROPAGATION OF LIGHT
By A.Einstein

In a memoir publish four years ago ( en l'occurence 1907 ) I tried to answer the questions whether the propagation of light is influenced by gravitation. I return to this theme, because my previous presentation of the subject does not satisfied me , and for a stronger reason, because I now see one of the most importante consequences of my former treatement is capable of being tested experimentally. For it follows from the theory here to be brought forward, that rays of light, passing close to the Sun, are deflected by its gravitational field, so that the angular distance between the Sun and a fixed star appearing near to it is apparently increased by a nearly second of arc.
In the course of these reflexions further results are yielded which relate to gravitation. But as the exposition of the entire group of considerations would be rather difficult to follow, only a few quite elementary reflexions will be given in the following pages, from which the reader will readily be able to inform himself as to the suppositions of the theory and its line of thought. The relations here deduced, even the theorical foudation is sound, are valid only to a first approximation.

$ 1. A Hypothesis as to the Physical Nature of the Gravitational Field

In a homogeneous gravitational field ( acceleration of gravity [ que j'ai noté g dans ma démo ] let there be a stationnary system of co-ordinates K, oriented so that the lines of force of the gravitational field run in the negative direction of the axis of . In a space free of gravitational field let there be a second system of co-ordinates K', moving with uniform acceleration () in the positive direction of the axis of z.

[ Initialement j'avais choisi des " pour décrire le rédférentiel de l'astronaute pour le différentier des ' qui son couramment utilisés pour des référentiels en MRU par rapport au référentiel supposé fixe. J' ai inversé K et K' pour que vous puissiez directement en avoir la correspondance dans le texte

Je vais donc reposer les conditions
Référentiel K ( t, z ) l'astronaute
Référentiel K' ( t' , z') le cosmonaute
L'origine O de K accélère vers O' de K' avec une accélération uniforme a
L'origine O' de K' est en chute libre vers O de K dans un champ de gravitationuniforme

Dans le texte:
Référentiel K ( t, z ) l'astronaute stationnaire dans un champ de gravitation
Référentiel K' ( t' , z') l'astronaute accélérant vers le cosmonaute]

Relatively to K, as well as relatively to K', matérial points which are not subjected to the action of other material points, move in keeping with the equations


for the accelerated system K' this follows directly from Galileo's principle, but for the system K, at rest in a homogeneous field, from the experience that all bodies in such a field a equally and uniformly accelerated. This experience, of the equal falling of all bodies in the gravitation field, is one of the most universal which the observation of nature are yielded; but in spite of that the law has not found any place in the foundations of our edifice of the physical universe [Ah? Et F = ma alors? ]


But we arrive at a very satisfactory interpretation if this law of experience, if we assume that the systems K and K' are physically exactly equivalent, that is, if we assume that we may just as well regard the system K as being in a space free from gravitationnal field, if we then regard K as uniformly accelerated.

[C'est à priori l'énnoncé du principe d'equivalence que j'ai également et succintement décrit dans ma démo :

Soit, à l'instant initial , un astronaute (t, z) accélère uniformément (avec une accélération a) en direction d'un cosmonaute (t' , x') supposé fixe; du point de vue de l'astronaute, à t=0, z = h.

Les conditions sont équivalentes à celle du cosmonaute en chute libre dans un champ de gravitation uniforme avec une accélération g = a. depuis la position z = h. ]

This assumption of exact physical equivalence makes it impossible for us to speak of the absolute acceleration of the system de reference, just as the usual theory [appelée plus tard special theory] of relativity forbids us to talk the absolute velocity of a system*; and it makes the equal falling of all bodies in a gravitationnal field seem a matter of course.

Of course we cannot replace any arbitrary gravitationnal field by a state of motion of the system without the gravitationnal field, any more than, by a transformation of relativiy, we cannot transform all points of a medium [???] in any kind of motion to rest.

[ Je pense cependant que ce point est important même si je ne suis pas certain du sens de la dernière phrase, il me semble que c'est ce qui fait la différence entre l'accélération gravitationnelle et une accération dans un espace-temps de Minkowski.
L'accélération gravitationnelle est induite par la courbure de l'espace-temps (en première approximation), elle est extrinsèque. L'observateur supposé fixe ressent l'accélération gravitationnelle.
L'accélération dans un espace-temps de Minkowsk est induite par le mobile (l'astronaute dans le référentiel K ) et est intrinsèque, l'observateur supposé fixe ne ressent pas l'accélération.

J'ai une interrogation à ce propos concernant l'accélération alectrostatique: elle est extrinsèque mais courbe t'elle l'espace-temps? ]

As long as we restrict ourselves to purely mechanical processes in the realm where Newton's mechanics holds sway, we are certains of the equivalence of the systems K and K'. But this view of ours will not have any deeper significance unless the systems K and K' are equivalent with respect to all physicals processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K'. by assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. For by theoritical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitation field. We shall now show, first at all, from the standpoint of the ordinary theory of relativity, what degree of probability is inherent in our hypothesis.

$2. On the Gravitation of Energy

One result yielded by the theory of relativity is that the inertia mass of a body increases with the energy in contains [nous sommes en 1911]; if the increase of energy amounts to E, the increase in inertia mass is equal to E/c² , when c denote the velocity of light.

[ pour être plus précis, m est la masse au repos]

Now is there an increase of gravitating mass corresponding to this increase of inertia mass? If not, then a body would fall in the same gravitational field with warying acceleration according to the energy it contained. That highly satisfactory result of the theory of relativity by which the law of the
conservation of mass is merged in the law of the conservation of energy could not be maintened, because it would compel us to abandon the law of the conservation of mass in its old form for inertia mass, and maintain it for gravitating mass.
But this must be regarded as very improbable. On the other hand, the usual theory of relativity does not provide us with any argument from wich to infer that the weight of the body depends on the energy contained in it. But we shall show that our hypothesis of the equivalence of the systems K and K' and K' gives us gravitation of energy as a necessary consequence.

Let the two materiels systems S1 and S2, provided with instruments of measurement, be situated on the z-axis of K at a distance form each other * so that the gravitation potential in S2 is greateur than than in S1 by
The dimensions of S1 and S2 are regarded as infinitely small in comparison with
[ dans ma démo, le gamma étant réservé au facteur de Lorentz ]

Let a definite quantity of energy E be be emitted from S2 towards S1. Let the quantities of energy in S1 and S2 be measured by contrivances wich - brought to one place in the system z and there compared – shall be perfectly alike. As to process of this conveyance of energy by radiation we can make no a priori assertion, because we do not know the influence of the gravitational field on the radiation and the measuring instruments in S1 and S2.
But by our postulate of the equivalence of K and K' we are able, in place of the system K in a homogeneous gravitational field to set the gravitation-free system K', which moves with uniform acceleration in the direction of positive , and with the z-axis of wich the materiel system S1 and S2 are rigidly connected.

We judge of the process of the transference of energy by radiation from S2 to S1 from a system Ko, which is to be free of acceleration. A t moment when the radiation energy E2 is emitted from S2 toward S1, let the vélocity of K' relatively to Ko be zero.

[ dans ma démo, la radiation d'énergie est le cosmonaute de masse m. À t = t'=0 la vitesse v' de l'astronaute est nulle ]

The radiation will arrive at S1 when the time has elapse (to a first approximation ).
[ Ici gros problème, comme nous le verrons dans un instant, puisque dans ma démo j'ai fait en sorte de lever l'approximation, autant le temps écoulé ne semble pas influer sur la démo, mais par contre il y a une réflexion nécessaire sur la distance à prendre en compte.]

But at this moment the velocity of S1 (astronaute) relatively to Ko is .

[Sachant qu'au final, Einstein nous donne une formule où c dans le référentiel K est fonction de c' dans le référentel K', je pose pour ma part
dans le référentiel de l'astronaute K
a est l'accélérétation gravitationnel du cosmonaute , h est la distance qui sépare le cosmonaute de sa position initiale à t'= t = 0 quand l'astronaute et le cosmonaute se croisent à une vitesse v du point de vue de l'astronaute v' du point de vue du cosmonaute.
Et donc pour le cosmonaute K'

Du point de vue du cosmonaute stationnaire en apesenteur, l'astronaute accélère vers lui avec une accélération a' donc
Du point de vue de l'astronaute stationnaire dans un champ de gravitation uniforme, le cosmonaute est en chute libre et est soumis à une accélération a = g. Pour plus de clarté, nous dirons donc que ]

Therefore by the ordinary theory of relativity the radiation arriving at S1 does not pocess the energy E2, but a greater energy E1, which is relatedto E2 to a first approximation by ve equations*

see above pp 69-71 [j'espère ne ppas devoir me taper les trois pages à copier sinon je devrais aussi refaire le bouquin]
[pour résumer ces pages correspondent au chapître DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT
La réponse étant : If a body gives off the energy L in a form of radiation, its mass diminishes by L/c² ]

(1)

[Was ???
réponse à l'intéressé de http://forums.futura-sciences.com/as...ml#post3439586

Salut,

Bienvenue sur le forum.

Expérimentalement aucune variation de la vitesse de la lumière (dans le vide) n'a jamais été constatée, ni avec la gravitation ni avec autre chose.

Tu manipules des équations newtonienne dans un contexte (RR) où la gravité newtonienne est fausse. Paaaaaas bieeeeen !

Il y a beaucoup d'autres erreurs dans le texte mais inutile de continuer au-delà de ça.

Je salue quand même l'effort

Mais pourquoi ne pas lire un bon bouquin sur la relativité générale et la géométrie de Schwartzchild ? Par exemple le livre Gravitation de Misner, Thorn et Wheeler. Je l'ai trouvé super agréable à lire.

C'est peut être par Gravitation mais The principle of relativity est pas mal non plus et en plus il est en Anglais.
-----

[Zefram Cochrane]

partie 2

Initialement, pour l'astronaute dans le référentiel K, le cosmonaute a une énergie Eo = mc²
lors que le cosmonaute croise l'astronaute, le cosmonaute a une énergie
Pour le cosmonaute dans le référentiel K' , l'astronaute a une énergie E'o = mc'² ( l'astronaute et le cosmonaute ont la même masse) et il a une énergie
Comme ils on la même masse, le travail fourni lors de l'accélération gravitationnelle ou inertielle selon les point de vue est le même.
Initialement Eo = E'o parce que c=c'
mais, lorsque l'astronaute arrive au niveau du cosmonaute, la vitesse de la lumière (ici c) dans le référentiel K a varié.Dans le référentiel K', c' est resté le même.
Nous avons donc parce que ]

By our assumtpion exactly the same K, which is not accelerated, but is provided with a gravitationnal field. In that case we may replace by the potential of the gravitation vector in S2, if the arbitrary constant of in S1 is equated to zero.

When we have the equation
(1a)

[En mécanique classique, on aurait les considération suivantes :



[B] Parce que l'astronaute accélère dans espace-temps de Minkovski Je peux me permettre d'utiliser le TEC [/TEX] Sinon je ne le pourrais pas.

D'après le TEC,


Comme pour les champs faibles , sachant que :


En combinant l'approximation des champs faibles avec le TEC j'obtiens que :


Soit :
v est la vitesse relative de l'astronaute lorsqu'il arrive en O'

http://forums.futura-sciences.com/as...ml#post4164579
La first approximation est levée]

This equation expresses the law of energy for the process under observation. The energy E1 arriving at S1 is greater than the E2, measured by the same means, which was emitted in S2, the excess being the potential energy of the mass E2/c² in the gravitational field. It thus proves that for the fulfilment of the principle of energy we have to ascribe to the energy E, before its emission in S2, a potential energy due to gravity, which corresponds to a gravitationnal masse E/c². Our assumption of the equivalence of K and K' thus removes the difficulty mentionned at the beginning of this paragraph which is left unsolved by the ordinary theory of relativity.
The meaning of this result is shown particuly clearly if we consider the following cyrcle of operations :

1. The Energy E, as measured in S2, is emitted in the form of radiation in S2 toward S1, where, by the result just obtained, the energy , as mesured in S1, is absorbed.
2. A body W of mass M is lowered from S2 to S1, work in the process.
3. The energy E is transferred from S1 to the body W while W is in S1. Let the gravitationnal mass M be thereby changed so that it acquire the Value M'.
4. let W be again raised to S2, work being done in the process.
5. Let E be transferred from W to S2.

The effect of this circle is simply that S1 has undergone the increase of energy and the energy has been conveyed to the system in form of mechanichal work. By the principle of energy, we must therefore have


or

The increase in gravitationnal mass is thus equal to E/c², and therfore equal to the increase in inertia mass as given by the theory of relativity.
The result emerges still more directly from the equivalence of the system K and K', according to Which the gravitationnal mass in respect of K is exactly equal to the inertia mass in respect of K', the balance will indicate the apparent weight on account of the inertia mass of Mo. If the quantity of energy E be transferred to Mo, the spring balance, by the law of the inertia of energy, will indicate . By reason of our foudamental assumption exactly the same thing must occur when the experiment is repeated in the system K, that is, in the gravitation field.


[ Pour ma part, j'avais tenu le raisonnement suivant :
Etant donné que du point de vue du cosmonaute (K'), son énergie n'a pas varié d'un iota et est égale à mc'². Donc lorsque le cosmonaute arrive en O on peut écrire :


->

Du point de vue de l'astronaute l'énergie gravitationnelle accumulée par le cosmonaute est égale à l'énergie correspondant au travail, du point de vue du cosmonaute, accumulée par l'astronaute lors de son accélération pour aller de sa position initial à la sienne (cosmonaute) qu'il atteindra avec une vitesse v'.

on a donc les relations suivantes :










v et v' étant les vitesses relatives de K et K' lors que O et O' sont confondues.

J'avais comme loi de vairation de la lumière :

C' est la vitesse de la lumière dans le référentiel K' fixe.
C est la vitesse de la lumière dans le référentiel K mobile.

Mais Albert Einstein donne plus loin comme relation :


Les gammas sont ici des facteurs de Lorentz respectivement du point de vue de K et K' et sont égaux.
Soit K référentiel fixe ou la vitesse de la lumière est c . L'énergie au repos Eo d'un mobile de masse m est mc² . Après accélérétion si E' = \gamma Eo [/TEX] si ]

.3 Time and the Velocity of Light in the Gravitationnal Field.

If the radiation emitted in the uniformly accelerated system K' in S2 toward S1 had the frequency relatively to the clock in S2, then relatively to S1, at its arrival in S1, it no longer as the frequency relatively to an identical clock in S1 but a greater frequency such to a first approximation

(2)

For if we again introduce the unaccelerated system of reference Ko, relatively to wich, at the time of emission of light, K' has no velocity, then S1, at the time of arrival of the radiation at S1, has relatively to Ko; the velocity from wich, by doppler principle, the relation as given results immediately.

In agreement with our assumption of the equivalence of the system K' and K, this equation also holds for the stationnary system of co-ordinate K, provided with a uniform gravitational field, if in it, the tranference by radiation takes place as described. It follows, then, that a ray of light emitted in S2 - will, at its arrival in S1, posses a different frequency - measured by a identical clock in S1. For we subtitute the gravitational potential of S2 – that of S1 being taken as zero- and assume that the relation which we have deduced for the homogeneous gravitational field also hold for other forms of field. Then
(2a)

[ ]

This result (which by our deduction is valid to a first approximation) permits, in the first place, of the following application. Let be the vibration-number of an elementary light-generator measured by a delicate clock at the same place. Let us imagine them both at a place on the surface of the Sun ( where our S2 is located). Of the light there emitted, a protion reach the Earth (S1), where we measure the frequency of the arriving light with a clock U in all respects resembling the one mentioned, the by (2a)



Where in the (negative) of gravitational potential between the surface of the Sun
and the Earth.

[
ici le potentiel gravitationnel est considéré égal à 0]

Thus according to our view the spectral lines of sunlight, as compared with the corresponding spectral lines of terrestrial sources of light, must be somewhat displaced toward the red, in fact by the relative amount



If the conditions under which the solar bands arise were exactly known, this schifting would be suceptible of measurement. But as other influences ( Pressure, temperature) affect the position of the centres of the spectral lines, it is difficult to discover whether the inferred influence of the gravitational potential really exist*

L.F.Jewell (Journ, de Phys , 6, 1897, p84 ) and particularly Ch. Fabry and H. Boisson ( Comptes rendus , 148, 1909 , pp, 688-690) have actually found such displacements of fine specral lines toward the red end of spectrum, of the order of magnitude here calculated, but have ascribed thel to an effect of pressure in the absorbing layer.

On a superficial consideration equation (2), or (2a), respectively, seems to assert an absurdity. If there is constant transmission of light from S2 to S1, how can any other number of periods per second arrive in S1, than than is emitted in S2? But the answer is simple. We cannot regard or respectively simply as frequencies (as the number of periods per second) since we have not yet determined the time in the system K. What \nu_2 denotes is the number with reference to the time-unit of the clock U in S2, while denotes the number of periods per second with reference to identical clock in S1. Nothing compels us that the clocks U in different gravitation potentials must be regarded as going at the same rate. On the contrary, we must define the time K in such a way that the number of wave creast and throughs between S2 and S1 is independent of the absolute value of time; for the process under observation is by nature a stationnary one. If we did not satisfy this condition of time by the application of which time would merge explicitly into the laws of nature, and this would certainly be unnatural and unpractical. Therfore the two clocks in S1 and S2 do not both give the "time correctly. If we measure time in S2 with a clock which goes more slowly than the clock U in the same place. For when measured by such a clock the frequency of the ray of light which is considered above is a its emission in S2

and is therfore, by (2a) equal to the frequency of the same ray of light on its arrival in S1.

[ Ici, Einstein vient de dire que dans un champ de gravitation, l'énergie d'un photon reste constante.
Il est émis en S2 avec l'énergie
Hors :
si

alors

->
->

La fréquence du photon émis en S2 et égal à la fréquence de ce photon à sa réception en S1.
Le fait est que cette énergie est perçue différemment parce que dans le référentiel K la vitesse de la lumière est c et que dans le référentiel K' la vitesse de la lumière est c' ]

This has a consequence which is fundamental importance for our theory. For if we measure the velocity of light at different places in the accelerate gravitation-free.system K', employing clocks U of identical constitution, we obtain the same magnitude at all these places. The same holds good, by our foudamental assumption, for the system K as well. But from what has just been said we must use clock of unlike constitution, for measuring time at place which, relatively to the origin of the co-ordinates, has the gravitationnal potential , we must employ a clock which – when removed to the origin of co-ordinates – goes
times more slowly than the clock used for measuring time at the origin of co-ordinates. If we call the velocity of light at the origine of co-ordinates then the velocity of light c at a place with the gravitation potential will be given by the relation

(3)

The principle of the constancy of the velocity of light holds good according to this theory in a different form from usually underlies the ordinary theory of relativity.

CONCLUSION
Après cela, vouloir faire varier c est anti-relativiste
Pour résumer :
Dans le cadre de l'espace-temps de Minkowski :
on peut définir un champ de gravitation dans lequel la vitesse de la lumière varie en :


L'énergie d'un objet matériel de masse m en chute libre dans le champ de gravitation se conserve.
L'énergie d'un photon suivant une trajectoire radiale dans un champ de gravitation reste constante et sa célérité est égale à la vitesse locale de la lumière c.

Cordialement,
Zefram

[Zefram Cochrane]

Re,
il faut lire dans la partie 2

et dans la conclusion,


petit problème de signe (le potentiel est une quantité négative)

Cordialement,
Zefram