The proof is quite similar. The law can also be put in the following form. This law can easily be generalized in the case of covariant and contravariant tensors of any rank. Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind of tensor: Hence the proposition follows at once. The covariant fundamental tensor.
In the invariant expression of the square of the linear element. We call it the "fundamental tensor". Afterwards we shall deduce some properties of this tensor, which will also be true for any tensor of the second rank. But the special role of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravitation makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor.
The contravariant fundamental tensor. According to the law of multiplication of determinants, we have. This can never be the case; so that g can never change its sign; we would, according to the special relativity theory assume that g has a finite negative value. It is a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice of co-ordinates. If however - g remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one.
We would afterwards see that such a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature. It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate.
We do not seek those laws of nature which are covariants with regard to the transformations having the determinant 1, but we ask: First we get the law, and then we simplify its expression by a special choice of the system of reference. Building up of new tensors with the help of the fundamental tensor. Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed.
From this equation, we can in a well-known way deduce 4 total differential equations which define the geodetic line; this deduction is given here for the sake of completeness. Relying on the equation of the geodetic line, we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation. For this purpose, we would first establish the general covariant differential equations.
We achieve this through a repeated application of the following simple law. From which follows immediately that. Here however we can not at once deduce the existence of any tensor. We have thus got the result that out of the covariant tensor of the first rank. In order to see this we first remark that. This is also the case for a sum of four such terms: For this latter case, however, a glance on the right hand side of 26 will show that we have only to bring forth the proof for the case when.
Through addition follows the tensor character of. With the help of the extension of the four-vector, we can easily define "extension" of a covariant tensor of any rank. This is a generalisation of the extension of the four-vector.
We confine ourselves to the case of the extension of the tensors of the 2nd rank for which the law of formation can be clearly seen. It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to 26 we have the expressions. Their addition gives the tensor of the third rank.
It is clear that 26 and 24 are only special cases of 27 extension of the tensors of the first and zero rank. In general we can get all special laws of formation of tensors from 27 combined with tensor multiplication. A few auxiliary lemmas concerning the fundamental tensor. We shall first deduce some of the lemmas much used afterwards. Both the first members of that expression, and the second member of the expression above cancel each other, since the naming of the summation-indices is immaterial.
The last member of that can then be united with first expression above. Anti-symmetrical Extension of a Six-vector. Divergence of the Six-vector. This is the expression for the extension of a contravariant tensor of the second rank; extensions can also be formed for corresponding contravariant tensors of higher and lower ranks.
Divergence of the mixed tensor of the second rank. It is found easily. We can easily convince ourselves that this vanishes identically.
We prove it in the following way; we substitute in The same is true for the sum of the second and third members. Remarks upon the choice of co-ordinates. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplification. I shall give in the following pages all relations in the simplified form, with the above-named specialisation of the co-ordinates.
It is then very easy to go back to the general covariant equations, if it appears desirable in any special case. A freely moving body not acted on by external forces moves, according to the special relativity theory, along a straight line and uniformly.
They are the components of the gravitational field. In the following, we differentiate "gravitation-field" from "matter", in the sense that everything besides the gravitation-field will be signified as matter; therefore [ ] the term includes not only "matter" in the usual sense, but also the electro-dynamic field. Our next problem is to seek the field-equations of gravitation in the absence of matter. For this we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motion for material points.
These vanish then also in the region considered, with reference to every other co-ordinate system. But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed away by any choice of axes, i. Remembering 44 we see that in absence of matter the field-equations come out as follows; when referred to the special co-ordinate-system chosen. It can also be shown that the choice of these equations is connected with a minimum of arbitrariness.
It will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity, together with equations 46 , give us the Newton ian law of attraction as a first approximation, and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors.
My view is that these are convincing proofs of the physical correctness of the theory. In order to show that the field equations correspond to the laws of impulse and energy, it is most convenient to write it in the following Hamilton ian form: Here the variations vanish at the limits of the finite four-dimensional integration-space considered.
It is first necessary to show that the form 47a is equivalent to equations Remembering 31 , we thus have: This equation expresses the laws of conservation of impulse and energy in a gravitation-held.
In fact, the integration of this equation over a three-dimensional volume V leads to the four equations. We recognise in this the usual expression for the laws of conservation.
I will now put the equation 47 in a third form which will be very serviceable for a quick realisation of our object. If we remember that. The third member of this expression cancel with the second member of the field-equations In place of the second term of this expression, we can, on account of the relations 50 , put. The field-equations established in the preceding paragraph for spaces free from matter is to be compared with the equation.
We have now to find the equations which wall correspond to Poisson 's Equation. The special relativity theory has led to the conception that the inertial mass is no other than energy. It can also be fully expressed mathematically by a symmetrical tensor of the second rank, the energy-tensor. If we consider a complete system for example the Solar-system its total mass, as also its total gravitating action, will depend on the total energy of the system, ponderable as well as gravitational.
These are the general field-equations of gravitation in [ ] the mixed form. In place of 47 , we get by working backwards the system. It must be admitted, that this introduction of the energy-tensor of matter cannot be justified by means of the Relativity-Postulate alone; for we have in the foregoing analysis deduced it from the condition that the energy of the gravitation-field should exert gravitating action in the same way as every other kind of energy.
The strongest ground for the choice of the above equation however lies in this, that they lead, as their consequences, to equations expressing the conservation of the components of total energy the impulses and the energy which exactly correspond to the equations 49 and 49a. This shall be shown afterwards. The equations 52 can be easily so transformed that the second member on the right-hand side vanishes. The second term can be transformed according to So that we get.
The expression arising out of the last member within the round bracket vanishes according to 29 on account of the choice of axes. The two others can be taken together and give us on account of 31 , the expression. From the field equations of gravitation, it also follows that the conservation-laws of impulse and energy are satisfied. A comparison with 41b shows that these equations for the above choice of co-ordinates [ ] asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter.
The second member is an expression for impulse and energy which the gravitation-field exerts per time and per volume upon matter. This comes out clearer when instead of 57 we write it in the Form of The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations of gravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We get the equations of the material phenomena completely when the latter is characterised by four other differential equations independent of one another.
The Mathematical auxiliaries developed under B at once enables us to generalise, according to the generalised theory of relativity, the physical laws of matter Hydrodynamics, Maxwell 's Electro-dynamics as they lie already formulated according to the special-relativity-theory. The generalised Relativity Principle leads us to no further limitation of possibilities; but it enables us to know exactly the influence of gravitation on all processes without the introduction of any new hypothesis.
It is owing to this, that as regards the physical nature of matter in a narrow sense no definite necessary assumptions are to be introduced. The question may lie open whether the theories of the electro-magnetic field and the gravitational-field [ ] together, will form a sufficient basis fur the theory of matter.
The general relativity postulate can teach us no new principle. But by building up the theory it must be shown whether electro-magnetism and gravitation together can achieve what the former alone did not succeed in doing.
Let the contravariant symmetrical tensor. If we put the right-hand side of 58b in 57a , we get the general hydrodynamical equations of Euler according to the generalised relativity theory.
Now it is be noticed that the equation 57a is already contained in 53 , so that the latter only represents 7 independent equations. This indefiniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the problem is indefinite in the sense that three of the Space-functions can be arbitrarily chosen. This system 60 contains essentially four equations, which can be thus written: We see it at once if we put.
Instead of 60a we can therefore write according to the usual notation of three-dimensional vector-analysis: The first Maxwell ian system is obtained by a generalisation of the form given by Minkowski. Then remembering 40 we can establish the system of equations, which remains invariant for any substitution with determinant 1 according to our choice of co-ordinates. The equations 60 , 62 and 63 give thus a generalisation of Maxwell 's field-equations [ ] in vacuum, which remains true in our chosen system of co-ordinates.
With the help of 61 and 64 we can easily show that the energy-components of the electro-magnetic field, in the case of the special relativity theory, give rise to the well-known Maxwell-Poynting expressions. Thereby we achieve an important simplification in all our formulas and calculations, without renouncing the conditions of general covariance, as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations.
Still the question is not without formal interest, whether, when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates, the laws of conservation have the form of the equation 56 , and the field-equations of gravitation hold in the form 52 or 52a ; such that on the left-hand side, we have a divergence in the usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation.
I have found out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive work on this subject will not pay, for nothing essentially new comes out of it. This signifies, according to what has been said before, a total neglect of the influence of gravitation. We can assume that this approximation should lead to Newton 's theory. And thanks to the Internet such research into these biographical details now becomes much easier than for those who had to rely on libraries and library networks, taking weeks and months to get a tome and then having to excerpt it and return it swiftly.
This is good stuff. Hopefully people will now read this original material and note the differences between what Einstein said and what people say he said. For example, he spoke repeatedly about the speed of light varying with gravitational potential. See for example this Baez article where Don Koks says this: Einstein talked about the speed of light changing in his new theory.
A curvature of rays of light can only take place when the velocity [speed] of propagation of light varies with position. Thanks for replying Schmi; I shall choose to take your word for it.
No doubt the Internet has made a big difference. Makes it a lot harder to keep genies in bottles. But remember I said revolutionary theories such as the Higgs mechanism. How willing are the community of physicists to at least peruse theories that challenge the status quo? By the way Schmi means my name is in Hebrew. Do you have some examples of journals that do that? First, the Higgs mechanism was not nearly as revolutionary as you seem to think.
It was proposed independently by multiple researchers, including Phil Anderson two years prior to Higgs, in a slightly different setting.
Second, high-energy physics these days is in a desperate enough situation that any promising new model, no matter how outrageous, is considered seriously. When you plot the inhomogeneity using say light-clock elapsed times in an equatorial slice through the Earth and surrounding space, your space-time plot is curved, like this. After a while you notice its path curves left a little.
Now look at the surface of the sea where the wave is. I appreciated your comment. Thanks for the link to Prof. Guess I will share it off my blog, too. One that struck me was Einstein talking about electromgnetic and gravitational fields in In some situations an electromagnetic field is a gravitational field.
That must surely have some bearing on QED and quantum gravity. What it says is…. And the fact that that research was done by a women makes it an even more grievous crime. And to compound the perfidy, these two great geniuses showed it to her to get her approval. What happened was these two theories were totally hyped up in the media, giving the impression that they were theories worth taking seriously.
IMHO the obvious thing that comes out of that is that a clock clocks up some kind of regular cyclical motion. But despite what Einstein said, and despite Magueijo and Moffat pointing out the tautology in http: First, a blog comment is not the place to announce a revolutionary new theory. Actually, if a really good new theory comes along, people will notice it.
Actually, probably less, because today we know that many current theories are only approximations, whereas just before Einstein came on the scene physics was regarded by many as being essentially complete.
That almost every physicist has rejected such a paper as referee? Again, I think not. An endorsement is actually neither necessary nor sufficient, though it can often help. In some fields this is common, in others less so, or not the case at all. Keep in mind that arXiv moderation does keep the signal-to-noise ratio relatively high. However, sometimes they throw the baby out with the bathwater. Assuming that Relativity is more incisive, this is true. General Relativity is a massive achievement, but hard evidence that it is probably right came much later.
Even after having lived for more than 20 years in the USA, he still thought in German and spoke in German if his partner in conversation could understand German. While today there is too much information to read it all, in his day not all was distributed. The golden mean was probably in the s, when Feynman read every issue of the Physical Review cover to cover. Einstein made important contributions to statistical mechanics and thermodynamics, quantum theory, electromagnetism special relativity , mechanics general relativity , solid-state physics specific heat in quantum theory , particle physics photoelectric effect, Bose-Einstein condensation , atomic physics stimulated emission.
He was probably the last physicist who was much more than his contemporaries and perhaps the only other one besides Newton. The sentence structure reminds me of Kant or Hegel in particular. Where Have We Tested Gravity? This entry was posted in Science , Words. December 5, at December 5, at 2: December 5, at 8:
REPRINTED FROM: T H E C O L L E C T E D PA P E R S O F Albert Einstein VOLUME 6 THE BERLIN YEARS: WRITINGS, – A. J. Kox, Martin J. Klein, and Robert Schulmann EDITORS József Illy and Jean Eisenstaedt CONTRIBUTING EDITORS Rita Fountain and Annette Pringle EDITORIAL ASSISTANTS ENGLISH.
THE COLLECTED PAPERS OF Albert Einstein VOLUME 6 THE BERLIN YEARS: WRITINGS, – A. J. Kox, Martin J. Klein, and Robert Schulmann EDITORS Alfred Engel, TRANSLATOR Engelbert Schucking, CONSULTANT DOC. 30 The Foundation of the General Theory of Relativity (pp. – in translation volume) .
25 rows · Albert Einstein (–) was a renowned theoretical physicist of the . Einstein's Original General Relativity Paper, English, "Hamilton’s Variational Principle and General Relativity Theory", German, by Albert Einstein, Historical Account of Einstein's Review Paper for his final General Relativity Paper.
Einstein’s Papers Online. Ten years later he is putting the final touches on general relativity, whose centennial we will be celebrating next year. Here is the opening of one foundational paper from , The Formal Foundation of the General Theory of Relativity: In recent years I have worked, in part together with my friend Grossman. The Foundation of the Generalised Theory of Relativity By A. Einstein. The mathematical apparatus useful for the general relativity theory, lay already complete in the "Absolute Differential Calculus", a study of the mathematical literature is not necessary for an understanding of this paper.