On Friday, October 29, 2021 at 9:21:38 PM UTC-7, Richard Hertz wrote:
> I extracted a couple of posts lost in another thread's stack, to write them
> here, as an OP.
>
> -----------------------------------------------------------------------------------------------------------------
>
> χ Θᵤᵥ = Γᵤᵥ was proposed and developed by Marcel Grossmann in the 1913 Entwurf, co-authored with Einstein, while in Zurich:
>
> https://einsteinpapers.press.princeton.edu/vol4-doc/325
>
> Outline of a Generalized Theory of Relativity and of a Theory of Gravitation.
> I. Physical Part by A. Einstein
> II. Mathematical Part by M. Grossmann
>
> Published by: Teubner, Leipzig (1913) Reprinted in: Zeitschrift für Mathematik und Physik 62 (1914)
>
> It starts (Einstein's part) with:
> -------------------------------------------------------------------------------------------------------
> § 1. Equations of motion of the material point in the static gravitational field.
>
> According to the usual theory of relativity, a point that moves without any force moves according to the equation
>
> (1) δ { ∫ ds } = δ { ∫ √(- dx² - dy² - dz² + c² dt²) } = 0
>
> Because this equation says nothing else than that the material point moves in a straight line and uniformly. This is the equation
> of motion in the form of Hamilton's principle; because we can also say that
>
> (1a) δ { ∫ H dt } = 0 where H = ds/dt m
>
> is set if m is the rest mass of the material point. This results in the known impulse Jx, Jy, Jz and energy E of the moving point:
>
> (2) Jx = m ∂H/∂x = m ẋ /√(c² - q²) ; etc
>
> E = ∂H/∂ẋ ẋ + ∂H/∂ẏ ẏ + ∂H/∂ż ż - H = mc²/√(c² - q²)
>
> This method of representation differs from the usual one only in that Jx, Jy, Jz and E still have a factor c in the latter. But since c is constant in the ordinary theory of relativity, the system given here is equivalent to the one usually given. The only difference is that J and E have different dimensions than in the usual representation.
> ........................................................................................................................
>
> And, just like that, he introduced the modifications of momentum and energy following the 1905 papers (redefining Hamilton).
> The point § 5 introduces Riemann et. all through Poisson, with a lot of assumptions and zero subleties:
>
> --------------------------------------------------------------------------------------------------------------------
> § 5. The differential equations of the gravitational field.
>
> After we have established the momentum-energy equation for the material processes (mechanical, electrical and other processes)
> with reference to the gravitational field, we still have the following task. Let the tensor Θᵤᵥ be given for the material process.
>
> Which are the differential equations that define the quantities g_ik, and allow the gravitational field to be determined?
>
> In other words, we are looking for the generalization of Poisson's equation
>
> Δφ = 4 π k ρ
>
> We have not found such a perfectly inevitable method for solving this problem as for solving the problem discussed above.
>
> It was necessary to introduce some assumptions, the correctness of which appears plausible, but is not evident.
>
> The generalization you are looking for is likely to be of form
>
> (11) χ Θᵤᵥ = Γᵤᵥ
>
> where χ is a constant, Γᵤᵥ is a contravariant tensor of the second order, which is derived from the fundamental tensor gᵤᵥ by
> differential operations. According to Newton-Poisson's law one would be inclined to demand that these equations (11)
> should be of the second order.
> .........................................................................................................................
>
>
> In point 7 of the 1913 Entwurf, Einstein could see how deeply he and Grossmann went into muddy waters
> by using Riemann's non-euclidean geometry, so he ask (and warn to future generation of retarded relativists):
>
> -----------------------------------------------------------------------------------------------------------------
> § 7. Can the gravitational field be traced back to a scalar?
>
> In view of the undeniable complexity of the theory of gravitation represented here, we must seriously ask ourselves
> whether the view that has hitherto been exclusively held, according to which the gravitational field is reduced to a
> scalar Φ, is not the only obvious and justified one.
>
> I will briefly explain why we think we have to answer this question NEGATIVELY. Characterization of the gravitational
> field by a scalar presents a path which is completely analogous to the one taken above.
>
> One starts with the equation of motion of the material point in Hamiltonian form
>
> δ { ∫ Φ ds } = 0
>
> where ds is the four-dimensional line element of the usual relativity theory and Φ is a scalar, and then proceeds quite
> analogously to the previous one, without having to leave the usual relativity theory.
>
> Here, too, the material process of any kind is characterized by a stress-energy tensor Tᵤᵥ . But with this view a scalar is
> decisive for the interaction between the gravitational field and the material process. This scalar, as Mr. Laue pointed out
> to me, can only be
>
> Σᵤ Tᵤᵤ = P
>
> WHICH I want to call the "Laue Scalar". ONE can do justice to the principle of the equivalence of INERTIAL and
> GRAVITATIONAL mass here to a certain extent. Mr Laue pointed out to me that for a closed system it is CERTAIN that
>
> ∫ P dV = ∫ T₄₄ dτ
>
> *** HERE IS REMARKING THAT, IN THIS WAY, THERE ARE NO GUARANTIES THAT INERTIAL AND GRAVITATIONAL MASS BE EQUAL.
>
> *** FROM "§ 6. Influence of the gravitational field on physical processes, especially on electromagnetic processes.":
>
> *** "Because momentum and energy play a role in every physical process, the latter in turn determine the gravitational field
> *** and are influenced by it, the variables gᵤᵥ determining the gravitational field must appear in all physical systems of equations.."
>
> *** THIS REMARK IS KEY FOR THE USE OF RICCI AND LEVI-CIVITA METHOD TO INCORPORATE GRAVITATION IN GR. IN THE
> LINE ELEMENT ds² = Σᵤᵥ gᵤᵥ dxᵤ dxᵥ, 10 OUT OF 16 TERMS CAN BE ASSOCIATED WITH GRAVITATIONAL POTENTIALS
> OR DYNAMIC FIELDS.
>
> *** GROSSMAN, A MATHEMATICIAN EXPERT IN DIFFERENTIAL GEOMETRY FOR MORE THAN 12 YEARS (NOT A GENIUS, HOWEVER)
> APPRECIATED HOW COMPATIBLE LEVI-CIVITA GEOMETRY WAS WITH A FOUR DIMENSIONAL SPACETIME. IN FACT, THE ONLY
> PURE MATHEMATICAL THEORY AVAILABLE BY 1912 TO DEAL WITH THE LINE ELEMENT ds² = Σᵤᵥ gᵤᵥ dxᵤ dxᵥ.
>
>
> From this one can see that, according to this view, its total energy is decisive for the WEIGHT of a closed system.
> The WEIGHT of incomplete systems would depend on the orthogonal stresses T₁₁ etc. to which the system is subjected.
> This gives rise to consequences that seem unacceptable to me, as the example of cavity radiation will show.
> It is well known that the scalar P disappears for the radiation in a vacuum. If the radiation is enclosed in a massless reflecting box,
> its walls experience tensile stresses which cause the system - taken as a whole - to become a mass ∫ P dτ, which is the energy
> E corresponds to the radiation.
>
> *** HERE, EINSTEIN IS TRYING TO DEAL WITH GRAVITATIONAL POTENTIAL AND GRAVITATIONAL MASS IN THE SAME WAY HE DID
> IN HIS 1911 PAPER, WHERE HE STATED:
>
> *** M'.g.h - M.g.h = E.g.h/c² OR (M' - M) = E/c²
>
> *** "The increase in gravitational mass is thus equal to E/c², and therefore equal to the increase in inertial mass as given
> by the theory of relativity."
>
>
> Instead of enclosing the radiation in a hollow box, I imagine it to be limited
>
> 1. through the reflective walls of a fixed shaft S,
> 2. by two vertically movable reflective walls W1 and W2, which are firmly connected to one another by a rod.
>
> In this case, the mass ∫ P dτ of the movable system is only the third part of the value that occurs in a box that is movable as a
> whole. In order to raise the radiation against a gravity field, one would only have to spend the third part of the work than in the
> case just considered, where the radiation is enclosed in a box.
>
> *** REMEMBER THAT THE BOX IS MASSLESS. EINSTEIN IS PULLING OUT 1905 HASENHORL PAPER WHERE M = 4/3 E/c².
>
> *** OBSERVE THAT, ACCORDING TO HASERNHORL, THE INCREASE IN MASS BY RADIATION IS M = 4/3 E/c² = M = 1/3 E/c² + E/c².
>
> *** SO, 1/3 OF MASS E/c² IS "WASTED" IN WORK, IN ORDER TO EQUAL "HIS FORMULA" M = E/c² FOR A CLOSED SYSTEM.
>
> *** AND, IN THIS PURE SAMPLE OF SOPHISTRY AND FALLACIES, HE USE HASENHORL TO TORN DOWN VON LAUE ARGUMENTS.
>
>
> This seems unacceptable to me. I must admit, of course, that for me the most effective argument for rejecting such a theory is
> based on the conviction that relativity is not only against orthogonal linear substitutions, but against a much broader group of substitutions.
>
> But we are not justified to assert this argument because we were unable to find the (most general) substitution
> group that belongs to our gravitational equations.
>
> *** REPHRASING IT: GROSSMANN COULDN'T FIND AN EXPRESSION FOR THE GRAVITATIONAL FIELD, BUT WE HAVE TO PUBLISH
> THIS CRAP ANYWAY. WE DID A LOT OF P.R. AND PUBLICITY STUNTS TO DISCREDIT OTHER COMPETING THEORIES, SO WE
> ARE FORCED TO PUBLISH THIS SCRATCH (ENTWURF), TO HAVE A CHANCE AGAINST NORDSTROM AND MIE.
>
> *** WE CAN'T PUBLISH OUR "SECRET CALCULATIONS" WITH BESSO, ABOUT MERCURY, BECAUSE WE GOT THIS STUPID FORMULA:
>
> *** δθ = - 4π.K.M.Rs².(Tm/Ts)/(10.c².Rm³) = - 2.3"/century (ACTUALLY, AFTER CORRECTING ERRORS, GIVES -0.0012”/century)
>
> https://groups.google.com/g/sci.physics.relativity/c/oLRiO1XB0fU/m/dI7HBmg2AwAJ
>
> *** SO, WE ARE PUBLISHING 32 PAGES OF "STATE OF THE ART" CRAP. PLEASE, WAIT 30 MONTHS FOR HILBERT COMING IN.
>
> -------------------------------------------------------------------------------------------------------------------
> NOTE: It can be noticed that, without any proof about the Hasenhorl-like "thought experiment", assess a loss of 1/3 of the energy
> as work against gravity when attempts to raise E within an enclosed box or just as pure energy.
>
> *** Note that von Laue (Planck behind) was his ADVISOR on the theoretical part. What follows next is Grossmann part:
>
> https://einsteinpapers.press.princeton.edu/vol4-doc/346
> ---------------------------------------------------------------------------------------------------------------
> II. Mathematical part. By MARCEL GROSSMANN.
>
> The mathematical tools for the development of the vector analysis of a gravitational field, which is characterized by the
> invariance of the line element
>
> ds² = Σᵤᵥ gᵤᵥ dxᵤ dxᵥ
>
> go back to Christoffel's fundamental treatise on the transformation of quadratic differential forms. Based on Christoffel's
> results, Ricci and Levi-Civita developed their methods of absolute differential calculus independent of the coordinate system,
> which allow the differential equations of mathematical physics to be given an invariant form.
>
> But since the vector analysis of the Euclidean space related to any curvilinear coordinates is formally identical to the vector
> analysis of any manifold given by its line element, there are no difficulties in using the vector-analytical concept formations
> as they were in the last few years by Minkowski, Sommerfeld, Laue et al for the theory of relativity, to be extended to the
> general theory of Einstein above.
>
> .......... 15 pages of differential geometry, trying to marry the curvature of
> donuts to energy, motion, time and gravity.
>
> ---------------------------------------------------------------------------------------------------------------

Richard, you comments to the papers are complete, total, idiocies. Sorry.

--
Jan