A Help on some equations in Einstein's original papers

[PeterDonis]

There's no unimodular constraint there, but it doesn't rewrite the EFE.

It "doesn't rewrite the EFE" in the sense that the field equation obtained from Hamilton's principle is formally identical to the field equation obtained by a different route in the main body of the paper. But the fact that there is no unimodular constraint when the same equation is derived from Hamilton's principle tells you that that constraint is not required for the EFE--the EFE is valid in any coordinate chart, not just unimodular ones. So the answer to the question you asked in post #38 is exactly what I responded in post #39.

 

[Pyter]

@PeterDonis the weird thing is that in the main paper Einstein states that $R_{\mu\nu}$ has a symmetrical and a skew-symmetrical part which vanishes only under the unimodular constraints. Shouldn't it have a symmetrical part only by derivation from the Riemann-Christoffel curvature tensor?
In the Hamilton addendum there are only 4 equations, the EFE are 10. In that sense I said that is doesn't rewrite the EFE.

 

[PeterDonis]

In the Hamilton addendum there are only 4 equations

The field equation for gravitation in the addendum (equation 4, p. 228 of the translation) has ten components. The addendum rewrites the equation in various ways, but none of them change the number of components.

 

[PeterDonis]

in the main paper Einstein states that $R_{\mu \nu}$ has a symmetrical and a skew-symmetrical part which vanishes only under the unimodular constraints.

Where?

 

[Pyter]

Where?

Eq. (44). But now that I look at them better, the two parts B and S are both symmetrical.

 

[Pyter]

The field equation for gravitation in the addendum (equation 4, p. 228 of the translation) has ten components. The addendum rewrites the equation in various ways, but none of them change the number of components.

As I interpreted them, Eq. 4 are not the EFE but ten Euler-Lagrange equations which, after the subsequent discussion, reduce to the 4 components of 22).

 

[PeterDonis]

As I interpreted them, Eq. 4 are not the EFE but ten Euler-Lagrange equations

In the variational method, the Euler-Lagrange equations are the EFE.

the 4 components of 22

Equation 22 is a set of equations of motion for the matter, not the EFE.

 

[Pyter]

In the variational method, the Euler-Lagrange equations are the EFE.

Equation 22 is a set of equations of motion for the matter, not the EFE.

For sure the E-L equations are a starting point to derive the EFE, anyway in the subsequent discussion Einstein doesn't derive them, but only the 22) which, as you say, are equations of motion for the matter and not the EFE (they link the $g_{\mu\nu}$ to the components of the stress-energy tensor like the EFE, though).
I would really be interested in seeing all the passages to derive the EFE from the 4). Maybe checking out chapter X of the English book I'll get some new insights.

 

[PeterDonis]

For sure the E-L equations are a starting point to derive the EFE

No, they are the EFE. See below.

anyway in the subsequent discussion Einstein doesn't derive them

He doesn't "derive" them explicitly because he considers it to be obvious that the variation of equation (1a) (the action integral) with respect to $g^{\mu \nu}$ gives equation (4) (the field equation, i.e., the EFE) as the Euler-Lagrange equation. If it's not obvious to you, more detailed derivations (in more modern notation) are given in many GR textbooks; see, for example, section 21.2 of Misner, Thorne, and Wheeler. (Also see the discussion in item 2. of Box 17.2 of MTW.)

I would really be interested in seeing all the passages to derive the EFE from the 4

(4) is the EFE. See above.

 

[vanhees71]

I recommend to also read Landau+Lifshitz vol. 2, which gives a very clear and straight to the point derivation of the EFE using the action principle. Though all the arguments are already in Einsteins papers of 1915 and 1916 it's much easier to learn the theorx first from a modern textbook and then looking at the original papers, though they are master pieces not only in the physics content but also of very carefully formulated science prose which is true for almost all of Einstein's papers.