Help on some equations in Einstein's original papers

[PAllen]

This condition on metric determinant is typically called the unimodular coordinate condition. It can always be achieved, though you may need more coordinate patches to cover a manifold than with other coordinate choices. In his early years, Einstein very much liked to impose this condition.

 

[haushofer]

See also "The road to relativity " of Gutfreund and Renn, page 71. The unimodular restriction imposes a significant simplification in the field equations.

 

[haushofer]

@PeterDonis I mean if it's even possible to find a non-local coordinate chart where that condition is satisfied everywhere.
For instance, for the 2-sphere of radius r inheriting the $E^3$ ambient metric, we have $g_{\mu \nu} = diag(r^2 sin^2 \theta, r^2)$ which doesn't satisfy the condition. Does it mean that we can't use spherical coordinates as a chart? And what should we use?

Why not? You have four coordinates, and the unimodular restriction is just one condition. So it even gives you plenty of other freedom to pick your coordinates.

And no, spherical coordinates are not unimodular.

 

[Pyter]

Does this mean that all the equations in the original papers, including the EFE, are only valid for those special coordinate systems, and would be much more complicated in a really general case?
What if you solved the EFE and found $g_{\mu\nu}$ not satisfying the condition?
For instance the Schwarzschild metric uses spherical coordinates, I've seen it computed with the "simplified" EFE, but its $\sqrt {-g} \neq 1$.

 

[vanhees71]

It can't. The condition $\sqrt{-g} = 1$ is a restriction on what coordinate charts are allowed.

Can't one impose this as one gauge condition? I've somehow in mind that Einstein preferred this choice of a gauge condition for the choice of the coordinates, but I've not the time to look for the source. It even may be in the here discussed paper.

 

[haushofer]

Can't one impose this as one gauge condition? I've somehow in mind that Einstein preferred this choice of a gauge condition for the choice of the coordinates, but I've not the time to look for the source. It even may be in the here discussed paper.

Yes. but Einstein didn't regard it as a "gauge condition"; as far as I understand, Einstein regarded them for some time confusingly as part of the field equations. Only later did he realize his mistake.

The crucial identity here is

<br /> \Gamma^a_{ab} = \partial_b \Bigl( \log{\sqrt{|g|}} \Bigr)<br />

In the unimodular "gauge" one thus has

<br /> \Gamma^a_{ab} = 0<br />

and this simplifies your calculation quite a bit. E.g., the Ricci tensor becomes<br /> R_{ab} = \partial_c \Gamma^c_{ab} - \Gamma^c_{ad}\Gamma^d_{cb} \ \ \ (unimodular \ gauge!)<br />

In the weak field limit, the quadratic term can even be neglected. I guess it's not a far reach to see

<br /> R_{00} \approx \partial_c \Gamma^c_{00} \ \,,<br />

.i.e. a Poisson-like equation for the Einstein field equations appearing. The Newtonian limit is then obtained by the condition of static metrical components and slowly moving particles.

See also "How Einstein found his field equations" of Renn and Janssen in Physics Today. His confusion regarding the meaning of coordinate "restrictions" also led Einstein to his infamous "hole argument".

 

[haushofer]

Does this mean that all the equations in the original papers, including the EFE, are only valid for those special coordinate systems, and would be much more complicated in a really general case?

Well, the nice thing is that if you arrive at tensorial equations in a certain coordinate choice, you know it must be valid for all coordinate choices. The EFE are tensorial, and hence do not depend on the unimodular restriction. I haven't read the paper in detail, but you must be careful in drawing conclusions from certain coordinate (= gauge) choices when they are not general-covariant. E.g., if you want to derive the Newtonian limit as I hinted to, you must check the possible coordinate systems which are compatibel with the coordinate choice. Only for this restricted class of observers your equations will hold.

It's a nice exercise, which makes you appreciate general covariance even more ;) As far as I understand, Einstein proposed his EFE partly because they give the right Newtonian equations (i.e. "the correspondence principle"). But of course this does not mean that the general-covariant EFE are restricted by the unimodular choice of coordinates. You can compare the situation with the gauge choices one makes in electrodynamics.

 

[Pyter]

Well, the nice thing is that if you arrive at tensorial equations in a certain coordinate choice, you know it must be valid for all coordinate choices.

But that's the point. Eq. 66) and 66a) are not tensorial equations if the condition on det G is not satisfied, because the extra 4th term added to the 3 tensor terms is a non-tensor. Might this also be the case for the EFE as we know them? After all they're derived under the same restriction.

 

[PeterDonis]

Eq. 66) and 66a) are not tensorial equations if the condition on det G is not satisfied, because the extra 4th term added to the 3 tensor terms is a non-tensor. Might this also be the case for the EFE as we know them?

No.

After all they're derived under the same restriction.

No, they aren't. The final 1915 derivation of the EFE, the one that has formed the basis of GR ever since, did not have the unimodular restriction.

 

[haushofer]

I also don't see how you would derive general covariant equations from non-g.c. ones. You can only write down g.c. equations and impose coordinate restrictions to see if your equations have 'generalized' known physics, i.e. the correspondence principle.

 

[Pyter]

No, they aren't. The final 1915 derivation of the EFE, the one that has formed the basis of GR ever since, did not have the unimodular restriction.

Are we talking about the EFE contained in the original papers? Because after eq. 44), they state that the Ricci tensor R has a symmetric part B and a skew-symmetric part S that only vanishes when $\sqrt{-g}=1$. It's still a tensor, but I thought the EFE assumed that it should be symmetric.

 

[PeterDonis]

Are we talking about the EFE contained in the original papers?

I was talking about the final version of the EFE that Einstein published in late November 1915. That's the version that has been used ever since.

What you are calling "the original papers", at least as far as the unimodular restriction is concerned, are papers published by Einstein before he figured out the final version of the EFE described above. So reading papers in which the unimodular restriction appears will not tell you about the final version of the EFE. It will only tell you about previous versions that are now known to not be correct.

 

[Pyter]

I was talking about the final version of the EFE that Einstein published in late November 1915. That's the version that has been used ever since.

What you are calling "the original papers", at least as far as the unimodular restriction is concerned, are papers published by Einstein before he figured out the final version of the EFE described above.

All my references to the original papers in this thread concern the link in the first post, published in Annalen Der Physik, issue #7, in 1916 (manuscript received: 20 March 1916, according to the website).
That's the version commonly disseminated and translated in other languages today.
Do you happen to know the reference to the papers where Einstein generalizes the equations appearing in the 1916 papers I linked? I'm genuinely interested.

 

[fresh_42]

The paper "DIe Feldgleichungen der Gravitation" (11/25/1915) uses $\sqrt{-g}=1$ (3a), but I think parallel to the general case.
https://echo.mpiwg-berlin.mpg.de/EC...einstein/sitzungsberichte/6E3MAXK4/index.meta

 

[PeterDonis]

Do you happen to know the reference to the papers where Einstein generalizes the equations appearing in the 1916 papers I linked?

There is a discussion starting on p. 34 of the annotated manuscript linked to in post 7 (in the section "The 1916 Manuscript: Not The End Of The Story"), regarding the removal of the unimodular restriction in the addendum to the 1916 paper. The unrestricted version of the field equations in the addendum is actually the one that has been used ever since and appears in modern textbooks.

Also see the "November 25th" paragraph on p. 31 of the annotated manuscript, which discusses the paper that Einstein submitted on that date.

 

[vanhees71]

Yes. but Einstein didn't regard it as a "gauge condition"; as far as I understand, Einstein regarded them for some time confusingly as part of the field equations. Only later did he realize his mistake.

The crucial identity here is

<br /> \Gamma^a_{ab} = \partial_b \Bigl( \log{\sqrt{|g|}} \Bigr)<br />

In the unimodular "gauge" one thus has

<br /> \Gamma^a_{ab} = 0<br />

and this simplifies your calculation quite a bit. E.g., the Ricci tensor becomes<br /> R_{ab} = \partial_c \Gamma^c_{ab} - \Gamma^c_{ad}\Gamma^d_{cb} \ \ \ (unimodular \ gauge!)<br />

In the weak field limit, the quadratic term can even be neglected. I guess it's not a far reach to see

<br /> R_{00} \approx \partial_c \Gamma^c_{00} \ \,,<br />

.i.e. a Poisson-like equation for the Einstein field equations appearing. The Newtonian limit is then obtained by the condition of static metrical components and slowly moving particles.

See also "How Einstein found his field equations" of Renn and Janssen in Physics Today. His confusion regarding the meaning of coordinate "restrictions" also led Einstein to his infamous "hole argument".

Well, Einstein was not familiar with the modern view of gauge symmetry which was introduced only a few years later by Weyl (gauging the scale invariance of gravity) as an attempt to unify gravity and electromagnetism, which is not sensible as a physical model, which was immediately critisized by Einstein and very harshly by Pauli.

 

[vanhees71]

The paper "DIe Feldgleichungen der Gravitation" (11/25/1915) uses $\sqrt{-g}=1$ (3a), but I think parallel to the general case.
https://echo.mpiwg-berlin.mpg.de/EC...einstein/sitzungsberichte/6E3MAXK4/index.meta

Yes the paper starts with the generally covariant eqs. In fact to turn back to generally covariant equations from an errorneous counter argument against this principle was the breakthrough for the discovery of the correct version of the theory we know today as GR. He clearly states that one can impose $g=-1$ as a simplifying condition that can be fulfilled by a special choice of coordinates. Also in the here discussed later longer paper this correct view is given. He emphesizes that the constraint is not necessary and the theory can be formulated in fully generall covariant form.

 

[Sagittarius A-Star]

For all those interested, an accurate English edition of the original papers is available at this link (ISBN-13: 978-0486600819, ISBN-10: 0486600815). I couldn't find a free version unfortunately.
The text is translated, but the equations are scanned directly from the original papers.

A free version is here:
https://archive.org/details/principleofrelat00lore_0/mode/2up

 

[Pyter]

There is a discussion starting on p. 34 of the annotated manuscript linked to in post 7 (in the section "The 1916 Manuscript: Not The End Of The Story"), regarding the removal of the unimodular restriction in the addendum to the 1916 paper. The unrestricted version of the field equations in the addendum is actually the one that has been used ever since and appears in modern textbooks.

The addendum they are referring to must be HAMILTONsches Prinzip und allgemeine Relativitätstheorie. I've read that paper too, its translation in English is included in the book I've linked in post #6.
There's no unimodular constraint there, but it doesn't rewrite the EFE. Its main results are eq. (21), expressing the conservation of energy and momentum, and (22), four equations linking the stress-energy-momentum tensor to the $g_{\mu \nu}$, all derived from the Hamilton principle.

 

[Pyter]

A free version is here:
https://archive.org/details/principleofrelat00lore_0/mode/2up

Exactly, the addendum to the main corpus is right there at Chapter VIII. Chapter IX introduces the infamous cosmological constant, and I still have to read Chapter X, maybe that one will provide additional insights about the dropping of the unimodular constraint.

 

[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.