A Help on some equations in Einstein's original papers

[Pyter]

TL;DR Summary

Help on tensor equations in Einstein's original General Relativity papers

Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie, published in 1916's Annalen Der Physik, I came across some equations which I couldn't verify after doing the computations hinted at.

The first are equations 47b) regarding the gravity contribution to the stress-energy-momentum-tensor through the Hamiltonian:
$$ \begin{align} \frac{\partial}{\partial x_{\alpha}} \left( \frac{\partial H}{\partial g^{\mu\nu}_{,\alpha}} \right ) - \frac{\partial H}{\partial g^{\mu\nu}} = 0 \tag{47b} \end{align} $$
According to the discussion in the following paragraphs, this should be equivalent to:
$$ \frac{\partial}{\partial x_{\alpha}} \left( g^{\mu\nu}_{,\sigma}\frac{\partial H}{\partial g^{\mu\nu}_{,\alpha}} \right )
- \frac{\partial H}{\partial x_{\sigma}} = 0 $$
but you can see that in the original equation there is a term:
$$ - \frac{\partial H}{\partial g^{\mu\nu}} $$
not appearing in the modified equation, which doesn't seem to be vanishing.
Can you explain why?

The second are Equations 66) and 66a) regarding the electromagnetic contribution to the stress-energy-momentum-tensor:
$$ \begin{align} x_{\sigma} = \frac{\partial T_{\sigma}^{\;\nu }}{\partial x_{\nu}} - \frac{1}{2}g^{\tau\mu} \frac{\partial g_{\mu \nu}}{\partial x_{\sigma}}\,T^{\;\nu}_{\tau} \tag{66} \end{align} $$
$$ \begin{align} T^{\; \nu}_{\sigma} = -F_{\sigma \alpha}F^{\nu \alpha} + \frac 14 \delta_{\sigma}^{\; \nu}\; F_{\alpha \beta}F^{\alpha \beta} \tag{66a} \end{align} $$
According to the discussion in the previous pages, by substituting 66a) into 66) you should get three terms, but even before doing the actual computation, it's plain to see that you also get a fourth term which doesn't seem to vanish:
$$ -\frac 18\;F_{\alpha\beta}F^{\alpha\beta}\;g^{\mu\nu}\;g_{\mu\nu,\sigma} $$
Am I missing something?

P.S.: I've linked the original papers in German to make sure that the equations are indeed the original ones.

 

[Dale]

Are you learning this material for the first time or do you already understand it from modern textbooks and you are just exploring the history?

 

[Pyter]

I've studied it on modern textbooks but I like going directly to the source: Einstein, Minkowski, Lorentz. Often they're clearer, more concise and straight to the point.

 

[anuttarasammyak]

Summary:: Help on tensor equations in Einstein's original General Relativity papers

Can you explain why?

Multiply $\frac{\partial g^{\mu\nu}}{\partial x_\sigma}$ to (47b) and take sum for $\mu$ and $\nu$ . We get the formula in the paper.

When you would calculate $\frac{\partial H}{\partial x_\sigma}$, please be reminded that H is function of not only $g^{\mu\nu}$ but $g^{\mu\nu}_{\ \ \alpha}$ first derivative.

\frac{\partial H}{\partial x_\sigma}=\frac{\partial H}{\partial g^{\mu\nu}}\frac{\partial g^{\mu\nu}}{\partial x_\sigma}+\frac{\partial H}{\partial g^{\mu\nu}_{\ \ ,\alpha}}\frac{\partial g^{\mu\nu}_{\ \ ,\alpha}}{\partial x_\sigma }
clearer copies
1 http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1916_49_769-822.pdf
2 https://echo.mpiwg-berlin.mpg.de/EC.../Einst_Grund_de_1916/index.meta&mode=texttool

 

[Pyter]

@anuttarasammyak multiplying what?

 

[Pyter]

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.

 

[Sagittarius A-Star]

For all those interested, a freely available English translation of the original Einstein paper is available starting on document page 204 /258 (=printed page number 184) at
https://archive.org/details/einstein_relativity/page/n203/mode/2up

You can find the mentioned formula (47b) on document page 232 /258 (=printed page number 211).

You can find an annotated manuscript of the mentioned formula (47b) on document page 118 /258 (=printed page number 98).

 

[Sagittarius A-Star]

Here is a link to another English translation of the original Einstein paper. You can find the mentioned formula (47b) in §15.

The Foundation of the Generalised Theory of Relativity

Source:
https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

 

[Pyter]

Thanks @Sagittarius A-Star, nice addition.

 

[Pyter]

The annotated manuscript is very cool. On 47b) it doesn't seem to add extra info with respect to the printed edition though, the apparently missing term is still missing
And watch out for the second source, the 47b) has an error, the covariant sigma on the first term should be an alpha. I'd double check all the equations against the original.

 

[Pyter]

Multiply $\frac{\partial g^{\mu\nu}}{\partial x_\sigma}$ to (47b) and take sum for $\mu$ and $\nu$ . We get the formula in the paper.

When you would calculate $\frac{\partial H}{\partial x_\sigma}$, please be reminded that H is function of not only $g^{\mu\nu}$ but $g^{\mu\nu}_{\ \ \alpha}$ first derivative.

\frac{\partial H}{\partial x_\sigma}=\frac{\partial H}{\partial g^{\mu\nu}}\frac{\partial g^{\mu\nu}}{\partial x_\sigma}+\frac{\partial H}{\partial g^{\mu\nu}_{\ \ ,\alpha}}\frac{\partial g^{\mu\nu}_{\ \ ,\alpha}}{\partial x_\sigma }
clearer copies
1 http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1916_49_769-822.pdf
2 https://echo.mpiwg-berlin.mpg.de/EC.../Einst_Grund_de_1916/index.meta&mode=texttool

That's the problem, he multiplies only the first term of 47b) by $g^{\mu\nu}_{\sigma}$, then he equates it to 0, but this would only be legal for the whole 47b) multiplied by that factor.

 

[anuttarasammyak]

@Pyter Please investigate the attached calculation.

 

[Pyter]

@anuttarasammyak my calculations are attached, as per the original paper. It's pretty straightforward, there's no need to derive $\frac{\partial H}{\partial \sigma}$ .

 

[Pyter]

@anuttarasammyak I see your point. You get the $\frac{\partial H}{\partial \sigma}$ with the total derivative of H, but I get it by simply canceling out the $\partial g^{\mu \nu}_{\alpha}$ at the numerator and denominator.

 

[anuttarasammyak]

@Pyter

Formula of H in (47a) shows
H =g \Gamma \Gamma=H(g^{\mu\nu},g^{\mu\nu}_\alpha)
All the parameters should be considered for partial derivative calculation. See derivative formula in post #4. Both RHS terms are necessary to make $\frac{\partial H}{\partial x_\sigma}$ though you deals only the 2nd term. Einstein prepared the both.

 

[Pyter]

@anuttarasammyak absolutely right. Now I see my error: given $H = H \left( \xi^{\nu} \right)\text{, }\xi^{\nu} = \xi^{\nu} \left( x^{\sigma}\right)$, I calculated:
$$ \frac {\partial H}{\partial \xi_1} \frac {\partial \xi_1}{\partial x^\sigma} = \frac {\partial H}{\partial x^\sigma}, $$
while it's in fact:
$$ \frac {\partial H}{\partial \xi_\nu} \frac {\partial \xi_\nu}{\partial x^\sigma} = \frac {\partial H}{\partial x^\sigma} .$$
I guess tensor calculus tricked me again. Well spotted.
Do you happen to have any thought about the 66) and 66a) too?

 

[anuttarasammyak]

According to the discussion in the previous pages, by substituting 66a) into 66) you should get three terms, but even before doing the actual computation, it's plain to see that you also get a fourth term which doesn't seem to vanish:

Follow the calculations by your hand as Einstein taught and tell me what $\chi_\sigma$ you get.

 

[Pyter]

@anuttarasammyak using this link, as reference, starting from the (65a) I get the three terms mentioned right after in the paper, that is:
$$ x_\sigma = \frac{\partial}{\partial x_\nu} (F_{\sigma \mu}F^{\mu \nu})
+\frac 14 \frac{\partial}{\partial x_\sigma} (F^{\mu \nu} F_{\mu \nu})
+ \frac 12 F^{\mu \tau} F_{\mu \nu}g^{\nu \rho} \frac{\partial g_{\rho \tau}}{\partial x_\sigma}
$$
By substituting (66a) [the equation right after (66)] into the (66), I get the same three terms plus a fourth one which doesn't seem to vanish:
$$ -\frac 18\;F_{\alpha\beta}F^{\alpha\beta}\;g^{\mu\nu}\;g_{\mu\nu,\sigma} $$
and surely is not a tensor. Isn't $x_{\sigma}$ supposed to be a covariant vector?
I find this section really interesting because it gives the EMF components of the stress-energy-momentum tensor.

EDIT: watch out for the document I've linked, I've just noticed there's a misprint in the third term, there should be a rho instead of the sigma in the covariant index of the $\partial g$. I've just corrected it here.

 

[anuttarasammyak]

@Pyter
Thank you for showing calculation.
g^{\mu\nu}g_{\mu\nu,\sigma}=g^\mu_{\mu,\sigma}=0
because $g^\mu_\nu=1$ for $\mu=\nu$ otherwise 0, so constant anyway.

 

[Pyter]

@anuttarasammyak I thought so too at first, but apparently that contraction is not allowed.
You can verify by yourself with a random metric, i.e. $g_{\mu \nu} = diag[ 3x^1, 4x^2 ]$, that it doesn't vanish.
I guess it's because $g_{\mu \nu, \sigma }$ is not a tensor.

 

[kent davidge]

$g^{\mu \nu}g_{\mu \nu,\sigma} = (g^{\mu \nu}g_{\mu \nu})_{,\sigma} - g_{\mu \nu}g^{\mu \nu}{}_{,\sigma} = - g_{\mu \nu}g^{\mu \nu}{}_{,\sigma}$. Problem is that $g_{\mu \nu}g^{\mu \nu}{}_{,\sigma} \neq g^{\mu \nu}g_{\mu \nu,\sigma}$. As just shown one is minus the other. Therefore it doesn't vanish.

 

[anuttarasammyak]

@anuttarasammyak I thought so too at first, but apparently that contraction is not allowed.

Hum... Let us investigate more.
Following Einstein for $\chi_\sigma$, I got the same 1st and 2nd term with you that coincide with $T^{\ \ \ \nu}_{ \sigma,\nu}$ but the 3rd one is
-\frac{1}{4}F_{\alpha\beta}F_{\mu\nu}(g^{\mu\alpha}g^{\nu\beta})_{,\sigma}
that should coincide with
-\frac{1}{2}g^{\tau\mu}g_{\mu\nu,\sigma}T^{\ \ \nu}_\tau
accrording to (66).

Is it same with yours but different expression ?

 

[Pyter]

@anuttarasammyak if I follow the explanation up to 66) I get an expression with the 3 terms that are explicitly written in the paper (I did the math and they check out); if I substitute 66a) into 66) I get the same 3 terms, plus the one I wrote about that doesn't vanish.

 

[anuttarasammyak]

I am looking at (66) . It should say about covariant derivatives, so instead of $\frac{\partial g_{\mu\nu}}{\partial x^\sigma}$, Christoffel symbol $\Gamma_{\mu\nu\sigma}$ be there. So two more derivatives might be included. I do not know they happen to be zero here. This suspected difference might have caused your concern.

 

[Pyter]

@anuttarasammyak are you saying that there is a misprint in the equations of the original papers in German?

 

[Pyter]

Found it. The extra term vanishes because of equation (29):
$$
-\frac12 g^{\mu \nu}g_{\mu \nu,\alpha} = \frac {1}{\sqrt{-g}} \frac{\partial \sqrt {-g}}{\partial x^{\alpha}}
$$
with the additional hypothesis that $\sqrt{-g} = 1$.
That's another thing I didn't get about the papers, how can the fundamental tensor determinant be constant in every chart. Maybe I'll open another thread about it.

 

[PeterDonis]

how can the fundamental tensor determinant be constant in every chart

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

 

[Pyter]

@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?

 

[kent davidge]

@Pyter , you are right that in general $\sqrt{|g|} \neq 1$. In spherical coordinates $(ct,r,\theta,\phi)$ we have $\sqrt{|g|} = r^2\sin\theta$.

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

He was telling you that if you insist on having $\sqrt{|g|} = 1$, there is only a limited family of coordinate systems which will satisfy that.

 

[Pyter]

@kent davidge it's not me who's insisting, it's Einstein that is making this assumption from equation (18a) on. He mentions that eq. (49) is only valid for the coordinate systems where $\sqrt{-g}=1$. And apparently also the (66)-(66a), as I've just realized.
A limited family could reduce to the empty set. What would you use on a $S^2$, for instance?

 

[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.