By what method did Einstein derived his gravitational field equation?

[DOTDO]

Hi.

In class, the professor has tried to derive the equation by using the principle of least-action. (But not yet completed. Maybe next class...)

However I heard this method is used by Hilbert, who had derived the equation 5 days before Einstein derived it.

Then, what method did Einstein use to derive the gravitational field equation?

And why did Einstein choose to use Riemann geometry ?

 

[ShayanJ]

The book Gravitation by Misner,Thorne and Wheeler lists six different ways for deriving Einstein's equations. I can surely say which ones Einstein didn't use but can't be sure which one he actually used. But I guess he used the method that is based on the automatic conservation of source and correspondence with Newtonian gravity. Its explained in the book I mentioned.
About your last question, Einstein was led to GR in the light of equivalence principle and general covariance. So he needed a kind of mathematics that had those principles incorporated in it. The proper math turned out to be tensors on generally curved spaces which is actually Pseudo-Riemannian geometry.(Riemannian geometry isn't general enough for GR.)

 

[PeterDonis]

I guess he used the method that is based on the automatic conservation of source and correspondence with Newtonian gravity.

Yes, he did. Kip Thorne's Black Holes and Time Warps gives a good discussion of both Einstein's and Hilbert's procedure and the timing of the key events in 1915.

 

[Matterwave]

From my rough knowledge of this part of the history of physics, Einstein started by simply positing (educated guessing) the field equations $R_{\mu\nu}=0$ (which is true in a vacuum) but found that if he included stress-energy into the problem, $R_{\mu\nu}=\kappa T_{\mu\nu}$ doesn't work since the Ricci tensor, although symmetric like the Stress-energy tensor, is not locally covariantly conserved (i.e. it is not true in general that $\nabla_\mu R^\mu_{~~\nu}=0$). I think you can see him write down this equation in a pretty famous picture:

But I'm not sure if this picture was staged for publicity reasons. He then went on to work with the trace-reversed version of the Ricci tensor, the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ and found that due to the Bianchi identities, the local covariant conservation of the Einstein tensor is geometrically guaranteed, which made $G_{\mu\nu}=\kappa T_{\mu\nu}$ a very good candidate for the field equations. The only things left to do is to find the constant of proportionality $\kappa$ which can be done by dimensional analysis, and by matching predictions with Newtonian theory.

 

[PeterDonis]

Matterwave, yes, your summary is correct, and Kip Thorne's book describes the same process you describe, just in more detail. (I don't know if the picture was staged, though.)

 

[Ben Niehoff]

I'm pretty sure that photo has to be staged. I don't think he was that old yet, during the early stages of developing the theory.

Early researchers in GR had quite a lot of difficulty with the mathematics, Einstein included. The significance of general covariance was not well-appreciated, and you will see in early GR papers a great deal of obsession over coordinates and their physical meaning. This misguided obsession is the source of pretty much all confusion regarding the Schwarzschild horizon, since in Schwarzschild's original paper, the horizon is located at a coordinate singularity which is an artifact of the chart chosen, and not a genuine feature of the geometry.

I cringe when people on this forum want to go read old papers by Einstein and Schwarzschild for any purpose other than historical curiosity. :) The best introduction to a subject is not usually a chronologically-ordered one, because the research community's understanding of a subject tends to increase over time. A modern book can give an introduction with much more clarity and insight (not MTW, though, I think that book goes out of its way to make things sound more complicated than they are).

 

[ShayanJ]

I cringe when people on this forum want to go read old papers by Einstein and Schwarzschild for any purpose other than historical curiosity. :) The best introduction to a subject is not usually a chronologically-ordered one, because the research community's understanding of a subject tends to increase over time. A modern book can give an introduction with much more clarity and insight (not MTW, though, I think that book goes out of its way to make things sound more complicated than they are).

I think its not futile to read the old papers. Of course the advantages doesn't lie in understanding the theory itself better, but in learning how physics itself develops and how physicists think.

 

[harrylin]

I think its not futile to read the old papers. Of course the advantages doesn't lie in understanding the theory itself better, but in learning how physics itself develops and how physicists think.

Yes indeed, and it's also important for proper referral. Without actually having read some books and/or papers of physicists, one cannot correctly describe their thinking - and we see plenty examples of that. ;)

 

[Matterwave]

I'm pretty sure that photo has to be staged. I don't think he was that old yet, during the early stages of developing the theory.

Haha I think you are right, he is far too old in that picture. :D