Deriving Einstein's Field Equations: A Comprehensive Guide for Laymen

[anvesh111]

can any 1 tell mme the derivation of Einstein's field equations? please

 

[Matterwave]

You're much better off looking at the wikipedia article, or reading a textbook...

http://en.wikipedia.org/wiki/Einstein_field_equations

 

[George Jones]

How is F = ma derived?

 

[anvesh111]

i already know about wikipedia...but i din't find any derivations of how did he get it?

 

[HallsofIvy]

It's hard to believe you are serious. Whole books are dedicated to deriving Einstein's field equations! How much to you know about tensor analysis to begin with?

 

[jtbell]

It seems that we're stumbling over the meanings of English words here. "Derivation" in physics and math means a strict logical/mathematical sequence of steps starting from some axioms (first principles). In general relativity, the Einstein field equations are axioms, so there's no real "derivation" of them.

It seems that you're asking about what inspired or led Einstein to postulate these specific field equations in the first place. I expect you'd have to look for a source that focuses on the historical and philosophical development of general relativity.

 

[George Jones]

It seems that we're stumbling over the meanings of English words here. "Derivation" in physics and math means a strict logical/mathematical sequence of steps starting from some axioms (first principles). In general relativity, the Einstein field equations are axioms, so there's no real "derivation" of them.

Yes, thanks. With my previous post, I tried to provoke anvesh111 into a discussion of this, but anvesh111 didn't take the bait. Because Einstein's equation seem so unfamiliar, sometimes people think that they should be derived, and these same people sometimes overlook the fact that some more elementary and familiar physics equations are also taken as axioms and not derived.

It seems that you're asking about what inspired or led Einstein to postulate these specific field equations in the first place. I expect you'd have to look for a source that focuses on the historical and philosophical development of general relativity.

anvesh111, are you ooking for modern motivations for Einstein's equation? For Einstein's route, including dead ends, to Einstein's equation? For the meaning of Einstein's equation?

For Einstein's route, see the book Subtle Is the Lord: The Science and the Life of Albert Einstein by Abraham Pais,

https://www.amazon.com/dp/0192806726/?tag=pfamazon01-20.

For the meaning of Einstein's Equation, see

http://arxiv.org/abs/gr-qc/0103044

or see any modern introductory general relativity text. For modern motivations for Einstein's equation, see any modern introductory general relativity text.

Everything that I have referenced, including the biography by Pais, assumes a math and physics background of two or three years of university physics and math courses.

 

[Matterwave]

That biography looks intriguing...I may get it when i have time <_<

 

[jtbell]

anvesh111, are you ooking for modern motivations for Einstein's equation? For Einstein's route, including dead ends, to Einstein's equation?

I hadn't thought of there being different ways to motivate the Einstein field equations, but I'm not surprised. Schrödinger was motivated towards his formulation of quantum mechanics by making an analogy between mechanics and optics (classical and quantum mechanics have a similar relationship as geometric and wave optics); later Dirac arrived at QM by starting with the Hamiltonian formulation of classical mechanics and replacing Poisson brackets with commutators.

 

[atyy]

What is the derivation of special relativity?
1) Principle of relativity
2) Constancy of the speed of light
Both are experimental facts, ie. SR can be derived as saying we use experiments to constrain theory uniquely.

What is the derivation of "Entropy increases"?
1) Conservation of energy
2) Kelvin and Clausius statements about not being able to transfer energy from cold to hot without doing work etc.
Again experimental facts constrain theory uniquely.

Also appealing about these two derivations are that all the facts (except the constancy of the speed of light) are almost common sense in modern times, where people have ridden cars, ships, airplanes moving at constant velocity. One imagines that general relativity should have an equivalent "derivation". Usually the Principle of Equivalence (EP) is used in textbooks - unfortunately, unlike SR and classical thermodynamics, the EP does not lead unqiuely to Einstein's equations, although it is consistent with it.

A fascinating theory is Nordstrom's second theory which was the first theory of gravity consistent with some form of the EP and special relativity. It predicts red shift and perihelion weirdness (but the perihelion prediction is numerically wrong).
http://arxiv.org/abs/gr-qc/0405030

Also Brans-Dicke theory which came after Einstein, and respects some form of the EP, but not "ultra-strong" forms, and so are severely constrained by Nordtvedt sort of experiments.
http://relativity.livingreviews.org/Articles/lrr-2006-3/

Another alternative theory that survived many tests:
http://arxiv.org/abs/gr-qc/0611006
On the Multiple Deaths of Whitehead's Theory of Gravity
Gary Gibbons (DAMTP), Clifford M. Will

 

[atyy]

Matthias Blau's notes have a "derivation" http://www.blau.itp.unibe.ch/Lecturenotes.html

MTW state by now EFE should be axiom, but for fun they give 5 or 6 different "derivations" of the EFE.

 

[atyy]

How is F = ma derived?

Yeah, why not F=m1x'+m2x''+m3x'''+...?

I guess the Lorentz force law sneaks in x' on the LHS.

 

[Nabeshin]

One could say the field equations are derived from the Einstein-Hilbert action, but this seems a lot more like shifting the topic of discussion to "How is the Einstein-Hilbert action derived?" than anything else.

 

[Matterwave]

One could say the field equations are derived from the Einstein-Hilbert action, but this seems a lot more like shifting the topic of discussion to "How is the Einstein-Hilbert action derived?" than anything else.

I'm not familiar with this Einstein-Hilbert action. Is it similar to how you can "derive" Newton's law using the principle of least action?

 

[Nabeshin]

Yup, completely analogous.

 

[AEM]

If you are curious about Einstein's path to his field equations, probably the best place to start is with Volume 1 of the "Einstein Stidies" series, "Einstein and the History of General Relativity", edited by Don Howard and John Stachel. Chapter 4 is titled "How Einstein Found His Field Equations, 1912 - 1915 by John Norton. This and the other books in the series are a direct result of the editing of Einstein's papers. Do to the work on Einstein's papers it was discovered in a 1912 notebook the Einstein actually had come very close to the final theory at that time, but was struggling with the concept of general covariance.

 

[Mentz114]

I can't believe that no one has mentioned the Einstein-Hilbert action. The field equations follow if this action is extremised wrt variation of the metric.

 

[George Jones]

I can't believe that no one has mentioned the Einstein-Hilbert action.

Look at posts #13, #14, and #15 . Also, in post#11, atyy mentions that Misner, Thorne, and Wheeler gives a number of "derivations" (or "routes") to Einstein's equation. One of MTW's routes is via the Einstein-Hilbert action.

 

[heldervelez]

SR & GR by "www.bartleby.com/173"[/URL], +- 30 pages to layman
The GR by Einstein (1916) Ann. d. Phys. 49, (there must be a free translation available) full math treatment.
The best IMO is the 1920 one because it is very clear.