From Equivalnce Principle to General Relativity

[Goldbeetle]

Dear all,
is there a book that by consulting the original sources (articles, letters etc etc) traces the evolution of Einstein's thought that lead him from the Equivalence Principle to postulate General Relativity as curved space.
Thanks,
Goldbeetle

 

[bcrowell]

For a general outline of the physical motivation, the first few sections of "The foundation of the general theory of relativity" are very readable. There is a public-domain translation by Perrett and Jeffery, which is available in various places on the web, including an appendix to my book: http://www.lightandmatter.com/genrel/ These sections consist of *motivation*, not a derivation.

If you want to read something that literally "traces the evolution of Einstein's thought," then it's going to get messy. Einstein spent a frustrating decade trying to figure out GR, and a lot of what he did ended up being wrong.

-Ben

 

[Goldbeetle]

Ben,
I've downloaded the Einstein's article, but I do not understand the example of the two non-interacting fluid body system. Maybe I'm missing a point in the reasoning...

 

[bcrowell]

I've downloaded the Einstein's article, but I do not understand the example of the two non-interacting fluid body system. Maybe I'm missing a point in the reasoning...

Tell us where you're stuck.

 

[Goldbeetle]

Ben,
I've seen that your book explains that example on pages 28, 104 and seq.. I'll read them first and then get back to you in a couple of days if I still have questions. In the meantime, thank you for your help!
Godlbeetle

 

[Cleonis]

Dear all,
is there a book that by consulting the original sources (articles, letters etc etc) traces the evolution of Einstein's thought that lead him from the Equivalence Principle to postulate General Relativity as curved space.

The homepage of the historian of science http://www.tc.umn.edu/~janss011/" , who specializes in history of relativistic physics in the formative years (1905-1920), has links to various articles by him.

For instance, the article: https://netfiles.umn.edu/xythoswfs/webui/_xy-15267453_1-t_ycAqaW0A"

As Benjamin Crowell has pointed out, from that you will gain historical knowledge, but not necessarily understanding of relativistic physics. Many of Einstein's explorations were blind alleys, and several times during the process Einstein had severe misconceptions as to the nature of the problem. (His eventual triumph shows both his extraordinary tenacity and his genius.)

Also a very interesting read, I think: http://www.pitt.edu/~jdnorton/jdnorton.html". Throughout Nordström was in communication with Einstein. Nordström issued several versions, each time urged by Einstein to push towards more complete implementation of the principle of equivalence. (The choices were never clearcut: all the time it was educated guesses on how push harder at implementing the principle of equivalence.)
John Norton writes that Einstein pushing Nordström shows how committed Einstein was to the principle of equivalence.

Nordström's theory was a dead end too, but no doubt Einstein learned a lot from the process. (For example, Einstein collaborated with another physicist to write an article in which a version of Nordström's theory was presented in a different mathematical form.)


By the way, it may be tempting to think that the principle of equivalence itself is clearcut, but that is not the case. In a sense you can say that only after GR reached its final form the insight was there to formulate the Principle of equivalence in such a way that it slots in with GR.

 

[Goldbeetle]

Cleonis: thanks for the links.

For everybody: a question I have is, Is it true that from the principle of equivalence we can infer directly that space-time is curved? Or is it just a strong argument that suggests curved space-time? Also, is the following reasoning correct: SR has flat geometry (no gravity). If gravity enters the picture, then one realizes that SR can be regained locally by the principle of equivalence. (Additionally, according to said principle of equivalence gravity is a property of space-time and not of the objects because free-falling objects move all the same way if only gravitation is present.) However, no SR inertial frame can be used to cover the whole of space-time because of gravitational tidal effects. Since SR space-time is flat, it then follows necessarily that for a space-time with gravity, the geometry must be globally "non flat", that is curved. What I find not convincing in this reasoning (please help!) is that it somehow assumes that SR space-time is the only flat space-time. Could one imagine a space-time that is globally flat but with a different metric from the SR metric? That is, a globally flat metric that is only locally the SR metric? Restating in terms of curvature tensor: is the curvature tensor globally null if and only if the metric is globally the SR metric?

I know I'm missing something but what?

 

[atyy]

For a given metric signature, only the SR metric is globally flat.

http://arxiv.org/abs/0707.2748

 

[K^2]

Ben,
I've downloaded the Einstein's article, but I do not understand the example of the two non-interacting fluid body system. Maybe I'm missing a point in the reasoning...

The gravity must be related to inertial mass of the body, which is equivalent to total energy. However, that quantity is not Lorentz invariant. The four-momentum, containing both energy and momentum terms, undergoes hyper-rotations with Lorentz boost.

By considering a simple 2-flux, Einstein shows that the relevant invariant quantity is the flux-stress tensor. It can be used to derive a 2-flux in any coordinate system. The quantity similarly associated with 4-momentum is the stress-energy tensor.

Using stress-energy tensor, you'll always get the correct total energy of the body, and therefore, you'll get correct classical gravity in any coordinate system. It's a natural thing to consider when trying to build a general theory.

As far as I can tell, that's the entire motivation for considering the 2-flow problem.

 

[Cleonis]

Also, is the following reasoning correct: SR has flat geometry (no gravity). If gravity enters the picture, then one realizes that SR can be regained locally by the principle of equivalence.

I have to say, the word 'regained' is particularly ill-fitting here.

The spacetime of SR and the spacetime of GR have the following in common: the signature of the metric is (+,-,-,-)

GR ushered in unification of the phenomena inertia and gravitation. In Newtonian dynamics theory of motion and theory of gravitation are distinct, GR is a single theory of inertia and gravitation.

The following three properties of SR spacetime are relevant here: SR spacetime is immutable, SR spacetime is geometrically flat, and the signature of the metric is (+,-,-,-).

To unify inertia and gravitation one keeps the (+,-,-,-) signature, and one relaxes the property of spacetime being geometrically flat.

The (+,-,-,-) signature of the metric is not "regained", it's better to say that the metric's signature is what you keep (in moving from SR to GR).


There is no need to formulate things in terms of 'locally you get so-and-so'. That 'locally' is ambiguous, and unnecessary.

 

[Mueiz]

Dear all,
is there a book that by consulting the original sources (articles, letters etc etc) traces the evolution of Einstein's thought that lead him from the Equivalence Principle to postulate General Relativity as curved space.
Thanks,
Goldbeetle

Equivalence Principle ,even thought played an heuristic role during the construction of GR,
is logically a conequence of GR and not a postulate .. it can be used before discovering GR to introduce the idea of the relation betweem gravitational field and geometry
P A M Dirac in his book The General Theory of Relativity fomulated the basis of GR without EP
We should distinguish the historical formulation of a theory from the logical one
The fact that the metric tensor can be made to be like that of SR in a small region by a suitable frame of reference is a simple geometrical feature of smooth curved space-time.

So I think it would be more exiting question to ask'' is there any book that introduce GR without Equivalence Principle?''

 

[atyy]

MTW devotes a section to this. However, this seems still controversial.

http://arxiv.org/abs/0906.0926

 

[PAllen]

MTW devotes a section to this. However, this seems still controversial.

http://arxiv.org/abs/0906.0926

J. L. Synge rejected the equivalence principle as objectively false, nothing but historical heuristice used to arive at GR. His book on GR consistently takes this position (which I disagree with). However, it is a mainstream GR book, with several unique features I've found in no other book, that never mentions equivalence principle except to proclaim that it is false.

 

[atyy]

J. L. Synge rejected the equivalence principle as objectively false, nothing but historical heuristice used to arive at GR. His book on GR consistently takes this position (which I disagree with). However, it is a mainstream GR book, with several unique features I've found in no other book, that never mentions equivalence principle except to proclaim that it is false.

Yes, the EP is false or rather ill-defined. To make it true, we take GR, make a statement that is true in GR, and define that to be the EP. Synge's particular argument was that the curvature at every point in spacetime is measurable, and of course non-zero curvature is not locally special relativity. So Synge was right. So to make the EP true, we define local to mean, provided we don't look higher than first derivatives. Each derivative is non local in the sense that "physically" it takes the value of a quantity and more than one point, although mathematically, the limit exists at a point.

 

[sdhobbs]

I would of said true "distance" is flat, but due to motion and the deformity of apparent position as opposed to true position it is often modeled as curved, its important to remember the difference between a mathematical model to simulate the effect and reality.

 

[sdhobbs]

sorry, that reply was to

"is there a book that by consulting the original sources (articles, letters etc etc) traces the evolution of Einstein's thought that lead him from the Equivalence Principle to postulate General Relativity as curved space."

short answer: probably, you mean like an "Einstein's theories for Dummies"?