Resource that gives classical limit solution

[flatmaster]

Hey all. Greetings from Auburn Alabama. I'm a phys grad student at Auburn University. I've never yet taken a GR course, so I suppose i'd better learn some. My physical intuition is ahead of my math chops, so I'm having some trouble working through Einstein's new notation. When trying to work through any intro problem, I quickly get caught up in the math and keeping indicies straight that I loose sight of the physical situation.

What I'm looking for is some resource (book or web) that demonstrates how to work out the classical limit through relativity. IE. I think it would be usefull to put a point mass of mass M in the stress-energy tensor and arive at y=y0 + v0yt + .5gt^2. Any resourse with multiple classical limit problems worked out?


Thanks in advance

 

[atyy]

The equation y=y0 + v0yt + .5gt^2 does not describe the mass distribution causing the gravitational field. It describes the motion of a test mass in a uniform gravitational field. In Newtonian gravity, all test masses fall with the same acceleration, so an accelerometer in free fall reads zero. This suggests a new definition of "acceleration" so that free fall in gravity is "non-accelerated" motion. Hence, the equation of motion for a free-falling test mass in general relativity is the geodesic equation, a generalization of the equation for non-accelerated motion.

The Newtonian equation which describes how a mass distribution produces a gravitational field is Poisson's equation relating the second derivative of the gravitational potential to the mass distribution. In general relativity, the second derivative of the gravitational potential becomes the "second derivative" of the metric, and the mass distribution becomes the stress-energy-momentum tensor.

I like the books by J L Martin (1988, 1996) and Ohanian and Ruffini (1976, 1988) because they present quick routes to simple but meaningful calculations. I like the books by Ludvigsen (1999) and Rindler (2001, 2006) for their attention to conceptual details. Online there's Blandford and Thorne, Chapters 1, 23 , 24: http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html, Blau's http://www.unine.ch/phys/string/Lecturenotes.html , and van Holten's http://arxiv.org/abs/gr-qc/9704043.

 

[Naty1]

I'm a former engineering student with the same problem ...

These may be too basic for you...but they are short enough you'll know quickly!

http://arxiv.org/abs/gr-qc/9712019
http://arxiv.org/PS_cache/gr-qc/pdf/9712/9712019v1.pdf

At least we can understand how it took some twenty years for aspects of General Relativity to be fleshed out, understood and widely accepted by the physics community...Einstein himself did not solve many of the situations he would have liked...say the Schwarzschild solution for example...

 

[Naty1]

When trying to work through any intro problem, I quickly get caught up in the math and keeping indicies straight that I loose sight of the physical situation.

I have a closely related question I have been waiting to post hoping to find some background first...but this seems like a great opportunity..

What are the physical input insights versus output results in GR? I have not been able to find any discussion of this. Anybody got references?

Gravitational fields have some similarities and some significant differences from other force fields...say electromagnetism for example. Does anyone know which physical attributes formed the basis for inputs to Einstein's tensors and which, if any attributes popped out and were "discoveries".

For example, I have read that the two dimensional orthnormal nature of a gravitational wave told Einstein he needed a tensor description...fine. but how did anybody know that before the theory? Also, I have read that Einstein had several theories "ready to go" but he could not distinguish between them because experimental capability at the time was limited and the formulations rather close...he finally came across his "equivalence" principle, between acceleration and gravity, and supposedly that enabled him to pick the theory we all know and love...so somehow he apparently figured out a few more physcial relationships.

It appears others had figured out some physcial relationships:
Peter Bergmann, a former student of Einsteins, says in THE RIDDLE OF GRAVITATION, 1992,
"The twenty components+ of the curvature in a four dimensional space can be grouped into two sets of ten each in a manner that is independent of any coordinate system. One of these two sets involves the turning of vectors in the course of parallel transport in a surface that is spanned by the vector to be turned and on other, fixed vector; this set is usually called the Ricci tensor...The other ten components form the Weyl Tensor..."
+ These are the 20 of 36 components that are truly independent...

So these might be a good basis for locating the descriptions.

Wiki says: "In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space..."
I wonder what physcial insights Einstein used to pick these..."acceleration" equivalence?

 

[Naty1]

The wiki story on the Einstein Field Equations is a reasonable summary at

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

but as usual appears to have been written by a methematician ( a typo but inadvertenly funny,too) rather than a person interested in the physcial attributes, interpretations and consequences...

says " The EFE collectively form a tensor equation and equate the curvature of spacetime (as expressed using the Einstein tensor) with the energy and momentum within the spacetime (as expressed using the stress-energy tensor)..."

I think the Einstein tensor is the Ricci Tensor...then is the "stress energy" also called the Weyl tensor??

How did Einstein know energy and pressure were part of gravitational attraction akin to mass?? (I knew some students who were incredible guessors, but not THAT good!)