Stuff about the Einstein tensor

[Martin]

Wiki has stuff on the Einstein equation which is nice

(someone at Wiki is occasionally using index-free notation, which is neater and easier to write)

 

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

 

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

 

this has my favorite way of writing the einstein equation which is

simply (in Wiki's notation)

 

G = kappa T

 

where G is the Einstein tensor (expressed in curvature units)

and T is the stress-energy tensor (expressed in energydensity or pressure units)

and kappa is the reciprocal of a force, namely the force c⁴/8piG.

 

Sean Carroll and John Baez also have some intuitive handles on the Einstein eqn.

 

We should collect some links. It is worth understanding from various angles. Here is Sean Carroll "Lecture Notes on GR"

http://pancake.uchicago.edu/~carroll/notes/

Here is a PDF version of Carroll's notes

http://arxiv.org/gr-qc/9712019

 

Sean also has a nice short article on the cosmological constant here

http://relativity.livingreviews.org/Articles/lrr-2001-1/node1.html

 

John Baez and Ted Bunn have a fine intuitive introduction to the Einstein equation called "The Meaning of Einstein's Equation"

http://math.ucr.edu/home/baez/einstein/einstein.html

 

It has a different way of looking at the equation which Baez has found works well in teaching relativity to novices. In a kind of postscript at the end, Bunn goes back and shows how this intuitive description relates mathematically to the usual form of the equations.

 

I will add some more links as they come up.

[Martin]

the way Wikipedia writes the Einstein equation has got to be my favorite because it's so clean:

 

G = kappa T

 

Here G is the Einstein tensor (expressed in curvature units)

and T is the stress-energy tensor (expressed in energydensity or pressure units)

and kappa is the reciprocal of a force, namely the force c⁴/8piG.

 

When you can get one tensor by taking another and multiplying by a constant then I would say that what you multiply by is the RATIO of the two tensors. Some people may not like that use of language but all I mean is that you get one tensor from the other by multiplying.

 

Baez has an intuitive description of what the stress-energy tensor measures here

http://math.ucr.edu/home/baez/einstein/node3.html

and it is clear that the units it is written in (if you write out the individual components in a 4x4 matrix) must be units of pressure, which is the same unit as energy density.

 

If you multiply each of those terms by what Wiki (and quite a lot of relativists/quantum gravity people) call kappa, well that is the same

as dividing each term by a certain constant force

 

and dividing a pressure by a force gives a reciprocal area unit, also known as a curvature.

 

and well, in mathematics there are a lot of different types of curvature, there is Gauss curvature and Riemann curvature and various contractions of the Riemann tensor. I guess the issue came up whether it was OK to pick one of them (say Riemann curvature) and say that was THE curvature and all the rest were NOT "really" curvature, even tho they were mathematical expressions written in units of curvature. But my take on it is that this is too restrictive, especially in an informal discussion.

 

As I see it the Ricci tensor and the Ricci scalar are both curvature-type quantities (which are contracted from Riemann) and it is OK to call them curvature. And also the Einst. tensor is just a combination of Ricci tensor together with the scalar times the metric, so in a general sense I think of it as measuring curvature too.

 

But this is just semantics and I dont know whether it really helps to worry about what curvature-type quantities one is going to call or not call that.

 

Anyway the basic notion of the einstein eqn is that the LHS measures spacetime curvature and the RHS involves a measure of energy density and that the two are related by a force

 

and the force is telling you how much concentration of energy it will take to produce a given amount of curvature------how much mass it takes to bend spacetime.

 

We should spell out the einstein tensor some more, talk about the Ricci tensor etc.

[luc]

How many stress-enrgy tensors exist? i know that there's the stress energy tnesor for a perfect fluid, that is the tensor that is habitually used to represent our Universe. there's also a stress energy tensor for a void universe. But I would like to know how many there are, I'm thinking even about the possibility that there exist an infinite number of different stress-energy tensors

[Johnny5]

 

Baez has an intuitive description of what the stress-energy tensor measures here

http://math.ucr.edu/home/baez/einstein/node3.html

and it is clear that the units it is written in (if you write out the individual components in a 4x4 matrix) must be units of pressure' date=' which is the same unit as energy density.

[/quote']

 

What do they mean by flow of momentum?

They start out with a spherical ball of 'test particles' all initially at rest with respect to one another, and then say that the Einstein equation leads to the conclusion that after awhile the ball will take on the shape of an ellipsoid. Does the Einstein equation imply this?

 

What in the world is t-momentum?!?

 

I like this article, it's well worth discussing when you get a chance. This part here is almost understandable... but not quite.

 

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the direction at that point, plus the pressure in the direction, plus the pressure in the direction.

 

It says that the "energy density" at the center of the ball plus the direction at that point, pluss.... see they lose me there. The center of a ball is a point in three dimensional space. The volume of a point is zero, since the radius is zero, and mass density is mass divided by volume, energy density would be energy divided by volume, but since volume of a point is zero, energy density of a point is infinite.

Am I to assume they mean in some tiny region of volume?

[nikols]

About the meaning of Einstein equation

 

I found this

 

http://www.settheory.net/cosmology

http://www.settheory.net/general-relativity

 

This is better than the John baez ' article

 

The relation between energy and curvature is not only expressed but also justified

It is directly applied to an important example (universal expansion)

The expression is simpler (relating 1 componnt of the energy tensor to 3 components of the Riemann tensor)

Both diagonal space and time components of the relation are expressed and justified, resulting in showing their similarity like a coincidence

 

.......