Page 5242

ModPhy1/Unit1/GeneralRelativity/EinsteinGravity/

9/6/04

Einstein Tensor

---

Rather than begin this discussion with the Einstein tensor, let us begin with an arbitrarily curved Riemannian space and show how one can obtain the Einstein tensor from its geometry.

The first step in quantifying a Riemannian space is to select as system of coordinates that span the space. In n-dimensions, n coordinates are required. Let xⁱ = (x¹, x², …, xⁿ) represent such a coordinate system. (Note: the superscripts on these coordinates are not exponents but indices used only to distinguish one coordinate from another.)

Summation Convention

In the following equations, we will use Einstein’s summation convention such that whenever an index is repeated in a term, once as a subscript and once as a superscript, it signifies that that term should be summed over all possible values of the index. For example, the expression means

Displacement Vector

= displacement vector from one point to another dx^i = the difference in coordinates for the two points = the base vectors associated with the coordinate system

One of the most important vectors in any Riemannian space is the displacement vector shown above. It is expressed here in terms of a set of base vectors associated with the coordinate system. These base vectors are not necessarily unit vectors because they express how much and in what direction the displacement changes for a given change in one coordinate while the other coordinates are being held constant.

Minkowski Displacement Vector

= displacement in time dx^i = t, x, y, z = rectangular coordinates , , , = unit vectors in the coordinate directions = the unit imaginary number

An example of a displacement vector in Minkowski space is illustrated above. Notice that the base vectors in this example are imaginary because the displacement is real for a timelike displacement.

Reciprocal Base Vector

= the reciprocal base vectors associated with the coordinate system = the Kronecker delta = = the del operator = the gradient operator

For each set of base vectors, there exists a set of reciprocal base vectors such that (1) the dot product of each reciprocal vector with its corresponding base vector is unity and (2) each reciprocal vector is orthogonal (perpendicular) to all the other base vectors. This reciprocal relationship between the base and reciprocal base vectors greatly simplifies the evaluation of the dot product when the base vectors are not unit vectors.

The del operator takes the gradient of everything that follows it. It tells how fast the following quantity changes and in what direction that change occurs. If it operates on a scalar field, it produces a vector whose magnitude and direction are equal to the maximum rate of change of that field with respect to displacement. Therefore, the reciprocal base vector has a magnitude and direction equal to the maximum rate of change of its coordinate with respect to displacement.

Metric Tensor

ds = distance between neighboring points dx^i = the difference in coordinates for the two points gij = gji = components of the metric tensor = the metric tensor

The metric tensor expresses how the distance between neighboring points in space are related to one another.

Riemann Curvature Tensor

[¶k, ¶l ] = commutator of partial derivative operators R^ijkl = components of the Riemann tensor = Riemann curvature tensor

The fourth rank Riemann curvature tensor tells how a vector changes as it is parallel transported from one point to another along different paths. More specifically, take the base vector and displace it an amount dx^l along the x^l axis, then displace it an amount dx^k along the x^k axis, then do the same thing first along the x^k axis and then the x^l axis, then compare your results. If the space is flat, you will get the same thing from both routs. But if the space is curved, you will wind up with different vectors. Take the vector difference and dot (project) it onto the reciprocal base vector and you will have the Rⁱ_jkl component of the Riemann tensor. Repeat the process for all the other components and then express the invariant tensor in terms of its components and base vectors. When you get through doing all of this, let me know and we will celebrate together. For once you have obtained the components of the Riemann tensor, getting the Einstein tensor is rather straightforward.

Ricci Curvature Tensor

·(,..,,..) = contract the Riemann tensor on its first and third vectors. = Ricci curvature tensor

The Ricci tensor is the symmetrical second order tensor obtained by contracting the Riemann tensor on its first and third vectors. This means that you dot into the first of the ordered vectors and into the third and sum over the index k. Or you can shorten the whole process simply by dotting the Riemann tensor’s first ordered vectors into its third ordered vectors.

Curvature Scalar

g^ij = ·= contravarianat components of metric tensor ·(,) = contract the Ricci tensor R = Ricci curvature scalar

This Ricci scalar is a single invariant number that expresses the curvature of the space. However, since it is only one number, it does not contain the information that the Ricci and Riemann tensors contain. The more we contract the tensors, the simpler the description of curvature becomes, but the more information about curvature we lose.

Einstein Tensor

= Einstein tensor · = ·¶i = divergence of Einstein tensor

The Einstein tensor is a symmetrical second rank tensor whose divergence is identically zero. Like all the other tensors mentioned above, it is completely determined by the geometry of the space.

ModPhy1/Unit1/GeneralRelativity/EinsteinGravity/