- #1
jstrunk
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I am trying to understand the meaning of the Ricci Tensor. I tried to work it out in a way that was meaningful to me based on ideas from Baez and Loveridge. Unfortumately, the forum tool won't allow me to include the URLs to those documents in this post. Anyway, I get the wrong answer. Can someone tell me where I am going wrong?
Ricci Tensor
Is given by [itex]R_{jk} = R_{jki}^i[/itex].
Describes the evolution of the size of a volume element as each point in the element flows along its
geodesic curve.
Motivation:
1. Consider a small volume element in a 3-D Riemannian space. Each point in the element is
traveling along its geodesic curve. At t=0 the element is aligned with the local coordinate basis
so that it is described by the vectors [itex]\left\{ {dx^1 \vec e_1 ,dx^2 \vec e_2 ,dx^3 \vec e_3 } \right\}[/itex]. [itex]\vec v[/itex] is the velocity of the reference point at
the corner of the element at the origin of the local coordinate basis. We will use the Geodesic
Deviation formula to find how the edges of the volume element are changing and use that
information to determine how the volume of the element is changing.
2. The change to an edge of the element is given by the Geodesic Deviation formula
[itex]
\nabla _v \nabla _v u^\alpha = R_{\beta \gamma \delta }^\alpha v^\beta v^\gamma u^\delta
[/itex]
where [itex]\vec v[/itex] is the geodesic velocity vector of a reference point and [itex]\vec u[/itex] is the geodesic vector
connecting the reference point to another point at equal times.
3. Edge [itex]\vec u = dx^1 \vec e_1[/itex] could be rewritten as [itex]\vec u = dx^1 \vec e_1 + 0\vec e_2 + 0\vec e_3 [/itex] giving geodesic deviation
$$
\eqalign{
& \nabla _v \nabla _v u^1 = R_{\beta \gamma \delta }^1 v^\beta v^\gamma u^\delta \cr
& \nabla _v \nabla _v u^1 = R_{\beta \gamma 1}^1 v^\beta v^\gamma u^1 + R_{\beta \gamma 2}^1 v^\beta v^\gamma u^2 + R_{\beta \gamma 3}^1 v^\beta v^\gamma u^3 \cr
& \nabla _v \nabla _v u^1 = \left( {R_{\beta \gamma 1}^1 v^\beta v^\gamma } \right)dx^1 + \left( {R_{\beta \gamma 2}^1 v^\beta v^\gamma } \right)0 + \left( {R_{\beta \gamma 3}^1 v^\beta v^\gamma } \right)0 \cr
& \nabla _v \nabla _v u^1 = \left( {R_{\beta \gamma 1}^1 v^\beta v^\gamma } \right)dx^1 + \left( {R_{\beta \gamma 2}^2 v^\beta v^\gamma } \right)0 + \left( {R_{\beta \gamma 3}^3 v^\beta v^\gamma } \right)0 \cr}
$$
4. The last expression contracts the Riemann Tensor on symmetric indices giving the Ricci Tensor.
The expression can be rewritten as
$$
\eqalign{
& \nabla _v \nabla _v u^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 + R_{\beta \gamma } v^\beta v^\gamma 0 + R_{\beta \gamma } v^\beta v^\gamma 0 \cr
& \nabla _v \nabla _v u^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 \cr
& \nabla _v \nabla _v dx^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 \cr}
$$
Note that there could be non-zero values for [itex]\nabla _v \nabla _v u^2 [/itex] and [itex]\nabla _v \nabla _v u^3 [/itex]but we don't care about them.
They effect the shape or orientation of the volume element but not the length of this edge.
5. If we denote the signed magnitude of the first and second covariant derivatives of [itex]u^\alpha
[/itex] as [itex]\dot u^\alpha[/itex] and [itex]\ddot u^\alpha[/itex],
then we have
$$
\ddot u^1 = d\ddot x^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1
$$
6. By a similar analysis we obtain the the signed magnitude of the second covariant derivative of the
all the edges of the volume element.
$$
\eqalign{
& d\ddot x^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 \cr
& d\ddot x^2 = R_{\beta \gamma } v^\beta v^\gamma dx^2 \cr
& d\ddot x^3 = R_{\beta \gamma } v^\beta v^\gamma dx^3 \cr}
$$
7. The size of an volume element and its covariant derivatives in this notation are
$$
\eqalign{
& V = \sqrt g \left( {dx^1 dx^2 dx^3 } \right) \cr
& \dot V = \sqrt g \left( {d\dot x^1 dx^2 dx^3 + dx^1 d\dot x^2 dx^3 + dx^1 dx^2 d^3 } \right) \cr
& \ddot V = \sqrt g \left[ {\left( {d\ddot x^1 dx^2 dx^3 + d\dot x^1 d\dot x^2 dx^3 + d\dot x^1 dx^2 d\dot x^3 } \right) + \left( {dx^1 d\ddot x^2 dx^3 + d\dot x^1 d\dot x^2 dx^3 + dx^1 d\dot x^2 d\dot x^3 } \right) + \left( {dx^1 dx^2 d\ddot x^3 + d\dot x^1 dx^2 d\dot x^3 + dx^1 d\dot x^2 d\dot x^3 } \right)} \right] \cr}
$$
8. The terms containing two first covariant derivatives are not caused by the curvature of space. They
are non-zero even in some coordinate systems in Euclidean space. For instance, imagine a cube
moving along the x-axis of a rectangular coordinate system. Then superimpose a spherical
coordinate system on the same space with the same origin. As the cube moves to greater x, its
thickness [itex]dx = dr[/itex] will stay the same. But the angles [itex]d\theta[/itex] and [itex]d\varphi[/itex] subtended by the cube will change.
You have two components changing times one that is isnt, even when volume isn't changing. We
subtract out these terms to get the expression for the volume change due to the curvature of space,
which we designate with subscript C.
$$
\ddot V_C = \sqrt g \left[ {d\ddot x^1 dx^2 dx^3 + dx^1 d\ddot x^2 dx^3 + dx^1 dx^2 d\ddot x^3 } \right]
$$
9. Substitute [itex]d\ddot x^\alpha = R_{\beta \gamma } v^\beta v^\gamma dx^\alpha[/itex] into
the last equation.
$$
\eqalign{
& \ddot V_C = \sqrt g \left[ {d\ddot x^1 dx^2 dx^3 + dx^1 d\ddot x^2 dx^3 + dx^1 dx^2 d\ddot x^3 } \right] \cr
& \ddot V_C = \sqrt g \left[ {\left( {R_{\beta \gamma } v^\beta v^\gamma dx^1 } \right)dx^2 dx^3 + dx^1 \left( {R_{\beta \gamma } v^\beta v^\gamma dx^2 } \right)dx^3 + dx^1 dx^2 \left( {R_{\beta \gamma } v^\beta v^\gamma dx^3 } \right)} \right] \cr
& \ddot V_C = \left( {R_{\beta \gamma } v^\beta v^\gamma } \right)\sqrt g \left[ {dx^1 dx^2 dx^3 + dx^1 dx^2 dx^3 + dx^1 dx^2 dx^3 } \right] \cr
& \ddot V_C = 3\left( {R_{\beta \gamma } v^\beta v^\gamma } \right)\sqrt g \left[ {dx^1 dx^2 dx^3 } \right] \cr}
$$
10. Divide both sides by the size of the volume element, [itex]V = \sqrt g \left( {dx^1 dx^2 dx^3 } \right)[/itex].
$$
\eqalign{
& {{\ddot V_C } \over V} = {{3\left( {R_{\beta \gamma } v^\beta v^\gamma } \right)\sqrt g \left[ {dx^1 dx^2 dx^3 } \right]} \over {\sqrt g \left( {dx^1 dx^2 dx^3 } \right)}} \cr
& {{\ddot V_C } \over V} = 3\left( {R_{\beta \gamma } v^\beta v^\gamma } \right) \cr}
$$
11. The answer is supposed to be [itex]
{{\ddot V_C } \over V} = - \left( {R_{\beta \gamma } v^\beta v^\gamma } \right)
[/itex].
Ricci Tensor
Is given by [itex]R_{jk} = R_{jki}^i[/itex].
Describes the evolution of the size of a volume element as each point in the element flows along its
geodesic curve.
Motivation:
1. Consider a small volume element in a 3-D Riemannian space. Each point in the element is
traveling along its geodesic curve. At t=0 the element is aligned with the local coordinate basis
so that it is described by the vectors [itex]\left\{ {dx^1 \vec e_1 ,dx^2 \vec e_2 ,dx^3 \vec e_3 } \right\}[/itex]. [itex]\vec v[/itex] is the velocity of the reference point at
the corner of the element at the origin of the local coordinate basis. We will use the Geodesic
Deviation formula to find how the edges of the volume element are changing and use that
information to determine how the volume of the element is changing.
2. The change to an edge of the element is given by the Geodesic Deviation formula
[itex]
\nabla _v \nabla _v u^\alpha = R_{\beta \gamma \delta }^\alpha v^\beta v^\gamma u^\delta
[/itex]
where [itex]\vec v[/itex] is the geodesic velocity vector of a reference point and [itex]\vec u[/itex] is the geodesic vector
connecting the reference point to another point at equal times.
3. Edge [itex]\vec u = dx^1 \vec e_1[/itex] could be rewritten as [itex]\vec u = dx^1 \vec e_1 + 0\vec e_2 + 0\vec e_3 [/itex] giving geodesic deviation
$$
\eqalign{
& \nabla _v \nabla _v u^1 = R_{\beta \gamma \delta }^1 v^\beta v^\gamma u^\delta \cr
& \nabla _v \nabla _v u^1 = R_{\beta \gamma 1}^1 v^\beta v^\gamma u^1 + R_{\beta \gamma 2}^1 v^\beta v^\gamma u^2 + R_{\beta \gamma 3}^1 v^\beta v^\gamma u^3 \cr
& \nabla _v \nabla _v u^1 = \left( {R_{\beta \gamma 1}^1 v^\beta v^\gamma } \right)dx^1 + \left( {R_{\beta \gamma 2}^1 v^\beta v^\gamma } \right)0 + \left( {R_{\beta \gamma 3}^1 v^\beta v^\gamma } \right)0 \cr
& \nabla _v \nabla _v u^1 = \left( {R_{\beta \gamma 1}^1 v^\beta v^\gamma } \right)dx^1 + \left( {R_{\beta \gamma 2}^2 v^\beta v^\gamma } \right)0 + \left( {R_{\beta \gamma 3}^3 v^\beta v^\gamma } \right)0 \cr}
$$
4. The last expression contracts the Riemann Tensor on symmetric indices giving the Ricci Tensor.
The expression can be rewritten as
$$
\eqalign{
& \nabla _v \nabla _v u^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 + R_{\beta \gamma } v^\beta v^\gamma 0 + R_{\beta \gamma } v^\beta v^\gamma 0 \cr
& \nabla _v \nabla _v u^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 \cr
& \nabla _v \nabla _v dx^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 \cr}
$$
Note that there could be non-zero values for [itex]\nabla _v \nabla _v u^2 [/itex] and [itex]\nabla _v \nabla _v u^3 [/itex]but we don't care about them.
They effect the shape or orientation of the volume element but not the length of this edge.
5. If we denote the signed magnitude of the first and second covariant derivatives of [itex]u^\alpha
[/itex] as [itex]\dot u^\alpha[/itex] and [itex]\ddot u^\alpha[/itex],
then we have
$$
\ddot u^1 = d\ddot x^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1
$$
6. By a similar analysis we obtain the the signed magnitude of the second covariant derivative of the
all the edges of the volume element.
$$
\eqalign{
& d\ddot x^1 = R_{\beta \gamma } v^\beta v^\gamma dx^1 \cr
& d\ddot x^2 = R_{\beta \gamma } v^\beta v^\gamma dx^2 \cr
& d\ddot x^3 = R_{\beta \gamma } v^\beta v^\gamma dx^3 \cr}
$$
7. The size of an volume element and its covariant derivatives in this notation are
$$
\eqalign{
& V = \sqrt g \left( {dx^1 dx^2 dx^3 } \right) \cr
& \dot V = \sqrt g \left( {d\dot x^1 dx^2 dx^3 + dx^1 d\dot x^2 dx^3 + dx^1 dx^2 d^3 } \right) \cr
& \ddot V = \sqrt g \left[ {\left( {d\ddot x^1 dx^2 dx^3 + d\dot x^1 d\dot x^2 dx^3 + d\dot x^1 dx^2 d\dot x^3 } \right) + \left( {dx^1 d\ddot x^2 dx^3 + d\dot x^1 d\dot x^2 dx^3 + dx^1 d\dot x^2 d\dot x^3 } \right) + \left( {dx^1 dx^2 d\ddot x^3 + d\dot x^1 dx^2 d\dot x^3 + dx^1 d\dot x^2 d\dot x^3 } \right)} \right] \cr}
$$
8. The terms containing two first covariant derivatives are not caused by the curvature of space. They
are non-zero even in some coordinate systems in Euclidean space. For instance, imagine a cube
moving along the x-axis of a rectangular coordinate system. Then superimpose a spherical
coordinate system on the same space with the same origin. As the cube moves to greater x, its
thickness [itex]dx = dr[/itex] will stay the same. But the angles [itex]d\theta[/itex] and [itex]d\varphi[/itex] subtended by the cube will change.
You have two components changing times one that is isnt, even when volume isn't changing. We
subtract out these terms to get the expression for the volume change due to the curvature of space,
which we designate with subscript C.
$$
\ddot V_C = \sqrt g \left[ {d\ddot x^1 dx^2 dx^3 + dx^1 d\ddot x^2 dx^3 + dx^1 dx^2 d\ddot x^3 } \right]
$$
9. Substitute [itex]d\ddot x^\alpha = R_{\beta \gamma } v^\beta v^\gamma dx^\alpha[/itex] into
the last equation.
$$
\eqalign{
& \ddot V_C = \sqrt g \left[ {d\ddot x^1 dx^2 dx^3 + dx^1 d\ddot x^2 dx^3 + dx^1 dx^2 d\ddot x^3 } \right] \cr
& \ddot V_C = \sqrt g \left[ {\left( {R_{\beta \gamma } v^\beta v^\gamma dx^1 } \right)dx^2 dx^3 + dx^1 \left( {R_{\beta \gamma } v^\beta v^\gamma dx^2 } \right)dx^3 + dx^1 dx^2 \left( {R_{\beta \gamma } v^\beta v^\gamma dx^3 } \right)} \right] \cr
& \ddot V_C = \left( {R_{\beta \gamma } v^\beta v^\gamma } \right)\sqrt g \left[ {dx^1 dx^2 dx^3 + dx^1 dx^2 dx^3 + dx^1 dx^2 dx^3 } \right] \cr
& \ddot V_C = 3\left( {R_{\beta \gamma } v^\beta v^\gamma } \right)\sqrt g \left[ {dx^1 dx^2 dx^3 } \right] \cr}
$$
10. Divide both sides by the size of the volume element, [itex]V = \sqrt g \left( {dx^1 dx^2 dx^3 } \right)[/itex].
$$
\eqalign{
& {{\ddot V_C } \over V} = {{3\left( {R_{\beta \gamma } v^\beta v^\gamma } \right)\sqrt g \left[ {dx^1 dx^2 dx^3 } \right]} \over {\sqrt g \left( {dx^1 dx^2 dx^3 } \right)}} \cr
& {{\ddot V_C } \over V} = 3\left( {R_{\beta \gamma } v^\beta v^\gamma } \right) \cr}
$$
11. The answer is supposed to be [itex]
{{\ddot V_C } \over V} = - \left( {R_{\beta \gamma } v^\beta v^\gamma } \right)
[/itex].