The space-time metric, \(g_{\mu\nu}\), is a geometric object, specifically a rank-2 symmetric tensor, specified at every point in a particular space-time. As space-time has n=4 dimensions, it follows that the metric has \(\frac{n(n+1)}{2} = 10\) independent components which are functions of position in space and time. As position can be expressed in 4 coordinates, the differentials to these coordinates form a convenient basis for the identification of these components. The choice of coordinates here are not unique and indeed differ from those used by Kerr in his original 1963 paper. Today this choice of coordinates is referred to as the Boyer-Lindquist coordinates \((t,\, r,\,\theta,\,\phi)\). The Kerr vacuum solution to General Relativity is expressed in terms of the line element, a compact way of expressing the metric: \( \begin{eqnarray} c^2 (d\tau)^2 & = & g_{tt}(t,r,\theta,\phi) (dt)^2 + g_{rr}(t,r,\theta,\phi) (dr)^2 + g_{\theta\theta}(t,r,\theta,\phi) (d\theta)^2 + g_{\phi\phi}(t,r,\theta,\phi) (d\phi)^2 \\ & & + 2 g_{tr}(t,r,\theta,\phi) \, dt \, dr + 2 g_{t\theta}(t,r,\theta,\phi) \, dt \, d\theta + 2 g_{t\phi}(t,r,\theta,\phi) \, dt \, d\phi \\ & & + 2 g_{r\theta}(t,r,\theta,\phi) \, dr \, d\theta + 2 g_{r\phi}(t,r,\theta,\phi) \, dr \, d\phi + 2 g_{\theta\phi}(t,r,\theta,\phi) \, d\theta \, d\phi \\ & = & \left( c^2 - \frac{2 c^2 G M^3 r}{c^2 M^2 r^2+J^2 \cos^2 \theta } \right) \, (dt)^2 - \frac{c^2 M^2 r^2+J^2 \cos^2 \theta }{J^2-2 G M^3 r+c^2 M^2 r^2} \, (dr)^2 - \frac{c^2 M^2 r^2+J^2 \cos^2 \theta}{c^2 M^2} \, (d\theta)^2 \\ & & - \left( \frac{J^2}{c^2 M^2} + r^2 + \frac{2 G J^2 M r \sin^2 \theta}{c^4 M^2 r^2 + c^2 J^2 \cos^2 \theta} \right) \sin^2 \theta \, (d\phi)^2 + \frac{4 G J M^2 r \sin^2 \theta}{c^2 M^2 r^2 + J^2 \cos^2 \theta} \, dt \, d\phi \end{eqnarray} \) As rank-2 tensors may be identified with matrices, here are the 10 components identified in shorthand: \( g_{\mu\nu} = \begin{pmatrix} g_{tt} & g_{tr} & g_{t\theta} & g_{t\phi} \\ g_{rt} = g_{tr} & g_{rr} & g_{r\theta} & g_{r\phi} \\ g_{\theta t} = g_{t \theta} & g_{\theta r} = g_{r \theta} & g_{\theta\theta} & g_{\theta\phi} \\ g_{\phi t} = g_{t \phi} & g_{\phi r} = g_{r \phi} & g_{\phi\theta} = g_{\theta\phi} & g_{\phi\phi} \end{pmatrix} \\ = \begin{pmatrix} c^2 - \frac{2 c^2 G M^3 r}{c^2 M^2 r^2+J^2 \cos^2 \theta} & { 0 } & { 0 } & \frac{2 G J M^2 r \sin^2 \theta}{c^2 M^2 r^2 + J^2 \cos^2 \theta} \\ { 0 } & - \frac{c^2 M^2 r^2+J^2 \cos^2 \theta }{J^2-2 G M^3 r+c^2 M^2 r^2} & { 0 } & { 0 } \\ { 0 } & { 0 } & - \frac{c^2 M^2 r^2+J^2 \cos^2 \theta}{c^2 M^2} & { 0 } \\ \frac{2 G J M^2 r \sin^2 \theta}{c^2 M^2 r^2 + J^2 \cos^2 \theta} & { 0 } & { 0 } & - \left( \frac{J^2}{c^2 M^2} + r^2 + \frac{2 G J^2 M r \sin^2 \theta}{c^4 M^2 r^2 + c^2 J^2 \cos^2 \theta} \right) \sin^2 \theta \end{pmatrix}\) Only values on or above the diagonal are independent. Obviously if \(J \to 0\) then we have: \(\begin{pmatrix} g_{tt} & g_{tr} & g_{t\theta} & g_{t\phi} \\ g_{rt} & g_{rr} & g_{r\theta} & g_{r\phi} \\ g_{\theta t} & g_{\theta r} & g_{\theta\theta} & g_{\theta\phi} \\ g_{\phi t} & g_{\phi r} & g_{\phi\theta} & g_{\phi\phi} \end{pmatrix} = \begin{pmatrix} c^2 - \frac{2 G M}{r} & { 0 } & { 0 } & { 0 } \\ { 0 } & - \frac{c^2 r}{-2 G M+c^2 r} & { 0 } & { 0 } \\ { 0 } & { 0 } & - r^2 & { 0 } \\ { 0 } & { 0 } & { 0 } & - r^2 \sin^2 \theta \end{pmatrix}\) which is the Schwarzschild metric and if we further take the limit as \(M\to 0\), we have the metric of Minkowski vacuum in polar coordinates: \(\begin{pmatrix} g_{tt} & g_{tr} & g_{t\theta} & g_{t\phi} \\ g_{rt} & g_{rr} & g_{r\theta} & g_{r\phi} \\ g_{\theta t} & g_{\theta r} & g_{\theta\theta} & g_{\theta\phi} \\ g_{\phi t} & g_{\phi r} & g_{\phi\theta} & g_{\phi\phi} \end{pmatrix} = \begin{pmatrix} c^2 & { 0 } & { 0 } & { 0 } \\ { 0 } & - 1 & { 0 } & { 0 } \\ { 0 } & { 0 } & - r^2 & { 0 } \\ { 0 } & { 0 } & { 0 } & - r^2 \sin^2 \theta \end{pmatrix}\) So hopefully, any results we derive from the Kerr metric will have immediate applicability to these other well-known vacuum metrics. The first question, I think, should be: if this is the Kerr vacuum solution, to what is it a solution of? And the answer is a deceptively simple statement of geometry: \(G_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8 \pi G}{c^4} T_{\mu\nu} =_{\tiny \textrm{Vacuum}} 0\) where the left side is the Einstein curvature tensor which can be expressed in terms of the components of the metric, and the right side is the stress-energy-momentum tensor of space-time which for a vacuum is zero. Thus this thread is devoted to verifying that for this Kerr metric that the Einstein curvature tensor is in fact zero and along the way I will introduce additional symbols and calculations. Because coordinates label positions in space-time and the metric specifies the shape of space-time, geometry (and calculus) in curved spacetime is considerably different than working with the Cartesian coordinates of Euclidean space. We frequently need the metric tensor (or its inverse) to connect quantities in ways that are valid. Round one: The inverse metric. Related to the space-time metric, \(g_{\mu\nu}\) there is an inverse metric which has the unusual notational convention that winds up having utility: \(g^{\mu\nu} = \left( g^{-1} \right)^{\mu\nu}\). Since the Kerr metric only has 5 non-zero components, its inverse is particularly easy to compute: \( g^{\mu\nu} = \begin{pmatrix} g^{tt} & g^{tr} & g^{t\theta} & g^{t\phi} \\ g^{rt} = g^{tr} & g^{rr} & g^{r\theta} & g^{r\phi} \\ g^{\theta t} = g^{t \theta} & g^{\theta r} = g^{r \theta} & g^{\theta\theta} & g^{\theta\phi} \\ g^{\phi t} = g^{t \phi} & g^{\phi r} = g^{r \phi} & g^{\phi\theta} = g^{\theta\phi} & g^{\phi\phi} \end{pmatrix} = \begin{pmatrix} \frac{ g_{\phi\phi} }{ g_{tt} g_{\phi\phi} - (g_{t \phi})^2 } & 0 & 0 & \frac{ g_{t \phi} }{ (g_{t \phi})^2 - g_{tt} g_{\phi\phi} } \\ 0 & \frac{1}{g_{rr}} & 0 & 0 \\ 0 & 0 & \frac{1}{g_{\theta\theta}} & 0 \\ \frac{ g_{t \phi} }{ (g_{t \phi})^2 - g_{tt} g_{\phi\phi} } & 0 & 0 & \frac{ g_{tt} }{ g_{tt} g_{\phi\phi} - (g_{t \phi})^2 } \end{pmatrix} = { \tiny \begin{pmatrix} \frac{1}{c^2}\left( \frac{J^2 + c^2 M^2 r^2}{J^2 + c^2 M^2 r^2 -2 r G M^3} + \frac{2 G J^2 M^3 r \sin^2 \theta }{(J^2 + c^2 M^2 r^2 - 2 r G M^3)(J^2 \cos^2 \theta + c^2 r^2 M^2)} \right) & 0 & 0 & \frac{2 G J M^4 r}{(c^2 M^2 r^2 + J^2 \cos^2 \theta) (J^2 - 2 G M^3 r + c^2 r^2 M^2)} \\ 0 & - \frac{J^2-2 G M^3 r+c^2 M^2 r^2}{c^2 M^2 r^2+J^2 \cos^2 \theta } & 0 & 0 \\ 0 & 0 & - \frac{c^2 M^2}{c^2 M^2 r^2+J^2 \cos^2 \theta} & 0 \\ \frac{2 G J M^4 r}{(c^2 M^2 r^2 + J^2 \cos^2 \theta) (J^2 - 2 G M^3 r + c^2 r^2 M^2)} & 0 & 0 & -\frac{c^2 M^2 ( J^2 \cos^2 \theta -2 r G M^3 + c^2 r^2 M^2)}{(c^2 r^2 M^2+ J^2 -2 r G M^3) (J^2 \cos^2 \theta + c^2 r^2 M^2) \sin^2 \theta} \end{pmatrix} } \) OK, so that wasn't so easy. More importantly, it was error-prone. So for purposes of algebra, the following substitutions are useful to keep track of the various terms: \( \begin{eqnarray} \chi(\theta) &= & \cos \theta \\ 1 - \chi^2 & = & \sin^2 \theta \\ r_s & = & \frac{2 G M}{c^2} \\ \alpha & = & \frac{J}{Mc} \\ \rho^2(r, \theta) & = & r^2 + \alpha^2 \chi^2 & = & \frac{M^2 c^2 r^2 + J^2 \cos^2 \theta}{M^2 c^2} \\ \Delta(r) & = & r^2 + \alpha^2 - r_s r & = & \frac{M^2 c^2 r^2 + J^2 - 2 G M^3 r}{M^2 c^2} \\ \rho^2 - r_s r &= & \Delta - \alpha^2 ( 1 - \chi^2 ) & = & \frac{M^2 c^2 r^2 + J^2 \cos^2 \theta - 2 G M^3 r}{M^2 c^2} \end{eqnarray}\) And we have: \( g_{\mu\nu} = \begin{pmatrix} \frac{(\rho^2 - r_s r)c^2}{\rho^2} & 0 & 0 & \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} \\ 0 & - \frac{\rho^2}{\Delta} & 0 & 0 \\ 0 & 0 & - \rho^2 & 0 \\ \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} & 0 & 0 & - \left(r^2 + \alpha^2 + \frac{\alpha^2 r_s r (1 - \chi^2)}{\rho^2} \right) (1 - \chi^2) \end{pmatrix}\) \( g^{\mu\nu} = \begin{pmatrix} \frac{1}{c^2}\left(\frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r (1 - \chi^2)}{\rho^2 \Delta} \right) & 0 & 0 & \frac{\alpha r_s r}{c \rho^2 \Delta} \\ 0 & - \frac{\Delta}{\rho^2} & 0 & 0 \\ 0 & 0 & - \frac{1}{\rho^2} & 0 \\ \frac{\alpha r_s r}{c \rho^2 \Delta} & 0 & 0 & - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 - \chi^2)} \end{pmatrix} \) Finally, we can demonstrate that these are inverses in the normal sense by computing: \(g_{\mu a} g^{a \nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \delta_{\mu}^{\nu}\) Tensor contractions require summing over the shared slots (always paired upper and lower) so \(\delta_{t}^{t} = g_{t t} g^{t t} + g_{t r} g^{r t} + g_{t \theta} g^{\theta t} + g_{t \phi} g^{\phi t} = \frac{(\rho^2 - r_s r)c^2}{\rho^2} \times \left( \frac{r^2 + \alpha^2}{c^2 \Delta} + \frac{\alpha^2 r_s r (1 - \chi^2)}{c^2 \rho^2 \Delta} \right) + 0 \times 0 + 0 \times 0 + \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} \times \frac{\alpha r_s r}{c \rho^2 \Delta} = 1\) -- Coming Soon: All First and Second Derivatives of the Metric components, Connections, Curvature and more -- Thanks to: http://en.wikipedia.org/wiki/Kerr_metric (for the sign conventions) http://www.roma1.infn.it/teongrav/leonardo/bh/bhcap3.pdf (for 3.18) http://arxiv.org/abs/0706.0622 (for a history and any number of good ideas)
I have a question about this. Does the metric need all 10 components? Wouldn't it suffice to describe momentum in terms of each spatial dimension as they relate to time, and drop 3 components? Why would we care about, for example, X dimension as it relates to Y dimension? This seems redundant to me, yet we call them independent components..?
Part I might be special relativity. Part II might be linear algebra. Part III might be algebraic geometry. Part IV might be multi-variate calculus. Part V might be Riemannian geometry. Part VI might be tensor calculus. Any 4-dimensional metrical theory -- like General Relativity -- requires 10 components to fully specify all possible symmetrical rank-2 tensors. We aren't describing momentum (yet) -- we are talking about the metric of space-time. Your confusion might be because you don't know how a metric is used. From Part III: Linear Geometry and coordinate changes in 2-D Let's have a sheet of paper and mark off rectilinear coordinates with 1-cm spacings in the horizontal direction and 1-inch spacing in the vertical direction. Given \((h_1,v_1)\) and \((h_2,v_2)\) as coordinates of points along these spacings, which is not the only choice of coordinates, what is the distance between those points? \(d^2 = \begin{pmatrix}h_2 - h_1 & v_2 - v_1 \end{pmatrix} \begin{pmatrix} \frac{1}{10^4} \, \textrm{m}^2 & 0 \\ 0 & \frac{64516}{10^8} \, \textrm{m}^2\end{pmatrix} \begin{pmatrix}h_2 - h_1 \\ v_2 - v_1 \end{pmatrix} = ( h_2 - h_1 ) \left( \frac{1}{10^4} \, \textrm{m}^2 \right) ( h_2 - h_1 ) + ( v_2 - v_1 ) \left( \frac{64516}{10^8} \, \textrm{m}^2 \right) ( v_2 - v_1 ) = \frac{10^4 ( h_2 - h_1 )^2 + 64516 ( v_2 - v_1 )^2 }{10^8} \, \textrm{m}^2\) But, for the same sheet of paper, I could equally well have coordinates \(\alpha = 12 + \sqrt{\frac{1}{2}} h + \sqrt{\frac{1}{2}} v , \; \beta = 14 + \sqrt{\frac{1}{2}} h - \sqrt{\frac{1}{2}} v\), in which case \( \begin{pmatrix}h_2 - h_1 \\ v_2 - v_1 \end{pmatrix} = \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} \alpha_2 - \alpha_1 \\ \beta_2 - \beta_1 \end{pmatrix} \) and \(d^2 = \begin{pmatrix} \alpha_2 - \alpha_1 \\ \beta_2 - \beta_1 \end{pmatrix}^T \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix}^T \begin{pmatrix} \frac{1}{10^4} \, \textrm{m}^2 & 0 \\ 0 & \frac{64516}{10^8} \, \textrm{m}^2\end{pmatrix} \begin{pmatrix} \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \\ \sqrt{\frac{1}{2}} & - \sqrt{\frac{1}{2}} \end{pmatrix} \begin{pmatrix} \alpha_2 - \alpha_1 \\ \beta_2 - \beta_1 \end{pmatrix} = \begin{pmatrix} \alpha_2 - \alpha_1 & \beta_2 - \beta_1 \end{pmatrix} \begin{pmatrix} \frac{37258}{10^8} \, \textrm{m}^2 & -\frac{27258}{10^8} \, \textrm{m}^2 \\ -\frac{27258}{10^8} \, \textrm{m}^2 & \frac{37258}{10^8} \, \textrm{m}^2\end{pmatrix} \begin{pmatrix} \alpha_2 - \alpha_1 \\ \beta_2 - \beta_1 \end{pmatrix} = \frac{ 37258 (\alpha_2 - \alpha_1)^2-54516 (\alpha_2 - \alpha_1) (\beta_2 - \beta_1)+37258 (\beta_2 - \beta_1)^2 }{10^8} \, \textrm{m}^2 = \frac{10^4 ( h_2 - h_1 )^2 + 64516 ( v_2 - v_1 )^2 }{10^8} \, \textrm{m}^2\) So the principle that any non-singular linear combinations of our original coordinates allows for a admissible system of coordinates do to geometry and compute our paper metric generalizes to General Relativity for allowing any smooth system of non-singular coordinates whatsoever, with the geometric object of the metric having different representations in different coordinate systems. Just as we need \(\frac{2 (2+1)}{2} = 3\) components in 2-d paper, so we need \(\frac{4 (4+1)}{2} = 10\) components in 4-d space-time. Of course, zero components save us some effort, so we like Boyer-Lindquist coordinates for some purposes.
I see, so in order to be as general as possible we must allow for each dimension to have arbitrary coordinates, and therefore must supply a component to account for each combination of dimensional pairings?
That's the idea, to be as general as possible. This gives a lot of freedom to find the "best" coordinates although "best", like beauty, is often in the eye of the beholder.
Just for those that wanted to follow the algebra. \(\delta_{ t }^{ t } = g_{ t k } g^{ k t } = \left( \frac{(\rho^2 - r_s r)c^2}{\rho^2} \right) \left( \frac{1}{c^2} \left( \frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r ( 1 - \chi^2 )}{\rho^2 \Delta} \right) \right) \; + \; \left( \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} \right) \left( \frac{\alpha r_s r }{c \rho^2 \Delta} \right) \quad \quad \quad = \frac{(\rho^2 - r_s r)(r^2 + \alpha^2) \rho^2 + (\rho^2 - r_s r)(1 - \chi^2) \alpha^2 r_s r + \alpha^2 r_s^2 r^2 (1- \chi^2)}{\left( \rho^2 \right)^2 \Delta} \quad \quad \quad = \frac{ r^4 + \alpha^2 r^2 + \alpha^2 r^2 \chi^2 + \alpha^4 \chi^2 - r_s r^3 - \alpha^2 r_s r + \alpha^2 r_s r - \alpha^2 r_s r \chi^2}{r^4 + \alpha^2 r^2 + \alpha^2 r^2 \chi^2 + \alpha^4 \chi^2 - r_s r^3 - \alpha^2 r_s r \chi^2}= 1 \delta_{ t }^{ r } = \delta_{ r }^{ t } = g_{ t k } g^{ k r } = g_{ r k } g^{ k t } = 0 \delta_{ t }^{ \theta } = \delta_{ \theta }^{ t } = g_{ t k } g^{ k \theta } = g_{ \theta k } g^{ k t } = 0 \delta_{ t }^{ \phi } = \delta_{ \phi }^{ t } = g_{ t k } g^{ k \phi } = g_{ \phi k } g^{ k t } = \left( \frac{(\rho^2 - r_s r)c^2}{\rho^2} \right) \left( \frac{\alpha r_s r}{c \rho^2 \Delta} \right) \; + \; \left( \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} \right) \left( - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 -\chi^2)} \right) = 0 \delta_{ r }^{ r } = g_{ r k } g^{ k r } = \left( -\frac{\rho^2}{\Delta} \right) \left( -\frac{\Delta}{\rho^2} \right) = 1 \delta_{ r }^{ \theta } = \delta_{ \theta }^{ r } = g_{ r k } g^{ k \theta } = g_{ \theta k } g^{ k r } = 0 \delta_{ r }^{ \phi } = \delta_{ \phi }^{ r } = g_{ r k } g^{ k \phi } = g_{ \phi k } g^{ k r } = 0 \delta_{ \theta }^{ \theta } = g_{ \theta k } g^{ k \theta } = \left( -\rho^2 \right) \left( -\frac{1}{\rho^2} \right) = 1 \delta_{ \theta }^{ \phi } = \delta_{ \phi }^{ \theta } = g_{ \theta k } g^{ k \phi } = g_{ \phi k } g^{ k \theta } = 0 \delta_{ \phi }^{ \phi } = g_{ \phi k } g^{ k \phi } = \left( \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} \right) \left( \frac{\alpha r_s r}{c \rho^2 \Delta} \right) \; + \; \left( -(r^2 + \alpha^2)(1-\chi^2) - \frac{\alpha^2 r_s r \left( 1 - \chi^2 \right)^2}{\rho^2} \right) \left( - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 -\chi^2)} \right) \quad \quad \quad = \frac{\alpha^2 r_s^2 r^2 ( 1 - \chi^2 ) + \left(\rho^2 \right)^2 (r^2 + \alpha^2) - r_s r^3 \rho^2 - \alpha^2 r_s r \rho^2 + \rho^2 \alpha^2 r_s r (1 - \chi^2 ) - \alpha^2 r_s^2 r^2 ( 1 - \chi^2 ) }{\left( \rho^2 \right)^2 \Delta} \quad \quad \quad = \frac{\rho^2 (r^2 + \alpha^2) - r_s r^3 - \alpha^2 r_s r + \alpha^2 r_s r (1 - \chi^2 ) }{ \rho^2 \Delta} \quad \quad \quad = \frac{r^4 + \alpha^2 r^2 + \alpha^2 r^2 \chi^2 + \alpha^4 \chi^2 - r_s r^3 - \alpha^2 r_s r + \alpha^2 r_s r - \alpha^2 r_s r \chi^2 }{r^4 + \alpha^2 r^2 + \alpha^2 r^2 \chi^2 + \alpha^4 \chi^2 - r_s r^3 - \alpha^2 r_s r \chi^2} = 1\)
Next topic... Round two: Coordinate-based derivatives. Because tensors are geometric objects, the only natural concept of the derivative of a tensor is a geometrically-informed one. Having written the coordinate-based components of the tensor in terms of those components, this is something we always need to keep in mind. But if we do, we will be able to build natural geometric derivatives \(\nabla_a T\) from the metric, it's inverse and coordinate-based derivatives, \(\partial_\mu g_{\nu\xi}\). So lets start with that. Here we are assisted, because none of these terms has any dependence on \(t\) or \(\phi\). So let's compute all first derivatives with respect to coordinates. ( A important thing to remember about this coordinates is that \(0 \leq \theta \leq \pi \) so \(\sin \theta \geq 0\) and therefore \(\sin \theta = + \sqrt{ 1 - \cos^2 \theta}\). Otherwise we could not hide all the dependence on trignometric functions of \(\theta\)) So by starting with the components and some related expressions, \(\begin{eqnarray} \chi(\theta) &= & \cos \theta \\ 1 - \chi^2 & = & \sin^2 \theta \\ r_s & = & \frac{2 G M}{c^2} \\ \alpha & = & \frac{J}{Mc} \\ \rho^2(r, \theta) & = & r^2 + \alpha^2 \chi^2 & = & \frac{M^2 c^2 r^2 + J^2 \cos^2 \theta}{M^2 c^2} \\ \Delta(r) & = & r^2 + \alpha^2 - r_s r & = & \frac{M^2 c^2 r^2 + J^2 - 2 G M^3 r}{M^2 c^2} \\ \rho^2 - r_s r &= & \Delta - \alpha^2 ( 1 - \chi^2 ) & = & \frac{M^2 c^2 r^2 + J^2 \cos^2 \theta - 2 G M^3 r}{M^2 c^2} \\ g_{tt} & = & \frac{(\rho^2 - r_s r)c^2}{\rho^2} \\ g_{t\phi} & = & \frac{c \alpha r_s r (1 - \chi^2)}{\rho^2} & = & g_{\phi t} \\ g_{rr} & = & - \frac{\rho^2}{\Delta} \\ g_{\theta\theta} & = & - \rho^2 \\ g_{\phi\phi} & = & - \left(r^2 + \alpha^2 + \frac{\alpha^2 r_s r (1 - \chi^2)}{\rho^2} \right) (1 - \chi^2) & = & - (r^2 + \alpha^2 ) (1 - \chi^2) - \frac{\alpha^2 r_s r (1 - \chi^2)^2}{\rho^2} \end{eqnarray}\) We apply repeated the chain-rule, the product rule and the quotient rule, keeping a close eye on those signs to get the following: \(\begin{array}{c|c|c} \textrm{Term} & \partial_r \, \textrm{Term} & \partial_{\theta} \, \textrm{Term} \\ \hline \chi & 0 & - \sqrt{1 - \chi^2} \sqrt{1 - \chi^2} & 0 & \chi \chi^2 & 0 & -2 \chi \sqrt{1 - \chi^2} 1 - \chi^2 & 0 & 2 \chi \sqrt{1 - \chi^2} \chi \sqrt{1 - \chi^2} & 0 & 2 \chi^2 - 1 \chi \left(1 - \chi^2\right)^{ \tiny \frac{3}{2} } & 0 & (4 \chi^2 - 1)(1 - \chi^2) \chi \left(1 - \chi^2\right)^{ \tiny \frac{5}{2} } & 0 & (6 \chi^2 - 1)(1- \chi^2)^2 \rho^2 & 2r & -2 \alpha^2 \chi \sqrt{1 - \chi^2} \Delta & 2r - r_s & 0 \rho^2 - r_s r & 2r - r_s & -2 \alpha^2 \chi \sqrt{1 - \chi^2} \alpha^2 r_s r & \alpha^2 r_s & 0 \alpha^2 r_s r (1 - \chi^2) & \alpha^2 r_s (1 - \chi^2) & 2 \alpha^2 r_s r \chi \sqrt{1 - \chi^2} \alpha^2 r_s r (1 - \chi^2)^2 & \alpha^2 r_s (1 - \chi^2)^2 & 4 \alpha^2 r_s r \chi \left(1 - \chi^2\right)^{\tiny \frac{3}{2}} \frac{\alpha^2 r_s r (1 - \chi^2)^2}{\rho^2} & - \frac{ ( r^2 - \alpha^2 \chi^2) \alpha^2 r_s (1 - \chi^2)^2}{\left( \rho^2 \right)^2} & \frac{2 \left( 2 r^2 + \alpha^2 (1 + \chi^2 ) \right) \alpha^2 r_s r \chi \left(1 - \chi^2\right)^{\tiny \frac{3}{2}} }{\left( \rho^2 \right)^2} (r^2 + \alpha^2 ) (1 - \chi^2) & 2 r (1 - \chi^2) & 2 (r^2 + \alpha^2 ) \chi \sqrt{1 - \chi^2} g_{tt} & \frac{c^2 r_s ( r^2 - \alpha^2 \chi^2 )}{\left( \rho^2 \right)^2} & - \frac{2 c^2 \alpha^2 r_s r \chi \sqrt{1 - \chi^2}}{\left( \rho^2 \right)^2} g_{t\phi} & - \frac{c \alpha r_s (r^2 - \alpha^2 \chi^2 ) (1 - \chi^2) }{\left( \rho^2 \right)^2} & - \frac{2 c \alpha r_s ( r^2 + 2 \alpha^2 \chi^2 - \alpha^2 ) r \chi \sqrt{1 - \chi^2}}{\left( \rho^2 \right)^2} g_{rr} & \frac{r_s (r^2 - \alpha^2 \chi^2) - 2 \alpha r ( 1 - \chi^2 )}{\Delta^2} & \frac{2 \alpha^2 \chi \sqrt{1 - \chi^2}}{\Delta} g_{\theta\theta} & - 2r & 2 \alpha^2 \chi \sqrt{1 - \chi^2} g_{\phi\phi} & \frac{ ( r^2 - \alpha^2 \chi^2) \alpha^2 r_s (1 - \chi^2)^2}{\left( \rho^2 \right)^2} - 2 r (1 - \chi^2) & - \frac{2 \left( 2 r^2 + \alpha^2 (1 + \chi^2 ) \right) \alpha^2 r_s r \chi \left(1 - \chi^2\right)^{\tiny \frac{3}{2}} }{\left( \rho^2 \right)^2} - 2 (r^2 + \alpha^2 ) \chi \sqrt{1 - \chi^2} \end{array}\) http://en.wikipedia.org/wiki/Derivative#Rules_for_finding_the_derivative http://en.wikipedia.org/wiki/Partial_derivative
Round Three: Connections (As you may have guessed, the last three posts have been saved on my laptop for many months. Things just kept getting in the way of proofing every calculation. Sorry for the delay.) Define \(\Gamma^{\mu}_{\nu\xi} = \frac{1}{2} \sum_{o} g^{\mu o} \left( \partial_{\xi} g_{o \nu} + \partial_{\nu} g_{o \xi} - \partial_{o} g_{\nu\xi}\right)\). If it weren't for symmetry, this would be 64 complicated sums of products of sums. But because \(g_{\mu\nu} = g_{\nu\mu}\) it follows that \(\Gamma^{\mu}_{\nu\xi} = \Gamma^{\mu}_{\xi\nu}\). Further, because the metric has only 5 non-zero components, which only exhibit variation as functions of \(\theta\) and \(r\) then this calculation is quite a lot bit easier than the general case. (Because this is part X, the importance of the connection in making a "connection" between coordinate-based algebra on the metric and the geometry of space-time has presumably already been discussed. But, if you need help with this concept, let me know.) http://en.wikipedia.org/wiki/Levi-Civita_connection Here are the five non-zero components of the inverse metric: \(g^{tt} = \frac{1}{c^2}\left(\frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r (1 - \chi^2)}{\rho^2 \Delta} \right) g^{t\phi} = \frac{\alpha r_s r}{c \rho^2 \Delta} g^{rr} = - \frac{\Delta}{\rho^2} g^{\theta\theta} = - \frac{1}{\rho^2} g^{\phi\phi} = - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 - \chi^2)} \) Which we apply to simplify the general expression in various ways \( \Gamma^{\mu}_{\nu\xi} = \Gamma^{\mu}_{\xi\nu} = \frac{1}{2} \left( g^{\mu t} \left( \partial_{\xi} g_{t \nu} + \partial_{\nu} g_{t \xi} \right) + g^{\mu r} \left( \partial_{\xi} g_{r \nu} + \partial_{\nu} g_{r \xi} - \partial_{r} g_{\nu\xi}\right) + g^{\mu \theta} \left( \partial_{\xi} g_{\theta \nu} + \partial_{\nu} g_{\theta \xi} - \partial_{\theta} g_{\nu\xi}\right) + g^{\mu \phi} \left( \partial_{\xi} g_{\phi \nu} + \partial_{\nu} g_{\phi \xi} \right) \right) \Gamma^{t}_{\nu\xi} = \Gamma^{t}_{\xi\nu} = \frac{1}{2} \left( g^{t t} \left( \partial_{\xi} g_{t \nu} + \partial_{\nu} g_{t \xi} \right) + g^{t \phi} \left( \partial_{\xi} g_{\phi \nu} + \partial_{\nu} g_{\phi \xi} \right) \right) \Gamma^{r}_{\nu\xi} = \Gamma^{r}_{\xi\nu} = \frac{1}{2} g^{r r} \left( \partial_{\xi} g_{r \nu} + \partial_{\nu} g_{r \xi} - \partial_{r} g_{\nu\xi}\right) \Gamma^{\theta}_{\nu\xi} = \Gamma^{\theta}_{\xi\nu} = \frac{1}{2} g^{\theta \theta} \left( \partial_{\xi} g_{\theta \nu} + \partial_{\nu} g_{\theta \xi} - \partial_{\theta} g_{\nu\xi}\right) \Gamma^{\phi}_{\nu\xi} = \Gamma^{\phi}_{\xi\nu} = \frac{1}{2} \left( g^{\phi t} \left( \partial_{\xi} g_{t \nu} + \partial_{\nu} g_{t \xi} \right) + g^{\phi \phi} \left( \partial_{\xi} g_{\phi \nu} + \partial_{\nu} g_{\phi \xi} \right) \right) \Gamma^{\mu}_{\nu\nu} = g^{\mu t} \partial_{\nu} g_{t \nu} + g^{\mu r} \left( \partial_{\nu} g_{r \nu} - \frac{1}{2} \partial_{r} g_{\nu\nu}\right) + g^{\mu \theta} \left( \partial_{\nu} g_{\theta \nu} - \frac{1}{2} \partial_{\theta} g_{\nu\nu}\right) + g^{\mu \phi} \partial_{\nu} g_{\phi \nu} \Gamma^{t}_{\nu\nu} = g^{t t} \partial_{\nu} g_{t \nu} + g^{t \phi} \partial_{\nu} g_{\phi \nu} \Gamma^{r}_{\nu\nu} = g^{r r} \left( \partial_{\nu} g_{r \nu} - \frac{1}{2} \partial_{r} g_{\nu\nu}\right) \Gamma^{\theta}_{\nu\nu} = g^{\theta \theta} \left( \partial_{\nu} g_{\theta \nu} - \frac{1}{2} \partial_{\theta} g_{\nu\nu}\right) \Gamma^{\phi}_{\nu\nu} = g^{\phi t} \partial_{\nu} g_{t \nu} + g^{\phi \phi} \partial_{\nu} g_{\phi \nu} \) With specific gory calculation: \( \Gamma_{ t t }^{ t } = 0 \Gamma_{ t r }^{ t } = \Gamma_{ r t }^{ t } = \left( \frac{1}{2} \right) \left( \frac{1}{c^2} \left( \frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r ( 1 - \chi^2 )}{\rho^2 \Delta} \right) \right) \left( \frac{c^2 r_s ( r^2 - \alpha^2 \chi^2 )}{\left( \rho^2 \right)^2 } \right) \; + \; \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r}{c \rho^2 \Delta} \right) \left( - \frac{ c \alpha r_s (r^2 - \alpha^2 \chi^2)(1 - \chi^2)}{ \left( \rho^2 \right)^2 \right) \quad \quad \quad = \frac{r_s (r^2 + \alpha^2)(r^2 - \alpha^2 \chi^2)}{2 \left( \rho^2 \right)^2 \Delta} \Gamma_{ t \theta }^{ t } = \Gamma_{ \theta t }^{ t } = \left( \frac{1}{2} \right) \left( \frac{1}{c^2} \left( \frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r ( 1 - \chi^2 )}{\rho^2 \Delta} \right) \right) \left( - \frac{2 c^2 \alpha^2 r_s r \chi \sqrt{1-\chi^2}}{\left( \rho^2 \right)^2} \right) \; + \; \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r}{c \rho^2 \Delta} \right) \left( -\frac{2 c \alpha r_s (r^2 + 2 \alpha^2 \chi^2 - \alpha^2) r \chi \sqrt{1 -\chi^2}}{\left( \rho^2 \right)^2} \right) \quad \quad \quad = - \frac{\alpha^2 r_s ( r^2 + \alpha^2 + r_s r ) r \chi \sqrt{1-\chi^2}}{\left( \rho^2 \right)^2 \Delta} \Gamma_{ t \phi }^{ t } = \Gamma_{ \phi t }^{ t } = 0\) \(\Gamma_{ r r }^{ t } = 0 \Gamma_{ r \theta }^{ t } = \Gamma_{ \theta r }^{ t } = 0 \Gamma_{ r \phi }^{ t } = \Gamma_{ \phi r }^{ t } = \left( \frac{1}{2} \right) \left( \frac{1}{c^2} \left( \frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r ( 1 - \chi^2 )}{\rho^2 \Delta} \right) \right) \left( - \frac{ c \alpha r_s (r^2 - \alpha^2 \chi^2)(1 - \chi^2)}{ \left( \rho^2 \right)^2 \right) \\ \quad \quad \quad \quad \quad \quad + \; \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r}{c \rho^2 \Delta} \right) \left( \frac{(r^2 - \alpha^2 \chi^2) \alpha^2 r_s \left(1 - \chi^2 \right)^2}{\left( \rho^2 \right)^2 } - 2 r ( 1 - \chi^2 ) \right) \quad \quad \quad = - \frac{\alpha r_s ( 3 r^4 + \alpha^2 r^2 + \alpha^2 r^2 \chi^2 - \alpha^4 \chi^2 ) ( 1 - \chi^2 )}{2 c \left( \rho^2 \right)^2 \Delta} \Gamma_{ \theta \theta }^{ t } = 0 \Gamma_{ \theta \phi }^{ t } = \Gamma_{ \phi \theta }^{ t } = \left( \frac{1}{2} \right) \left( \frac{1}{c^2} \left( \frac{r^2 + \alpha^2}{\Delta} + \frac{\alpha^2 r_s r ( 1 - \chi^2 )}{\rho^2 \Delta} \right) \right) \left( -\frac{2 c \alpha r_s (r^2 + 2 \alpha^2 \chi^2 - \alpha^2) r \chi \sqrt{1 -\chi^2}}{\left( \rho^2 \right)^2} \right) \\ \quad \quad \quad \quad \quad \quad + \; \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r}{c \rho^2 \Delta} \right) \left( -\frac{2 \left( 2 r^2 + \alpha^2 + \alpha^2 \chi^2 \right) \alpha^2 r_s r \chi \left( 1 - \chi^2 \right)^{\tiny \frac{3}{2}}}{\left( \rho^2 \right)^2} - 2 ( r^2 + \alpha^2 ) \chi \sqrt{1 - \chi^2} \right) \\ \quad \quad \quad = - \frac{\alpha r_s r \chi \sqrt{1 - \chi^2} \left( 3 \alpha^2 r_s r ( 1 - \chi^2 ) + (r^2 + \alpha^2)(2 r^2 + 3 \alpha^2 \chi^2 - \alpha^2 ) \right)}{c \left( \rho^2 \right)^2 \Delta} \) \( \Gamma_{ \phi \phi }^{ t } = 0 \Gamma_{ t t }^{ r } = \left( - \frac{1}{2} \right) \left( -\frac{\Delta}{\rho^2} \right) \left( \frac{c^2 r_s ( r^2 - \alpha^2 \chi^2 )}{\left( \rho^2 \right)^2 } \right) = \frac{c^2 r_s ( r^2 - \alpha^2 \chi^2 ) \Delta}{2 \left( \rho^2 \right)^3} \Gamma_{ t r }^{ r } = \Gamma_{ r t }^{ r } = 0 \Gamma_{ t \theta }^{ r } = \Gamma_{ \theta t }^{ r } = 0 \Gamma_{ t \phi }^{ r } = \Gamma_{ \phi t }^{ r } = \left( - \frac{1}{2} \right) \left( -\frac{\Delta}{\rho^2} \right) \left( - \frac{ c \alpha r_s (r^2 - \alpha^2 \chi^2)(1 - \chi^2)}{ \left( \rho^2 \right)^2 \right) = - \frac{c \alpha r_s (r^2 - \alpha^2 \chi^2)(1 - \chi^2) \Delta}{2 \left( \rho^2 \right)^3} \Gamma_{ r r }^{ r } = \left( \frac{1}{2} \right) \left( -\frac{\Delta}{\rho^2} \right) \left( \frac{r_s ( r^2 - \alpha^2 \chi^2) - 2 \alpha r ( 1 - \chi^2 )}{\Delta^2} \right) = - \frac{r_s ( r^2 - \alpha^2 \chi^2) - 2 \alpha r ( 1 - \chi^2 )}{2 \rho^2 \Delta} \Gamma_{ r \theta }^{ r } = \Gamma_{ \theta r }^{ r } = \left( \frac{1}{2} \right) \left( -\frac{\Delta}{\rho^2} \right) \left( \frac{2 \alpha^2 \chi \sqrt{1 - \chi^2}}{\Delta} \right) = -\frac{\alpha^2 \chi \sqrt{1 - \chi^2}}{\rho^2} \Gamma_{ r \phi }^{ r } = \Gamma_{ \phi r }^{ r } = 0 \Gamma_{ \theta \theta }^{ r } = \left( - \frac{1}{2} \right) \left( -\frac{\Delta}{\rho^2} \right) \left( -2r \right) = - \frac{r \Delta}{\rho^2} \Gamma_{ \theta \phi }^{ r } = \Gamma_{ \phi \theta }^{ r } = 0 \) \( \Gamma_{ \phi \phi }^{ r } = \left( - \frac{1}{2} \right) \left( -\frac{\Delta}{\rho^2} \right) \left( \frac{(r^2 - \alpha^2 \chi^2) \alpha^2 r_s \left(1 - \chi^2 \right)^2}{\left( \rho^2 \right)^2 } - 2 r ( 1 - \chi^2 ) \right) = \frac{\Delta(1-\chi^2)\left( \alpha^2 r_s (r^2 - \alpha^2 \chi^2) - 2 r \left( \rho^2 \right)^2 \right)}{2 \left( \rho^2 \right)^3 } \Gamma_{ t t }^{ \theta } = \left( - \frac{1}{2} \right) \left( -\frac{1}{\rho^2} \right) \left( - \frac{2 c^2 \alpha^2 r_s r \chi \sqrt{1-\chi^2}}{\left( \rho^2 \right)^2} \right) = - \frac{c^2\alpha^2 r_s r \chi \sqrt{1 - \chi^2}}{\left( \rho^2 \right)^3 } \Gamma_{ t r }^{ \theta } = \Gamma_{ r t }^{ \theta } = 0 \Gamma_{ t \theta }^{ \theta } = \Gamma_{ \theta t }^{ \theta } = 0 \Gamma_{ t \phi }^{ \theta } = \Gamma_{ \phi t }^{ \theta } = \left( - \frac{1}{2} \right) \left( -\frac{1}{\rho^2} \right) \left( -\frac{2 c \alpha r_s (r^2 + 2 \alpha^2 \chi^2 - \alpha^2) r \chi \sqrt{1 -\chi^2}}{\left( \rho^2 \right)^2} \right) = -\frac{c \alpha r_s (r^2 + 2 \alpha^2 \chi^2 - \alpha^2) r \chi \sqrt{1 -\chi^2}}{\left( \rho^3 \right)^2} \) \( \Gamma_{ r r }^{ \theta } = \left( - \frac{1}{2} \right) \left( -\frac{1}{\rho^2} \right) \left( \frac{2 \alpha^2 \chi \sqrt{1 - \chi^2}}{\Delta} \right) = \frac{\alpha^2 \chi \sqrt{1 - \chi^2}}{\rho^2 \Delta} \Gamma_{ r \theta }^{ \theta } = \Gamma_{ \theta r }^{ \theta } = \left( \frac{1}{2} \right) \left( -\frac{1}{\rho^2} \right) \left( -2r \right) = \frac{r}{\rho^2} \Gamma_{ r \phi }^{ \theta } = \Gamma_{ \phi r }^{ \theta } = 0 \Gamma_{ \theta \theta }^{ \theta } = \left( \frac{1}{2} \right) \left( -\frac{1}{\rho^2} \right) \left( 2 \alpha^2 \chi \sqrt{1 - \chi^2} \right) = \frac{\alpha^2 \chi \sqrt{1 - \chi^2}}{\rho^2} \Gamma_{ \theta \phi }^{ \theta } = \Gamma_{ \phi \theta }^{ \theta } = 0 \Gamma_{ \phi \phi }^{ \theta } = \left( - \frac{1}{2} \right) \left( -\frac{1}{\rho^2} \right) \left( -\frac{2 \left( 2 r^2 + \alpha^2 + \alpha^2 \chi^2 \right) \alpha^2 r_s r \chi \left( 1 - \chi^2 \right)^{\tiny \frac{3}{2}}}{\left( \rho^2 \right)^2} - 2 ( r^2 + \alpha^2 ) \chi \sqrt{1 - \chi^2} \right) \\ \quad \quad \quad = - \frac{\chi \sqrt{1-\chi^2} \left( \alpha r_s r ( 2 r^2 + \alpha^2 + \alpha^2 \chi^2 )( 1 - \chi^2) + ( r^2 + \alpha^2) \left( \rho^2 \right)^2 \right)}{ \left( \rho^2 \right)^3 } \) \( \Gamma_{ t t }^{ \phi } = 0 \Gamma_{ t r }^{ \phi } = \Gamma_{ r t }^{ \phi } = \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r }{c \rho^2 \Delta} \right) \left( \frac{c^2 r_s ( r^2 - \alpha^2 \chi^2 )}{\left( \rho^2 \right)^2 } \right) \; + \; \left( \frac{1}{2} \right) \left( - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 -\chi^2)} \right) \left( - \frac{ c \alpha r_s (r^2 - \alpha^2 \chi^2)(1 - \chi^2)}{ \left( \rho^2 \right)^2 \right) \\ \quad \quad \quad = \frac{c \alpha r_s (r^2 - \alpha^2 \chi^2) }{ 2 \left( \rho^2 \right)^2 \Delta } \Gamma_{ t \theta }^{ \phi } = \Gamma_{ \theta t }^{ \phi } = \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r }{c \rho^2 \Delta} \right) \left( - \frac{2 c^2 \alpha^2 r_s r \chi \sqrt{1-\chi^2}}{\left( \rho^2 \right)^2} \right) \; + \; \left( \frac{1}{2} \right) \left( - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 -\chi^2)} \right) \left( -\frac{2 c \alpha r_s (r^2 + 2 \alpha^2 \chi^2 - \alpha^2) r \chi \sqrt{1 -\chi^2}}{\left( \rho^2 \right)^2} \right) \\ \quad \quad \quad = \frac{c \alpha r_s r \chi}{\left( \rho^2 \right)^2 \sqrt{1 -\chi^2}} \Gamma_{ t \phi }^{ \phi } = \Gamma_{ \phi t }^{ \phi } = 0 \)
Last terms to do: \( \Gamma_{ r r }^{ \phi } = 0 \Gamma_{ r \theta }^{ \phi } = \Gamma_{ \theta r }^{ \phi } = 0 \Gamma_{ r \phi }^{ \phi } = \Gamma_{ \phi r }^{ \phi } = \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r }{c \rho^2 \Delta} \right) \left( - \frac{ c \alpha r_s (r^2 - \alpha^2 \chi^2)(1 - \chi^2)}{ \left( \rho^2 \right)^2 \right) \; + \; \left( \frac{1}{2} \right) \left( - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 -\chi^2)} \right) \left( \frac{(r^2 - \alpha^2 \chi^2) \alpha^2 r_s \left(1 - \chi^2 \right)^2}{\left( \rho^2 \right)^2 } - 2 r ( 1 - \chi^2 ) \right) = ? \Gamma_{ \theta \theta }^{ \phi } = 0 \Gamma_{ \theta \phi }^{ \phi } = \Gamma_{ \phi \theta }^{ \phi } = \left( \frac{1}{2} \right) \left( \frac{\alpha r_s r }{c \rho^2 \Delta} \right) \left( -\frac{2 c \alpha r_s (r^2 + 2 \alpha^2 \chi^2 - \alpha^2) r \chi \sqrt{1 -\chi^2}}{\left( \rho^2 \right)^2} \right) \; + \; \left( \frac{1}{2} \right) \left( - \frac{\rho^2 - r_s r}{\rho^2 \Delta (1 -\chi^2)} \right) \left( -\frac{2 \left( 2 r^2 + \alpha^2 + \alpha^2 \chi^2 \right) \alpha^2 r_s r \chi \left( 1 - \chi^2 \right)^{\tiny \frac{3}{2}}}{\left( \rho^2 \right)^2} - 2 ( r^2 + \alpha^2 ) \chi \sqrt{1 - \chi^2} \right) = ? \Gamma_{ \phi \phi }^{ \phi } = 0 \)
A nice analysis, RP. I read all of your links with interest. Very classical. One of my colleagues enjoys this kind of tortured analysis as well, and for similar reasons. He thinks the Cavendish experiment has more to tell us about the exact centers of gravitating bodies. His math, like yours, just goes on for pages and pages. Some of his results are, however, extremely interesting. The Kerr metric description of a rotating black hole does seem to have a definite origin, which betrays an underlying mental assumption of absolute space and time. Do not deny it. Time travel is also possible in these bubbles; exactly the same result you would get from Special Relativity when v >> c, and without any math beyond ordinary algebra. Too bad, no such absolute space, time, or origin (even at the geometric centers of perfectly isotropic, rotating BHs) exists. This is true whether you are using only two dimensions or ten. How much really can an analysis of something like a black hole treated as though it were any other rotating solid tell us about the vacuum in which it is immersed, when we don't even understand where a factor like G comes from? Not very much, I'd wager. About as much as Euclid knew about the Higgs field, not to put too fine a point on it. A mathematically or otherwise justifiable cause for the assertion that the center is the preferred coordinate system is also absent, just like Newton's "divine hand" always guide rocks to fall toward the center of the Earth, even if it is too dark for us to see which way they fell when our hands released them. No amount of mathematics in any coordinate system obfuscates the fact that our minds impose such order when we know not its true cause. It might as well be magic. As I told my classical physics colleague, your math (and our science) is missing a critical ingredient.
