We will compute all the components of the curvature tensor for Schwarzschild metric, and derive that for r=0 there is a singularity, while r = 2MG is only an apparent singularity, but the curvature is well defined.
We start from the Schwarzschild metric:
\mathrm d\tau^2 = \left(1 - \frac{2MG}{r}\right) \mathrm dt^2 - \left(\frac{1}{1 - \frac{2MG}{r}}\right) \mathrm dr^2 - r^2 \mathrm d\Omega^2
with:
\mathrm d\Omega^2 = \mathrm d\theta^2 + \sin^2(\theta) \mathrm \phi^2
We relates this interval to the metric tensor g_{\mu\nu} as follows:
\mathrm d\tau^2 = - g_{\mu\nu} \mathrm dx^\mu \mathrm dx^\nu
When we expand the right-hand side for the coordinates (x^0, x^1, x^2, x^3) = (t, r, \theta, \phi), assuming a diagonal metric, we get:
\mathrm d\tau^2 = - g_{tt} (\mathrm dt)^2 - g_{rr} (\mathrm dr)^2 - g_{\theta\theta} (\mathrm d\theta)^2 - g_{\phi\phi} (\mathrm d\phi)^2
To find the components of g_{\mu\nu}, we equate the coefficients of the differential elements (\mathrm dt^2, \mathrm dr^2, \mathrm d\theta^2, \mathrm d\phi^2) between the two expressions for \mathrm d\tau^2.
Comparing the coefficients for the \mathrm dt^2 term, gives the time component of the metric:
g_{tt} = -\left(1 - \frac{2MG}{r}\right)
Comparing the coefficients for the \mathrm dr^2 term gives the radial component:
g_{rr} = \left(1 - \frac{2MG}{r}\right)^{-1}
Comparing the coefficients for the \mathrm d\theta^2 term provides the first angular component:
g_{\theta\theta} = r^2
Finally, comparing the coefficients for the \mathrm d\phi^2 term gives the second angular component:
g_{\phi\phi} = r^2 \sin^2(\theta)
We can now assemble these components into the 4 \times 4 matrix for the metric tensor g_{\mu\nu}:
g_{\mu\nu} = \begin{bmatrix} -\left(1 - \frac{2MG}{r}\right) & 0 & 0 & 0 \\ 0 & \left(1 - \frac{2MG}{r}\right)^{-1} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta) \end{bmatrix}
The Christoffel symbols {\Gamma^\sigma}_{\mu\nu}, are computed from the derivatives of the metric tensor. The general formula is:
{\Gamma^\sigma}_{\mu\nu} = \frac{1}{2} g^{\sigma\rho} \left(\partial_\mu g_{\rho\nu} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu} \right)
To proceed, we need the inverse metric tensor, g^{\mu\nu}, and the partial derivatives of the metric components The inverse metric tensor g^{\mu \nu} is defined as the matrix inverse of the metric tensor g_{\mu\nu}. It satisfies the relation:
g^{\mu\sigma} g_{\rho\nu} = {\delta^\mu}_\nu
Since the metric tensor is a diagonal matrix, its inverse is found by taking the reciprocal of each element on the main diagonal:
g^{\mu\nu} = \begin{bmatrix} -\left(1 - \frac{2MG}{r}\right)^{-1} & 0 & 0 & 0 \\ 0 & \left(1 - \frac{2MG}{r}\right) & 0 & 0 \\ 0 & 0 & \frac{1}{r^2} & 0 \\ 0 & 0 & 0 & \frac{1}{r^2 \sin^2(\theta)} \end{bmatrix}
Next, we calculate the partial derivatives of the metric tensor components with respect to the coordinates:
\begin{aligned} & \partial_r g_{tt} = -\frac{2MG}{r^2} \\ & \partial_r g_{rr} = -\frac{2MG}{(r-2MG)^2} \\ & \partial_r g_{\theta\theta} = 2r \\ & \partial_r g_{\phi\phi} = 2r \sin^2(\theta) \\ & \partial_\theta g_{\phi\phi} =2 r^2 \cos(\theta)\sin(\theta) \end{aligned}
All the other components are zero.
The Christoffel symbol is denoted as {\Gamma^\sigma}_{\mu\nu} and each of the three indices, \sigma, \mu, and \nu, can take on any value from the set of coordinates, which in this case is \{0, 1, 2, 3\} corresponding to \{t, r, \theta, \phi\}.
Therefore we could assume that there are 4 \times 4 \times 4 = 64 components in total. However, the definition of the Christoffel symbol has a symmetry in the lower indices:
{\Gamma^\sigma}_{\mu\nu} = {\Gamma^\sigma}_{\nu\mu}
This symmetry reduces the number of independent components we need to calculate. We can count the number of unique pairs for the lower indices (\mu, \nu):
There are 4 cases where \mu = \nu:
(0,0), (1,1), (2,2), (3,3)
There are \binom{4}{2} = \frac{4 \times 3}{2} = 6 cases where \mu \neq \nu:
(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)
The total number of unique pairs for the lower indices is 4 + 6 = 10.
Since the upper index \sigma can still be any of the 4 coordinate indices, the total number of unique Christoffel symbols is:
4 \times 10 = 40
It is important to note that while there are 40 unique symbols to consider, many of them will be zero for the Schwarzschild metric. The symmetries of the spacetime (it is static and spherically symmetric) and the diagonal nature of the metric cause most of the derivatives of the metric components to be zero, which in turn makes a large number of the Christoffel symbols zero.
We compute the t components:
{\Gamma^t}_{\mu\nu} = \frac{1}{2} g^{t\rho} \left(\partial_\mu g_{\rho\nu} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu} \right)
Computation of {\Gamma^t}_{\mu \nu}
Since the metric is diagonal, the only non-zero term in the sum over \rho is for \rho = t. The component g^{tt} is equal to -\left(1 - \frac{2MG}{r}\right)^{-1}:
\begin{aligned} {\Gamma^t}_{\mu\nu} & = \frac{1}{2} g^{tt} \left(\partial_\mu g_{\nu t} + \partial_\nu g_{\mu t} - \partial_t g_{\mu\nu} \right) \\ & = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_\mu g_{\nu t} + \partial_\nu g_{\mu t} - \partial_t g_{\mu\nu} \right) \end{aligned}
We now calculate each component of {\Gamma^t}_{\mu \nu} individually.
Component {\Gamma^t}_{tt}
For this component, we set \mu = t and \nu = t:
{\Gamma^t}_{tt} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_t g_{tt} + \partial_t g_{tt} - \partial_t g_{tt} \right) = 0
This gives the result:
{\Gamma^t}_{tt} = 0
Component {\Gamma^t}_{tr}
For this component, we set \mu = t and \nu = r:
\begin{aligned} {\Gamma^t}_{tr} & = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_t g_{rt} + \partial_r g_{tt} - \partial_t g_{tr} \right) \\ & = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(0 - \frac{2MG}{r^2} - 0 \right) \\ & = \frac{1}{\left(1 - \frac{2MG}{r}\right)} \frac{MG}{r^2} \\ & = \frac{MG}{r(r - 2MG)} \end{aligned}
This gives the result:
{\Gamma^t}_{tr} = \frac{MG}{r(r - 2MG)}
Component {\Gamma^t}_{t\theta}
For this component, we set \mu = t and \nu = \theta:
{\Gamma^t}_{t\theta} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_t g_{\theta t} + \partial_\theta g_{tt} - \partial_t g_{t\theta} \right) = 0
This gives the result:
{\Gamma^t}_{t\theta} = 0
Component {\Gamma^t}_{t\phi}
For this component, we set \mu = t and \nu = \phi:
{\Gamma^t}_{t\phi} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_t g_{\phi t} + \partial_\phi g_{tt} - \partial_t g_{t\phi} \right) = 0
This gives the result:
{\Gamma^t}_{t\phi} = 0
Component {\Gamma^t}_{rr}
For this component, we set \mu = r and \nu = r:
{\Gamma^t}_{rr} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_r g_{tr} + \partial_r g_{tr} - \partial_t g_{rr} \right) = 0
This gives the result:
{\Gamma^t}_{rr} = 0
Component {\Gamma^t}_{r\theta}
For this component, we set \mu = r and \nu = \theta:
{\Gamma^t}_{r\theta} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_r g_{\theta t} + \partial_\theta g_{rt} - \partial_t g_{r\theta} \right) = 0
This gives the result:
{\Gamma^t}_{r\theta} = 0
Component {\Gamma^t}_{r\phi}
For this component, we set \mu = r and \nu = \phi:
{\Gamma^t}_{r\phi} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_r g_{\phi t} + \partial_\phi g_{rt} - \partial_t g_{r\phi} \right) = 0
This gives the result:
{\Gamma^t}_{r\phi} = 0
Component {\Gamma^t}_{\theta\theta}
For this component, we set \mu = \theta and \nu = \theta:
{\Gamma^t}_{\theta\theta} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_\theta g_{t\theta} + \partial_\theta g_{t\theta} - \partial_t g_{\theta\theta} \right) = 0
This gives the result:
{\Gamma^t}_{\theta\theta} = 0
Component {\Gamma^t}_{\theta\phi}
For this component, we set \mu = \theta and \nu = \phi:
{\Gamma^t}_{\theta\phi} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_\theta g_{\phi t} + \partial_\phi g_{t\theta} - \partial_t g_{\theta\phi} \right) = 0
This gives the result:
{\Gamma^t}_{\theta\phi} = 0
Component {\Gamma^t}_{\phi\phi}
For this component, we set \mu = \phi and \nu = \phi:
{\Gamma^t}_{\phi\phi} = -\frac{1}{2\left(1 - \frac{2MG}{r}\right)} \left(\partial_\phi g_{t\phi} + \partial_\phi g_{t\phi} - \partial_t g_{\phi\phi} \right) = 0
This gives the result:
{\Gamma^t}_{\phi\phi} = 0
Components {\Gamma^t}_{\mu\nu} summary
The only non-zero components is:
{\Gamma^t}_{tr} = \frac{MG}{r(r - 2MG)}
We compute the r components:
{\Gamma^r}_{\mu\nu} = \frac{1}{2} g^{r\rho} \left(\partial_\mu g_{\rho\nu} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu} \right)
Computation of {\Gamma^r}_{\mu \nu}
Since the metric is diagonal, the only non-zero term in the sum over \rho is for \rho = r. The component g^{rr} is equal to \left(1 - \frac{2MG}{r}\right):
\begin{aligned} {\Gamma^r}_{\mu\nu} & = \frac{1}{2} g^{rr} \left(\partial_\mu g_{\nu r} + \partial_\nu g_{\mu r} - \partial_r g_{\mu\nu} \right) \\ & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_\mu g_{\nu r} + \partial_\nu g_{\mu r} - \partial_r g_{\mu\nu} \right) \end{aligned}
We now calculate each component of {\Gamma^r}_{\mu \nu} individually.
Component {\Gamma^r}_{tt}
For this component, we set \mu = t and \nu = t:
\begin{aligned} {\Gamma^r}_{tt} & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_t g_{rt} + \partial_t g_{rt} - \partial_r g_{tt} \right) \\ & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(0 + 0 + \frac{2MG}{r^2} \right) \\ & = \left(1 - \frac{2MG}{r}\right) \frac{MG}{r^2} \\ & = \frac{(r-2MG)MG}{r^3} \end{aligned}
This gives the result:
{\Gamma^r}_{tt} = \frac{(r-2MG)MG}{r^3}
Component {\Gamma^r}_{tr}
For this component, we set \mu = t and \nu = r:
{\Gamma^r}_{tr} = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_t g_{rr} + \partial_r g_{tr} - \partial_r g_{tr} \right) = 0
This gives the result:
{\Gamma^r}_{tr} = 0
Component {\Gamma^r}_{t\theta}
For this component, we set \mu = t and \nu = \theta:
{\Gamma^r}_{t\theta} = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_t g_{r\theta} + \partial_\theta g_{rt} - \partial_r g_{t\theta} \right) = 0
This gives the result:
{\Gamma^r}_{t\theta} = 0
Component {\Gamma^r}_{t\phi}
For this component, we set \mu = t and \nu = \phi:
{\Gamma^r}_{t\phi} = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_t g_{r\phi} + \partial_\phi g_{rt} - \partial_r g_{t\phi} \right) = 0
This gives the result:
{\Gamma^r}_{t\phi} = 0
Component {\Gamma^r}_{rr}
For this component, we set \mu = r and \nu = r:
\begin{aligned} {\Gamma^r}_{rr} & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_r g_{rr} + \partial_r g_{rr} - \partial_r g_{rr} \right) \\ & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(-\frac{2MG}{(r-2MG)^2} \right) \\ & = -\left(1 - \frac{2MG}{r}\right) \left(\frac{MG}{(r-2MG)^2} \right) \\ & = -\frac{(r-2MG)MG}{r(r-2MG)^2} = -\frac{MG}{r(r-2MG)} \end{aligned}
This gives the result:
{\Gamma^r}_{rr} = -\frac{MG}{r(r-2MG)}
Component {\Gamma^r}_{r\theta}
For this component, we set \mu = r and \nu = \theta:
{\Gamma^r}_{r\theta} = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_r g_{\theta r} + \partial_\theta g_{rr} - \partial_r g_{r\theta} \right) = 0
This gives the result:
{\Gamma^r}_{r\theta} = 0
Component {\Gamma^r}_{r\phi}
For this component, we set \mu = r and \nu = \phi:
{\Gamma^r}_{r\phi} = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_r g_{\phi r} + \partial_\phi g_{rr} - \partial_r g_{r\phi} \right) = 0
This gives the result:
{\Gamma^r}_{r\phi} = 0
Component {\Gamma^r}_{\theta\theta}
For this component, we set \mu = \theta and \nu = \theta:
\begin{aligned} {\Gamma^r}_{\theta\theta} & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_\theta g_{r\theta} + \partial_\theta g_{r\theta} - \partial_r g_{\theta\theta} \right) \\ & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(0 + 0 - 2r \right) \\ & = -\left(1 - \frac{2MG}{r}\right) r = -(r - 2MG) \end{aligned}
This gives the result:
{\Gamma^r}_{\theta\theta} = -(r - 2MG)
Component {\Gamma^r}_{\theta\phi}
For this component, we set \mu = \theta and \nu = \phi:
{\Gamma^r}_{\theta\phi} = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_\theta g_{r\phi} + \partial_\phi g_{r\theta} - \partial_r g_{\theta\phi} \right) = 0
This gives the result:
{\Gamma^r}_{\theta\phi} = 0
Component {\Gamma^r}_{\phi\phi}
For this component, we set \mu = \phi and \nu = \phi:
\begin{aligned} {\Gamma^r}_{\phi\phi} & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(\partial_\phi g_{r\phi} + \partial_\phi g_{r\phi} - \partial_r g_{\phi\phi} \right) \\ & = \frac{1}{2}\left(1 - \frac{2MG}{r}\right) \left(0 + 0 - 2r \sin^2(\theta) \right) \\ & = -\left(1 - \frac{2MG}{r}\right) r \sin^2(\theta) = - (r - 2MG)\sin^2(\theta) \end{aligned}
This gives the result:
{\Gamma^r}_{\phi\phi} = - (r - 2MG)\sin^2(\theta)
Components {\Gamma^r}_{\mu\nu} summary
The only non-zero components are:
\begin{aligned} & {\Gamma^r}_{tt} = \frac{(r-2MG)MG}{r^3} \\ & {\Gamma^r}_{rr} = -\frac{MG}{r(r-2MG)} \\ & {\Gamma^r}_{\theta\theta} = -(r - 2MG) \\ & {\Gamma^r}_{\phi\phi} = - (r - 2MG)\sin^2(\theta) \end{aligned}
We compute the \theta components:
{\Gamma^\theta}_{\mu\nu} = \frac{1}{2} g^{\theta\rho} \left(\partial_\mu g_{\rho\nu} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu} \right)
Computation of {\Gamma^\theta}_{\mu \nu}
Since the metric is diagonal, the only non-zero term in the sum over \rho is for \rho = \theta. The component g^{\theta\theta} is equal to r^{-2}:
\begin{aligned} {\Gamma^\theta}_{\mu\nu} & = \frac{1}{2} g^{\theta\theta} \left(\partial_\mu g_{\nu \theta} + \partial_\nu g_{\mu \theta} - \partial_\theta g_{\mu\nu} \right) \\ & = \frac{1}{2r^2} \left(\partial_\mu g_{\nu \theta} + \partial_\nu g_{\mu \theta} - \partial_\theta g_{\mu\nu} \right) \end{aligned}
We now calculate each component of {\Gamma^\theta}_{\mu \nu} individually.
Component {\Gamma^\theta}_{tt}
For this component, we set \mu = t and \nu = t:
{\Gamma^\theta}_{tt} = \frac{1}{2r^2} \left(\partial_t g_{\theta t} + \partial_t g_{\theta t} - \partial_\theta g_{tt} \right) = 0
This gives the result:
{\Gamma^\theta}_{tt} = 0
Component {\Gamma^\theta}_{tr}
For this component, we set \mu = t and \nu = r:
{\Gamma^\theta}_{tr} = \frac{1}{2r^2} \left(\partial_t g_{\theta r} + \partial_r g_{\theta t} - \partial_\theta g_{tr} \right) = 0
This gives the result:
{\Gamma^\theta}_{tr} = 0
Component {\Gamma^\theta}_{t\theta}
For this component, we set \mu = t and \nu = \theta:
{\Gamma^\theta}_{t\theta} = \frac{1}{2r^2} \left(\partial_t g_{\theta\theta} + \partial_\theta g_{\theta t} - \partial_\theta g_{t\theta} \right) = 0
This gives the result:
{\Gamma^\theta}_{t\theta} = 0
Component {\Gamma^\theta}_{t\phi}
For this component, we set \mu = t and \nu = \phi:
{\Gamma^\theta}_{t\phi} = \frac{1}{2r^2} \left(\partial_t g_{\theta\phi} + \partial_\phi g_{\theta t} - \partial_\theta g_{t\phi} \right) = 0
This gives the result:
{\Gamma^\theta}_{t\phi} = 0
Component {\Gamma^\theta}_{rr}
For this component, we set \mu = r and \nu = r:
{\Gamma^\theta}_{rr} = \frac{1}{2r^2} \left(\partial_r g_{\theta r} + \partial_r g_{\theta r} - \partial_\theta g_{rr} \right) = 0
This gives the result:
{\Gamma^\theta}_{rr} = 0
Component {\Gamma^\theta}_{r\theta}
For this component, we set \mu = r and \nu = \theta:
\begin{aligned} {\Gamma^\theta}_{r\theta} & = \frac{1}{2r^2} \left(\partial_r g_{\theta\theta} + \partial_\theta g_{r\theta} - \partial_\theta g_{r\theta} \right) \\ & = \frac{1}{2r^2} \left(2r + 0 - 0 \right) = \frac{1}{r} \end{aligned}
This gives the result:
{\Gamma^\theta}_{r\theta} = \frac{1}{r}
Component {\Gamma^\theta}_{r\phi}
For this component, we set \mu = r and \nu = \phi:
{\Gamma^\theta}_{r\phi} = \frac{1}{2r^2} \left(\partial_r g_{\theta\phi} + \partial_\phi g_{r\theta} - \partial_\theta g_{r\phi} \right) = 0
This gives the result:
{\Gamma^\theta}_{r\phi} = 0
Component {\Gamma^\theta}_{\theta\theta}
For this component, we set \mu = \theta and \nu = \theta:
{\Gamma^\theta}_{\theta\theta} = \frac{1}{2r^2} \left(\partial_\theta g_{\theta\theta} + \partial_\theta g_{\theta\theta} - \partial_\theta g_{\theta\theta} \right) = 0
This gives the result:
{\Gamma^\theta}_{\theta\theta} = 0
Component {\Gamma^\theta}_{\theta\phi}
For this component, we set \mu = \theta and \nu = \phi:
{\Gamma^\theta}_{\theta\phi} = \frac{1}{2r^2} \left(\partial_\theta g_{\phi\theta} + \partial_\phi g_{\theta\theta} - \partial_\theta g_{\theta\phi} \right) = 0
This gives the result:
{\Gamma^\theta}_{\theta\phi} = 0
Component {\Gamma^\theta}_{\phi\phi}
For this component, we set \mu = \phi and \nu = \phi:
\begin{aligned} {\Gamma^\theta}_{\phi\phi} & = \frac{1}{2r^2} \left(\partial_\phi g_{\theta\phi} + \partial_\phi g_{\theta\phi} - \partial_\theta g_{\phi\phi} \right) \\ & = \frac{1}{2r^2} \left(0 + 0 - 2 r^2 \cos(\theta)\sin(\theta) \right) \\ & = -\cos(\theta)\sin(\theta) \end{aligned}
This gives the result:
{\Gamma^\theta}_{\phi\phi} = -\cos(\theta)\sin(\theta)
Components {\Gamma^\theta}_{\mu\nu} summary
The only non-zero components are:
\begin{aligned} & {\Gamma^\theta}_{r\theta} = \frac{1}{r} \\ & {\Gamma^\theta}_{\phi\phi} = -\cos(\theta)\sin(\theta) \end{aligned}
We compute the \phi components:
{\Gamma^\phi}_{\mu\nu} = \frac{1}{2} g^{\phi\rho} \left(\partial_\mu g_{\rho\nu} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu} \right)
Computation of {\Gamma^\phi}_{\mu \nu}
Since the metric is diagonal, the only non-zero term in the sum over \rho is for \rho = \phi. The component g^{\phi\phi} is equal to [r^2 \sin^2(\theta)]^{-1}:
\begin{aligned} {\Gamma^\phi}_{\mu\nu} & = \frac{1}{2} g^{\phi\phi} \left(\partial_\mu g_{\nu \phi} + \partial_\nu g_{\mu \phi} - \partial_\phi g_{\mu\nu} \right) \\ & = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_\mu g_{\nu \phi} + \partial_\nu g_{\mu \phi} - \partial_\phi g_{\mu\nu} \right) \end{aligned}
We now calculate each component of {\Gamma^\phi}_{\mu \nu} individually.
Component {\Gamma^\phi}_{tt}
For this component, we set \mu = t and \nu = t:
{\Gamma^\phi}_{tt} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_t g_{\phi t} + \partial_t g_{\phi t} - \partial_\phi g_{tt} \right) = 0
This gives the result:
{\Gamma^\phi}_{tt} = 0
Component {\Gamma^\phi}_{tr}
For this component, we set \mu = t and \nu = r:
{\Gamma^\phi}_{tr} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_t g_{\phi r} + \partial_r g_{\phi t} - \partial_\phi g_{tr} \right) = 0
This gives the result:
{\Gamma^\phi}_{tr} = 0
Component {\Gamma^\phi}_{t\theta}
For this component, we set \mu = t and \nu = \theta:
{\Gamma^\phi}_{t\theta} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_t g_{\phi\theta} + \partial_\theta g_{\phi t} - \partial_\phi g_{t\theta} \right) = 0
This gives the result:
{\Gamma^\phi}_{t\theta} = 0
Component {\Gamma^\phi}_{t\phi}
For this component, we set \mu = t and \nu = \phi:
{\Gamma^\phi}_{t\phi} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_t g_{\phi\phi} + \partial_\phi g_{\phi t} - \partial_\phi g_{t\phi} \right) = 0
This gives the result:
{\Gamma^\phi}_{t\phi} = 0
Component {\Gamma^\phi}_{rr}
For this component, we set \mu = r and \nu = r:
{\Gamma^\phi}_{rr} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_r g_{\phi r} + \partial_r g_{\phi r} - \partial_\phi g_{rr} \right) = 0
This gives the result:
{\Gamma^\phi}_{rr} = 0
Component {\Gamma^\phi}_{r\theta}
For this component, we set \mu = r and \nu = \theta:
{\Gamma^\phi}_{r\theta} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_r g_{\phi\theta} + \partial_\theta g_{\phi r} - \partial_\phi g_{r\theta} \right) = 0
This gives the result:
{\Gamma^\phi}_{r\theta} = 0
Component {\Gamma^\phi}_{r\phi}
For this component, we set \mu = r and \nu = \phi:
\begin{aligned} {\Gamma^\phi}_{r\phi} & = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_r g_{\phi\phi} + \partial_\phi g_{r\phi} - \partial_\phi g_{r\phi} \right) \\ & \frac{1}{2r^2 \sin^2(\theta)} \left(2r \sin^2(\theta) + 0 - 0 \right) = \frac{1}{r} \end{aligned}
This gives the result:
{\Gamma^\phi}_{r\phi} = \frac{1}{r}
Component {\Gamma^\phi}_{\theta\theta}
For this component, we set \mu = \theta and \nu = \theta:
{\Gamma^\phi}_{\theta\theta} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_\theta g_{\phi\theta} + \partial_\theta g_{\phi\theta} - \partial_\phi g_{\theta\theta} \right) = 0
This gives the result:
{\Gamma^\phi}_{\theta\theta} = 0
Component {\Gamma^\phi}_{\theta\phi}
For this component, we set \mu = \theta and \nu = \phi:
\begin{aligned} {\Gamma^\phi}_{\theta\phi} & = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_\theta g_{\phi\phi} + \partial_\phi g_{\theta\phi} - \partial_\phi g_{\theta\phi} \right) \\ & = \frac{1}{2r^2 \sin^2(\theta)} \left(2 r^2 \cos(\theta)\sin(\theta) + 0 - 0 \right) \\ & = \cot(\theta) \end{aligned}
This gives the result:
{\Gamma^\phi}_{\theta\phi} = \cot(\theta)
Component {\Gamma^\phi}_{\phi\phi}
For this component, we set \mu = \phi and \nu = \phi:
{\Gamma^\phi}_{\phi\phi} = \frac{1}{2r^2 \sin^2(\theta)} \left(\partial_\phi g_{\phi\phi} + \partial_\phi g_{\phi\phi} - \partial_\phi g_{\phi\phi} \right) = 0
This gives the result:
{\Gamma^\phi}_{\phi\phi} = 0
Components {\Gamma^\phi}_{\mu\nu} summary
The only non-zero components are:
\begin{aligned} & {\Gamma^\phi}_{r\phi} = \frac{1}{r} \\ & {\Gamma^\phi}_{\theta\phi} = \cot(\theta) \end{aligned}
The calculations show that for the given metric tensor, most of the {\Gamma^\sigma}_{\mu \nu} components are zero. The only non-zero components are:
\begin{aligned} & {\Gamma^t}_{tr} = {\Gamma^t}_{rt} = \frac{MG}{r(r - 2MG)} \\ & {\Gamma^r}_{tt} = \frac{(r-2MG)MG}{r^3} \\ & {\Gamma^r}_{rr} = -\frac{MG}{r(r-2MG)} \\ & {\Gamma^r}_{\theta\theta} = -(r - 2MG) \\ & {\Gamma^r}_{\phi\phi} = - (r - 2MG)\sin^2(\theta) \\ & {\Gamma^\theta}_{r\theta} = {\Gamma^\theta}_{\theta r} = \frac{1}{r} \\ & {\Gamma^\theta}_{\phi\phi} = -\cos(\theta)\sin(\theta) \\ & {\Gamma^\phi}_{r\phi} = {\Gamma^\phi}_{\phi r} = \frac{1}{r} \\ & {\Gamma^\phi}_{\theta\phi} = {\Gamma^\phi}_{\phi \theta} = \cot(\theta) \end{aligned}
For the component \sigma = t, the following symbols are zero:
{\Gamma^t}_{tt} = {\Gamma^t}_{t\theta} = {\Gamma^t}_{t\phi} = {\Gamma^t}_{rr}={\Gamma^t}_{r\theta}={\Gamma^t}_{r\phi} = {\Gamma^t}_{\theta\theta} ={\Gamma^t}_{\theta\phi} = {\Gamma^t}_{\phi\phi} = 0
For the component \sigma = r, the following symbols are zero:
{\Gamma^r}_{tr} = {\Gamma^r}_{t\theta} = {\Gamma^r}_{t\phi} = {\Gamma^r}_{r\theta} = {\Gamma^r}_{r\phi} = {\Gamma^r}_{\theta\phi} = 0
For the component \sigma = \theta, the following symbols are zero:
{\Gamma^\theta}_{tt} = {\Gamma^\theta}_{tr} = {\Gamma^\theta}_{t\theta} = {\Gamma^\theta}_{t\phi} = {\Gamma^\theta}_{rr} = {\Gamma^\theta}_{r\phi} = {\Gamma^\theta}_{\theta\theta} = {\Gamma^\theta}_{\theta\phi} = 0
For the component \sigma = \phi, the following symbols are zero:
{\Gamma^\phi}_{tt} = {\Gamma^\phi}_{tr} = {\Gamma^\phi}_{t\theta} = {\Gamma^\phi}_{t\phi} = {\Gamma^\phi}_{rr} = {\Gamma^\phi}_{r\theta} = {\Gamma^\phi}_{\theta\theta} = {\Gamma^\phi}_{\phi\phi} = 0
The Riemann curvature tensor, {R^\mu}_{\nu\rho\sigma}, is defined in terms of the Christoffel symbols and their derivatives:
{R^\mu}_{\nu\rho\sigma} =\partial_\rho{\Gamma^\mu}_{\nu\sigma}-\partial_\sigma{\Gamma^\mu}_{\nu\rho} +{\Gamma^\mu}_{\lambda\rho}{\Gamma^\lambda}_{\nu\sigma} -{\Gamma^\mu}_{\lambda\sigma}{\Gamma^\lambda}_{\nu\rho}
Notes: p,\lambda are dummy; products commute; ensure the (\rho,\sigma) slot order is consistent across all four terms.
Note that there is a summation over the index \lambda for the last two terms, where \lambda takes on all coordinate values \{t, r, \theta, \phi\}.
In a 4-dimensional spacetime, a count of the components of the tensor {R^\mu}_{\nu\rho\sigma} would suggest there are 4 \times 4 \times 4 \times 4 = 256 components. However, the tensor possesses a number of symmetries that significantly reduce the number of unique, independent components.
Antisymmetry in the last two indices:
{R^\mu}_{\nu\rho\sigma} = -{R^\mu}_{\nu\sigma\rho}
Symmetry related to the first Bianchi identity, which states that the sum over a cyclic permutation of the last three indices is zero:
{R^\mu}_{\nu\rho\sigma} + {R^\mu}_{\sigma\nu\rho} + {R^\mu}_{\rho\sigma\nu} = 0
The symmetries are more easily analyzed using the fully covariant form of the tensor, R_{\mu\nu\rho\sigma}, which has additional symmetries.
Antisymmetry in the first two indices:
R_{\mu\nu\rho\sigma} = -R_{\nu\mu\rho\sigma}
Symmetry under the exchange of the first and second pairs of indices:
R_{\mu\nu\rho\sigma} = R_{\rho\sigma\mu\nu}
Due to these extensive symmetries, the number of independent components is much smaller than 256. A detailed analysis of these constraints leads to a general formula for the number of independent components of the Riemann tensor in a space of dimension N:
\frac{1}{12} N^2(N^2 - 1)
In the case of a 4-dimensional spacetime, we set N=4:
\frac{1}{12} (4^2)(4^2 - 1) = \frac{1}{12} (16)(15) = \frac{240}{12} = 20
So, in a 4-dimensional spacetime, the Riemann curvature tensor has 20 independent components.
Component {R^t}_{rtr}
For this component, we set \mu=t, \nu=r, \rho=t, \sigma=r:
\begin{aligned} {R^t}_{rtr} & = \partial_t {\Gamma^t}_{rr} - \partial_r {\Gamma^t}_{tr} + {\Gamma^\lambda}_{rr} {\Gamma^t}_{t\lambda} - {\Gamma^\lambda}_{tr} {\Gamma^t}_{r\lambda} \\ & = 0 - \partial_r {\Gamma^t}_{tr} + {\Gamma^r}_{rr} {\Gamma^t}_{tr} - {\Gamma^t}_{tr} {\Gamma^t}_{rt} \\ & = 0 - \partial_r {\Gamma^t}_{tr} - {\Gamma^r}_{rr} {\Gamma^t}_{tr} - \left({\Gamma^t}_{tr}\right)^2\\ & = 0 - \partial_r \left(\frac{MG}{r(r - 2MG)}\right) + \left(-\frac{MG}{r(r-2MG)}\right) \frac{MG}{r(r - 2MG)} - \left(\frac{MG}{r(r-2MG)}\right)^2\\ & = 0 - \partial_r \left(\frac{MG}{r(r - 2MG)}\right) - \left(\frac{MG}{r(r - 2MG)}\right)^2 - \left(\frac{MG}{r(r-2MG)}\right)^2 \\ & = 0 + \partial_r \left(\frac{MG}{r(r - 2MG)}\right) - 2 \left(\frac{MG}{r(r - 2MG)}\right)^2\\ & = -\frac{2MG(MG - r)}{r^2(r-2MG)^2} - 2 \left(\frac{MG}{r(r - 2MG)}\right)^2 \\ & = \frac{2MG(r - MG) + 2M^2 G^2}{r^2(r - 2MG)^2} = \frac{-2MGr + 2M^2G^2 + 2M^2G^2}{r^2(r - 2MG)^2} \\ & = \frac{-2MGr + 4M^2G^2}{r^2(r - 2MG)^2} = \frac{2MG(r-2MG)}{r^2(r - 2MG)^2} \\ & = \frac{2MG}{r^2(r - 2MG)} \end{aligned}
This gives the result:
{R^t}_{rtr} = \frac{2MG}{r^2(r - 2MG)}
Component {R^t}_{\theta t \theta}
For this component, we set \mu=t, \nu=\theta, \rho=t, \sigma=\theta:
\begin{aligned} {R^t}_{\theta t\theta} & = \partial_t {\Gamma^t}_{\theta\theta} - \partial_\theta {\Gamma^t}_{t\theta} {\Gamma^\lambda}_{\theta\theta} + {\Gamma^t}_{t\lambda} - {\Gamma^\lambda}_{t\theta} {\Gamma^t}_{\theta\lambda} \\ & = 0 - 0 - {\Gamma^r}_{\theta\theta} {\Gamma^t}_{tr} - 0 \cdot {\Gamma^t}_{\theta\lambda} \\ & = -(-(r - 2MG))\left(\frac{MG}{r(r - 2MG)}\right) \\ & = -\frac{MG}{r} \end{aligned}
This gives the result:
{R^t}_{\theta t\theta} = -\frac{MG}{r}
Component {R^t}_{\phi t\phi}
\begin{aligned} {R^t}_{\phi t\phi} & = \partial_t {\Gamma^t}_{\phi\phi} - \partial_\phi {\Gamma^t}_{t\phi} + {\Gamma^\lambda}_{\phi\phi}{\Gamma^t}_{t\lambda} - {\Gamma^\lambda}_{t\phi}{\Gamma^t}_{\phi\lambda} \\ & = 0 + 0 + {\Gamma^r}_{\phi\phi}{\Gamma^t}_{tr} - 0\cdot {\Gamma^t}_{\phi\lambda} \\ & = \left(- (r - 2MG)\sin^2 (\theta)\right)\left(\frac{MG}{r(r - 2MG)}\right) \\ & = -\frac{MG}{r}\sin^2 (\theta) \end{aligned}
This gives the result:
{R^t}_{\phi t\phi} = -\frac{MG}{r}\sin^2 (\theta)
Component {R^r}_{\theta r \theta}
For this component, we set \mu=r, \nu=\theta, \rho=r, \sigma=\theta:
\begin{aligned} {R^r}_{\theta r \theta} & = \partial_r {\Gamma^r}_{\theta \theta} - \partial_\theta {\Gamma^r}_{r \theta} + {\Gamma^\lambda}_{\theta \theta} {\Gamma^r}_{r\lambda} - {\Gamma^\lambda}_{r \theta} {\Gamma^r}_{\theta \lambda} \\ & = \partial_r \left(- (r-2MG)\right) - 0 + {\Gamma^r}_{\theta \theta} {\Gamma^r}_{rr} - {\Gamma^\theta}_{r \theta} {\Gamma^r}_{\theta \theta} \\ & = -1 - 0 + \left(-(r-2MG)\right)\left(-\frac{MG}{r(r-2MG)}\right) - \frac{1}{r}\left(-(r-2MG)\right) \\ & = -1 + \frac{MG}{r} + \frac{r-2MG}{r} \\ & = -1 + \frac{MG}{r} + 1 - \frac{2MG}{r} \\ & = -\frac{MG}{r} \end{aligned}
This gives the result:
{R^r}_{\theta r \theta} = -\frac{MG}{r}
Component {R^r}_{\phi r \phi}
For this component, we set \mu=r, \nu=\phi, \rho=r, \sigma=\phi:
\begin{aligned} {R^r}_{\phi r \phi} & = \partial_r {\Gamma^r}_{\phi \phi} - \partial_\phi {\Gamma^r}_{r \phi} + {\Gamma^\lambda}_{\phi \phi}{\Gamma^r}_{r\lambda} - {\Gamma^\lambda}_{r \phi}{\Gamma^r}_{\phi \lambda} \\ & = \partial_r \left(- (r-2MG)\sin^2 (\theta)\right) - 0 + {\Gamma^r}_{\phi\phi}{\Gamma^r}_{rr} - {\Gamma^\phi}_{r\phi}{\Gamma^r}_{\phi\phi} \\ & = -\sin^2 (\theta) + \left(- (r - 2MG)\sin^2(\theta)\right)\left(-\frac{MG}{r(r-2MG)}\right) - \frac{1}{r} \left(- (r - 2MG)\sin^2(\theta)\right) \\ & = -\sin^2 (\theta) + \frac{MG}{r}\sin^2 (\theta) + \frac{r-2MG}{r}\sin^2 (\theta) \\ & = -\sin^2 (\theta) + \frac{MG}{r}\sin^2 (\theta) + \sin^2 (\theta) - \frac{2MG}{r}\sin^2 (\theta) \\ & = -\frac{MG}{r}\sin^2 (\theta) \end{aligned}
This gives the result:
{R^r}_{\phi r \phi} = -\frac{MG}{r}\sin^2 (\theta)
Component {R^\theta}_{\phi \theta \phi}
For this component, we set \mu=\theta, \nu=\phi, \rho=\theta, \sigma=\phi:
\begin{aligned} {R^\theta}_{\phi\theta\phi} & = \partial_\theta {\Gamma^\theta}_{\phi\phi} - \partial_\phi {\Gamma^\theta}_{\theta\phi} + {\Gamma^\lambda}_{\phi\phi} {\Gamma^\theta}_{\theta\lambda} - {\Gamma^\lambda}_{\theta\phi} {\Gamma^\theta}_{\phi\lambda} \\ & = \partial_\theta {\Gamma^\theta}_{\phi\phi} - 0 + {\Gamma^\lambda}_{\phi\phi} {\Gamma^\theta}_{\theta\lambda} - {\Gamma^\lambda}_{\theta\phi} {\Gamma^\theta}_{\phi\lambda} \\ & = \partial_\theta\left(-\cos(\theta)\sin(\theta)\right) + {\Gamma^\phi}_{\theta\phi} {\Gamma^\theta}_{\phi \phi} - {\Gamma^r}_{\phi\phi} {\Gamma^\theta}_{\theta r} \\ & = \sin^2(\theta) - \cos^2(\theta) + \left(- (r - 2MG)\sin^2(\theta)\right) \frac{1}{r} -\cot(\theta) (-\cos(\theta)\sin(\theta)) \\ & = \sin^2(\theta) - \cos^2(\theta) -\left(1 - \frac{2MG}{r}\right)\sin^2(\theta) + \cos(\theta)^2 \\ & = \sin^2(\theta) - \left(1 - \frac{2MG}{r}\right)\sin^2(\theta) \\ & = \frac{2MG}{r}\sin^2(\theta) \end{aligned}
This gives the result:
{R^\theta}_{\phi \theta \phi} = \frac{2MG}{r}\sin^2 (\theta)
Component {R^t}_{r\theta\phi}
For this component, we set \mu=t, \nu=r, \rho=\theta, \sigma=\phi:
\begin{aligned} {R^t}_{r\theta\phi} & = \partial_\theta {\Gamma^t}_{r\phi} - \partial_\phi {\Gamma^t}_{r\theta} + {\Gamma^t}_{\lambda\theta}{\Gamma^\lambda}_{r\phi} - {\Gamma^t}_{\lambda\phi}{\Gamma^\lambda}_{r\theta} \\ & = 0 - 0 + {\Gamma^t}_{\lambda\theta}\cdot 0 - 0 \cdot {\Gamma^\lambda}_{r\theta} = 0 \end{aligned}
This gives the result:
{R^t}_{r\theta\phi} = 0
Component {R^t}_{\theta\phi r}
For this component, we set \mu=t, \nu=\theta, \rho=\phi, \sigma=r:
\begin{aligned} {R^t}_{\theta\phi r} & = \partial_\phi {\Gamma^t}_{\theta r} - \partial_r {\Gamma^t}_{\theta\phi} + {\Gamma^t}_{\lambda\phi}{\Gamma^\lambda}_{\theta r} - {\Gamma^t}_{\lambda r}{\Gamma^\lambda}_{\theta\phi} \\ & = 0 - 0 + 0 \cdot {\Gamma^\lambda}_{\theta r} - {\Gamma^t}_{t r}{\Gamma^t}_{\theta\phi} = 0 \end{aligned}
This gives the result:
{R^t}_{\theta\phi r} = 0
Component {R^t}_{\phi r\theta}
For this component, we set \mu=t, \nu=\phi, \rho=r, \sigma=\theta:
\begin{aligned} {R^t}_{\phi r\theta} & = \partial_r {\Gamma^t}_{\phi\theta} - \partial_\theta {\Gamma^t}_{\phi r} + {\Gamma^t}_{\lambda r}{\Gamma^\lambda}_{\phi\theta} - {\Gamma^t}_{\lambda\theta}{\Gamma^\lambda}_{\phi r}\\ & = 0 - 0 + {\Gamma^t}_{t r}{\Gamma^t}_{\phi\theta} - 0 \cdot{\Gamma^\lambda}_{\phi r} = 0 \end{aligned}
This gives the result:
{R^t}_{\phi r\theta} = 0
Component {R^r}_{t\theta\phi}
For this component, we set \mu=r, \nu=t, \rho=\theta, \sigma=\phi:
\begin{aligned} {R^r}_{t\theta\phi} & = \partial_\theta {\Gamma^r}_{t\phi} - \partial_\phi {\Gamma^r}_{t\theta} + {\Gamma^r}_{\lambda\theta}{\Gamma^\lambda}_{t\phi} - {\Gamma^r}_{\lambda\phi}{\Gamma^\lambda}_{t\theta} \\ & = 0 - 0 + {\Gamma^r}_{\lambda\theta}\cdot 0 - {\Gamma^r}_{\lambda\phi}\cdot 0 = 0 \end{aligned}
This gives the result:
{R^r}_{t\theta\phi} = 0
Component {R^r}_{\theta\phi t}
For this component, we set \mu=r, \nu=\theta, \rho=\phi, \sigma=t:
\begin{aligned} {R^r}_{\theta\phi t} & = \partial_\phi {\Gamma^r}_{\theta t} - \partial_t {\Gamma^r}_{\theta\phi} + {\Gamma^r}_{\lambda\phi}{\Gamma^\lambda}_{\theta t} - {\Gamma^r}_{\lambda t}{\Gamma^\lambda}_{\theta\phi} \\ & = 0 - 0 + {\Gamma^r}_{\lambda\phi}\cdot 0 - {\Gamma^r}_{r t}{\Gamma^r}_{\theta\phi} = 0 \end{aligned}
This gives the result:
{R^r}_{\theta\phi t} = 0
Component {R^r}_{\phi t\theta}
For this component, we set \mu=r, \nu=\phi, \rho=t, \sigma=\theta:
\begin{aligned} {R^r}_{\phi t\theta} & = \partial_t {\Gamma^r}_{\phi\theta} - \partial_\theta {\Gamma^r}_{\phi t} + {\Gamma^r}_{\lambda t}{\Gamma^\lambda}_{\phi\theta} - {\Gamma^r}_{\lambda\theta}{\Gamma^\lambda}_{\phi t} \\ & = 0 - 0 + {\Gamma^r}_{r t}{\Gamma^r}_{\phi\theta} - {\Gamma^r}_{\lambda\theta}\cdot 0 = 0 \end{aligned}
This gives the result:
{R^r}_{\phi t\theta} = 0
Component {R^\theta}_{t r\phi}
For this component, we set \mu=\theta, \nu=t, \rho=r, \sigma=\phi:
\begin{aligned} {R^\theta}_{t r\phi} & = \partial_r {\Gamma^\theta}_{t\phi} - \partial_\phi {\Gamma^\theta}_{tr} + {\Gamma^\theta}_{\lambda r}{\Gamma^\lambda}_{t\phi} - {\Gamma^\theta}_{\lambda\phi}{\Gamma^\lambda}_{tr} \\ & = 0 - 0 + {\Gamma^\theta}_{\lambda r}\cdot 0 - {\Gamma^\theta}_{t\phi}{\Gamma^t}_{tr} = 0 \end{aligned}
This gives the result:
{R^\theta}_{t r\phi} = 0
Component {R^\theta}_{r\phi t}
For this component, we set \mu=\theta, \nu=r, \rho=\phi, \sigma=t:
\begin{aligned} {R^\theta}_{r\phi t} & = \partial_\phi {\Gamma^\theta}_{rt} - \partial_t {\Gamma^\theta}_{r\phi} + {\Gamma^\theta}_{\lambda\phi}{\Gamma^\lambda}_{rt} - {\Gamma^\theta}_{\lambda t}{\Gamma^\lambda}_{r\phi} \\ & = 0 - 0 + {\Gamma^\theta}_{t\phi}{\Gamma^t}_{rt} - 0 \cdot {\Gamma^\lambda}_{r\phi} = 0 \end{aligned}
This gives the result:
{R^\theta}_{r\phi t} = 0
Component {R^\theta}_{\phi t r}
For this component, we set \mu=\theta, \nu=\phi, \rho=t, \sigma=r:
\begin{aligned} {R^\theta}_{\phi t r} & = \partial_t {\Gamma^\theta}_{\phi r} - \partial_r {\Gamma^\theta}_{\phi t} + {\Gamma^\theta}_{\lambda t}{\Gamma^\lambda}_{\phi r} - {\Gamma^\theta}_{\lambda r}{\Gamma^\lambda}_{\phi t} \\ & = 0 - 0 + 0 \cdot {\Gamma^\lambda}_{\phi r} - {\Gamma^\theta}_{\lambda r} \cdot 0 = 0 \end{aligned}
This gives the result:
{R^\theta}_{\phi t r} = 0
Component {R^\phi}_{t r\theta}
For this component, we set \mu=\phi, \nu=t, \rho=r, \sigma=\theta:
\begin{aligned} {R^\phi}_{t r\theta} & = \partial_r {\Gamma^\phi}_{t\theta} - \partial_\theta {\Gamma^\phi}_{tr} + {\Gamma^\phi}_{\lambda r}{\Gamma^\lambda}_{t\theta} - {\Gamma^\phi}_{\lambda\theta}{\Gamma^\lambda}_{tr} \\ & = 0 - 0 + {\Gamma^\phi}_{\lambda r}\cdot 0 - {\Gamma^\phi}_{t\theta}{\Gamma^t}_{tr} = 0 \end{aligned}
This gives the result:
{R^\phi}_{t r\theta} = 0
Component {R^\phi}_{r\theta t}
For this component, we set \mu=\phi, \nu=r, \rho=\theta, \sigma=t:
\begin{aligned} {R^\phi}_{r\theta t} & = \partial_\theta {\Gamma^\phi}_{rt} - \partial_t {\Gamma^\phi}_{r\theta} + {\Gamma^\phi}_{\lambda\theta}{\Gamma^\lambda}_{rt} - {\Gamma^\phi}_{\lambda t}{\Gamma^\lambda}_{r\theta} \\ & = 0 - 0 + {\Gamma^\phi}_{t\theta}{\Gamma^t}_{rt} - 0 \cdot {\Gamma^\lambda}_{r\theta} = 0 \end{aligned}
This gives the result:
{R^\phi}_{r\theta t} = 0
Component {R^\phi}_{\theta t r}
For this component, we set \mu=\phi, \nu=\theta, \rho=t, \sigma=r:
\begin{aligned} {R^\phi}_{\theta t r} & = \partial_t {\Gamma^\phi}_{\theta r} - \partial_r {\Gamma^\phi}_{\theta t} + {\Gamma^\phi}_{\lambda t}{\Gamma^\lambda}_{\theta r} - {\Gamma^\phi}_{\lambda r}{\Gamma^\lambda}_{\theta t} \\ & = 0 - 0 + 0 \cdot {\Gamma^\lambda}_{\theta r} - {\Gamma^\phi}_{\lambda r} \cdot 0 = 0 \end{aligned}
This gives the result:
{R^\phi}_{\theta t r} = 0
Component {R^t}_{r t\theta}
For this component, we set \mu=t, \nu=r, \rho=t, \sigma=\theta:
\begin{aligned} {R^t}_{r t\theta} & = \partial_t {\Gamma^t}_{r\theta} - \partial_\theta {\Gamma^t}_{rt} + {\Gamma^t}_{\lambda t}{\Gamma^\lambda}_{r\theta} - {\Gamma^t}_{\lambda\theta}{\Gamma^\lambda}_{rt} \\ & = 0 - 0 + {\Gamma^t}_{\lambda t}\cdot 0 - {\Gamma^t}_{t\theta}{\Gamma^t}_{rt} = 0 \end{aligned}
This gives the result:
{R^t}_{r t\theta} = 0
Component {R^t}_{r t\phi}
For this component, we set \mu=t , \nu=r, \rho=t , \sigma=\phi:
\begin{aligned} {R^t}_{r t\phi} & = \partial_t {\Gamma^t}_{r\phi} - \partial_\phi {\Gamma^t}_{rt} + {\Gamma^t}_{\lambda t}{\Gamma^\lambda}_{r\phi} - {\Gamma^t}_{\lambda\phi}{\Gamma^\lambda}_{rt} \\ & = 0 - 0 + {\Gamma^t}_{r t}{\Gamma^r}_{r\phi} - {\Gamma^t}_{t\phi}{\Gamma^t}_{rt} = 0 \end{aligned}
This gives the result:
{R^t}_{r t\phi} = 0
The calculations show that for the curvature tensor, most of the {\Gamma^\sigma}_{\mu \nu} components are zero. The only six non-zero components are:
\begin{aligned} & {R^t}_{rtr} = \frac{2MG}{r^2(r - 2MG)} \\ & {R^t}_{\theta t\theta} = -\frac{MG}{r} \\ & {R^t}_{\phi t\phi} = -\frac{MG}{r}\sin^2 (\theta) \\ & {R^r}_{\theta r \theta} = -\frac{MG}{r} \\ & {R^r}_{\phi r \phi} = -\frac{MG}{r}\sin^2 (\theta) \\ & {R^\theta}_{\phi \theta \phi} = \frac{2MG}{r}\sin^2 (\theta) \end{aligned}
The remaining fourteen are zero:
\begin{aligned} & {R^t}_{r\theta\phi} = {R^t}_{\theta\phi r} = {R^t}_{\phi r\theta} = {R^r}_{t\theta\phi} \\ & = {R^r}_{\theta\phi t} = {R^r}_{\phi t\theta} = {R^\theta}_{t r\phi} = {R^\theta}_{r\phi t} \\ & = {R^\theta}_{\phi t r} = {R^\phi}_{t r\theta} = {R^\phi}_{r\theta t} = {R^\phi}_{\theta t r} \\ & = {R^t}_{r t\theta} = {R^t}_{r t\phi} = 0 \end{aligned}
From the six mixed components of the Riemann tensor already computed, the fully covariant form follows directly by lowering the free index with the metric tensor
R_{\alpha\beta\gamma\delta} = g_{\alpha\mu} {R^\mu}_{\beta\gamma\delta}
We apply the metric tensor computed earlier and we obtains the six independent covariant components:
\begin{aligned} & R_{trtr} = g_{tt} {R^t}_{rtr} = -\left(1-\frac{2MG}{r}\right) \frac{2MG}{r^2(r-2MG)} = -\frac{2MG}{r^3}\\ & R_{t\theta t\theta} = g_{tt} {R^t}_{\theta t\theta} = -\left(1-\frac{2MG}{r}\right) \left(-\frac{MG}{r}\right) = \frac{MG(r-2MG)}{r^2} \\ & R_{t\phi t\phi} = g_{tt} {R^t}_{\phi t\phi} = -\left(1-\frac{2MG}{r}\right) \left(-\frac{MG}{r}\sin^2\theta\right) = \frac{MG(r-2MG)}{r^2} \sin^2\theta\\ & R_{r\theta r\theta} = g_{rr} {R^r}_{\theta r\theta} = \left(1-\frac{2MG}{r}\right)^{-1} \left(-\frac{MG}{r}\right) = -\frac{MG}{r-2MG}\\ & R_{r\phi r\phi} = g_{rr} {R^r}_{\phi r\phi} = \left(1-\frac{2MG}{r}\right)^{-1} \left(-\frac{MG}{r}\sin^2\theta\right) = -\frac{MG}{r-2MG} \sin^2\theta \\ & R_{\theta\phi\theta\phi} = g_{\theta\theta} {R^\theta}_{\phi\theta\phi} = r^2\cdot\frac{2MG}{r}\sin^2\theta = 2MG r \sin^2\theta \end{aligned}
These six components are the standard independent covariant entries of the Riemann tensor for the Schwarzschild spacetime, from which all others follow by antisymmetry and pair-exchange symmetries.
The Ricci scalar R is obtained by a two-step contraction of the Riemann tensor, first, contracting on one pair of indices to form the Ricci tensor, and then contracting again with the metric:
R = g^{\mu\nu}R_{\mu\nu} = g^{\mu\nu}{R^\rho}_{\mu\rho\nu}
For the Schwarzschild geometry, the explicit calculation starts from the six independent covariant components of the Riemann tensor. Each of these contributes to the Ricci tensor under the contraction, but the structure of the Schwarzschild solution is such that the contributions cancel exactly. This cancellation is a direct reflection of the vacuum Einstein field equations:
R_{\mu\nu} = 0
which state that in empty space the Ricci tensor must vanish.
Let’s compute the Ricci tensor R_{\mu\nu} = g^{\rho\sigma}R_{\rho\mu\sigma\nu}.
Component R_{tt}
For this component, we set \mu = t, \nu = t:
\begin{aligned} R_{tt} = & g^{\rho\sigma}R_{\rho t \sigma t} =g^{rr}R_{trrt} + g^{\theta\theta}R_{\theta t\theta t} +g^{\phi\phi}R_{\phi t\phi t} \\ = & g^{rr}R_{trtr}+g^{\theta\theta}R_{t\theta t\theta} + g^{\phi\phi}R_{t\phi t\phi} \\ = & \left(1-\frac{2MG}{r}\right) \left(-\frac{2MG}{r^3}\right) +\frac{1}{r^2} \frac{MG(r-2MG)}{r^2} \\ & + \frac{1}{r^2\sin^2\theta} \frac{MG(r-2MG)}{r^2}\sin^2\theta\\ = & \left(-\frac{2MG}{r^3}+\frac{4M^2G^2}{r^4}\right) +\left(\frac{MG}{r^3} \\- \frac{2M^2G^2}{r^4}\right) \\ & + \left(\frac{MG}{r^3}-\frac{2M^2G^2}{r^4}\right)= 0 \end{aligned}
This gives the result:
R_{tt} = 0
Component R_{rr}
For this component, we set \mu = r, \nu = r:
\begin{aligned} R_{rr} = & g^{\rho\sigma}R_{\rho r \sigma r} = g^{tt}R_{trtr}+g^{\theta\theta}R_{\theta r\theta r} + g^{\phi\phi}R_{\phi r\phi r} \\ = & g^{tt}R_{trtr} + g^{\theta\theta}R_{r\theta r\theta} + g^{\phi\phi}R_{r\phi r\phi} \\ = & -\left(1-\frac{2MG}{r}\right)^{-1} \left(-\frac{2MG}{r^3}\right) +\frac{1}{r^2} \left(-\frac{MG}{r-2MG}\right) \\ & + \frac{1}{r^2\sin^2\theta} \left(-\frac{MG}{r-2MG}\sin^2\theta\right)\\ = & \frac{2MG}{r^3}\frac{r}{r-2MG} - \frac{MG}{r^2(r-2MG)} - \frac{MG}{r^2(r-2MG)}=0 \end{aligned}
This gives the result:
R_{rr} = 0
Component R_{\theta\theta}
For this component, we set \mu = \theta, \nu = \theta:
\begin{aligned} R_{\theta\theta} = & g^{\rho\sigma}R_{\rho\theta\sigma\theta} = g^{tt}R_{t\theta t\theta}+g^{rr}R_{r\theta r\theta} + g^{\phi\phi}R_{\phi\theta\phi\theta} \\ = & g^{tt}R_{t\theta t\theta} + g^{rr}R_{r\theta r\theta} + g^{\phi\phi}R_{\theta\phi\theta\phi} \\ = & -\left(1-\frac{2MG}{r}\right)^{-1} \frac{MG(r-2MG)}{r^2} + \left(1-\frac{2MG}{r}\right) \left(-\frac{MG}{r-2MG}\right) \\ & + \frac{1}{r^2\sin^2\theta} (2MG r \sin^2\theta) \\ & = -\frac{MG}{r}-\frac{MG}{r}+\frac{2MG}{r}=0 \end{aligned}
This gives the result:
R_{tt} = 0
Component R_{\phi\phi}
For this component, we set \mu = \phi, \nu = \phi:
\begin{aligned} R_{\phi\phi} = & g^{\rho\sigma}R_{\rho\phi\sigma\phi} = g^{tt}R_{t\phi t\phi} + g^{rr}R_{r\phi r\phi} + g^{\theta\theta}R_{\theta\phi\theta\phi} \\ = & -\left(1-\frac{2MG}{r}\right)^{-1} \frac{MG(r-2MG)}{r^2}\sin^2\theta \\ & + \left(1-\frac{2MG}{r}\right) \left(-\frac{MG}{r-2MG}\sin^2\theta\right) + \frac{1}{r^2} (2MG r \sin^2\theta)\\ & = -\frac{MG}{r}\sin^2\theta - \frac{MG}{r}\sin^2\theta + \frac{2MG}{r}\sin^2\theta = 0 \end{aligned}
This gives the result:
R_{\phi\phi} = 0
All off-diagonal Ricci components vanish by symmetry and the absence of mixed R_{\rho\mu\sigma\nu} with distinct \mu\neq\nu among the six entries, so the Ricci scalar R=g^{\mu\nu}R_{\mu\nu}=0.
The Kretschmann scalar, K, is a curvature invariant, which means its value is the same for all observers at a specific point in spacetime, regardless of their coordinate system.
This makes it a tool for identifying true physical singularities, where spacetime curvature becomes infinite, as opposed to coordinate singularities, which are merely artifacts of a poorly chosen coordinate system.
For the Schwarzschild metric, the Ricci tensor is zero, so invariants like the Ricci scalar are also zero everywhere, providing no information about the curvature.
The Kretschmann scalar, however, is not zero and quantifies the tidal forces present in the vacuum spacetime around the massive object:
K = R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}
The first step requires to raise the indices of the Riemann tensor.
The formula requires the fully contravariant Riemann tensor, R^{\alpha\beta\gamma\delta}.
We have already calculated the fully covariant form, R_{\mu\nu\rho\sigma}. To get the contravariant form, we’ll raise all four indices using the inverse metric tensor, g^{\mu\nu}:
R^{\alpha\beta\gamma\delta} = g^{\alpha\mu} g^{\beta\nu} g^{\gamma\rho} g^{\delta\sigma} R_{\mu\nu\rho\sigma}
Component R^{trtr}
For this component, we set \alpha = t, \beta = r, \gamma =t, \delta = r:
\begin{aligned} R^{trtr} &= g^{tt} g^{rr} g^{tt} g^{rr} R_{trtr} \\ &= \left[-\left(1-\frac{2MG}{r}\right)^{-1}\right] \left(1-\frac{2MG}{r}\right) \left[-\left(1-\frac{2MG}{r}\right)^{-1}\right] \left(1-\frac{2MG}{r}\right) \left(-\frac{2MG}{r^3}\right) \\ &= (-1) \cdot (-1) \cdot \left(-\frac{2MG}{r^3}\right) \end{aligned}
This gives the result:
R^{trtr} = -\frac{2MG}{r^3}
Component R^{t\theta t\theta}
For this component, we set \alpha = t, \beta = \theta, \gamma =t, \delta = \theta:
\begin{aligned} R^{t\theta t\theta} &= g^{tt} g^{\theta\theta} g^{tt} g^{\theta\theta} R_{t\theta t\theta} \\ &= \left[-\left(1-\frac{2MG}{r}\right)^{-1}\right] \left(\frac{1}{r^2}\right) \left[-\left(1-\frac{2MG}{r}\right)^{-1}\right] \left(\frac{1}{r^2}\right) \left(\frac{MG(r-2MG)}{r^2}\right) \\ &= \left(1-\frac{2MG}{r}\right)^{-2} \left(\frac{1}{r^4}\right) \left(\frac{MG(r-2MG)}{r^2}\right) \\ &= \left(\frac{r}{r-2MG}\right)^2 \left(\frac{1}{r^4}\right) \left(\frac{MG(r-2MG)}{r^2}\right) \\ &= \frac{r^2}{(r-2MG)^2} \frac{1}{r^4} \frac{MG(r-2MG)}{r^2} \\ &= \frac{MG}{r^4(r-2MG)} \end{aligned}
This gives the result:
R^{t\theta t\theta} = \frac{MG}{r^4(r-2MG)}
Component R^{t\phi t\phi}
For this component, we set \alpha = t, \beta = \phi, \gamma =t, \delta = \phi:
\begin{aligned} R^{t\phi t\phi} &= g^{tt} g^{\phi\phi} g^{tt} g^{\phi\phi} R_{t\phi t\phi} \\ &= \left[-\left(1-\frac{2MG}{r}\right)^{-1}\right] \left(\frac{1}{r^2\sin^2\theta}\right) \left[-\left(1-\frac{2MG}{r}\right)^{-1}\right] \left(\frac{1}{r^2\sin^2\theta}\right) \left(\frac{MG(r-2MG)}{r^2} \sin^2\theta\right) \\ &= \left(1-\frac{2MG}{r}\right)^{-2} \left(\frac{1}{r^4\sin^4\theta}\right) \left(\frac{MG(r-2MG)}{r^2} \sin^2\theta\right) \\ &= \left(\frac{r}{r-2MG}\right)^2 \left(\frac{1}{r^4\sin^4\theta}\right) \left(\frac{MG(r-2MG)}{r^2} \sin^2\theta\right) \\ &= \frac{MG}{r^4(r-2MG)\sin^2\theta} \end{aligned}
This gives the result:
R^{t\phi t\phi} = \frac{MG}{r^4(r-2MG)\sin^2\theta}
Component R^{r\theta r\theta}
For this component, we set \alpha = r, \beta = \theta, \gamma =r, \delta = \theta:
\begin{aligned} R^{r\theta r\theta} &= g^{rr} g^{\theta\theta} g^{rr} g^{\theta\theta} R_{r\theta r\theta} \\ &= \left(1-\frac{2MG}{r}\right) \left(\frac{1}{r^2}\right) \left(1-\frac{2MG}{r}\right) \left(\frac{1}{r^2}\right) \left(-\frac{MG}{r-2MG}\right) \\ &= \left(\frac{r-2MG}{r}\right)^2 \left(\frac{1}{r^4}\right) \left(-\frac{MG}{r-2MG}\right) \\ &= \frac{(r-2MG)^2}{r^2} \frac{1}{r^4} \left(-\frac{MG}{r-2MG}\right) \\ &= -\frac{MG(r-2MG)}{r^6} \end{aligned}
This gives the result:
R^{r\theta r\theta} = -\frac{MG(r-2MG)}{r^6}
Component R^{r\phi r\phi}
For this component, we set \alpha = r, \beta = \phi, \gamma =r, \delta = \phi:
\begin{aligned} R^{r\phi r\phi} &= g^{rr} g^{\phi\phi} g^{rr} g^{\phi\phi} R_{r\phi r\phi} \\ &= \left(1-\frac{2MG}{r}\right) \left(\frac{1}{r^2\sin^2\theta}\right) \left(1-\frac{2MG}{r}\right) \left(\frac{1}{r^2\sin^2\theta}\right) \left(-\frac{MG}{r-2MG} \sin^2\theta\right) \\ &= \left(\frac{r-2MG}{r}\right)^2 \left(\frac{1}{r^4\sin^4\theta}\right) \left(-\frac{MG}{r-2MG} \sin^2\theta\right) \\ &= -\frac{MG(r-2MG)}{r^6\sin^2\theta} \end{aligned}
This gives the result:
R^{r\phi r\phi} = -\frac{MG(r-2MG)}{r^6\sin^2\theta}
Component R^{\theta\phi\theta\phi}
For this component, we set \alpha = \theta, \beta = \phi, \gamma =\theta, \delta = \phi:
\begin{aligned} R^{\theta\phi\theta\phi} &= g^{\theta\theta} g^{\phi\phi} g^{\theta\theta} g^{\phi\phi} R_{\theta\phi\theta\phi} \\ &= \left(\frac{1}{r^2}\right) \left(\frac{1}{r^2\sin^2\theta}\right) \left(\frac{1}{r^2}\right) \left(\frac{1}{r^2\sin^2\theta}\right) (2MG r \sin^2\theta) \\ &= \frac{1}{r^8\sin^4\theta} (2MG r \sin^2\theta) \\ &= \frac{2MG}{r^7\sin^2\theta} \end{aligned}
This gives the result:
R^{\theta\phi\theta\phi} = \frac{2MG}{r^7\sin^2\theta}
The full expression for the Kretschmann scalar is the sum over all 256 components:
K = \sum_{\alpha,\beta,\gamma,\delta} R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}
The expanded sum, considering only the non-zero components and their symmetries, simplifies to:
K = 4 \left( R_{trtr}R^{trtr} + R_{t\theta t\theta}R^{t\theta t\theta} + R_{t\phi t\phi}R^{t\phi t\phi} + R_{r\theta r\theta}R^{r\theta r\theta} + R_{r\phi r\phi}R^{r\phi r\phi} + R_{\theta\phi\theta\phi}R^{\theta\phi\theta\phi} \right)
Let’s calculate these products:
\begin{aligned} & R_{trtr}R^{trtr} = \left(-\frac{2MG}{r^3}\right) \left(-\frac{2MG}{r^3}\right) = \frac{4M^2G^2}{r^6} \\ & R_{t\theta t\theta}R^{t\theta t\theta} = \left(\frac{MG(r-2MG)}{r^2}\right) \left(\frac{MG}{r^4(r-2MG)}\right) = \frac{M^2G^2}{r^6} \\ & R_{t\phi t\phi}R^{t\phi t\phi} = \left(\frac{MG(r-2MG)}{r^2} \sin^2\theta\right) \left(\frac{MG}{r^4(r-2MG)\sin^2\theta}\right) = \frac{M^2G^2}{r^6} \\ & R_{r\theta r\theta}R^{r\theta r\theta} = \left(-\frac{MG}{r-2MG}\right) \left(-\frac{MG(r-2MG)}{r^6}\right) = \frac{M^2G^2}{r^6}\\ & R_{r\phi r\phi}R^{r\phi r\phi} = \left(-\frac{MG}{r-2MG} \sin^2\theta\right) \left(-\frac{MG(r-2MG)}{r^6\sin^2\theta}\right) = \frac{M^2G^2}{r^6} \\ & R_{\theta\phi\theta\phi}R^{\theta\phi\theta\phi} = (2MG r \sin^2\theta) \left(\frac{2MG}{r^7\sin^2\theta}\right) = \frac{4M^2G^2}{r^6} \end{aligned}
Now, we substitute these products back into the sum for the Kretschmann scalar:
\begin{aligned} K &= 4 \left[ \frac{4M^2G^2}{r^6} + \frac{M^2G^2}{r^6} + \frac{M^2G^2}{r^6} + \frac{M^2G^2}{r^6} + \frac{M^2G^2}{r^6} + \frac{4M^2G^2}{r^6} \right] \\ &= 4 \left[ \frac{(4 + 1 + 1 + 1 + 1 + 4)M^2G^2}{r^6} \right] \\ &= 4 \left[ \frac{12M^2G^2}{r^6} \right] = \frac{48M^2G^2}{r^6} \end{aligned}
This gives the the Kretschmann scalar in Schwarzschild spacetime:
K = \frac{48M^2G^2}{r^6}
This result proves that the curvature of spacetime becomes infinite as r \to 0, confirming that this is the true physical singularity of a Schwarzschild black hole.