Riemann Tensor

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Notation

g_{ab} is the metric, \nabla_a is the covariant derivative, and \partial_a is the partial derivative with respect to x^a.

Square brackets surrounding indices denote antisymmetrization, and round brackets denote symmetrization. For example, U_{[a} V_{b]} \equiv (U_a V_b - U_b V_a)/2 and U_{(a} V_{b)} \equiv (U_a V_b + U_b V_a)/2.

Christoffel symbols


\begin{array}{rcl}
\Gamma^a{}_{bc} & \equiv &  g^{ad}(\partial_b g_{cd} + \partial_c g_{bd} - \partial_d g_{bc})/2 \\
\Gamma_{abc} & \equiv & g_{ad}\Gamma^d{}_{bc} \\
    & = & (\partial_b g_{ca} + \partial_c g_{ba} - \partial_a g_{bc})/2 \\
\partial_a g_{bc} & = & 2 \Gamma_{(bc)a}
\end{array}

Definition of Riemann tensor

The Riemann tensor or the Riemann-Christoffel curvature tensor is a four-index tensor describing the curvature of Riemannian manifolds.

It is often used to derive other relativistic tensors such as the Ricci curvature tensor or the curvature scalar.


\begin{array}{rcl}
2 \nabla_{[a} \nabla_{b]} V^c & = & R^c{}_{dab} V^d \\
2 \nabla_{[a} \nabla_{b]} W_c & = &  R_{abc}{}^d W_d 
\end{array}

Riemann, Ricci, curvature scalar, Einstein tensor, and Weyl tensor:

Ricci and curvature scalar are both contractions of the Riemann tensor, simplifying it.

The scalar curvature is the trace of the Ricci curvature.


\begin{array}{rcl}
R^a{}_{bcd} & = & \partial_c \Gamma^a_{bd} - \partial_d \Gamma^a_{bc} + \Gamma^e_{bd}\Gamma^a_{ec} 
    - \Gamma^e_{bc}\Gamma^a_{ed}  \\
R_{bd} & \equiv & R^a{}_{bad} = \partial_a \Gamma^a_{bd} - \partial_d \Gamma^a_{ba} + 
        \Gamma^e_{bd}\Gamma^a_{ea}   - \Gamma^e_{ba}\Gamma^a_{ed} \\
R & \equiv & g^{bd} R_{bd}  \\
G_{ab} &  \equiv & R_{ab} - \frac{1}{2} R g_{ab}   \\ 
W_{abcd} & \equiv & R_{abcd} - g_{a[c}R_{d]b} + g_{b[c}R_{d]a} + \frac{1}{3} R g_{a[c} g_{d]b} 
\end{array}

Weyl curvature tensor represents the traceless component of the Riemann curvature tensor.

(The definition of the Weyl tensor depends on the number of spacetime dimensions n. Here it is assumed that n=4.)

Symmetries

For Riemann, the three symmetries of the curvature tensor are:


\begin{array}{rcl}
   R_{bacd} & = & -R_{abcd} \\
   R_{abdc} & = & -R_{abcd} \\
   R_{cdab} & = & R_{abcd} \\
   R_{a[bcd]} & = & 0 
\end{array}

The last symmetry, discovered by Ricci is called the first Bianchi identity or algebraic Bianchi identity. Given any tensor which satisfies these symmetries, one can completely describe a Riemannian manifold with the indicated curvature tensor at any point. The Ricci and Einstein tensors are symmetric.

Bianchi identities


\begin{array}{rcl}
   \nabla_{[a} R_{bc]de} & = & 0 \\
  \nabla_a G^{ab} & = & 0 
\end{array}
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