The Ricci[mu, nu], displayed as $\mathrm{R\_\_\μ,\ν}$, is a computational representation for the Ricci tensor, defined in terms of the Riemann tensor as

From this definition, and because of the symmetries of the Riemann tensor with respect to interchanging the positions of its indices the Ricci tensor is symmetric with respect to interchanging the position of its indices.

Expressing the Riemann tensor in terms of the Christoffel symbols and its derivatives, an equivalent definition is

During a Maple session, the value of any component of $\mathrm{R\_\_\μ,\ν}$ is automatically determined by the value of the spacetime metric. When Physics is loaded, the spacetime is set to galilean, of Minkowski type, and so all the elements of Ricci are automatically zero. To set the spacetime metric to something different use Setup or g_. Also, at least one system of coordinates must be set in order to compute the derivatives entering the definition of the Christoffel symbols, used to construct the Ricci tensor. For that purpose see Setup or Coordinates.

When the indices of Ricci assume integer values they are expected to be between 0 and the spacetime dimension, prefixed by ~ when they are contravariant, and the corresponding value of Ricci is returned. The values 0 and 4, or for the case any dimension set for the spacetime, represent the same object. When the indices have symbolic values Ricci returns unevaluated after normalizing its indices taking into account their symmetry property.

Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices - see `.` and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.

Besides being indexed with two indices, Ricci accepts other indexations (keywords) with special meaning:

matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in Ricci[]) returns a Matrix that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of Ricci. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting matrix takes into account the character of the indices.

~: returns an Array with the all-contravariant components of the Ricci tensor

definition: returns the definition of the Ricci tensor in terms of the Christoffel symbols and their derivatives.

nonzero: returns a set of equations, with the left-hand-side as a sequence of two positive numbers identifying the element of $\mathrm{R\_\_\μ,\ν}$ and the corresponding value on the right-hand-side. Note that this set is actually the output of the ArrayElems command when passing to it the Array obtained with the keyword array.

scalar: returns $R_{\mathrm{\mu }}^{\mathrm{\mu }}$, the trace of the Ricci tensor, also known as the Ricci scalar, an invariant.

scalars: returns the Ricci scalars of the Newman-Penrose formalism (see [2], sec. 2.7), ${\mathrm{\Phi }}_{\mathrm{00}},{\mathrm{\Phi }}_{\mathrm{01}},{\mathrm{\Phi }}_{\mathrm{02}},{\mathrm{\Phi }}_{\mathrm{11}},{\mathrm{\Phi }}_{\mathrm{12}},{\mathrm{\Phi }}_{\mathrm{22}},\mathrm{\Lambda }$, relevant in the classification of the SegreType of a spacetime. The spinor components ${\mathrm{\Phi }}_{\mathrm{10}},{\mathrm{\Phi }}_{\mathrm{20}},{\mathrm{\Phi }}_{\mathrm{21}}$ are respectively obtained from ${\mathrm{\Phi }}_{\mathrm{01}},{\mathrm{\Phi }}_{\mathrm{02}},{\mathrm{\Phi }}_{\mathrm{12}}$ by exchanging $m_{\mathrm{\mu }}$ and ${\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}$ in the definitions shown when using scalarsdefinition. For electrovacuum solutions, the NP curvature scalar $\mathrm{\Lambda }=0$.

scalarsdefinition: returns the definition of the Ricci scalars in terms of the null vectors of the Newman-Penrose formalism, $l_{\mathrm{\mu }},n_{\mathrm{\mu }},m_{\mathrm{\mu }},{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}$. Note that a general sign in this definition depends on the signature, that you can query entering Setup(signature) and change it using Setup to any of the four possible signatures predefined.

Some automatic checking and normalization are carried out each time you enter Ricci[...]. The checking is concerned with possible syntax errors. The automatic normalization takes into account the symmetry of Ricci[mu,nu] with respect to interchanging the positions of the indices mu and nu.

The %Ricci command is the inert form of Ricci, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

Sometimes it is convenient to rewrite tensorial expressions involving the Einstein tensor in terms of the Ricci tensor. For this purpose use convert(expression, Ricci) - see the Examples section.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Setup}\left(\mathrm{coordinatesystems}=\mathrm{cartesian}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\right]$

$\mathrm{g\_}\left[\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{-1}& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& 1\end{array}\right]$

$\mathrm{Christoffel}\left[\mathrm{nonzero}\right]$

${\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\varnothing$

$\mathrm{Ricci}\left[\mathrm{definition}\right]$

$R_{\mathrm{\mu },\mathrm{\nu }}={\partial }_{\mathrm{\alpha }}\left({\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\mu },\mathrm{\nu }}}\right)-{\partial }_{\mathrm{\nu }}\left({\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\mu },\mathrm{\alpha }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\mu },\mathrm{\alpha }}}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\beta }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\beta }\phantom{\mathrm{\mu },\mathrm{\nu }}}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\beta },\mathrm{\alpha }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\beta },\mathrm{\alpha }}}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\beta }}\mathrm{\mu },\mathrm{\alpha }}^{\phantom{}\mathrm{\beta }\phantom{\mathrm{\mu },\mathrm{\alpha }}}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\nu },\mathrm{\beta }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\nu },\mathrm{\beta }}}$

$\mathrm{Ricci}\left[\mathrm{\mu },\mathrm{\nu }\right]$

$0$

$\mathrm{ds2}≔r^2{\mathrm{dtheta}}^2+r^2{\sin \left(\mathrm{\theta }\right)}^2{\mathrm{dphi}}^2-2\mathrm{dt}\mathrm{dr}-2{k\left(r\,t\right)}^2{\mathrm{dt}}^2$

$\mathrm{ds2}≔r^2{\mathrm{d\θ}}^2+r^2{\sin \left(\mathrm{\theta }\right)}^2{\mathrm{d\φ}}^2-2\mathrm{dt}\mathrm{dr}-2{k\left(r\,t\right)}^2{\mathrm{dt}}^2$

$\mathrm{Setup}\left(\mathrm{coordinates}=\mathrm{spherical}\,\mathrm{metric}=\mathrm{ds2}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Coordinates:}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\mathrm{.\; Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-1}\\ 0& r^2& 0& 0\\ 0& 0& r^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ \mathrm{-1}& 0& 0& -2{k\left(r\,t\right)}^2\end{array}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\,\mathrm{metric}=\left\{\left(1\,4\right)=\mathrm{-1}\,\left(2\,2\right)=r^2\,\left(3\,3\right)=r^2{\sin \left(\mathrm{\theta }\right)}^2\,\left(4\,4\right)=-2{k\left(r\,t\right)}^2\right\}\,\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right]$

$\mathrm{CompactDisplay}\left(\right)$

$k\left(r\,t\right)\mathrm{will\; now\; be\; displayed\; as}k$

$\mathrm{Ricci}\left[\mathrm{\alpha },\mathrm{\beta }\right]$

$R_{\mathrm{\alpha },\mathrm{\beta }}$

$\mathrm{Ricci}\left[\mathrm{\beta },\mathrm{\alpha }\right]$

$R_{\mathrm{\alpha },\mathrm{\beta }}$

$-$

$0$

$\mathrm{Ricci}\left[\mathrm{scalarsdefinition}\right]$

$\mathrm{\Φ\_\_00}=-\frac{R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{l}_{\mathrm{\mu }}{l}_{\mathrm{\nu }}}{2},\mathrm{\Φ\_\_01}=-\frac{R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{l}_{\mathrm{\mu }}{m}_{\mathrm{\nu }}}{2},\mathrm{\Φ\_\_02}=-\frac{R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{m}_{\mathrm{\mu }}{m}_{\mathrm{\nu }}}{2},\mathrm{\Φ\_\_11}=-\frac{R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\left(l_{\mathrm{\mu }}{n}_{\mathrm{\nu }}+m_{\mathrm{\mu }}{\stackrel{\&conjugate0;}{m}}_{\mathrm{\nu }}\right)}{4},\mathrm{\Φ\_\_12}=-\frac{R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{m}_{\mathrm{\mu }}{n}_{\mathrm{\nu }}}{2},\mathrm{\Φ\_\_22}=-\frac{R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{n}_{\mathrm{\mu }}{n}_{\mathrm{\nu }}}{2},\mathrm{\Lambda }=\frac{R_{\mathrm{\mu }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\mu }}\mathrm{\mu }}}{24}$

$\mathrm{Ricci}\left[\mathrm{scalars}\right]$

$\mathrm{\Φ\_\_00}=0,\mathrm{\Φ\_\_01}=0,\mathrm{\Φ\_\_02}=0,\mathrm{\Φ\_\_11}=\frac{2k_{r,r}k{r}^2+2{k_r}^2{r}^2-2k^2+1}{4r^2},\mathrm{\Φ\_\_12}=0,\mathrm{\Φ\_\_22}=\frac{2\stackrel{\mathbf{.}}{k}}{kr},\mathrm{\Lambda }=\frac{-2k_{r,r}k{r}^2-2{k_r}^2{r}^2-8kr{k}_r-2k^2+1}{12r^2}$

$=\mathrm{convert}\left(\,\mathrm{Christoffel}\right)$

$R_{\mathrm{\alpha },\mathrm{\beta }}={\partial }_{\mathrm{\mu }}\left({\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\alpha },\mathrm{\beta }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}\right)-{\partial }_{\mathrm{\beta }}\left({\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\mu },\mathrm{\nu }}}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\beta },\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\beta },\mathrm{\nu }}}$

$\mathrm{Einstein}\left[\mathrm{\alpha },\mathrm{\beta }\right]$

$G_{\mathrm{\alpha },\mathrm{\beta }}$

$\mathrm{convert}\left(\,\mathrm{Ricci}\right)$

$R_{\mathrm{\alpha },\mathrm{\beta }}-\frac{g_{\mathrm{\alpha },\mathrm{\beta }}{R}_{\mathrm{\mu }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\mu }}\mathrm{\mu }}}{2}$

$\mathrm{convert}\left(\,\mathrm{Ricci}\,\mathrm{evaluatetrace}\right)$

$R_{\mathrm{\alpha },\mathrm{\beta }}+\frac{g_{\mathrm{\alpha },\mathrm{\beta }}\left(2k_{r,r}k{r}^2+2{k_r}^2{r}^2+8kr{k}_r+2k^2-1\right)}{r^2}$

$\mathrm{Ricci}\left[1,1\right]$

$0$

$\mathrm{Ricci}\left[\mathrm{~1},1\right]$

$-\frac{2\left(k{k}_{r,r}r+{k_r}^{2}r+2k{k}_r\right)}{r}$

$\mathrm{\%Ricci}\left[\mathrm{~1},1\right]$

$R_{\phantom{}\phantom{1}1}^{\phantom{}1\phantom{1}}$

$\mathrm{value}\left(\right)$

$-\frac{2\left(k{k}_{r,r}r+{k_r}^{2}r+2k{k}_r\right)}{r}$

$\mathrm{Ricci}\left[\right]$

$R_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}0& 0& 0& \frac{2k{k}_{r,r}r+2{k_r}^{2}r+4k{k}_r}{r}\\ 0& -4kr{k}_r-2k^2+1& 0& 0\\ 0& 0& \left(-4kr{k}_r-2k^2+1\right){\sin \left(\mathrm{\theta }\right)}^2& 0\\ \frac{2k{k}_{r,r}r+2{k_r}^{2}r+4k{k}_r}{r}& 0& 0& \frac{4k\left(k^2{k}_{r,r}r+k{k_r}^{2}r+2k^2{k}_r+\stackrel{\mathbf{.}}{k}\right)}{r}\end{array}\right]$

$\mathrm{Ricci}\left[\mathrm{`~`}\right]$

$R_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{4k\left(-k^2{k}_{r,r}r-k{k_r}^{2}r-2k^2{k}_r+\stackrel{\mathbf{.}}{k}\right)}{r}& 0& 0& \frac{2k{k}_{r,r}r+2{k_r}^{2}r+4k{k}_r}{r}\\ 0& \frac{-4kr{k}_r-2k^2+1}{r^4}& 0& 0\\ 0& 0& \frac{\left(-4kr{k}_r-2k^2+1\right){\csc \left(\mathrm{\theta }\right)}^2}{r^4}& 0\\ \frac{2k{k}_{r,r}r+2{k_r}^{2}r+4k{k}_r}{r}& 0& 0& 0\end{array}\right]$

$\mathrm{Ricci}\left[\mathrm{scalar}\right]$

$-\frac{2\left(2k_{r,r}k{r}^2+2{k_r}^2{r}^2+8kr{k}_r+2k^2-1\right)}{r^2}$

$\mathrm{Ricci}\left[\mathrm{nonzero}\right]$

$R_{\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\,4\right)=\frac{2k{k}_{r,r}r+2{k_r}^{2}r+4k{k}_r}{r}\,\left(2\,2\right)=-4kr{k}_r-2k^2+1\,\left(3\,3\right)=\left(-4kr{k}_r-2k^2+1\right){\sin \left(\mathrm{\theta }\right)}^2\,\left(4\,1\right)=\frac{2k{k}_{r,r}r+2{k_r}^{2}r+4k{k}_r}{r}\,\left(4\,4\right)=\frac{4k\left(k^2{k}_{r,r}r+k{k_r}^{2}r+2k^2{k}_r+\stackrel{\mathbf{.}}{k}\right)}{r}\right\}$

$\mathrm{Ricci}\left[\mathrm{~mu},\mathrm{\nu },\mathrm{matrix}\right]$

$R_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}}=\left[\begin{array}{cccc}\frac{-2k{k}_{r,r}r-2{k_r}^{2}r-4k{k}_r}{r}& 0& 0& -\frac{4k\stackrel{\mathbf{.}}{k}}{r}\\ 0& \frac{-4kr{k}_r-2k^2+1}{r^2}& 0& 0\\ 0& 0& \frac{-4kr{k}_r-2k^2+1}{r^2}& 0\\ 0& 0& 0& \frac{-2k{k}_{r,r}r-2{k_r}^{2}r-4k{k}_r}{r}\end{array}\right]$

$R≔\mathrm{rhs}\left(\right)\:$

$R\left[1,1\right]$

$\frac{-2k{k}_{r,r}r-2{k_r}^{2}r-4k{k}_r}{r}$

$\mathrm{normal}\left(-\right)$

$0$

$\mathrm{Tetrads}:-\mathrm{PetrovType}\left(\right)$

$''D''$

$\mathrm{Tetrads}:-\mathrm{SegreType}\left(\right)$

$''D'',''[2,(11)]''$

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

[2] Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E.  Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. Cambridge University Press, 2003.

[3] Hawking, S.W., Israel, W., Chandrasekhar, S. General Relativity: an Einstein Centenary Survey, Chapter 7, Cambridge University Press, 2010.

[4] Letniowski, F.W., McLenaghan, R.G. An Improved Algorithm for Quartic Equation Classification and Petrov Classification, General Relativity and Gravitation, Vol.20, No. 5, 1988.