Enhancements in the simplification of tensorial expressions.

$\mathrm{restart}\;\mathrm{with}\left(\mathrm{Physics}\right)\: \mathrm{Setup}\left(\mathrm{notation} \= \mathrm{true}\, \mathrm{coordinates} \= X\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}notation\text{'}\; against\; keyword\; \text{'}mathematicalnotation\text{'}}$

$\mathrm{*\; Partial\; match\; of\; \text{'}coordinates\text{'}\; against\; keyword\; \text{'}coordinatesystems\text{'}}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{Systems\; of\; spacetime\; Coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

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

Define a tensor $l_{\mathrm{\mu }}\left(X\right)$ and define the metric in terms of the components of this tensor so that $g_{\mathrm{\mu },\mathrm{\nu }}={\mathrm{\eta }}_{\mathrm{\mu },\mathrm{\nu }}+l_{\mathrm{\mu }}{l}_{\mathrm{\nu }}$ and ${\mathrm{\eta }}_{\mathrm{\mu },\mathrm{\nu }}$ is a Minkowski metric.

$l_{\mathrm{\mu }}\,{\mathrm{\eta }}_{\mathrm{\mu },\mathrm{\nu }}=-\mathrm{rhs}\left({\mathrm{g\_}}_{\mathrm{Minkowski}}\right)$

$\mathrm{The\; Minkowski\; metric\; in\; cartesian\; coordinates}$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,X_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,{\mathrm{\η}}_{\mathrm{\mu },\mathrm{\nu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,l_{\mathrm{\mu }}\,{\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

Avoid redundant display:

$\mathrm{PDEtools}:-\mathrm{declare}\left(l\left(X\right)\right)$

$l\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}l$

The metric:

$\mathrm{g\_}\left[\mathrm{mu}\,\mathrm{nu}\right] \= {\mathrm{\eta }}_{\mathrm{\mu },\mathrm{\nu }}+l_{\mathrm{\mu }}\left(X\right)l_{\mathrm{\nu }}\left(X\right)$

$g_{\mathrm{\mu },\mathrm{\nu }}={\mathrm{\η}}_{\mathrm{\mu },\mathrm{\nu }}+l_{\mathrm{\mu }}{l}_{\mathrm{\nu }}$

New: you can define it directly using a tensorial equation like (4), using either Define or Setup.

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

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{{𝒟}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,X_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,{\mathrm{\η}}_{\mathrm{\mu },\mathrm{\nu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,l_{\mathrm{\mu }}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

Verify the resulting $g_{\mathrm{\mu },\mathrm{\nu }}$:

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}1+{l_1}^2& l_1{l}_2& l_1{l}_3& l_1{l}_4\\ l_1{l}_2& 1+{l_2}^2& l_2{l}_3& l_2{l}_4\\ l_1{l}_3& l_2{l}_3& 1+{l_3}^2& l_3{l}_4\\ l_1{l}_4& l_2{l}_4& l_3{l}_4& -1+{l_4}^2\end{array}\right]$

New: you can query about the definition you gave for any tensor directly from the tensor itself.

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

$g_{\mathrm{\mu },\mathrm{\nu }}={\mathrm{\η}}_{\mathrm{\mu },\mathrm{\nu }}+l_{\mathrm{\mu }}{l}_{\mathrm{\nu }}$

Enhanced simplification capabilities: show that this expression involving derivatives of $l_{\mathrm{\mu }}\left(X\right)$ is equal to zero.

$-\left({\mathrm{d\_}}_{\mathrm{\alpha }}\left(l_{\mathrm{\lambda }}\left(X\right)\right){\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}{\mathrm{d\_}}_{\mathrm{\beta }}\left(l_{\mathrm{\kappa }}\left(X\right)\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}\left(X\right)\right)+\left({\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}{\mathrm{d\_}}_{\mathrm{\beta }}\left(l_{\mathrm{\kappa }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\lambda }}\left(l_{\mathrm{\alpha }}\left(X\right)\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}\left(X\right)\right)+\left({\mathrm{d\_}}_{\mathrm{\lambda }}\left(l_{\mathrm{\kappa }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right)\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}\left(X\right)\right)+\left({\mathrm{d\_}}_{\mathrm{\beta }}\left({\mathrm{d\_}}_{\mathrm{\lambda }}\left(l_{\mathrm{\kappa }}\left(X\right)\right)\right)l_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}\left(X\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\mu }}\left(l_{\mathrm{\nu }}\left(X\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right){\mathrm{d\_}}_{\mathrm{\rho }}\left({\mathrm{d\_}}_{\mathrm{\mu }}\left(l_{\mathrm{\nu }}\left(X\right)\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\nu }}\left(l_{\mathrm{\mu }}\left(X\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right){\mathrm{d\_}}_{\mathrm{\rho }}\left({\mathrm{d\_}}_{\mathrm{\nu }}\left(l_{\mathrm{\mu }}\left(X\right)\right)\right)+\left(\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right){\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}{\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\alpha }}\left(l_{\mathrm{\nu }}\left(X\right)\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right){\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}{\mathrm{d\_}}_{\mathrm{\mu }}\left(l_{\mathrm{\alpha }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\left(X\right)\right)+\left(\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right){\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}{\mathrm{d\_}}_{\mathrm{\alpha }}\left(l_{\mathrm{\mu }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\left(X\right)\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right){\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}{\mathrm{d\_}}_{\mathrm{\nu }}\left(l_{\mathrm{\alpha }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\left(X\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\alpha }}\left(l_{\mathrm{\mu }}\left(X\right)\right)+\left(\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right){\mathrm{d\_}}_{\mathrm{\mu }}\left(l_{\mathrm{\alpha }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\left(X\right)\right)\right)-\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\alpha }}\left(l_{\mathrm{\nu }}\left(X\right)\right)+\left(\frac{1}{2}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(X\right){\mathrm{d\_}}_{\mathrm{\nu }}\left(l_{\mathrm{\alpha }}\left(X\right)\right){\mathrm{d\_}}_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\left(X\right)\right)\right)$

$-\left({\partial }_{\mathrm{\alpha }}\left(l_{\mathrm{\lambda }}\right)\right){\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}\left({\partial }_{\mathrm{\beta }}\left(l_{\mathrm{\kappa }}\right)\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{l}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}+{\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}\left({\partial }_{\mathrm{\beta }}\left(l_{\mathrm{\kappa }}\right)\right)\left({\partial }_{\mathrm{\lambda }}\left(l_{\mathrm{\alpha }}\right)\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{l}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}+\left({\partial }_{\mathrm{\lambda }}\left(l_{\mathrm{\kappa }}\right)\right)\left({\partial }_{\mathrm{\rho }}\left(l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\right)\right)l_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{l}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}+\left({\partial }_{\mathrm{\beta }}\left({\partial }_{\mathrm{\lambda }}\left(l_{\mathrm{\kappa }}\right)\right)\right)l_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}{l}_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{l}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left({\partial }_{\mathrm{\rho }}\left(l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\right)\right)\left({\partial }_{\mathrm{\mu }}\left(l_{\mathrm{\nu }}\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{l}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left({\partial }_{\mathrm{\rho }}\left({\partial }_{\mathrm{\mu }}\left(l_{\mathrm{\nu }}\right)\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left({\partial }_{\mathrm{\rho }}\left(l_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\right)\right)\left({\partial }_{\mathrm{\nu }}\left(l_{\mathrm{\mu }}\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{l}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left({\partial }_{\mathrm{\rho }}\left({\partial }_{\mathrm{\nu }}\left(l_{\mathrm{\mu }}\right)\right)\right)}{2}+\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\right)\right)\left({\partial }_{\mathrm{\alpha }}\left(l_{\mathrm{\nu }}\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}\left({\partial }_{\mathrm{\mu }}\left(l_{\mathrm{\alpha }}\right)\right)\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\right)\right)}{2}+\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}\left({\partial }_{\mathrm{\alpha }}\left(l_{\mathrm{\mu }}\right)\right)\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\η}}_{\phantom{}\phantom{\mathrm{\alpha }}\phantom{,}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}\left({\partial }_{\mathrm{\nu }}\left(l_{\mathrm{\alpha }}\right)\right)\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{l}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{l}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\right)\right)\left({\partial }_{\mathrm{\alpha }}\left(l_{\mathrm{\mu }}\right)\right)}{2}+\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{l}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{l}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left({\partial }_{\mathrm{\mu }}\left(l_{\mathrm{\alpha }}\right)\right)\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\nu }}\right)\right)}{2}-\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{l}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{l}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\right)\right)\left({\partial }_{\mathrm{\alpha }}\left(l_{\mathrm{\nu }}\right)\right)}{2}+\frac{l_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{l}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{l}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left({\partial }_{\mathrm{\nu }}\left(l_{\mathrm{\alpha }}\right)\right)\left({\partial }_{\mathrm{\rho }}\left(l_{\mathrm{\mu }}\right)\right)}{2}$

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

$0$

A new tryhard option significantly improves zero recognition, which also got improved regardless of the new option.

$\mathrm{Setup}\left(\mathrm{spacetimeindices} \= \mathrm{lowercaselatin}\right)$

$\left[\mathrm{spacetimeindices}=\mathrm{lowercaselatin}\right]$

$\mathrm{Define}\left(A\left[a\right]\, B\left[a\, b\right]\, F\left[a\, b\, c\right]\, H\left[a\, b\, c\, d\right]\, J\left[a\, b\, c\, d\, e\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_a\,B_{a,b}\,{𝒟}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{a,b,c}\,H_{a,b,c,d}\,J_{a,b,c,d,e}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,X_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,{\mathrm{\η}}_{\mathrm{\mu },\mathrm{\nu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,l_{\mathrm{\mu }}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{e\_\_1}≔A_c{H}_{d,h,c,b}{F}_{h,d,f}{H}_{a,e,e,f}-A_h{H}_{a,c,c,f}{F}_{e,d,f}{H}_{d,e,h,b}$

$\mathrm{e\_\_1}≔H_{a,e\phantom{e}f}^{\phantom{a}\phantom{,}\phantom{e}e\phantom{f}}{H}_{d,h,c,b}{F}_{\phantom{}\phantom{h}\phantom{,}\phantom{d}\phantom{,}\phantom{f}}^{\phantom{}h,d,f}{A}_{\phantom{}\phantom{c}}^{\phantom{}c}-H_{a,c\phantom{c}f}^{\phantom{a}\phantom{,}\phantom{c}c\phantom{f}}{H}_{d,e,h,b}{F}_{\phantom{}\phantom{e}\phantom{,}\phantom{d}\phantom{,}\phantom{f}}^{\phantom{}e,d,f}{A}_{\phantom{}\phantom{h}}^{\phantom{}h}$

$\mathrm{e\_\_2}≔J_{f,a,b,g,c}{F}_{e,f,d}{F}_{e,d,g}-J_{e,a,b,f,c}{F}_{d,e,g}{F}_{d,g,f}$

$\mathrm{e\_\_2}≔-J_{e,a,b,f,c}{F}_{d\phantom{e}g}^{\phantom{d}e\phantom{g}}{F}_{\phantom{}\phantom{d}\phantom{,}\phantom{g}\phantom{,}\phantom{f}}^{\phantom{}d,g,f}+J_{f,a,b,g,c}{F}_{e,d\phantom{g}}^{\phantom{e}\phantom{,}\phantom{d}g}{F}_{\phantom{}\phantom{e}\phantom{,}\phantom{f}\phantom{,}\phantom{d}}^{\phantom{}e,f,d}$

$\mathrm{e\_\_3}≔J_{g,c,k,j,e}{H}_{i,h,a,h}{H}_{e,b,i,g}{F}_{d,k,j}-H_{h,b,g,j}{F}_{d,k,e}{J}_{j,c,k,e,h}{H}_{g,i,a,i}$

$\mathrm{e\_\_3}≔-J_{j,c,k,e,h}{H}_{g,i,a\phantom{i}}^{\phantom{g}\phantom{,}\phantom{i}\phantom{,}\phantom{a}i}{H}_{\phantom{}\phantom{h}b\phantom{g}\phantom{,}\phantom{j}}^{\phantom{}h\phantom{b}g,j}{F}_{d\phantom{k}\phantom{,}\phantom{e}}^{\phantom{d}k,e}+J_{g,c,k,j,e}{H}_{\phantom{}\phantom{e}b,i\phantom{g}}^{\phantom{}e\phantom{b}\phantom{,}\phantom{i}g}{H}_{\phantom{}\phantom{i}h,a\phantom{h}}^{\phantom{}i\phantom{h}\phantom{,}\phantom{a}h}{F}_{d\phantom{k}\phantom{,}\phantom{j}}^{\phantom{d}k,j}$

$\mathrm{map}\left(u \to \left[\mathrm{Check}\left(u\,\mathrm{all}\,\mathrm{quiet}\right)\right]\, \left[\mathrm{e\_\_1}\,\mathrm{e\_\_2}\,\mathrm{e\_\_3}\right]\right)$

$\left[\left[\left[\left\{c\,d\,e\,f\,h\right\}\,\left\{c\,d\,e\,f\,h\right\}\right]\,\left\{a\,b\right\}\right]\,\left[\left[\left\{d\,e\,f\,g\right\}\,\left\{d\,e\,f\,g\right\}\right]\,\left\{a\,b\,c\right\}\right]\,\left[\left[\left\{e\,g\,h\,i\,j\,k\right\}\,\left\{e\,g\,h\,i\,j\,k\right\}\right]\,\left\{a\,b\,c\,d\right\}\right]\right]$

$\mathrm{Simplify}\left(\mathrm{e\_\_1}\,\mathrm{tryhard}\right)$

$0$

$\mathrm{Simplify}\left(\mathrm{e\_\_2}\,\mathrm{tryhard}\right)$

$0$

$\mathrm{Simplify}\left(\mathrm{e\_\_3}\right)$

$0$

Improve the simplification with respect to side relations (equations) in the presence of quantum vectorial equations.

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

$\mathrm{Setup}\left(\mathrm{redo}\, \mathrm{hermitianoperators} \= \left\{L\,\mathrm{r\_}\, \mathrm{p\_}\, x\,y\,z\, p\right\}\, \mathrm{quantumoperators} \= \left\{A\, B\right\}\right)$

$\left[\mathrm{hermitianoperators}=\left\{L\,p\,\overrightarrow{p}\,\overrightarrow{r}\,x\,y\,z\right\}\,\mathrm{quantumoperators}=\left\{A\,B\,L\,p\,\overrightarrow{p}\,\overrightarrow{r}\,x\,y\,z\right\}\right]$

The identification of the vectorial character of expressions got improved. For example, this commutator is actually a non-projected vector.

$\mathrm{Commutator}\left(\mathrm{p\_}·\mathrm{p\_}\,\mathrm{r\_}\right)$

${\left[{‖\overrightarrow{p}‖}^2,\overrightarrow{r}\right]}_{-}$

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

$5$

Consider an expression involving vectorial quantum operators.

$\mathrm{interface}\left(\mathrm{imaginaryunit} \= i\right)$

$I$

$\mathrm{\_i}\cdot \left(\mathrm{i}x{p}_z-\mathrm{i}z{p}_x\right)+\mathrm{\_j}\cdot \left(\mathrm{i}y{p}_z-\mathrm{i}z{p}_y\right)$

$\stackrel{\wedge }{i}\left(\mathrm{i}x{p}_z-\mathrm{i}z{p}_x\right)+\stackrel{\wedge }{j}\left(\mathrm{i}y{p}_z-\mathrm{i}z{p}_y\right)$

Simplify (23) taking into account the definition of angular momentum.

$\left\{y{p}_z-z{p}_y=L_x\,z{p}_x-x{p}_z=L_y\right\}$

$\left\{y{p}_z-z{p}_y=L_x\,z{p}_x-x{p}_z=L_y\right\}$

$\mathrm{simplify}\left(\,\right)$

$-\mathrm{i}\stackrel{\wedge }{i}{L}_y+\mathrm{i}\stackrel{\wedge }{j}{L}_x$

Enhancements in the simplification and integration of spherical harmonics (SphericalY) relevant in quantum mechanics.

$\frac{1}{8648640}\frac{\sqrt{7293}{\ⅇ}^{6\mathrm{I}\mathrm{theta}}{\left(1-{\cos \left(\mathrm{phi}\right)}^2\right)}^3\left(\frac{2027025}{2}{\cos \left(\mathrm{phi}\right)}^2-\frac{135135}{2}\right)}{\sqrt{\mathrm{\pi }}}\+\mathrm{SphericalY}\left(8\,6\,\mathrm{phi}\,\mathrm{theta}\right)$

$\frac{\sqrt{7293}{\ⅇ}^{6\mathrm{i}\mathrm{\theta }}{\left(1-{\cos \left(\mathrm{\phi }\right)}^2\right)}^3\left(\frac{2027025{\cos \left(\mathrm{\phi }\right)}^2}{2}-\frac{135135}{2}\right)}{8648640\sqrt{\mathrm{\pi }}}+Y_8^6\left(\mathrm{\phi }\,\mathrm{\theta }\right)$

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

$0$

$\mathrm{conjugate}\left(\mathrm{SphericalY}\left(8\,6\,\mathrm{phi}\,\mathrm{theta}\right)\right)\*\left(1\+1\/5\*\sin \left(6\*\mathrm{theta}\right)\*\sin \left(5\*\mathrm{phi}\right)\right)\*\sin \left(\mathrm{phi}\right)$

$\stackrel{\&conjugate0;}{Y_8^6\left(\mathrm{\phi }\,\mathrm{\theta }\right)}\left(1+\frac{\sin \left(6\mathrm{\theta }\right)\sin \left(5\mathrm{\phi }\right)}{5}\right)\sin \left(\mathrm{\phi }\right)$

$\mathrm{Int}\left(\mathrm{Int}\left(\,\mathrm{theta}\=0..2 \mathrm{Pi}\right)\, \mathrm{phi}\=0..\mathrm{Pi}\right)$

${\int }_0^{\mathrm{\pi }}{\int }_0^{2\mathrm{\pi }}\stackrel{\&conjugate0;}{Y_8^6\left(\mathrm{\phi }\,\mathrm{\theta }\right)}\left(1+\frac{\sin \left(6\mathrm{\theta }\right)\sin \left(5\mathrm{\phi }\right)}{5}\right)\sin \left(\mathrm{\phi }\right)\ⅆ\mathrm{\theta }\ⅆ\mathrm{\phi }$

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

$-\frac{37\mathrm{i}}{81920}{\mathrm{\pi }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{7293}$

Minkowski spacetime: four different conventions.

$\mathrm{restart}\; \mathrm{with}\left(\mathrm{Physics}\right)\:\mathrm{Coordinates}\left(X\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{Systems\; of\; spacetime\; Coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\left\{X\right\}$

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

$\mathrm{x0}$

$\mathrm{x4}$

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

$\mathrm{Changing\; the\; signature\; of\; the\; tensor\; spacetime\; to:\; +\; +\; +\; +}$

$\mathrm{The\; Euclidean\; metric\; in\; coordinates}\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]$

$\mathrm{g\_}\left[\mathrm{`+++-`}\right]$

$\mathrm{The\; Minkowski\; metric,\; with\; signature\; +\; +\; +\; -,\; in\; coordinates}\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]$

$\mathrm{Setup}\left(\mathrm{signature} \= \mathrm{`-\; +++`}\right)$

$\left[\mathrm{signature}=\mathrm{-\; +\; +\; +}\right]$

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

The position of the different sign in a Minkowski spacetime always, either the first or the last symbol in the signature, refers to the time variable. So if you indicate $t$ as one of the coordinates, say in the first position, and the signature is not in agreement with time in the first position, it is now automatically corrected and the metric set. For example, in this moment, due to (33), time is implicitly assumed to be in the first position; set coordinates with $t$ in the last (fourth) position.

$\mathrm{Coordinates}\left(X\=\left[x\,y\,z\,t\right]\right)$

$\mathrm{Detected\; `t`,\; the\; time\; variable,\; in\; position\; 4.\; Changing\; the\; signature\; of\; the\; spacetime\; metric\; accordingly,\; to:\; +\; +\; +\; -}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

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

$\left\{X\right\}$

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

New shortcut for the matrix form of the all contravariant components of a tensor.

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

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

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

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

It is now possible to also use the keyword matrix passing only space (not spacetime) indices, resulting in a matrix with only the space components.

$\mathrm{Setup}\left(\mathrm{spaceindices}\=\mathrm{lowercase\_is}\right)$

$\left[\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right]$

$\mathrm{g\_}\left[~i\,~j\, \mathrm{matrix}\right]$

$g_{\phantom{}\phantom{i}\phantom{,}\phantom{j}}^{\phantom{}i,j}=\left[\begin{array}{ccc}\frac{-r+2m}{r}& 0& 0\\ 0& -\frac{1}{r^2}& 0\\ 0& 0& -\frac{1}{r^2{\sin \left(\mathrm{\theta }\right)}^2}\end{array}\right]$

All the tensor predefined in the Physics package, or defined using defining equations and the Define command, accept a new keyword, definition, that returns their definition. For example:

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

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

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

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

$\mathrm{PDEtools}:-\mathrm{declare}\left(A\left(X\right)\right)$

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

$F\left[\mathrm{mu}\,\mathrm{nu}\right] \= \mathrm{d\_}\left[\mathrm{mu}\right]\left(A\left[\mathrm{nu}\right]\left(X\right)\right) - \mathrm{d\_}\left[\mathrm{nu}\right]\left(A\left[\mathrm{mu}\right]\left(X\right)\right)$

$F_{\mathrm{\mu },\mathrm{\nu }}={\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-\left({\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)\right)$

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

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_{\mathrm{\mu }}\,{𝒟}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,X_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$