gamma_ - Maple Help

Physics[Tetrads][gamma_] - represent and compute the Ricci rotation coefficients

Physics[Tetrads][lambda_] - represent a linear combination of the Ricci rotation coefficients - see reference

 Calling Sequence gamma_[a, b, c] gamma_[a, b, c, matrix] gamma_[keyword] lambda_[a, b, c] lambda_[a, b, c, matrix] lambda_[keyword]

Parameters

 _mu, nu_ - the spacetime indices related to a global system of references, these are names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves _a, b, c_ - the tetrad indices related to a local system of references, as names representing integer numbers the same way as the spacetime indices keyword - optional, it can be definition, matrix, nonzero

Description

 • The components of the gamma_[a, b, c] tensor are the Ricci rotation coefficients and the components of the lambda_[a, b, c] tensor are linear combinations of them, according to the definitions in the Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2 (definitions (98.9) and (98.10)). Thus,

${\mathrm{\gamma }}_{a,b,c}={𝔢}_{b}^{\mathrm{\mu }}{𝔢}_{c}^{\mathrm{\nu }}{▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right)$

 where ${▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right)$, represented in Maple by D_[nu](e_[a,mu]), is the covariant derivative of the tetrad, and

${\mathrm{\lambda }}_{a,b,c}={𝔢}_{b}^{\mathrm{\mu }}{𝔢}_{c}^{\mathrm{\nu }}\left({▿}_{\mathrm{\nu }}\left({𝔢}_{a,\mathrm{\mu }}\right)-\left({▿}_{\mathrm{\mu }}\left({𝔢}_{a,\mathrm{\nu }}\right)\right)\right)$

 From these definitions it follows that ${\mathrm{gamma}}_{a,b,c}$ is antisymmetric in its first pair of indices, and the covariant derivatives entering the definition of ${\mathrm{lambda}}_{a,b,c}$ can be replaced by non-covariant ones using the d_ operator. You can retrieve these definitions directly in the worksheet entering gamma_[definition] and lambda_[definition].
 • Both gamma_ and lambda_ accept the keywords accepted by the other tensors of the Physics package, these are definition, matrix and nonzero, that can be given with or without indices. If given with indices, the corresponding output takes their character (covariant or contravariant) into account.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$$\mathrm{with}\left(\mathrm{Tetrads}\right)$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_ah}\mathrm{letters to represent}\mathrm{tetrad}\mathrm{indices}$
 $\mathrm{Defined as tetrad tensors}\left(\mathrm{see ?Physics,tetrads}\right),{𝔢}_{a,\mathrm{\mu }},{\mathrm{\eta }}_{a,b},{\mathrm{\gamma }}_{a,b,c},{\mathrm{\lambda }}_{a,b,c}$
 $\mathrm{Defined as spacetime tensors representing the NP null vectors of the tetrad formalism}\left(\mathrm{see ?Physics,tetrads}\right),{l}_{\mathrm{\mu }},{n}_{\mathrm{\mu }},{m}_{\mathrm{\mu }},{\stackrel{&conjugate0;}{m}}_{\mathrm{\mu }}$
 ${}\mathrm{_______________________________________________________}$
 $\left[{\mathrm{IsTetrad}}{,}{\mathrm{NullTetrad}}{,}{\mathrm{OrthonormalTetrad}}{,}{\mathrm{PetrovType}}{,}{\mathrm{SegreType}}{,}{\mathrm{TransformTetrad}}{,}{\mathrm{WeylScalars}}{,}{\mathrm{e_}}{,}{\mathrm{eta_}}{,}{\mathrm{gamma_}}{,}{\mathrm{l_}}{,}{\mathrm{lambda_}}{,}{\mathrm{m_}}{,}{\mathrm{mb_}}{,}{\mathrm{n_}}\right]$ (1)
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (2)

Set the spacetime to be curved, for instance set the Schwarzschild metric in spherical coordinates (see g_):

 > ${\mathrm{g_}}_{\mathrm{sc}}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{space}\mathrm{indices}$
 ${}\mathrm{The Schwarzschild metric in coordinates}{}\left[r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right]$
 $\mathrm{Parameters:}\left[m\right]$
 $\mathrm{Signature:}\left(\mathrm{- - - +}\right)$
 ${}\mathrm{_______________________________________________________}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}\frac{r}{2{}m-r}& 0& 0& 0\\ 0& -{r}^{2}& 0& 0\\ 0& 0& -{r}^{2}{}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}& 0\\ 0& 0& 0& \frac{r-2{}m}{r}\end{array}\right]\right)$ (3)

The definition of the Ricci rotation coefficients and related ${\mathrm{\lambda }}_{a,b,c}$

 > ${\mathrm{gamma_}}_{\mathrm{definition}}$
 ${{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}{=}{{▿}}_{{\mathrm{\nu }}}{}\left({{𝔢}}_{{a}{,}{\mathrm{\mu }}}\right){}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}{}{{𝔢}}_{{c}\phantom{{\mathrm{\nu }}}}^{\phantom{{c}}{\mathrm{\nu }}}$ (4)
 > ${\mathrm{lambda_}}_{\mathrm{definition}}$
 ${{\mathrm{\lambda }}}_{{a}{,}{b}{,}{c}}{=}\left({{▿}}_{{\mathrm{\nu }}}{}\left({{𝔢}}_{{a}{,}{\mathrm{\mu }}}\right){-}{{▿}}_{{\mathrm{\mu }}}{}\left({{𝔢}}_{{a}{,}{\mathrm{\nu }}}\right)\right){}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}{}{{𝔢}}_{{c}\phantom{{\mathrm{\nu }}}}^{\phantom{{c}}{\mathrm{\nu }}}$ (5)

Isolating the covariant derivative that appears on the right-hand side, of the definitions of ${\mathrm{\gamma }}_{a,b,c}$, we have

 > ${\mathrm{e_}}_{b,\mathrm{α}}{\mathrm{e_}}_{c,\mathrm{β}}$
 ${{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}{}{{𝔢}}_{\phantom{{}}\phantom{{b}}{\mathrm{\alpha }}}^{\phantom{{}}{b}\phantom{{\mathrm{\alpha }}}}{}{{𝔢}}_{\phantom{{}}\phantom{{c}}{\mathrm{\beta }}}^{\phantom{{}}{c}\phantom{{\mathrm{\beta }}}}{=}{{▿}}_{{\mathrm{\nu }}}{}\left({{𝔢}}_{{a}{,}{\mathrm{\mu }}}\right){}{{𝔢}}_{{b}{,}{\mathrm{\alpha }}}{}{{𝔢}}_{{c}{,}{\mathrm{\beta }}}{}{{𝔢}}_{\phantom{{}}\phantom{{b}{,}{\mathrm{\mu }}}}^{\phantom{{}}{b}{,}{\mathrm{\mu }}}{}{{𝔢}}_{\phantom{{}}\phantom{{c}{,}{\mathrm{\nu }}}}^{\phantom{{}}{c}{,}{\mathrm{\nu }}}$ (6)
 > $\mathrm{Simplify}\left(\mathrm{rhs}\left(\right)\right)=\mathrm{lhs}\left(\right)$
 ${{▿}}_{{\mathrm{\beta }}}{}\left({{𝔢}}_{{a}{,}{\mathrm{\alpha }}}\right){=}{{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}{}{{𝔢}}_{\phantom{{}}\phantom{{b}}{\mathrm{\alpha }}}^{\phantom{{}}{b}\phantom{{\mathrm{\alpha }}}}{}{{𝔢}}_{\phantom{{}}\phantom{{c}}{\mathrm{\beta }}}^{\phantom{{}}{c}\phantom{{\mathrm{\beta }}}}$ (7)

The ${\mathrm{\gamma }}_{a,b,c}$ and ${\mathrm{\lambda }}_{a,b,c}$ tensors are related by formulas (98.10) and (98.11) of Landau's book referenced at the end, this relationship can be obtained as follows

 > $\mathrm{SubstituteTensor}\left(,\right)$
 ${{\mathrm{\lambda }}}_{{a}{,}{b}{,}{c}}{=}\left({-}{{\mathrm{\gamma }}}_{{a}{,}{d}{,}{e}}{}{{𝔢}}_{\phantom{{}}\phantom{{d}}{\mathrm{\nu }}}^{\phantom{{}}{d}\phantom{{\mathrm{\nu }}}}{}{{𝔢}}_{\phantom{{}}\phantom{{e}}{\mathrm{\mu }}}^{\phantom{{}}{e}\phantom{{\mathrm{\mu }}}}{+}{{𝔢}}_{\phantom{{}}\phantom{{f}}{\mathrm{\mu }}}^{\phantom{{}}{f}\phantom{{\mathrm{\mu }}}}{}{{𝔢}}_{\phantom{{}}\phantom{{g}}{\mathrm{\nu }}}^{\phantom{{}}{g}\phantom{{\mathrm{\nu }}}}{}{{\mathrm{\gamma }}}_{{a}{,}{f}{,}{g}}\right){}{{𝔢}}_{{b}\phantom{{\mathrm{\mu }}}}^{\phantom{{b}}{\mathrm{\mu }}}{}{{𝔢}}_{{c}\phantom{{\mathrm{\nu }}}}^{\phantom{{c}}{\mathrm{\nu }}}$ (8)

Simplifying the repeated indices,

 > $\mathrm{Simplify}\left(\right)$
 ${{\mathrm{\lambda }}}_{{a}{,}{b}{,}{c}}{=}{-}{{\mathrm{\gamma }}}_{{a}{,}{c}{,}{b}}{+}{{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}$ (9)

Substituting the free indices in this expression and adding as follows, then isolating ${\mathrm{gamma}}_{a,b,c}$, we obtain the inverse of this relationship

 > $+\mathrm{subs}\left(\left[a=b,b=c,c=a\right],\right)-\mathrm{subs}\left(\left[a=c,b=a,c=b\right],\right)$
 ${{\mathrm{\lambda }}}_{{a}{,}{b}{,}{c}}{-}{{\mathrm{\lambda }}}_{{b}{,}{a}{,}{c}}{-}{{\mathrm{\lambda }}}_{{c}{,}{a}{,}{b}}{=}{2}{}{{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}$ (10)
 > $\mathrm{isolate}\left(,{\mathrm{gamma_}}_{a,b,c}\right)$
 ${{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}{=}\frac{{{\mathrm{\lambda }}}_{{a}{,}{b}{,}{c}}}{{2}}{-}\frac{{{\mathrm{\lambda }}}_{{b}{,}{a}{,}{c}}}{{2}}{-}\frac{{{\mathrm{\lambda }}}_{{c}{,}{a}{,}{b}}}{{2}}$ (11)

The nonzero Ricci coefficients and related ${\mathrm{\lambda }}_{a,b,c}$ components for the Schwarzschild metric

 > ${\mathrm{gamma_}}_{\mathrm{nonzero}}$
 ${{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}{=}\left\{\left({1}{,}{2}{,}{2}\right){=}\frac{{I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({1}{,}{3}{,}{3}\right){=}\frac{{I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({1}{,}{4}{,}{4}\right){=}\frac{{I}{}{m}}{{{r}}^{{3}}{{2}}}{}\sqrt{{2}{}{m}{-}{r}}}{,}\left({2}{,}{1}{,}{2}\right){=}\frac{{-I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({2}{,}{3}{,}{3}\right){=}\frac{{\mathrm{cot}}{}\left({\mathrm{\theta }}\right)}{{r}}{,}\left({3}{,}{1}{,}{3}\right){=}\frac{{-I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({3}{,}{2}{,}{3}\right){=}{-}\frac{{\mathrm{cot}}{}\left({\mathrm{\theta }}\right)}{{r}}{,}\left({4}{,}{1}{,}{4}\right){=}\frac{{-I}{}{m}}{{{r}}^{{3}}{{2}}}{}\sqrt{{2}{}{m}{-}{r}}}\right\}$ (12)
 > ${\mathrm{lambda_}}_{\mathrm{nonzero}}$
 ${{\mathrm{\lambda }}}_{{a}{,}{b}{,}{c}}{=}\left\{\left({2}{,}{1}{,}{2}\right){=}\frac{{-I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({2}{,}{2}{,}{1}\right){=}\frac{{I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({3}{,}{1}{,}{3}\right){=}\frac{{-I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({3}{,}{2}{,}{3}\right){=}{-}\frac{{\mathrm{cot}}{}\left({\mathrm{\theta }}\right)}{{r}}{,}\left({3}{,}{3}{,}{1}\right){=}\frac{{I}{}\sqrt{{2}{}{m}{-}{r}}}{{{r}}^{{3}}{{2}}}}{,}\left({3}{,}{3}{,}{2}\right){=}\frac{{\mathrm{cot}}{}\left({\mathrm{\theta }}\right)}{{r}}{,}\left({4}{,}{1}{,}{4}\right){=}\frac{{-I}{}{m}}{{{r}}^{{3}}{{2}}}{}\sqrt{{2}{}{m}{-}{r}}}{,}\left({4}{,}{4}{,}{1}\right){=}\frac{{I}{}{m}}{{{r}}^{{3}}{{2}}}{}\sqrt{{2}{}{m}{-}{r}}}\right\}$ (13)

The value of ${\mathrm{\gamma }}_{1,1}^{1}$

 > ${\mathrm{gamma_}}_{\mathrm{~1},1,1}$
 ${0}$ (14)

The components of ${\mathrm{\gamma }}_{a,b}^{1}$

 > ${\mathrm{gamma_}}_{\mathrm{~1},a,b,\mathrm{matrix}}$
 ${{\mathrm{gamma_}}}_{{\mathrm{~1}}{,}{a}{,}{b}}{=}\left(\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& -\frac{I{}\sqrt{2{}m-r}}{{r}^{3/2}}& 0& 0\\ 0& 0& -\frac{I{}\sqrt{2{}m-r}}{{r}^{3/2}}& 0\\ 0& 0& 0& -\frac{I{}m}{{r}^{3/2}{}\sqrt{2{}m-r}}\end{array}\right]\right)$ (15)

The Riemann tensor in the tetrad system of coordinates can be expressed in terms of the Ricci rotation coefficients ${\mathrm{\gamma }}_{a,b,c}$ using formula (98.13) of Landau's book

 > ${\mathrm{Riemann}}_{a,b,c,d}={\mathrm{d_}}_{d}\left({\mathrm{gamma_}}_{a,b,c}\right)-{\mathrm{d_}}_{c}\left({\mathrm{gamma_}}_{a,b,d}\right)+{\mathrm{gamma_}}_{a,b,f}\left({\mathrm{gamma_}}_{f,c,d}-{\mathrm{gamma_}}_{f,d,c}\right)+{\mathrm{gamma_}}_{a,f,c}{\mathrm{gamma_}}_{f,b,d}-{\mathrm{gamma_}}_{a,f,d}{\mathrm{gamma_}}_{f,b,c}$
 ${{R}}_{{a}{,}{b}{,}{c}{,}{d}}{=}{{\partial }}_{{d}}{}\left({{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}\right){-}{{\partial }}_{{c}}{}\left({{\mathrm{\gamma }}}_{{a}{,}{b}{,}{d}}\right){+}\left({-}{{\mathrm{\gamma }}}_{{c}{,}{f}{,}{d}}{+}{{\mathrm{\gamma }}}_{{d}{,}{f}{,}{c}}\right){}{{\mathrm{\gamma }}}_{{a}{,}{b}\phantom{{f}}}^{\phantom{{a}}\phantom{{,}{b}}{f}}{-}{{\mathrm{\gamma }}}_{{a}{,}{f}{,}{c}}{}{{\mathrm{\gamma }}}_{{b}\phantom{{f}}{d}}^{\phantom{{b}}{f}\phantom{{d}}}{+}{{\mathrm{\gamma }}}_{{a}{,}{f}{,}{d}}{}{{\mathrm{\gamma }}}_{{b}\phantom{{f}}{c}}^{\phantom{{b}}{f}\phantom{{c}}}$ (16)

You can verify identities like this one by taking left-hand side minus right-hand side and computing an Array of components of this tensorial expression using TensorArray, then get its elements (components) using ArrayElems; recalling, ArrayElems returns a set with the elements of an Array, omitting all those elements equal to zero, so we expect an empty set here:

 > $\mathrm{ArrayElems}\left(\mathrm{TensorArray}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right),\mathrm{simplifier}=\mathrm{simplify}\right)\right)$
 ${\varnothing }$ (17)

Let's verify in the same way the expression of the covariant derivative in terms of the Ricci rotation coefficients

 > 
 ${{▿}}_{{\mathrm{\beta }}}{}\left({{𝔢}}_{{a}{,}{\mathrm{\alpha }}}\right){=}{{\mathrm{\gamma }}}_{{a}{,}{b}{,}{c}}{}{{𝔢}}_{\phantom{{}}\phantom{{b}}{\mathrm{\alpha }}}^{\phantom{{}}{b}\phantom{{\mathrm{\alpha }}}}{}{{𝔢}}_{\phantom{{}}\phantom{{c}}{\mathrm{\beta }}}^{\phantom{{}}{c}\phantom{{\mathrm{\beta }}}}$ (18)
 > $\mathrm{ArrayElems}\left(\mathrm{TensorArray}\left(\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\right),\mathrm{simplifier}=\mathrm{simplify}\right)\right)$
 ${\varnothing }$ (19)