tensor(deprecated)/connexF - Help

tensor

 connexF
 compute the covariant components of the connection coefficients in a rigid frame
 RiemannF
 compute the covariant Riemann curvature tensor in a rigid frame

 Calling Sequence connexF(coord, g, h) RiemannF(coord, ginv, hinv, gamma)

Parameters

 coord - list of names of coordinate variables g - constant covariant metric in the rigid frame; it must have character [-1,-1] and indexing function symmetric in its component array. ginv - constant contravariant metric in the rigid frame; it must have character [1,1] and indexing function symmetric in its component array h - covariant tetrad that transforms the metric in the natural basis to the one specified by the provided constant g (or ginv); h has the character [1,-1]. hinv - corresponding contravariant tetrad, in other words, the inverse of h; it has the character [-1,1] gamma - covariant components of the connection coefficients in the rigid frame and this can be computed using connexF(..)

Description

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

 • These two functions computes respectively the covariant connection coefficients and the covariant Riemann tensor in a rigid frame which may be null or orthonormal, given the constant metric and its inverse, and the tetrad (and its inverse) that transforms the natural-basis metric to the given constant metric.
 • These two routines can be used on spaces of arbitrary dimensions and signature.
 • For a diagonal constant metric, the tensor package function tensor[frame] can be used to obtain the required tetrad (and its inverse).
 • Simplification:
 – connexF() has one simplifier, tensor/connexF/simp, for simplifying algebraic expressions during the course of its computation.
 – RiemannF() has also only one simplifier, tensor/RiemannF/simp, for simplifying algebraic expressions.
 – Both simplifiers are initialized to tensor/simp, but it is recommended that each be customized to suit the needs of a particular problem.
 • These two functions are part of the tensor package, and can be used in the form connexF(..), RiemannF(..) only after performing the command with(tensor), or with(tensor, connexF), etc.  The functions can always be accessed in the long form tensor[connexF], and tensor[RiemannF].

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

Entering the coordinates and covariant tetrad of the Plane wave metric

 > $\mathrm{coord}≔\left[u,x,y,v\right]:$
 > $\mathrm{h_compts}≔\mathrm{array}\left(1..4,1..4,\left[\left(2,3\right)=0,\left(2,4\right)=0,\left(3,1\right)=0,\left(3,2\right)=0,\left(3,3\right)=-1,\left(1,2\right)=0,\left(4,1\right)=0,\left(2,1\right)=1,\left(4,2\right)=-1,\left(4,4\right)=0,\left(1,1\right)=a{x}^{2}+byx+c{y}^{2},\left(1,3\right)=0,\left(1,4\right)=1,\left(4,3\right)=0,\left(3,4\right)=0,\left(2,2\right)=0\right]\right):$
 > $h≔\mathrm{create}\left(\left[1,-1\right],\mathrm{op}\left(\mathrm{h_compts}\right)\right)$
 ${h}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cccc}{a}{}{{x}}^{{2}}{+}{b}{}{x}{}{y}{+}{c}{}{{y}}^{{2}}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {-}{1}& {0}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{-}{1}\right]\right]\right)$ (1)
 > $\mathrm{hinv}≔\mathrm{invert}\left(h,\mathrm{DETh}\right)$
 ${\mathrm{hinv}}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cccc}{0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {-}{a}{}{{x}}^{{2}}{-}{b}{}{x}{}{y}{-}{c}{}{{y}}^{{2}}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {-}{1}& {0}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-}{1}{,}{1}\right]\right]\right)$ (2)

Now specify the constant metric in the rigid frame.

 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},1..4,1..4\right):$
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}4\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{from}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}4\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{g_compts}}_{i,j}≔0\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}:$${\mathrm{g_compts}}_{1,2}≔1:$${\mathrm{g_compts}}_{3,4}≔-1:$$g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{op}\left(\mathrm{g_compts}\right)\right)$
 ${g}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {-}{1}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-}{1}{,}{-}{1}\right]\right]\right)$ (3)
 > $\mathrm{ginv}≔\mathrm{invert}\left(g,\mathrm{DETg}\right)$
 ${\mathrm{ginv}}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {-}{1}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{1}\right]\right]\right)$ (4)
 > $\mathrm{Γ}≔\mathrm{connexF}\left(\mathrm{coord},g,h\right):$

Now use act( 'display', .. ) to view the non-vanishing components.

 > $\mathrm{act}\left('\mathrm{display}',\mathrm{Γ}\right)$
 NON-ZERO INDEPENDENT COMPONENTS :"[2, 3, 2] ="
 "[3, 2, 2] ="
 "[2, 4, 2] ="
 "[4, 2, 2] ="
 CHARACTER :
 ${\mathrm{_________________________________________________}}$
 ${b}{}{x}{+}{2}{}{c}{}{y}$
 ${-}{b}{}{x}{-}{2}{}{c}{}{y}$
 ${2}{}{a}{}{x}{+}{b}{}{y}$
 ${-}{2}{}{a}{}{x}{-}{b}{}{y}$
 $\left[{-}{1}{,}{-}{1}{,}{-}{1}\right]$
 ${\mathrm{_________________________________________________}}$ (5)
 > $\mathrm{Rm}≔\mathrm{RiemannF}\left(\mathrm{coord},\mathrm{ginv},\mathrm{hinv},\mathrm{Γ}\right):$
 > $\mathrm{act}\left('\mathrm{display}',\mathrm{Rm}\right)$
 NON-ZERO INDEPENDENT COMPONENTS :"[2, 4, 2, 4] ="
 "[2, 3, 2, 3] ="
 "[2, 3, 2, 4] ="
 CHARACTER :
 INDEXING FUNCTION :
 ${\mathrm{_________________________________________________}}$
 ${-}{2}{}{a}$
 ${-}{2}{}{c}$
 ${-}{b}$
 $\left[{-}{1}{,}{-}{1}{,}{-}{1}{,}{-}{1}\right]$
 ${\mathrm{cov_riemann}}$
 ${\mathrm{_________________________________________________}}$ (6)