4.3. The determinant of the metric and the ε tensor

AbsDet                            Absolute value of the determinant of the metric in any given basis
epsilonToetaDown                  Transform an ε tensor into an etaUp tensor
epsilonToetaUp                    Transform an ε tensor into an etaDown tensor
etaDownToepsilon                Transform an etaDown tensor into an ε tensor
etaUpToepsilon                    Transform an etaUp tensor into an ε tensor
$epsilonSign                    Global sign of the ε tensor

Determinat of the metric; relation between η and ε tensors

The determinant of the metric is a weight 2 density

In[252]:=

DefMetric[-1, metric[-a, -b], cd, {"|", "D"}, PrintAs→ "g"]

** DefTensor: Defining symmetric metric tensor metric[-a, -b] .

** DefTensor: Defining antisymmetric tensor epsilonmetric[a, b, c] .

** DefCovD: Defining covariant derivative cd[-a] .

** DefTensor: Defining vanishing torsion tensor Torsioncd[a, -b, -c] .

** DefTensor: Defining symmetric Christoffel tensor Christoffelcd[a, -b, -c] .

** DefTensor: Defining Riemann tensor Riemanncd[-a, -b, -c, -d] .

** DefTensor: Defining symmetric Ricci tensor Riccicd[-a, -b] .

** DefCovD:  Contractions of Riemann automatically replaced by Ricci.

** DefTensor: Defining Ricci scalar RicciScalarcd[] .

** DefCovD:  Contractions of Ricci automatically replaced by RicciScalar.

** DefTensor: Defining symmetric Einstein tensor Einsteincd[-a, -b] .

** DefTensor: Defining vanishing Weyl tensor Weylcd[-a, -b, -c, -d] .

** DefTensor: Defining symmetric TFRicci tensor TFRiccicd[-a, -b] .

   Rules {1, 2} have been declared as DownValues for TFRiccicd .

** DefCovD:  Computing RiemannToWeylRules for dim 3

** DefCovD:  Computing RicciToTFRicci for dim 3

** DefCovD:  Computing RicciToEinsteinRules for dim 3

In[253]:=

AbsDet[metric, polar][]

** DefTensor: Defining weight +2 density AbsDetmetricpolar[] . Absolute value of determinant.

Out[253]=

((| Overscript[g, Overscript[~, ~]] |) _)^

Once we have a metric, we can build new tensors (with zero weight) from the etaUp and etaDown tensors. These new ε tensors depend only on the metric

In[254]:=

epsilonmetric[a, b, c]

Out[254]=

εg_   ^abc

In[255]:=

WeightOf[%]

Out[255]=

0

In[256]:=

etaUp[cartesian][a, b, c]/.etaUpToepsilon[cartesian, metric]

** DefTensor: Defining weight +2 density AbsDetmetriccartesian[] . Absolute value of determinant.

Out[256]=

-(((| Overscript[g, Overscript[~, ~]] |) _)^)^(1/2) εg_   ^abc

In[257]:=

%/.epsilonToetaDown[metric, cartesian]

Out[257]=

-((| Overscript[g, Overscript[~, ~]] |) _)^ Underscript[η, ~] _   ^abc

In[258]:=

%%/.epsilonToetaUp[metric, cartesian]

Out[258]=

Overscript[η, ~] _   ^abc

The global sign for ε tensors is controlled by the global variable $epsilonSign

In[259]:=

? $epsilonSign

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