7.7. Flat and Cartesian metrics

A flat metric is one without curvature. We define its Christoffel and epsilon objects, and then zero curvature tensors.

We can define a flat metric using the option FlatMetric:

In[627]:=

DefMetric[-1, gflat[-a, -b], Cdflat, {"%", "∂"}, FlatMetric→True]

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

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

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

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

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

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

** DefTensor: Defining vanishing Ricci tensor RicciCdflat[-a, -b] .

** DefTensor: Defining vanishing Ricci scalar RicciScalarCdflat[] .

** DefTensor: Defining vanishing Einstein tensor EinsteinCdflat[-a, -b] .

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

** DefTensor: Defining vanishing TFRicci tensor TFRicciCdflat[-a, -b] .

In[628]:=

UndefMetric[gflat]

** UndefTensor: Undefined antisymmetric tensor epsilongflat

** UndefTensor: Undefined symmetric Christoffel tensor ChristoffelCdflat

** UndefTensor: Undefined vanishing Einstein tensor EinsteinCdflat

** UndefTensor: Undefined vanishing Ricci tensor RicciCdflat

** UndefTensor: Undefined vanishing Ricci scalar RicciScalarCdflat

** UndefTensor: Undefined vanishing Riemann tensor RiemannCdflat

** UndefTensor: Undefined vanishing TFRicci tensor TFRicciCdflat

** UndefTensor: Undefined vanishing torsion tensor TorsionCdflat

** UndefTensor: Undefined vanishing Weyl tensor WeylCdflat

** UndefCovD: Undefined covariant derivative Cdflat

** UndefTensor: Undefined symmetric metric tensor gflat

A delicate issue is that of a Cartesian metric. The concept of a flat metric is covariantly defined and therefore perfectly fits in the current structure of xTensor`. However we say that a metric in a given basis of vectors is Cartesian if all metric components in that basis are constant (not necessarily of unit modulus). For a flat metric coordinate systems of that kind always exist. We introduce the possibility of associating PD as the covariant derivative of a flat metric. That means that PD is the partial derivative of one of those coordinate systems with respect to that metric and hence PD is not general anymore. This is an ugly trick for a problem which is only properly solved in the twin package xCoba`.

We redefine the flat metric. In this case not even the Christoffel symbol is defined. By definition it is zero.

In[629]:=

DefMetric[-1, gflat[-a, -b], PD, {",", "∂"}, FlatMetric→True]

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

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

In[630]:=

PD[-a][gflat[-b, -c]]

Out[630]=

0

In[631]:=

UndefMetric[gflat]

** UndefTensor: Undefined antisymmetric tensor epsilongflat

** UndefTensor: Undefined symmetric metric tensor gflat

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