7.1. Definition

Now we introduce metrics on the vbundles.

DefMetric                Define a metric on a single vbundle
DefProductMetric        Define a metric on a product vbundle
UndefMetric                Undefine a metric
$Metrics                List of defined metrics
$ProductMetrics            List of defined product metrics
MetricQ                    Validate name of metric

Definition of a metric.

We define a metric metricg with negative determinant. The associated covariant derivative will be called CD and will be represented with a semicolon under IndexForm. All the needed associated tensors are defined (note that epsilon has a suffix metricg and the other tensors have a suffix CD). Because we use indices of M3 it is understood that metricg will be a metric on M3.

In[536]:=

DefMetric[-1, metricg[-a, -b], CD, {";", "▽"}, PrintAs->"g"]

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

** DefTensor: Defining antisymmetric tensor epsilonmetricg[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[537]:=

RiemannCD[-a, b, -b, -c]

Out[537]=

-R[▽] _ac^  

In[538]:=

%//InputForm

Out[538]//InputForm=

-RicciCD[-a, -c]

In[539]:=

RiemannCD[-a, -b, b, a]

Out[539]=

-R[▽] _^

In[540]:=

%//InputForm

Out[540]//InputForm=

-RicciScalarCD[]

In[541]:=

CD[-c][metricg[-a, -b]]

Out[541]=

0

In[542]:=

Cd[-c][metricg[-a, -b]]

Out[542]=

▽_c^ g_ab^  

In[543]:=

CD[-c][epsilonmetricg[a, b, c]]

Out[543]=

0

In[544]:=

CD[-a][EinsteinCD[b, c]]

Out[544]=

▽_a^ G[▽] _  ^bc

In[545]:=

CD[-a][EinsteinCD[a, b]]

Out[545]=

0

Its Weyl tensor is zero because the manifold is 3d:

In[546]:=

WeylCD[-a, -b, -c, -d]

Out[546]=

0

There is also the traceless part of the Ricci tensor:

In[547]:=

TFRicciCD[-a, -b]

Out[547]=

S[▽] _ab^  

In[548]:=

TFRicciCD[a, -a]

Out[548]=

0

Now xTensor` accepts index structures that do not correspond to the original definition, as long as they only involve defined metrics:

In[549]:=

SlotsOfTensor[T]

Out[549]=

{TangentM3, TangentM3, -TangentM3}

In[550]:=

Validate[T[-a, -b, -c]]

Out[550]=

T_abc^   

In[551]:=

DefTensor[X[a, b, C], {M3, S2}]

** DefTensor: Defining tensor X[a, b, C] .

In[552]:=

SlotsOfTensor[X]

Out[552]=

{TangentM3, TangentM3, TangentS2}

In[553]:=

Validate[X[a, b, -C]]

Validate :: error : Invalid character of index in tensor X

Out[553]=

ERROR[X_ (  C)^ab ]

In[554]:=

UndefTensor[X]

** UndefTensor: Undefined tensor X

Expression of the Christoffel symbols (with respect to PD) in terms of the metric:

In[555]:=

CD[-a][v[-b]]//ChangeCovD

Out[555]=

-Γ[▽] _ ( ab)^c   v_c^ + ∂_a^ v_b^

In[556]:=

%//ChristoffelToGradMetric

Out[556]=

∂_a^ v_b^ - 1/2 g_  ^cd v_c^ (∂_a^ g_bd^   + ∂_b^ g_ad^   - ∂_d^ g_ab^  )

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