7.5. Curvature tensors of the metric

When we define a metric, a number of curvature tensors are defined. Currently they are: Riemann, Ricci, RicciScalar, Einstein, TFRici, Weyl. Note that they are all associated to the Levi-Civita connection of the metric, and not directly to the metric. There are a number of functions to change among them.

First we have some automatic contractions. These can be prevented with the option CurvatureRelations of DefCovD. By default, contractions of the Riemann tensor are automatically replaced by the Ricci tensor:

In[596]:=

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

Out[596]=

-R[▽] _bc^  

In[597]:=

%//InputForm

Out[597]//InputForm=

-RicciCD[-b, -c]

And contractions of the Ricci tensor are replaced by the Ricci scalar:

In[598]:=

metricg[c, d] RicciCD[-d, -c]

Out[598]=

R[▽] _^

In[599]:=

%//InputForm

Out[599]//InputForm=

RicciScalarCD[]

In[600]:=

RicciCD[a, -a]

Out[600]=

R[▽] _^

Then we can change among the tensors:

In[601]:=

EinsteinCD[-a, -b]//EinsteinToRicci

Out[601]=

R[▽] _ab^   - 1/2 g_ba^   R[▽] _^

In[602]:=

%//RicciToEinstein

Out[602]=

G[▽] _ab^  

In a 3d manifold the Weyl tensor is zero:

In[603]:=

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

Out[603]=

0

In[604]:=

RiemannCD[-a, -b, -c, -d]//RiemannToWeyl

Out[604]=

We can also change between Ricci and TFRicci:

In[605]:=

RicciCD[-a, -b]//RicciToTFRicci

Out[605]=

1/3 g_ab^   R[▽] _^ + S[▽] _ab^  

In[606]:=

%//TFRicciToRicci

Out[606]=

R[▽] _ab^   + 1/3 g_ab^   R[▽] _^ - 1/3 g_ba^   R[▽] _^

RiemannToWeyl            Expand Riemann tensors into Weyl, Ricci and RicciScalar tensors
WeylToRiemann            Expand Weyl tensors into Riemann, Ricci and RicciScalar tensors
RicciToEinstein            Expand Ricci tensors into Einstein and RicciScalar tensors
EinsteinToRicci            Expand Einstein tensors into Ricci and RicciScalar tensors
RicciToTFRicci            Expand Ricci tensors into TFRicci and RicciScalar tensors
TFRicciToRicci            Expand TFRicci tensors into Ricci and RicciScalar tensors

Relations among curvature objects.

As we said, we can work with connections which are compatible with a given metric field but have torsion too.

We define a new connection associated to our metric field, but this time with torsion. The Riemann tensor is antisymmetric in both pairs, but now those pairs cannot be exchanged. Hence the Ricci tensor is not symmetric. Currently Weyl is not even defined:

In[607]:=

DefCovD[CDT[-a], {"#", "D"}, Torsion→True, FromMetric→metricg]

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

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

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

** DefTensor: Defining Riemann tensor RiemannCDT[-a, -b, -c, -d] . Antisymmetric pairs cannot be exchanged.

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

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

** DefTensor: Defining Ricci scalar RicciScalarCDT[] .

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

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

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

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

** DefCovD:  Computing RicciToTFRicci for dim 3

** DefCovD:  Computing RicciToEinsteinRules for dim 3

The difference of covariant derivatives

In[608]:=

CDT[-a][v[b]] - CD[-a][v[b]]

Out[608]=

-(▽_a^ v_ ^b) + D_a^ v_ ^b

is given by a Christoffel tensor

In[609]:=

ChangeCovD[%, CDT, CD]

** DefTensor: Defining tensor ChristoffelCDCDT[a, -b, -c] .

Out[609]=

-Γ[▽, D] _ ( ac)^b   v_ ^c

which can be expressed uniquely in terms of torsion and metric fields:

In[610]:=

%//ChristoffelToGradMetric

Out[610]=

-1/2 g_  ^bd (T[D] _acd^    + T[D] _cad^    - T[D] _dac^   ) v_ ^c

In[611]:=

UndefCovD[CDT]

** UndefTensor: Undefined tensor ChristoffelCDCDT

** UndefTensor: Undefined non-symmetric Christoffel tensor ChristoffelCDT

** UndefTensor: Undefined non-symmetric Einstein tensor EinsteinCDT

** UndefTensor: Undefined non-symmetric Ricci tensor RicciCDT

** UndefTensor: Undefined Ricci scalar RicciScalarCDT

** UndefTensor: Undefined Riemann tensor RiemannCDT

** UndefTensor: Undefined non-symmetric TFRicci tensor TFRicciCDT

** UndefTensor: Undefined torsion tensor TorsionCDT

** UndefCovD: Undefined covariant derivative CDT

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