Atlas Paclet Symbol

Metric - declaration of a metric tensor

Metric[id]

Metric[id→expr]
allows one to declare a metric tensor

MORE INFORMATION

id - variable - metric identifier expr - expression - metric declaration.

The Metric procedure can be used in two ways:

- Metric[id] - sets id as a metric tensor identifier (the metric tensor components is indefinite).

- Metric[id→expr] - sets id as a metric tensor identifier and expr as a metric tensor.

In the Atlas package the metric declared is a procedure. Use ToBasis procedure to see metric as a tensor.

Declare forms:

Declare vectors:

Declare functions:

Declare coframe with 1-forms: e₁, e₂, ..e_n (see Coframe)

In[5]:=

Out[5]=

Declare frame vectors (see Frame):

In[6]:=

Out[6]=

Declare indefinite metric tensor g:

In[7]:=

Out[7]=

Interior product of

[j] and g (see iota):

In[8]:=

Out[8]=

Linear property of metric tensor (see ToBasis):

In[9]:=

Out[9]=

Conformally flat metric of sphere

(1)

Declare forms:

Declare vectors:

Declare functions:

Declare constant :

Declare coframe:

Declare frame:

Declare conformally flat metric of S³ :

Simple calculations:

Connection calculation (see Connection):

In[13]:=

Out[13]=

Now connection 1-forms

_{i, j} are calculated:

Verify that vector Z=xE₂-yE₁ is Killing one (see L):

In[16]:=

Out[16]=

Verify that there are no Killing vectors among frame ones:

In[17]:=

Out[17]=

Curvature calculation (see Curvature):

In[18]:=

Out[18]=

Now curvature 2-forms _{i, j} are calculated:

Schwarzchild metric (1)

Declare vectors:

Declare functions:

Declare constant r_g:

Declare forms:

Declare frame:

Declare coframe:

Declare metric:

Simple calculations:

Connection calculation:

In[13]:=

Out[13]=

Now connection 1-forms are calculated:

In[14]:=

Out[14]//MatrixForm=

Simple calculation:

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Metric - declaration of a metric tensor

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