atlas[Metric] - declaration of a metric tensor 

Calling Sequence: 

    Metric(Id)
    Metric(Id=expr)
    Metric(expr) 

Parameters: 

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

Description: 

The Metric procedure allows one to declare a metric tensor. 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 atlas[ToBasis] procedure to see metric as a tensor. 

Examples: 

Example 1 

Abstract metric 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,xi=1);

 

(2.1.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.1.2)

 

Declare functions: 

>	Functions(f=f(x,y,z));

 

(2.1.3)

 

Declare coframe with 1-forms: (see atlas[Coframe]) 

>	Coframe(e[j],j=1..n);

 

(2.1.4)

 

Declare frame vectors (see atlas[Frame]): 

>	Frame(E[i]);

 

(2.1.5)

 

Declare indefinite metric tensor g: 

>	Metric(g);

 

(2.1.6)

 

Interior product of E[j] and g (see atlas[iota]): 

>	'iota[E[j]](g)'=iota[E[j]](g);

 

(2.1.7)

 

Linear property of metric tensor (see atlas[ToBasis]): 

>	g(X,Y)=g(ToBasis(X),Y);

 

(2.1.8)

 

Example 2 

Conformally flat metric of sphere -  

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,xi=1);

 

(2.2.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.2.2)

 

Declare functions: 

>	Functions(f=f(x,y,z));

 

(2.2.3)

 

Declare constant :
Constants(lambda); 

(2.2.4)

 

Declare coframe:
Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z)); 

(2.2.5)

 

Declare frame:
Frame(E[i]); 

(2.2.6)

 

Declare conformally flat metric of  :
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z))/(1+lambda*(x^2+y^2+z^2))^2); 

(2.2.7)

 

Simple calculations:
'iota[E[k]](g)'=iota[E[k]](g); 

(2.2.8)

 

>	'g(E[i],E[j])'=g(E[i],E[j]);

 

(2.2.9)

 

>	'g(E[j],X)'=g(E[j],ToBasis(X));

 

(2.2.10)

 

>	g=ToBasis(g);

 

(2.2.11)

 

Connection calculation (see atlas[Connection]): 

>	Connection(omega);

 

(2.2.12)

 

Now connection 1-forms are calculated: 

>	'omega[3,2]'=omega[3,2];

 

(2.2.13)

 

>	'iota[E[i]](omega[1,2])'=iota[E[i]](omega[1,2]);

 

(2.2.14)

 

Verify that vector is Killing one (see atlas[L]):  

>	Z:=x*E[2]-y*E[1]; 'L[Z](g)'=L[Z](g);

 

 

	(2.2.15)

 

Verify that there are no Killing vectors among frame ones: 

>	'L[E[k]](g)'=L[E[k]](g);

 

(2.2.16)

 

Curvature calculation (see atlas[Curvature]): 

>	Curvature(Omega);

 

(2.2.17)

 

Now curvature 2-forms are calculated: 

>	'Omega[1,3]'=Omega[1,3];

 

(2.2.18)

 

>	'iota[E[k]](Omega[2,1])'=iota[E[k]](Omega[2,1]);

 

(2.2.19)

 

Example 3 

Schwarzchild metric 

>	restart: with(atlas):

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.3.1)

 

Declare functions: 

>	Functions(f=f(x,y,z));

 

(2.3.2)

 

Declare constant  

>	Constants(r[g]);

 

(2.3.3)

 

Declare vectors:
Vectors(L1,N,M,K); 

(2.3.4)

 

Declare forms:
Forms(e[j]=1,l=1,n=1,m=1,k=1,x=1,y=1,z=1); 

(2.3.5)

 

Using alias for vectors and forms:
alias(L1=E[1],N=E[2],M=E[3],K=E[4],l=e[1],n=e[2],m=e[3],k=e[4]); 

(2.3.6)

 

Now 1-forms are: 

>	[seq(e[i],i=1..4)];

 

(2.3.7)

 

Declare coframe:
Coframe(l=1/2*((1-r[g]/r)*d(t)+d(r)),
       n=d(t)-1/(1-r[g]/r)*d(r),
       m=r/sqrt(2)*(d(theta)-I*sin(theta)*d(phi)),
       k=r/sqrt(2)*(d(theta)+I*sin(theta)*d(phi))); 

(2.3.8)

 

Declare frame:
Frame(E[j]); 

(2.3.9)

 

Declare metric:
Metric(g=e[1]&.e[2]+e[2]&.e[1]-e[3]&.e[4]-e[4]&.e[3] ); 

(2.3.10)

 

Simple calculations: 

>	'g(E[i],E[j])'=g(E[i],E[j]);

 

(2.3.11)

 

>	'g(E[j],X)'=g(E[j],ToBasis(X));

 

(2.3.12)

 

>	'iota[E[k]](g)'=iota[E[k]](g);

 

(2.3.13)

 

>	g=ToBasis(g);

 

(2.3.14)

 

Connection calculation: 

>	Connection(Gamma);

 

(2.3.15)

 

Now connection 1-forms are calculated: 

>	'Gamma[4,2]'=Gamma[4,2];

 

(2.3.16)

 

Simple calculation: 

>	'd(r)'=d(r);

 

(2.3.17)

 

>	'd(m)'=d(m);

 

(2.3.18)

 

>	'L[M](E[i])'=L[M](E[i]);

 

(2.3.19)

 

>

 

See Also:  

atlas, atlas[Frame], atlas[Coframe], atlas[Connection].