E Examples of the Use of the Package NOVAMANIFOLDA

441

doricci

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Now I calculate the Ricci tensor 11 non-zero

22 non-zero

33 non-zero

44 non-zero

I have finished the calculation

The tensor ricten[[a,b]] giving the Ricci tensor is ready for storing on hard disk

Store it in your preferred directory with the name you choose

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{Null}

MatrixForm[ricten]

	3λ2	3λ2	0	0
	4
		0
	0	0	0	0
	0		3λ2	0
		4
			4	3λ	2
	0	0	0
				4

The above expression of the Ricci tensor is provided in the flat indices. Indeed the Ricci tensor is the trace of the Riemann tensor calculated in the flat basis as components of the curvature two-form along the vielbein basis.

Calculation of the (Pseudo-)Riemannian Geometry of a Kasner Metric in Vielbein Formalism

As an example of calculation of the pseudo-Riemannian geometry in vielbein formalism we consider in this section the case of a Kasner cosmological metric

We initialize this general package by typing contorgen

contorgen

Give me the dimension of your space Your space has dimension n = 4

Now I stop and you give me three vectors of dimension 4 vector fform = vector of 1-form vielbeins

vector coordi = vector of coordinates vector diffe = vector of differentials

Then resume the calculation typing: contorgenresume If you already have the contorsion type

spinpackgen {Null}

Next we supply the information required by the computer, namely, vielbein, coordinates and coordinate differentials

e1**e2s1[t] s1[t]

e1**e3s2[t] s2[t]

e1**e4s3[t] s3[t]

442	10 Conclusion of Volume 2

fform = {dt, s1[t]dx1, s2[t]dx2, s3[t]dx3}; coordi = {t, x1, x2, x3};

diffe = {dt, dx1, dx2, dx3};

We proceed the calculation evaluating the external differential of the vielbein

contorgenresume

I calculate the exterior differentials of the vielbeins

——————–

I finished!

Next I calculate the inverse vielbein Done!

I resume the calculation of the contorsion I calculate the contorsion c[i,j,k] for

i = 1 i = 2 i = 3 i = 4

I have finished!

The result, encoded in a vector dE[i] is the following: dE[1] = 0

dE[2] =

dE[3] =

dE[4] =

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The contorsion is encoded in tensor named contens

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Now you can begin the calculation of the spin connection by typing spinpackgen {Null}

We initialize the calculation of the spin connection

spinpackgen

I start

now give me the contorsion tensor by writing cont = ?

and give me the signature a vector of +/- 1 by writing signat = ?

then resume the calculation by typing spinresumegen {Null}

Requested by the computer we indicate the file containing the contorsion and we specify the signature

cont = contens; signat = {−1, 1, 1, 1};

We conclude the calculation of the spin connection

E Examples of the Use of the Package NOVAMANIFOLDA			443
spinresumegen
I resume the calculation of the spin connection
—————–
the result is
ω[12] =	e2s1[t]
	s1[t]
ω[13] =	e3s2[t]
	s2[t]
ω[14] =	e4s3[t]
	s3[t]

ω[23] = 0 ω[24] = 0 ω[34] = 0 Task finished

The result is encoded in a tensor omega[i,j]

Its components are encoded in a tensor ometen[i,j,m]

If you want the curvature, type curvapack for quasi-homogeneous manifolds

Otherwise, type curvapackgen for general manifolds {Null}

Next we calculate the curvature two form and the Ricci tensor

curvapackgen

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I calculate the Riemann tensor I tell you my steps:

a = 1 b = 1 b = 2 b = 3 b = 4 a = 2 b = 1 b = 2 b = 3 b = 4 a = 3 b = 1 b = 2 b = 3 b = 4 a = 4 b = 1 b = 2 b = 3 b = 4

Finished

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