SageNP: Newman-Penrose calculations for SageMath.

This work is funded by TUBITAK 1001 Program, Grant Number 123R114.

The class SageNP includes functions for some calculations defined in the Newman-Penrose formalism. The code is based on SageManifolds.

Coded by:

• Tolga Birkandan (Corr.: birkandant@itu.edu.tr)

• Onur Arman

• Emir Baysazan

• Selinay Sude Binici

• Pelin Ozturk

• Special thanks to Eric Gourgoulhon

FILES:

• SageNP.py: Main file to import in SageMath.

• SageNP_Tutorial.ipynb: Tutorial (ipynb file) - Definitions and calculations for the Schwarzschild (with covariant null-tetrad vectors) and Reissner-Nordstrom (with contravariant null-tetrad vectors) spacetimes.

• SageNP_Tutorial.pdf: Tutorial (PDF file)

REFERENCE:

Main reference for all definitions and calculations:

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, "Exact Solutions of Einstein’s Field Equations", 2nd ed. Cambridge: Cambridge University Press, 2003.

BASIC DEFINITIONS AND NOTATION:

• We will use the Metric signature: (- + + +)

• For the null-tetrad vector names, the ref. book uses (k,l,m,mbar). However, in the code we will use (l,n,m,mbar) like the rest of the literature. Therefore one should set k->l, l->n in the ref. book

• Products of the vectors are given by: ln = -1, mmbar = 1, all others zero.

• The metric is found using the covariant null-tetrad vectors as:

  g = -2ln + 2mmbar

  and,

  g = [[0 1 0 0], [1 0 0 0], [0 0 0 -1], [0 0 -1 0]]

• Please check the reference book for the details and further definitions.

INSTRUCTIONS WITH EXAMPLES:

• Import the class:

  from SageNP import NewmanPenrose

• Define your manifold:

  MyManifold = Manifold(4 , 'MyManifold', r'\mathcal{Man}')

• Define your coordinates:

  MyCoordinates.<t,r,th,ph> = MyManifold.chart(r't r th:\theta ph:\phi')

• Define the metric functions (if needed):

  var('M') Delta=r^2-2Mr

• Enter null tetrad elements:

  lvec=[1,-(r^2)/Delta,0,0]

  nvec=[Delta/(2*r^2),1/2,0,0]

  mvec=[0,0,(-r/sqrt(2)),(-I*r/sqrt(2))*sin(th)]

  mbarvec=[0,0,(-r/sqrt(2)),(I*r/sqrt(2))*sin(th)]

  • Here, the element ordering is the same as the coordinate ordering. (The first element is the t element, the second is the r element, etc.)

• Define an object of the class:

  schw=SageNP(MyManifold,MyCoordinates,lvec,nvec,mvec,mbarvec,'covariant')

  • Here, our null-tetrad vectors lvec, nvec, mvec and mbarvec are covariant. Thus we used the keyword 'covariant'.

  • If they were contravariant, then we should use the keyword 'contravariant'.

• Once the object is defined, the code calculates the metric and displays it on the screen. It is recommended that you check your metric.

FUNCTIONS:

• All page and equation numbers belong to the reference book.

• test_nulltetrad(): Checks the products of the vectors ln = -1, mmbar = 1, all others zero.

• Spin coefficients (Page 75-76, Eq.(7.2)):

  • calculate_spincoefficients(): Calculates the spin coefficients.

  • show_spincoefficients(): Displays the spin coefficients

  • All spin coefficients are available under their names:

    kappaNP, kappabarNP, tauNP, taubarNP, sigmaNP, sigmabarNP, rhoNP, rhobarNP, piNP, pibarNP, nuNP, nubarNP, muNP, mubarNP, lambdaNP, lambdabarNP, epsilonNP, epsilonbarNP, gammaNP, gammabarNP, betaNP, betabarNP, alphaNP, alphabarNP

• Directional derivatives (Page 43, Eq.(3.82)):

  • DlNP(X): Given X, calculates the D derivative (l direction).

  • DeltanNP(X): Given X, calculates the Delta derivative (n direction)

  • deltamNP(X): Given X, calculates the delta derivative (m direction)

  • deltambarNP(X): Given X, calculates the deltabar derivative (mbar direction)

• Commutators (Page 77, Eq.(7.6)):

  • The right-hand sides of the commutation relations are calculated.

  • Deltan_Dl_commNP(X): Given X, calculates the [Delta,D] commutator.

  • deltam_Dl_commNP(X): Given X, calculates the [delta,D] commutator.

  • deltam_Deltan_commNP(X): Given X, calculates the [delta,Delta] commutator.

  • deltambar_deltam_commNP(X): Given X, calculates the [deltabar,delta] commutator.

• Weyl tensor components (Page 38, Eq.(3.59)):

  • calculate_Weyl(): Calculates the Weyl tensor components.

  • show_Weyl(): Displays the Weyl tensor components.

  • All Weyl tensor components are available under their names:

    Psi0NP, Psi1NP, Psi2NP, Psi3NP, Psi4NP

• Ricci components (Page 78, Eq.(7.10-7.15)):

  • calculate_Ricci(): Calculates the Ricci tensor components.

  • show_Ricci(): Displays the Ricci tensor components.

  • All Ricci tensor components are available under their names:

    Phi00NP, Phi01NP, Phi10NP, Phi02NP, Phi20NP, Phi11NP, Phi12NP, Phi21NP, Phi22NP, LambdaNP

• Ricci (Newman-Penrose) equations (Page 79, Eq.(7.21)):

  • All Newman-Penrose equations are defined as 0 = -(left hand side)+(right hand side) of the equations.

  • calculate_NPeq(): Calculates the Newman-Penrose equations

  • show_NPeq(): Displays the Newman-Penrose equations

  • All Newman-Penrose equations are available under their names in the order they are given in the reference:

    NPeq1, NPeq2, NPeq3, NPeq4, NPeq5, NPeq6, NPeq7, NPeq8, NPeq9, NPeq10, NPeq11, NPeq12, NPeq13, NPeq14, NPeq15, NPeq16, NPeq17, NPeq18

• Bianchi identities (Page 81, Eq.(7.32)):

  • All Bianchi identities are defined as 0 = -(left hand side)+(right hand side) of the equations.

  • calculate_Bianchi(): Calculates the Bianchi identities

  • show_Bianchi(): Displays the Bianchi identities

  • All Bianchi identities are available under their names in the order they are given in the reference:

    BI1, BI2, BI3, BI4, BI5, BI6, BI7, BI8, BI9, BI10, BI11

• Petrov invariants I, J, K, L, N (Kramer p.121, 9.6; p.54, 4.19):

  (also check diagram Fig. 9.1 on p. 122)

  • calculate_PetrovinvINP(): Calculates the Petrov invariant I

  • calculate_PetrovinvJNP(): Calculates the Petrov invariant J

  • calculate_PetrovinvKNP(): Calculates the Petrov invariant K

  • calculate_PetrovinvLNP(): Calculates the Petrov invariant L

  • calculate_PetrovinvNNP(): Calculates the Petrov invariant N

  • All Petrov invariants are available under their names:

    PetrovinvINP, PetrovinvJNP, PetrovinvKNP, PetrovinvLNP, PetrovinvNNP

• Petrov type of the spacetime:

  • Petrov_frominvariants(): Calculates the Petrov type using I, J, K, L, N.

  • Petrov_fromWeyl(): Calculates the Petrov type using the Weyl components

• Massive Klein-Gordon equation:

  kleingordon(Phi,M2): Calculates the Klein-Gordon equation for a massive scalar field where Phi is a scalar field on the manifold and M2 is the mass of the scalar field. (Ref.: G. Silva-Ortigoza, Rev. Mex. Fis. 4, 543 (1996))

  • The result is available under the name: kgNP

• Massive Dirac equation:

  dirac(f1,f2,g1,g2,M2): Calculates the Dirac equation for a massive spinor field where f1,f2,g1,g2 are the components of the spinor field (defined as scalar fields on the manifold) and M2 is the mass of the spinor field. (Ref.: S. Chandrasekhar, "Mathematical Theory of Black Holes", Oxford Univ. Press, New York (1983), p.544.)

  • The result is available under the name: diracNP

• SL(2,C) Transformations: SL(2,C) transformations as defined in Carmeli and Kaye, Annals of Physics 99, 188 (1976).

  type_A_transformation(z): Calculates the Type A transformations where z is a complex variable.

  show_type_A_transformation(): Shows the results of the Type A transformations.

  type_B_transformation(z): Calculates the Type B transformations where z is a complex variable.

  show_type_B_transformation(): Shows the results of the Type B transformations.

  type_C_transformation(z): Calculates the Type C transformations where z is a complex variable.

  show_type_C_transformation(): Shows the results of the Type C transformations.

  The results are available under their names: lNP_trA, nNP_trA, mNP_trA, mbarNP_trA, kappaNP_trA, rhoNP_trA, etc., Psi0NP_trA, Psi1NP_trA, etc., Phi00NP_trA, Phi01NP_trA, etc. and the same notation for lNP_trB, nNP_trB, etc. and lNP_trC, nNP_trC, etc. for other types.

• calculate_allNP(): Runs the following functions:

  • calculate_spincoefficients()

  • calculate_Weyl()

  • calculate_Ricci()

  • calculate_NPeq()

  • calculate_Bianchi()

  • Petrov_frominvariants()

  • Petrov_fromWeyl()

• show_allNP(): Runs the following functions:

  • show_spincoefficients()

  • show_Weyl()

  • show_Ricci()

  • show_NPeq()

  • show_Bianchi()

  • Petrov_frominvariants()

  • Petrov_fromWeyl()