# SageNP: Newman-Penrose calculations for SageMath.
The class **SageNP** includes functions for some calculations defined in the Newman-Penrose formalism. The code is based on SageManifolds.
# Coded by:
- [Tolga Birkandan](https://web.itu.edu.tr/birkandant/) (Corr.: birkandant@itu.edu.tr)
- [Onur Arman](https://www.linkedin.com/in/onur-arman-709478337/)
- [Emir Baysazan](https://scholar.google.com/citations?user=kq9ia_oAAAAJ&hl=en)
- [Selinay Sude Binici](https://scholar.google.com/citations?user=OjiOpogAAAAJ&hl=tr)
- [Pelin Ozturk] (https://www.linkedin.com/in/pelin-%C3%B6zt%C3%BCrk-3904572b2/?utm_source=share&utm_campaign=share_via&utm_content=profile&utm_medium=ios_app)
- **Special thanks to [Eric Gourgoulhon](https://luth.obspm.fr/~luthier/gourgoulhon/en/)**
# FILES:
- **SageNP.py**: Main file to import in SageMath.
- **[SageNP_Tutorial.ipynb](https://github.com/tbirkandan/SageNP/blob/main/Notebooks/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](https://github.com/tbirkandan/SageNP/blob/main/Notebooks/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: l*n = -1, m*mbar = 1, all others zero.
- The metric is found using the covariant null-tetrad vectors as:
g = -2*l*n + 2*m*mbar
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^2-2*M*r
- **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 l*n = -1, m*mbar = 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()
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"description": "# SageNP: Newman-Penrose calculations for SageMath. \n\nThe class **SageNP** includes functions for some calculations defined in the Newman-Penrose formalism. The code is based on SageManifolds.\n\n# Coded by: \n\n- [Tolga Birkandan](https://web.itu.edu.tr/birkandant/) (Corr.: birkandant@itu.edu.tr)\n\n- [Onur Arman](https://www.linkedin.com/in/onur-arman-709478337/)\n\n- [Emir Baysazan](https://scholar.google.com/citations?user=kq9ia_oAAAAJ&hl=en)\n\n- [Selinay Sude Binici](https://scholar.google.com/citations?user=OjiOpogAAAAJ&hl=tr)\n\n- [Pelin Ozturk] (https://www.linkedin.com/in/pelin-%C3%B6zt%C3%BCrk-3904572b2/?utm_source=share&utm_campaign=share_via&utm_content=profile&utm_medium=ios_app)\n\n- **Special thanks to [Eric Gourgoulhon](https://luth.obspm.fr/~luthier/gourgoulhon/en/)**\n\n\n# FILES:\n\n- **SageNP.py**: Main file to import in SageMath.\n\n- **[SageNP_Tutorial.ipynb](https://github.com/tbirkandan/SageNP/blob/main/Notebooks/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.\n\n- **[SageNP_Tutorial.pdf](https://github.com/tbirkandan/SageNP/blob/main/Notebooks/SageNP_Tutorial.pdf)**: Tutorial (PDF file) \n\n\n# REFERENCE:\n\nMain reference for all definitions and calculations:\n\nH. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, \"Exact Solutions of Einstein\u2019s Field Equations\", 2nd ed. Cambridge: Cambridge University Press, 2003.\n\n\n# BASIC DEFINITIONS AND NOTATION:\n\n* We will use the Metric signature: (- + + +) \n\n* 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\n\n- Products of the vectors are given by: l*n = -1, m*mbar = 1, all others zero. \n \n- The metric is found using the covariant null-tetrad vectors as:\n \n g = -2*l*n + 2*m*mbar\n \n and,\n\n g = [[0 1 0 0], [1 0 0 0], [0 0 0 -1], [0 0 -1 0]]\n\n- Please check the reference book for the details and further definitions.\n\n# INSTRUCTIONS WITH EXAMPLES:\n\n- **Import the class:**\n \n from SageNP import NewmanPenrose\n \n- **Define your manifold:**\n \n MyManifold = Manifold(4 , 'MyManifold', r'\\mathcal{Man}')\n\n- **Define your coordinates:**\n \n MyCoordinates.<t,r,th,ph> = MyManifold.chart(r't r th:\\theta ph:\\phi')\n\n- **Define the metric functions (if needed):**\n\n var('M')\n Delta=r^2-2*M*r\n \n- **Enter null tetrad elements:**\n\n lvec=[1,-(r^2)/Delta,0,0]\n \n nvec=[Delta/(2*r^2),1/2,0,0]\n\n mvec=[0,0,(-r/sqrt(2)),(-I*r/sqrt(2))*sin(th)]\n\n mbarvec=[0,0,(-r/sqrt(2)),(I*r/sqrt(2))*sin(th)]\n\n - *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.)*\n\n- **Define an object of the class:**\n \n schw=SageNP(MyManifold,MyCoordinates,lvec,nvec,mvec,mbarvec,'covariant')\n \n - *Here, our null-tetrad vectors lvec, nvec, mvec and mbarvec are covariant. Thus we used the keyword 'covariant'.*\n \n - *If they were contravariant, then we should use the keyword 'contravariant'.*\n\n- **Once the object is defined, the code calculates the metric and displays it on the screen. It is recommended that you check your metric.**\n\n# FUNCTIONS:\n\n- *All page and equation numbers belong to the reference book.*\n\n\n- **test_nulltetrad()**: Checks the products of the vectors l*n = -1, m*mbar = 1, all others zero.\n\n\n- **Spin coefficients (Page 75-76, Eq.(7.2))**:\n \n - **calculate_spincoefficients()**: Calculates the spin coefficients.\n \n - **show_spincoefficients()**: Displays the spin coefficients\n \n - All spin coefficients are available under their names:\n \n **kappaNP, kappabarNP, tauNP, taubarNP, sigmaNP, sigmabarNP,\n rhoNP, rhobarNP, piNP, pibarNP, nuNP, nubarNP, muNP, mubarNP,\n lambdaNP, lambdabarNP, epsilonNP, epsilonbarNP, gammaNP, gammabarNP,\n betaNP, betabarNP, alphaNP, alphabarNP**\n\n \n- **Directional derivatives (Page 43, Eq.(3.82))**:\n\n - **DlNP(X)**: Given X, calculates the D derivative (l direction).\n\n - **DeltanNP(X)**: Given X, calculates the Delta derivative (n direction)\n \n - **deltamNP(X)**: Given X, calculates the delta derivative (m direction)\n \n - **deltambarNP(X)**: Given X, calculates the deltabar derivative (mbar direction)\n\n\n- **Commutators (Page 77, Eq.(7.6))**:\n\n - *The right-hand sides of the commutation relations are calculated.*\n \n - **Deltan_Dl_commNP(X)**: Given X, calculates the [Delta,D] commutator.\n \n - **deltam_Dl_commNP(X)**: Given X, calculates the [delta,D] commutator.\n \n - **deltam_Deltan_commNP(X)**: Given X, calculates the [delta,Delta] commutator.\n \n - **deltambar_deltam_commNP(X)**: Given X, calculates the [deltabar,delta] commutator.\n\n\n- **Weyl tensor components (Page 38, Eq.(3.59))**:\n \n - **calculate_Weyl()**: Calculates the Weyl tensor components.\n \n - **show_Weyl()**: Displays the Weyl tensor components.\n \n - All Weyl tensor components are available under their names:\n \n **Psi0NP, Psi1NP, Psi2NP, Psi3NP, Psi4NP**\n\n\n- **Ricci components (Page 78, Eq.(7.10-7.15))**:\n \n - **calculate_Ricci()**: Calculates the Ricci tensor components.\n \n - **show_Ricci()**: Displays the Ricci tensor components.\n \n - All Ricci tensor components are available under their names:\n \n **Phi00NP, Phi01NP, Phi10NP, Phi02NP, Phi20NP, \n Phi11NP, Phi12NP, Phi21NP, Phi22NP, LambdaNP**\n\n\n- **Ricci (Newman-Penrose) equations (Page 79, Eq.(7.21))**:\n \n - *All Newman-Penrose equations are defined as 0 = -(left hand side)+(right hand side) of the equations.*\n \n - **calculate_NPeq()**: Calculates the Newman-Penrose equations\n \n - **show_NPeq()**: Displays the Newman-Penrose equations\n \n - All Newman-Penrose equations are available under their names\n in the order they are given in the reference:\n \n **NPeq1, NPeq2, NPeq3, NPeq4, NPeq5, NPeq6, NPeq7, NPeq8, NPeq9, NPeq10, \n NPeq11, NPeq12, NPeq13, NPeq14, NPeq15, NPeq16, NPeq17, NPeq18**\n\n\n- **Bianchi identities (Page 81, Eq.(7.32))**:\n \n - *All Bianchi identities are defined as 0 = -(left hand side)+(right hand side) of the equations.*\n \n - **calculate_Bianchi()**: Calculates the Bianchi identities\n \n - **show_Bianchi()**: Displays the Bianchi identities\n \n - All Bianchi identities are available under their names\n in the order they are given in the reference:\n \n **BI1, BI2, BI3, BI4, BI5, BI6, BI7, BI8, BI9, BI10, BI11**\n\n\n- **Petrov invariants I, J, K, L, N (Kramer p.121, 9.6; p.54, 4.19)**:\n\n *(also check diagram Fig. 9.1 on p. 122)*\n \n - **calculate_PetrovinvINP()**: Calculates the Petrov invariant I\n \n - **calculate_PetrovinvJNP()**: Calculates the Petrov invariant J\n \n - **calculate_PetrovinvKNP()**: Calculates the Petrov invariant K\n \n - **calculate_PetrovinvLNP()**: Calculates the Petrov invariant L\n \n - **calculate_PetrovinvNNP()**: Calculates the Petrov invariant N\n\n - All Petrov invariants are available under their names:\n \n **PetrovinvINP, PetrovinvJNP, PetrovinvKNP, PetrovinvLNP, PetrovinvNNP**\n\n\n- **Petrov type of the spacetime**:\n - **Petrov_frominvariants()**: Calculates the Petrov type using I, J, K, L, N.\n \n - **Petrov_fromWeyl()**: Calculates the Petrov type using the Weyl components\n\n\t\n- **Massive Klein-Gordon equation**:\n\n\t**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))\n\t\n\t- The result is available under the name: **kgNP**\n\n\n- **Massive Dirac equation**:\n\n\t**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.)\n\t\n\t- The result is available under the name: **diracNP**\n\n\t\n- **SL(2,C) Transformations**: SL(2,C) transformations as defined in Carmeli and Kaye, Annals of Physics 99, 188 (1976).\n\n\t**type_A_transformation(z)**: Calculates the Type A transformations where z is a complex variable.\n\n\t**show_type_A_transformation()**: Shows the results of the Type A transformations.\n\n\t**type_B_transformation(z)**: Calculates the Type B transformations where z is a complex variable.\n\n\t**show_type_B_transformation()**: Shows the results of the Type B transformations.\n\n\t**type_C_transformation(z)**: Calculates the Type C transformations where z is a complex variable.\n\n\t**show_type_C_transformation()**: Shows the results of the Type C transformations.\n\t\n\tThe results are available under their names:\n\t**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.\n\n\n- **calculate_allNP()**: Runs the following functions:\n \n - calculate_spincoefficients()\n\n - calculate_Weyl()\n\n - calculate_Ricci()\n\n - calculate_NPeq()\n\n - calculate_Bianchi()\n\n - Petrov_frominvariants()\n\n - Petrov_fromWeyl()\n\n- **show_allNP()**: Runs the following functions:\n\n - show_spincoefficients()\n\n - show_Weyl()\n\n - show_Ricci()\n\n - show_NPeq()\n\n - show_Bianchi()\n\n - Petrov_frominvariants()\n\n - Petrov_fromWeyl()\n",
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