(* coordinate 벡터의 크기로 차원 결정한다: n=Length[vars]; *)

InverseMetric[g_] := Simplify[Inverse[g]]

ChristoffelSymbol[g_, vars_] := Block[{n, ig, res},

  n = Length[vars]; ig = InverseMetric[g];

  res = Table[(1/2)*

     Sum[ig[[i, s]]*(-D[g[[j, k]], vars[[s]]] + 

         D[g[[j, s]], vars[[k]]] + D[g[[s, k]], vars[[j]]]), {s, 1, 

       n}], {i, 1, n}, {j, 1, n}, {k, 1, n}];

  Simplify[res]]

RiemannTensor[g_, vars_] := Block[{n, Chr, res},

  n = Length[vars]; Chr = ChristoffelSymbol[g, vars];

  res = Table[

    D[Chr[[i, k, m]], vars[[l]]] - D[Chr[[i, k, l]], vars[[m]]] + 

     Sum[Chr[[i, s, l]]*Chr[[s, k, m]], {s, 1, n}] - 

     Sum[Chr[[i, s, m]]*Chr[[s, k, l]], {s, 1, n}], {i, 1, n}, {k, 1, 

     n}, {l, 1, n}, {m, 1, n}];

  Simplify[res]]

RicciTensor[g_, vars_] := Block[{Rie, res, n},

  n = Length[vars]; Rie = RiemannTensor[g, vars];

  res = Table[Sum[Rie[[s, i, s, j]], {s, 1, n}], {i, 1, n}, {j, 1, n}];

  Simplify[res]]

RicciScalar[g_, vars_] := Block[{Ricc, ig, res, n},

  n = Length[vars]; Ricc = RicciTensor[g, vars]; ig = InverseMetric[g];

  res = Sum[ig[[s, i]] Ricc[[s, i]], {s, 1, n}, {i, 1, n}];

  Simplify[res]]




예제:



vars = {t, x, \[Theta], \[Phi]};

g = {{-E^(2 \[Nu][x]), 0, 0, 0}, {0, E^(2 \[Lambda][x]), 0, 0}, {0, 0,

     x^2, 0}, {0, 0, 0, x^2 Sin[\[Theta]]^2}};

RicciTensor[g, vars]


결과:



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f(x)의 오른쪽 경계조건은 f(infinity)=1 이나, x가 증가하면 빠르게 1로 수렴하므로 f(15)->1로 잡아도 충분하다.





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In relaxation methods ODEs are replaced by approximate finite difference equations (FDEs) on a grid or mesh of points that spans the domain of interest.


알고리즘 및 사용예제 관련문서들:

NumericalRecipiesInC/c17-3.pdf

NumericalRecipiesInC/c17-4.pdf


Numerical Recipes 에서 얻을 수 있는 필요한 코드들:

1) solvde() : main routine; 

2) bksub ()  : supplementary routine

3) pinvs ()  : supplementary routine

4) red ()  : supplementary routine

5) difeq ()  : user supplied  routine in which differential equations, boundary conditions, and their jacobians are defined. called by solvde().


주의점: 경계조건 설정에서 주의해야 한다;

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별로 도움은 안되지만,
http://www.phys.huji.ac.il/~barak_kol/HDGR/proceedings/Brustein.pps 
  

f(R) gravity에서 Wald entropy 계산:
D.N. Vollick, Phys. Rev. D76 (2007) 124001, "Noether charge and black hole entropy in modified theories of gravity". http://arxiv.org/abs/0710.1859
 ==> based on the Palatini formalism. connection이 일반적으로 metric compatible 하지 않음. 따라서 covariant derivative에 compatible인 새로운 metric을 이용해서 connection을 표현해야 한다. 이 새로운 metric을 이용하면 metric formalism을 그대로 적용가능함.


R. Brustein, D. Gorbonos, M. Hadad and A.J.M. Medved, Phys.Rev. D84 (2011) 064011, "Evaluating the Wald entropy from two-derivative terms in quadratic actions". http://arxiv.org/abs/1106.4394   

horizon에서 벗어난 영역에서의 Wald entropy 계산: 
R. Brustein, and A.J.M. Medved, "Gravitational entropy and thermodynamics away from the horizon", http://arxiv.org/abs/1201.5754

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http://www.ihes.fr/~vanhove/Slides/deruelle-ihes-apr2010.pdf
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http://homepages.mcs.vuw.ac.nz/~visser/Seminars/NZ-seminars/sotiriou-at-canterbury.pdf
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  1. Shooting Methods for 2-Point Boundary Value Problems  
    Jonathan Thornburg,  Max Planck Institut für Gravitationsphysik, Berlin, Germany
  2. Two Point Boundary Value Problems. Solution Methods.  
    R.D. Poshusta, Chemistry Dept., Washington State University, Pullman, WA
  3. The Shooting Method  
    Reza Fotouhi-Chahouki, Dept. of Mechanical Engineering, Univ. of Saskatchewan, Canada
  4. The Shooting Method for 2-Point BVP's  
    J. R. White, Chemical & Nuclear Engineering Dept., University of Massachusetts, Lowell, MA
  5. A Overview of the Shooting Method  
    Adept Scientific plc, Herts, SG6 1ZA, UK
  6. The Shooting Method  
    Juan Restrepo, Math. Dept, Univ. of Arizona, Tucson, AZ
  7. The Shooting Method, Nonlinear Case  
    Juan Restrepo, Math. Dept, Univ. of Arizona, Tucson, AZ
  8. Boundary Value Problems ... Shooting Method  
    Paul Heckbert, Computer Science Dept., Carnegie Mellon University, Pittsburgh PA
  9. Boundary Value problems ... Shooting methods  
    Jun Ni, Department of Mechanical Engineering, Department of Civil Engineering, The University of Iowa
  10. Shooting Method   
    Heinrich Kirchauer, Institute for Microelectronics, Technical University of Vienna
  11. Shooting  
    E. Bruce Pitman, Math. Depat. State University of New York, Buffalo, NY
  12. Shooting method     
    Stuart Dalziel, University of Cambridge, Cambridge, England    
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André Fischer and Richard J. Szabo:
UV/IR duality in noncommutative quantum field theory
arXiv:1001.3776 [hep-th]
(Talk by R.J.S. at three workshops/schools in 2009, to appear in General Relativity and Gravitation)
Olaf Lechtenfeld and Marco Maceda:
The noncommutative Ward metric
arXiv:1001.3416 [hep-th] (SIGMA 6 (2010) 045)
Theodora Ioannidou and Olaf Lechtenfeld:
Noncommutative baby Skyrmions
arXiv:0905.4077 [hep-th] (Phys. Lett. B 678 (2009) 508)
Derek Harland and Seçkin Kürkçüoglu:
Equivariant reduction of Yang-Mills theory over the fuzzy sphere and the emergent vortices
arXiv:0905.2338 [hep-th] (Nucl. Phys. B 821 (2009) 380-398)
André Fischer and Olaf Lechtenfeld
The noncommutative sine-Gordon breather
arXiv:0811.1048 [hep-th] (J. Math. Phys. 50 (2009) 095201)
André Fischer and Richard J. Szabo
Duality covariant quantum field theory on noncommutative Minkowski space
arXiv:0808.1195 [hep-th] (JHEP 0902 (2009) 031)
Seçkin Kürkçüoglu
Noncommutative Q-lumps
arXiv:0808.3745 [hep-th] (Phys. Rev. D 78 (2008) 105018)
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo
SU(3)-equivariant quiver gauge theories and nonabelian vortices
arXiv:0806.2791 [hep-th] (JHEP 0808(2008) 093)
Seçkin Kürkçüoglu:
Noncommutative nonlinear sigma models and integrability
arXiv:0804.3782 [hep-th] (Phys. Rev. D 78 (2008) 065020)
Alexander D. Popov:
Explicit non-abelian monopoles in SU(N) pure Yang-Mills theory
arXiv:0803.3320 [hep-th] ()
Sergei V. Ketov and Olaf Lechtenfeld
Non-anticommutative solitons
arXiv:0802.2867 [hep-th] (Phys. Lett. B 663 (2008) 353-359)
Alexander D. Popov:
Non-abelian vortices on Riemann surfaces: an integrable case
arXiv:0801.0808 [hep-th] ()
Alexander D. Popov:
Integrability of vortex equations on Riemann surfaces
arXiv:0712.1756 [hep-th] ()
Olaf Lechtenfeld:
Noncommutative solitons:
arXiv:0710.2074 [hep-th]
(Talk at the "Third Mexican Meeting on Mathematical and Experimental Physics'' at El Colegio Nacional, Mexico City, Mexico, 14-17 September 2007;
AIP Conference Proceedings Vol. 977, pp. 37-51)
Christian Gutschwager, Tatiana A. Ivanova and Olaf Lechtenfeld:
Scattering of noncommutative waves and solitons in a supersymmetric chiral model in 2+1 dimensions
arXiv:0710.0079 [hep-th] (JHEP 0711 (2007) 052)
I.L. Buchbinder, E.A. Ivanov, O. Lechtenfeld, I.B. Samsonov and B.M. Zupnik:
Gauge theory in deformed N=(1,1) superspace
arXiv:0709.3770 [hep-th] (Phys. Part. Nucl. 39 (2008) 759-797)
Seçkin Kürkçüoglu and Olaf Lechtenfeld:
Quantum aspects of the noncommutative sine-Gordon model
arXiv:0708.1310 [hep-th] (JHEP 0709 (2007) 020)
Olaf Lechtenfeld:
Supersymmetric noncommutative solitons
arXiv:0707.3522 [hep-th]
(Talk at "Noncommutative Spacetime Geometries" in Alessandria, Italy, 26-31 March 2007, and "Noncommutative Geometry and Physics" in Orsay, France, 23-27 April 2007;
Journal of Physics: Conference Series 103 (2008) 012016)
Aiyalam P. Balachandran, Kumar S. Gupta and Seçkin Kürkçüoglu:
Interacting quantum topologies and the quantum Hall effect
arXiv:0707.1219 [hep-th] ()
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo:
Quiver gauge theory and noncommutative vortices
arXiv:0706.0979 [hep-th]
(Talk by O.L. at the 21st Nishinomiya-Yukawa Memorial Symposium in Nishinomiya/Kyoto, Japan, 11-15 November 2006;
Proceedings: Prog. Theor. Phys. Suppl. 171 (2007) 258-268)
Stefan Petersen:
Abelian solitons in the noncommutative Ward model
(Talk by S.P. at the 21st Nishinomiya-Yukawa Memorial Symposium in Nishinomiya/Kyoto, Japan, 11-15 November 2006;
Proceedings: Prog. Theor. Phys. Suppl. 171 (2007) 223-227)
Olaf Lechtenfeld and Alexander D. Popov:
Noncommutative solitons in a supersymmetric chiral model in 2+1 dimensions
arXiv:0704.0530 [hep-th] (JHEP 0706 (2007) 065)
Johannes Brödel, Tatiana Ivanova and Olaf Lechtenfeld:
Construction of noncommutative instantons in 4k dimensions
hep-th/0703009 (Mod. Phys. Lett. A 23 (2008) 179-189)
I.L. Buchbinder, O. Lechtenfeld and I.B. Samsonov:
Vector-mutliplet effective action in the non-anticommutative charged hypermultiplet model
hep-th/0608048 (Nucl. Phys. B 758 (2006) 185-203)
Alexandra De Castro and Leonardo Quevedo:
Non-singlet Q-deformed N=(1,1) and N=(1,1/2) U(1) actions
hep-th/0605187 (Phys. Lett. B 639 (2006) 117-123)
Olaf Lechtenfeld:
Noncommutative solitons
hep-th/0605034
(150 minutes of lectures given 01-04 November 2005 at the International Sendai-Beijing Joint Workshop on Noncommutative Geometry and Physics;
Proceedings: pp. 175-200, World Scientific, 2007)
Michael Klawunn, Olaf Lechtenfeld and Stefan Petersen:
Moduli-space dynamics of noncommutative abelian sigma-model solitons
hep-th/0604219 (JHEP 0606 (2006) 028)
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo:
Rank two quiver gauge theory, graded connections and noncommutative vortices
hep-th/0603232 ()
Tatiana A. Ivanova and Olaf Lechtenfeld:
Noncommutative instantons on CPn
hep-th/0603125 (Phys. Lett. B 639 (2006) 407-410)
I. Buchbinder, E. Ivanov, O. Lechtenfeld, I. Samsonov and B. Zupnik:
Renormalizability of non-anticommutative N=(1,1) theories with singlet deformation
hep-th/0511234 (Nucl. Phys. B 740 (2006) 358-385)
Alexandra De Castro, Evgeny Ivanov, Olaf Lechtenfeld and Leonardo Quevedo:
Non-singlet Q-deformation of the N=(1,1) vector multiplet in harmonic superspace
hep-th/0510013 (Nucl. Phys. B 747 (2006) 1-24)
Chong-Sun Chu and Olaf Lechtenfeld:
Emergence of time from dimensional reduction in noncommutative geometry
hep-th/0508055 (Mod. Phys. Lett. A 21 (2006) 639-647)
Chong-Sun Chu and Olaf Lechtenfeld:
Time-space noncommutative abelian solitons
hep-th/0507062 (Phys. Lett. B 625 (2005) 145-155)
Matthias Ihl and Christian Sämann:
Drinfeld-twisted supersymmetry and non-anticommutative superspace
hep-th/0506057 (JHEP 0601 (2006) 065)
Alexander D. Popov and Richard J. Szabo
Quiver gauge theory of nonabelian vortices and noncommutative instantons in higher dimensions
hep-th/0504025 (J. Math. Phys. 47 (2006) 012306)
Robert Wimmer:
D0-D4 brane tachyon condensation to a BPS state and its excitation spectrum in noncommutative super Yang-Mills theory
hep-th/0502158 (JHEP 0505 (2005) 022)
Tatiana A. Ivanova and Olaf Lechtenfeld:
Noncommutative instantons in 4k dimensions
hep-th/0502117 (Phys. Lett. B 612 (2005) 65-74)
Andrei V. Domrin, Olaf Lechtenfeld and Stefan Petersen:
Sigma-model solitons in the noncommutative plane: construction and stability analysis
hep-th/0412001 (JHEP 0503 (2005) 045)
Olaf Lechtenfeld:
Noncommutative sine-Gordon model
hep-th/0409108
(Talk at the XIIIth International Colloquium Integrable Systems and Quantum Groups, Prague, 17-19 June 2004, and at the 37th International Symposium Ahrenshoop on the Theory of Elementary Particles, Berlin-Schmöckwitz, 23-27 August 2004:
Czech. J. Phys. 54 (2004) 1351-1357; Fortsch. Phys. 53 (2005) 500-505)
Evgenyi Ivanov, Olaf Lechtenfeld and Boris Zupnik:
Non-anticommutative deformation of N=(1,1) hypermultiplets
hep-th/0408146 (Nucl. Phys. B 707 (2005) 69-86)
Olaf Lechtenfeld, Liuba Mazzanti, Silvia Penati, Alexander D. Popov and Laura Tamassia:
Integrable noncommutative sine-Gordon model
hep-th/0406065 (Nucl. Phys. B 705 (2005) 477-503)
Sergio Ferrara, Evgeny Ivanov, Olaf Lechtenfeld, Emery Sokatchev and Boris Zupnik:
Non-anticommutative chiral singlet deformation of N=(1,1) gauge theory
hep-th/0405049 (Nucl. Phys. B 704 (2005) 154-180)
Tatiana A. Ivanova, Olaf Lechtenfeld and and Helge Müller-Ebhardt:
Noncommutative moduli for multi-instantons
hep-th/0404127 (Mod. Phys. Lett. A 19 (2004) 2419-2430)
Evgeny Ivanov, Olaf Lechtenfeld and Boris Zupnik:
Non-anticommutative N=(1,1) Euclidean superspace
hep-th/0402062
(Talk by E.I. at JINR workshop ``Supersymmetries and Quantum Symmetries'', Dubna, Russia, 24-29 July 2003;
Proceedings 2004: SQS'03, pp. )
Olaf Lechtenfeld:
Noncommutative instantons and solitons
hep-th/0401158
(talk at the 27th Johns Hopkins Workshop on Current Problems in Particle Theory, Göteborg, Sweden, 24-26 August 2003, and at the 36th International Symposium Ahrenshoop on the Theory of Elementary Particles, Berlin, Germany, 26-30 August 2003:
short version: Fortsch. Phys. 52 (2004) 596-605; long version: JHEP PRHEP-jhw2003/017)
Christian Sämann and Martin Wolf:
Constraint and super Yang-Mills equations on the deformed superspace Rh(4|16)
hep-th/0401147 (JHEP 0403 (2004) 048)
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo:
Noncommutative instantons in higher dimensions, vortices and topological K-cycles
hep-th/0310267 (JHEP 0312 (2003) 022)
Evgenyi Ivanov, Olaf Lechtenfeld and Boris Zupnik:
Nilpotent deformations of N=2 superspace
hep-th/0308012 (JHEP 0402 (2004) 012)
Olaf Lechtenfeld and Alexander D. Popov:
Noncommutative monopoles and Riemann-Hilbert problems
hep-th/0306263 (JHEP 0401 (2004) 069)
Tatiana A. Ivanova and Olaf Lechtenfeld:
Noncommutative multi-instantons on R2n x S2
hep-th/0305195 (Phys. Lett. B 567 (2003) 107-115)
Alexander D. Popov, Armen G. Sergeev and Martin Wolf:
Seiberg-Witten monopole equations on noncommutative R4
hep-th/0304263 (J. Math. Phys. 44 (2003) 4527-4554)
Matthias Ihl and Sebastian Uhlmann:
Noncommutative extended waves and soliton-like configurations in N=2 string theory
hep-th/0211263 (Int. J. Mod. Phys. A 18 (2003) 4889-4932)
Zalan Horvath, Olaf Lechtenfeld and Martin Wolf:
Noncommutative instantons via dressing and splitting approaches
hep-th/0211041 (JHEP 0212 (2002) 060)
Filip Franco-Sollova and Tatiana Ivanova:
On noncommutative merons and instantons
hep-th/0209153 (J. Phys. A 36 (2003) 4207-4219)
Martin Wolf:
Soliton-antisoliton scattering configurations in a noncommutative sigma model in 2+1 dimensions
hep-th/0204185 (JHEP 0206 (2002) 055)
Stig Bieling:
Interaction of noncommutative plane waves in 2+1 dimensions
hep-th/0203269 (J. Phys. A 35 (2002) 6281-6291)
Olaf Lechtenfeld and Alexander D. Popov:
Noncommutative 't Hooft instantons
hep-th/0109209 (JHEP 0203 (2002) 040)
Olaf Lechtenfeld and Alexander D. Popov:
Scattering of noncommutative solitons in 2+1 dimensions
hep-th/0108118 (Phys. Lett. B 523 (2001) 178-184)
Olaf Lechtenfeld and Alexander D. Popov:
Noncommutative multi-solitons in 2+1 dimensions
hep-th/0106213 (JHEP 0111 (2001) 040)
Olaf Lechtenfeld, Alexander D. Popov and Bernd Spendig:
Noncommutative solitons in open N=2 string theory
hep-th/0103196 (JHEP 0106 (2001) 011)
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D-Branes in Noncommutative Field Theory
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Forces from Connes' Geometry
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Quantum Field Theory on Noncommutative Spaces
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Noncommutative Field Theory
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Komaba Lectures on Noncommutative Solitons and D-Branes
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Introduction to M(atrix) Theory and Noncommutative Geometry
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Noncommutative Geometry Year 2000
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