From d1beabbdbeedab43f2ae372acdab26a2f32d1443 Mon Sep 17 00:00:00 2001 From: Anton Khirnov Date: Mon, 9 Jan 2012 15:41:26 +0100 Subject: Add boosting along the x-axis. --- interface.ccl | 2 +- param.ccl | 6 + src/mdefs.h | 41 ++ src/trumpet.c | 177 ++++++- trumpet.nb | 1306 +++++++++++++++++++++++++++++++++++++++++++++++++++ trumpet2.nb | 1441 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 6 files changed, 2947 insertions(+), 26 deletions(-) create mode 100644 src/mdefs.h create mode 100644 trumpet.nb create mode 100644 trumpet2.nb diff --git a/interface.ccl b/interface.ccl index 535d955..f451ae6 100644 --- a/interface.ccl +++ b/interface.ccl @@ -1,7 +1,7 @@ # Interface definition for thorn Trumpet implements: Trumpet -INHERITS: ADMBase StaticConformal grid CoordBase +INHERITS: ADMBase grid CoordBase CCTK_INT FUNCTION GetDomainSpecification \ (CCTK_INT IN size, \ diff --git a/param.ccl b/param.ccl index a009072..42ab1ba 100644 --- a/param.ccl +++ b/param.ccl @@ -16,3 +16,9 @@ EXTENDS KEYWORD initial_shift { "trumpet" :: "one maximal trumpet" } + +RESTRICTED: +CCTK_REAL boost_velocity "Boost the trumpet with this velocity parameter in x-axis direction" +{ + (-1:1) :: "" +} 0 diff --git a/src/mdefs.h b/src/mdefs.h new file mode 100644 index 0000000..dc5d0c1 --- /dev/null +++ b/src/mdefs.h @@ -0,0 +1,41 @@ +/************************************************************************* + + Mathematica source file + + Copyright 1986 through 1999 by Wolfram Research Inc. + + +*************************************************************************/ + +/* C language definitions for use with Mathematica output */ + + +#define Power(x, y) (pow((double)(x), (double)(y))) +#define Sqrt(x) (sqrt((double)(x))) + +#define Abs(x) (fabs((double)(x))) + +#define Exp(x) (exp((double)(x))) +#define Log(x) (log((double)(x))) + +#define Sin(x) (sin((double)(x))) +#define Cos(x) (cos((double)(x))) +#define Tan(x) (tan((double)(x))) + +#define ArcSin(x) (asin((double)(x))) +#define ArcCos(x) (acos((double)(x))) +#define ArcTan(x) (atan((double)(x))) + +#define Sinh(x) (sinh((double)(x))) +#define Cosh(x) (cosh((double)(x))) +#define Tanh(x) (tanh((double)(x))) + + +#define E 2.71828182845904523536029 +#define Pi 3.14159265358979323846264 +#define Degree 0.01745329251994329576924 + + +/** Could add definitions for Random(), SeedRandom(), etc. **/ + + diff --git a/src/trumpet.c b/src/trumpet.c index 7b65822..0933c04 100644 --- a/src/trumpet.c +++ b/src/trumpet.c @@ -8,6 +8,8 @@ #include "cctk_Arguments.h" #include "cctk_Parameters.h" +#include "mdefs.h" + #define MAX(x,y) (x) > (y) ? (x) : (y) #define SQR(x) ((x)*(x)) @@ -23,11 +25,13 @@ /* * isotropic/coordinate radius */ -#define ISO_R(index) (sqrt(SQR(x[index]) + SQR(y[index]) + SQR(z[index]) + EPS)) +#define ISO_R(x, y, z, index, gamma) (sqrt(SQR(gamma*x[index]) + SQR(y[index]) + SQR(z[index]) + EPS)) + +#define TRUMPET_ALPHA(R) (sqrt(1 - 2*MASS/R + SQR(TRUMPET_CONST)/SQR(SQR(R)))) static inline double r_areal_to_isotropic(double r, double m) { - double par = sqrt(4*r*r + 4*m*r + 3*m*m); + double par = sqrt(4*SQR(r) + 4*m*r + 3*SQR(m)); double term1 = (2*r + m + par)/4; double term2 = (4 + 3*M_SQRT2)*(2*r - 3*m)/(8*r + 6*m + 3*M_SQRT2*par); return term1*pow(term2, M_SQRT1_2); @@ -76,11 +80,101 @@ static void get_r_tables(double **pr_iso, double **pr_areal, CCTK_INT *size, CCT *size = i; } +/* those equations come from trumpet.nb */ +static long double sqrt_factor(long double R, long double r, long double x, long double alpha, long double beta) +{ + long double R2 = SQR(R); + return sqrtl((R2*(R + alpha*beta*r) - beta*TRUMPET_CONST*x)*(R2*(R - alpha*beta*r) - beta*TRUMPET_CONST*x)); +} + +static inline CCTK_REAL K_11(CCTK_REAL x, CCTK_REAL y, CCTK_REAL z, + CCTK_REAL R, CCTK_REAL r, + CCTK_REAL beta, CCTK_REAL M, + CCTK_REAL alpha) +{ + long double b2 = SQR(beta); + long double r2 = SQR(r); + long double R2 = SQR(R); + + long double n1 = TRUMPET_CONST*(1.0 - 3*SQR(x/r))*(beta*x*TRUMPET_CONST/R - R2)/r2; + long double n2 = b2*beta*M*(-Power(TRUMPET_CONST,2)/(R2*R2) + Power(alpha,2))*x; + long double n3 = b2*TRUMPET_CONST*M*(1.0 + 2*SQR(x/r))/R; + long double n4 = beta*x*(SQR(R/r)*(-2*M + (-1 + alpha)*alpha*R)); + + long double den = (b2 - 1)*sqrt_factor(R, r, x, alpha, beta); + + return (n1 + n2 + n3 + n4)/den; +} + +static inline CCTK_REAL K_22(CCTK_REAL x, CCTK_REAL y, CCTK_REAL z, + CCTK_REAL R, CCTK_REAL r, + CCTK_REAL beta, CCTK_REAL M, + CCTK_REAL alpha) +{ + long double r2 = SQR(r); + long double R2 = SQR(R); + + long double n1 = TRUMPET_CONST*(R2 - beta*TRUMPET_CONST*(x/R))*(1.0 - 3*SQR(y/r))/r2; + long double n2 = alpha*(alpha - 1)*beta*SQR(R/r)*R*x; + + long double sq = sqrt_factor(R, r, x, alpha, beta); + + return (n1 + n2)/sq; +} + +static inline CCTK_REAL K_33(CCTK_REAL x, CCTK_REAL y, CCTK_REAL z, + CCTK_REAL R, CCTK_REAL r, + CCTK_REAL beta, CCTK_REAL M, + CCTK_REAL alpha) +{ + return K_22(x, z, y, R, r, beta, M, alpha); +} + +static inline CCTK_REAL K_12(CCTK_REAL x, CCTK_REAL y, CCTK_REAL z, + CCTK_REAL R, CCTK_REAL r, + CCTK_REAL beta, CCTK_REAL M, + CCTK_REAL alpha) +{ + long double b2 = SQR(beta); + long double r2 = SQR(r); + long double R2 = SQR(R); + + long double n1 = TRUMPET_CONST*(b2*M/R + 3*SQR(R/r))*x - 3*beta*SQR(TRUMPET_CONST)*SQR(x/r)/R; + long double n2 = beta*SQR(R/r)*(-M + (-1 + alpha)*alpha*R); + long double den = -sqrt_factor(R, r, x, alpha, beta)*sqrtl(1 - b2); // really minus? + + return y*(n1/r2 + n2)/den; +} + +static inline CCTK_REAL K_13(CCTK_REAL x, CCTK_REAL y, CCTK_REAL z, + CCTK_REAL R, CCTK_REAL r, + CCTK_REAL beta, CCTK_REAL M, + CCTK_REAL alpha) +{ + return K_12(x, z, y, R, r, beta, M, alpha); +} + +static inline CCTK_REAL K_23(CCTK_REAL x, CCTK_REAL y, CCTK_REAL z, + CCTK_REAL R, CCTK_REAL r, + CCTK_REAL beta, CCTK_REAL M, + CCTK_REAL alpha) +{ + long double r2 = SQR(r); + long double R2 = SQR(R); + + long double num = -3*TRUMPET_CONST*(R2 - beta*TRUMPET_CONST*x/R)*(y/r)*(z/r); + long double den = r2*sqrt_factor(R, r, x, alpha, beta); + + return num/den; +} + void trumpet_data(CCTK_ARGUMENTS) { DECLARE_CCTK_ARGUMENTS; DECLARE_CCTK_PARAMETERS; + double gamma = 1.0/sqrt(1.0 - SQR(boost_velocity)); + gsl_interp_accel *acc; gsl_spline *spline; @@ -93,8 +187,6 @@ void trumpet_data(CCTK_ARGUMENTS) gsl_spline_init(spline, r_iso, r_areal, size); acc = gsl_interp_accel_alloc(); - memset(gxy, 0, sizeof(*gxy)*CCTK_GFINDEX3D(cctkGH, cctk_lsh[0]-1, cctk_lsh[1]-1, cctk_lsh[2]-1)); - memset(gxz, 0, sizeof(*gxy)*CCTK_GFINDEX3D(cctkGH, cctk_lsh[0]-1, cctk_lsh[1]-1, cctk_lsh[2]-1)); memset(gyz, 0, sizeof(*gxy)*CCTK_GFINDEX3D(cctkGH, cctk_lsh[0]-1, cctk_lsh[1]-1, cctk_lsh[2]-1)); #pragma omp parallel for @@ -102,21 +194,32 @@ void trumpet_data(CCTK_ARGUMENTS) for (int j = 0; j < cctk_lsh[1]; j++) for (int i = 0; i < cctk_lsh[0]; i++) { int index = CCTK_GFINDEX3D(cctkGH, i, j, k); - CCTK_REAL r = ISO_R(index); + CCTK_REAL xx = gamma*x[index], yy = y[index], zz = z[index]; + CCTK_REAL r = ISO_R(x, y, z, index, gamma); CCTK_REAL R = gsl_spline_eval(spline, r, acc); - CCTK_REAL psi2 = R/r, psi4 = psi2*psi2; - CCTK_REAL k_fact = TRUMPET_CONST/(r*r*r*r*R); - - gxx[index] = gyy[index] = gzz[index] = psi4; - - kxx[index] = -k_fact*(3*SQR(x[index]) - SQR(r)); - kyy[index] = -k_fact*(3*SQR(y[index]) - SQR(r)); - kzz[index] = -k_fact*(3*SQR(z[index]) - SQR(r)); - - kxy[index] = -k_fact*3*x[index]*y[index]; - kxz[index] = -k_fact*3*x[index]*z[index]; - kyz[index] = -k_fact*3*y[index]*z[index]; + CCTK_REAL alpha = TRUMPET_ALPHA(R); + + long double r2 = SQR(r); + long double R2 = SQR(R); + long double b2 = SQR(boost_velocity); + + kxx[index] = K_11(xx, yy, zz, R, r, boost_velocity, MASS, alpha); + kyy[index] = K_22(xx, yy, zz, R, r, boost_velocity, MASS, alpha); + kzz[index] = K_33(xx, yy, zz, R, r, boost_velocity, MASS, alpha); + kxy[index] = K_12(xx, yy, zz, R, r, boost_velocity, MASS, alpha); + kxz[index] = K_13(xx, yy, zz, R, r, boost_velocity, MASS, alpha); + kyz[index] = K_23(xx, yy, zz, R, r, boost_velocity, MASS, alpha); + + gxx[index] = (-SQR(R/r) + b2*(-SQR(TRUMPET_CONST/R2) + Power(alpha,2)) + 2*boost_velocity*TRUMPET_CONST*xx/(R*r2)) / (b2 - 1); + gyy[index] = SQR(R/r); + gzz[index] = SQR(R/r); + gxy[index] = -((boost_velocity*TRUMPET_CONST*yy)/(Sqrt(1 - b2)*r2*R)); + gxz[index] = -((boost_velocity*TRUMPET_CONST*zz)/(Sqrt(1 - b2)*r2*R)); } + free(r_iso); + free(r_areal); + gsl_interp_accel_free(acc); + gsl_spline_free(spline); } void trumpet_lapse(CCTK_ARGUMENTS) @@ -124,6 +227,8 @@ void trumpet_lapse(CCTK_ARGUMENTS) DECLARE_CCTK_ARGUMENTS; DECLARE_CCTK_PARAMETERS; + double gamma = 1.0/sqrt(1.0 - SQR(boost_velocity)); + gsl_interp_accel *acc; gsl_spline *spline; @@ -141,11 +246,21 @@ void trumpet_lapse(CCTK_ARGUMENTS) for (int j = 0; j < cctk_lsh[1]; j++) for (int i = 0; i < cctk_lsh[0]; i++) { int index = CCTK_GFINDEX3D(cctkGH, i, j, k); - CCTK_REAL r = sqrt(SQR(x[index]) + SQR(y[index]) + SQR(z[index]) + EPS); + CCTK_REAL xx = gamma*x[index], yy = y[index], zz = z[index]; + CCTK_REAL r = ISO_R(x, y, z, index, gamma); CCTK_REAL R = gsl_spline_eval(spline, r, acc); + CCTK_REAL alpha = TRUMPET_ALPHA(R); - alp[index] = sqrt(1 - 2*MASS/R + TRUMPET_CONST*TRUMPET_CONST/(R*R*R*R)); + long double r2 = SQR(r); + long double R2 = SQR(R); + long double b2 = SQR(boost_velocity); + + alp[index] = alpha*R2*R*sqrt((b2 - 1) / (Power(alpha,2)*b2*r2*R2*R2 - Power(R2*R - boost_velocity*TRUMPET_CONST*xx,2))); } + free(r_iso); + free(r_areal); + gsl_interp_accel_free(acc); + gsl_spline_free(spline); } void trumpet_shift(CCTK_ARGUMENTS) @@ -153,6 +268,8 @@ void trumpet_shift(CCTK_ARGUMENTS) DECLARE_CCTK_ARGUMENTS; DECLARE_CCTK_PARAMETERS; + double gamma = 1.0/sqrt(1.0 - SQR(boost_velocity)); + gsl_interp_accel *acc; gsl_spline *spline; @@ -170,14 +287,24 @@ void trumpet_shift(CCTK_ARGUMENTS) for (int j = 0; j < cctk_lsh[1]; j++) for (int i = 0; i < cctk_lsh[0]; i++) { int index = CCTK_GFINDEX3D(cctkGH, i, j, k); - CCTK_REAL r = sqrt(SQR(x[index]) + SQR(y[index]) + SQR(z[index]) + EPS); + CCTK_REAL xx = gamma*x[index], yy = y[index], zz = z[index]; + CCTK_REAL r = ISO_R(x, y, z, index, gamma); CCTK_REAL R = gsl_spline_eval(spline, r, acc); + CCTK_REAL alpha = TRUMPET_ALPHA(R); - betax[index] = TRUMPET_CONST*x[index]/(R*R*R); - betay[index] = TRUMPET_CONST*y[index]/(R*R*R); - betaz[index] = TRUMPET_CONST*z[index]/(R*R*R); - } -} + long double r2 = SQR(r); + long double R2 = SQR(R); + long double b2 = SQR(boost_velocity); + long double a2 = SQR(alpha); + + betax[index] = -((a2*boost_velocity*r2*R2*R2 + TRUMPET_CONST*Power(R,3)*xx + b2*TRUMPET_CONST*Power(R,3)*xx - boost_velocity*(Power(R,6) + Power(TRUMPET_CONST,2)*Power(xx,2)))/ + (a2*b2*r2*R2*R2 - Power(Power(R,3) - boost_velocity*TRUMPET_CONST*xx,2))); + betay[index] = (Sqrt(1 - b2)*TRUMPET_CONST*(-Power(R,3) + boost_velocity*TRUMPET_CONST*xx)*yy)/(a2*b2*r2*R2*R2 - Power(Power(R,3) - boost_velocity*TRUMPET_CONST*xx,2)); + betaz[index] = (Sqrt(1 - b2)*TRUMPET_CONST*(-Power(R,3) + boost_velocity*TRUMPET_CONST*xx)*zz)/(a2*b2*r2*R2*R2 - Power(Power(R,3) - boost_velocity*TRUMPET_CONST*xx,2)); } + free(r_iso); + free(r_areal); + gsl_interp_accel_free(acc); + gsl_spline_free(spline); } diff --git a/trumpet.nb b/trumpet.nb new file mode 100644 index 0000000..f117c9a --- /dev/null +++ b/trumpet.nb @@ -0,0 +1,1306 @@ +(* Content-type: application/mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 7.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 145, 7] +NotebookDataLength[ 58685, 1297] +NotebookOptionsPosition[ 57079, 1244] +NotebookOutlinePosition[ 57416, 1259] +CellTagsIndexPosition[ 57373, 1256] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ +Cell[BoxData[ + RowBox[{"psi", ":=", + RowBox[{"Sqrt", "[", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "/", "r"}], "]"}]}]], "Input"], + +Cell[BoxData[ + RowBox[{"beta1", ":=", + RowBox[{ + RowBox[{"psi", "^", "4"}], "*", " ", "x", "*", + RowBox[{"C", "/", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "^", "3"}]}]}]}]], "Input", + CellChangeTimes->{{3.541405061030883*^9, 3.541405081211104*^9}, { + 3.541568693038992*^9, 3.54156869591467*^9}, 3.541576353980927*^9, { + 3.541578112640421*^9, 3.541578113917652*^9}, 3.541650183093624*^9}], + +Cell[BoxData[ + RowBox[{"beta2", ":=", + RowBox[{ + RowBox[{"psi", "^", "4"}], "*", " ", "y", "*", + RowBox[{"C", "/", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "^", "3"}]}]}]}]], "Input", + CellChangeTimes->{{3.541405087435377*^9, 3.541405089672179*^9}, { + 3.5415686986023483`*^9, 3.5415687083941183`*^9}, 3.5415763576834927`*^9, { + 3.5415781173360577`*^9, 3.541578119110003*^9}, {3.541650185131393*^9, + 3.541650185952448*^9}}], + +Cell[BoxData[ + RowBox[{"beta3", ":=", + RowBox[{ + RowBox[{"psi", "^", "4"}], " ", "*", "z", "*", + RowBox[{"C", "/", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "^", "3"}]}]}]}]], "Input", + CellChangeTimes->{{3.541405093909099*^9, 3.541405096428109*^9}, { + 3.541568711652349*^9, 3.5415687140903873`*^9}, 3.5415763611864567`*^9, { + 3.541578120891581*^9, 3.541578122699366*^9}, {3.541650188191567*^9, + 3.541650188911887*^9}}], + +Cell[BoxData[ + RowBox[{"alpha", ":=", + RowBox[{"Sqrt", "[", + RowBox[{"1", "-", + RowBox[{"2", "*", + RowBox[{"M", "/", + RowBox[{"R", "[", "r", "]"}]}]}], "+", + RowBox[{ + RowBox[{"C", "^", "2"}], "/", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "^", "4"}]}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.5414051009800997`*^9, 3.541405117339251*^9}}], + +Cell[BoxData[ + RowBox[{"gamma", ":=", + RowBox[{"1", "/", + RowBox[{"Sqrt", "[", + RowBox[{"1", "-", + RowBox[{"beta", "^", "2"}]}], "]"}]}]}]], "Input", + CellChangeTimes->{{3.541405132291383*^9, 3.541405138684085*^9}, + 3.541568684612611*^9, 3.5415687173641*^9, {3.541573628926816*^9, + 3.5415736363091927`*^9}}], + +Cell[BoxData[ + RowBox[{"gorig", ":=", + RowBox[{"{", + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"-", + RowBox[{"alpha", "^", "2"}]}], "+", + RowBox[{ + RowBox[{"psi", "^", + RowBox[{"(", + RowBox[{"-", "4"}], ")"}]}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{"beta1", "^", "2"}], "+", + RowBox[{"beta2", "^", "2"}], "+", + RowBox[{"beta3", "^", "2"}]}], ")"}]}]}], ",", " ", "beta1", ",", + " ", "beta2", ",", " ", "beta3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"beta1", ",", + RowBox[{"psi", "^", "4"}], ",", "0", ",", "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"beta2", ",", "0", ",", + RowBox[{"psi", "^", "4"}], ",", "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"beta3", ",", "0", ",", "0", ",", + RowBox[{"psi", "^", "4"}]}], "}"}]}], "}"}]}]], "Input", + CellChangeTimes->{{3.541405161936751*^9, 3.541405263881534*^9}}], + +Cell[BoxData[ + RowBox[{"Lambda", ":=", + RowBox[{"{", + RowBox[{ + RowBox[{"{", + RowBox[{"gamma", ",", + RowBox[{ + RowBox[{"-", "gamma"}], "*", "beta"}], ",", "0", ",", "0"}], "}"}], + ",", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"-", "gamma"}], "*", "beta"}], ",", "gamma", ",", "0", ",", + "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"0", ",", "0", ",", "0", ",", "1"}], "}"}]}], "}"}]}]], "Input", + CellChangeTimes->{{3.541405309979129*^9, 3.5414053406975393`*^9}, { + 3.541576211047379*^9, 3.541576214954173*^9}, {3.5415766134805937`*^9, + 3.541576615866742*^9}, {3.541577989494315*^9, 3.541577993214196*^9}, { + 3.541659117039507*^9, 3.541659120385696*^9}}], + +Cell[BoxData[ + RowBox[{"g", ":=", + RowBox[{"FullSimplify", "[", + RowBox[{"Lambda", ".", "gorig", ".", "Lambda"}], "]"}]}]], "Input", + CellChangeTimes->{{3.5416502169868383`*^9, 3.5416502287092876`*^9}, { + 3.5416506915108023`*^9, 3.541650691651325*^9}}], + +Cell[BoxData[ + RowBox[{"g3d", ":=", + RowBox[{"g", "[", + RowBox[{"[", + RowBox[{ + RowBox[{"2", ";;", "4"}], ",", + RowBox[{"2", ";;", "4"}]}], "]"}], "]"}]}]], "Input", + CellChangeTimes->{{3.541405355823965*^9, 3.541405399740391*^9}, { + 3.5416502327337837`*^9, 3.541650240217704*^9}}], + +Cell[BoxData[ + RowBox[{"g3u", ":=", + RowBox[{"FullSimplify", "[", + RowBox[{"Inverse", "[", "g3d", "]"}], "]"}]}]], "Input", + CellChangeTimes->{{3.5414054223252373`*^9, 3.541405432235853*^9}}], + +Cell[BoxData[ + RowBox[{"g3dy", ":=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"g3d", ",", "y", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"R", "'"}], "[", "r", "]"}], "\[Rule]", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "*", + RowBox[{"alpha", "/", + RowBox[{"(", "r", ")"}]}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"r", ",", "y", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "\[Rule]", + RowBox[{"y", "/", "r"}]}]}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.541406634290625*^9, 3.541406800404162*^9}, { + 3.541406833684698*^9, 3.541406879015818*^9}, {3.5414069575741167`*^9, + 3.541406967124864*^9}, {3.5416532441838408`*^9, 3.541653248021184*^9}}], + +Cell[BoxData[ + RowBox[{"g3dz", ":=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"g3d", ",", "z", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"R", "'"}], "[", "r", "]"}], "\[Rule]", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "*", + RowBox[{"alpha", "/", + RowBox[{"(", "r", ")"}]}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"r", ",", "z", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "\[Rule]", + RowBox[{"z", "/", "r"}]}]}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.541406979031295*^9, 3.541406987189321*^9}, { + 3.541653251676744*^9, 3.541653254754826*^9}}], + +Cell[BoxData[ + RowBox[{"g3dx", ":=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"gamma", "*", + RowBox[{"D", "[", + RowBox[{"g3d", ",", "x", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}]}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"R", "'"}], "[", "r", "]"}], "\[Rule]", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "*", + RowBox[{"alpha", "/", + RowBox[{"(", "r", ")"}]}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"r", ",", "x", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "\[Rule]", + RowBox[{"x", "/", "r"}]}]}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.541406991078109*^9, 3.5414069986545753`*^9}, { + 3.541491207633357*^9, 3.5414912099279613`*^9}, {3.5415665727617483`*^9, + 3.541566574896678*^9}, {3.5415668117623043`*^9, 3.541566812953938*^9}, { + 3.5416532580164423`*^9, 3.541653261549081*^9}}], + +Cell[BoxData[ + RowBox[{"g3diff", ":=", + RowBox[{"{", + RowBox[{"g3dx", ",", "g3dy", ",", "g3dz"}], "}"}]}]], "Input", + CellChangeTimes->{{3.541407188389018*^9, 3.54140719780622*^9}, { + 3.54140730189473*^9, 3.541407302322214*^9}}], + +Cell[BoxData[ + RowBox[{"bbeta", ":=", + RowBox[{"g", "[", + RowBox[{"[", + RowBox[{"1", ",", + RowBox[{"2", ";;", "4"}]}], "]"}], "]"}]}]], "Input", + CellChangeTimes->{ + 3.541407735337234*^9, {3.5414077707003508`*^9, 3.541407781471999*^9}, { + 3.541408115610737*^9, 3.541408162402279*^9}, {3.541409085265256*^9, + 3.541409086933219*^9}, {3.54140912603513*^9, 3.54140913825906*^9}, { + 3.541650280572554*^9, 3.541650285777712*^9}}], + +Cell[BoxData[ + RowBox[{"aalpha", ":=", + RowBox[{"FullSimplify", "[", + RowBox[{"Sqrt", "[", + RowBox[{ + RowBox[{"-", "1"}], "/", + RowBox[{ + RowBox[{"Inverse", "[", "g", "]"}], "[", + RowBox[{"[", + RowBox[{"1", ",", "1"}], "]"}], "]"}]}], "]"}], "]"}]}]], "Input", + CellChangeTimes->{{3.541650300536737*^9, 3.541650308665511*^9}, { + 3.5416503687152977`*^9, 3.5416503804793386`*^9}, {3.541650532412928*^9, + 3.541650564370451*^9}, {3.5416505986603622`*^9, 3.54165060601678*^9}}], + +Cell[BoxData[ + RowBox[{"G", " ", ":=", " ", + RowBox[{"Table", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", "2"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"k", ",", "l"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "i", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"j", ",", "l"}], "]"}], "]"}], "+", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "j", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "l"}], "]"}], "]"}], "-", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "l", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ")"}], "*", + RowBox[{"bbeta", "[", + RowBox[{"[", "k", "]"}], "]"}]}], ",", + RowBox[{"{", + RowBox[{"k", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"l", ",", "3"}], "}"}]}], "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.541407259375214*^9, 3.541407310526156*^9}, + 3.541407527171105*^9, {3.541408223327443*^9, 3.541408315725686*^9}, { + 3.54140835840478*^9, 3.541408488907057*^9}, {3.541408591554949*^9, + 3.541408641405265*^9}, {3.541408706675763*^9, 3.541408716630164*^9}, { + 3.541408795207923*^9, 3.5414088035820436`*^9}, {3.5414088523415213`*^9, + 3.541408982237076*^9}, 3.5414090560364122`*^9, {3.54140915459509*^9, + 3.541409166780575*^9}, {3.54140921014863*^9, 3.541409244803606*^9}, { + 3.541411331885972*^9, 3.541411350959804*^9}, {3.541491196736491*^9, + 3.541491196874372*^9}}], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Dbeta", " ", "=", " ", + RowBox[{"FullSimplify", "[", + RowBox[{"Table", "[", + RowBox[{ + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"bbeta", "[", + RowBox[{"[", "i", "]"}], "]"}], ",", "j", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"R", "'"}], "[", "r", "]"}], "\[Rule]", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "*", + RowBox[{"alpha", "/", + RowBox[{"(", "r", ")"}]}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"r", ",", "z", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "\[Rule]", + RowBox[{"z", "/", "r"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"r", ",", "x", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "\[Rule]", + RowBox[{"x", "/", "r"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{"r", ",", "y", ",", + RowBox[{"NonConstants", "\[Rule]", + RowBox[{"{", "r", "}"}]}]}], "]"}], "\[Rule]", + RowBox[{"y", "/", "r"}]}]}], "}"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", + RowBox[{"{", + RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], "}"}]}], "]"}], + "]"}]}]], "Input", + CellChangeTimes->{{3.5414135046567793`*^9, 3.541413520732307*^9}, { + 3.541413560650477*^9, 3.541413627625822*^9}, {3.541413658200732*^9, + 3.5414136887418222`*^9}, {3.541413860975049*^9, 3.541413865563333*^9}, { + 3.541413895894105*^9, 3.541414042262004*^9}, 3.541491352776266*^9, { + 3.541653270790123*^9, 3.541653274140605*^9}}], + +Cell[OutputFormData["\<\ +{{(-2*beta*C^2*x*(-r^2 + x^2 + y^2 + z^2)*(1 + 2*Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]) + + (-((1 + beta^2)*C*(r^2 - x^2*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))) - \ +2*beta*M*r^2*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + 2*beta*x*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6)/((-1 + \ +beta^2)*r^4*R[r]^4), + (-2*beta*C^2*y*(-r^2 + x^2 + y^2 + z^2)*(1 + 2*Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]) + + y*R[r]^3*((1 + beta^2)*C*x*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +2*beta*M*r^2*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] + + 2*beta*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3))/((-1 + \ +beta^2)*r^4*R[r]^4), + (-2*beta*C^2*z*(-r^2 + x^2 + y^2 + z^2)*(1 + 2*Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]) + + z*R[r]^3*((1 + beta^2)*C*x*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +2*beta*M*r^2*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] + + 2*beta*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3))/((-1 + \ +beta^2)*r^4*R[r]^4)}, + {-((C*x*y*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))/(Sqrt[1 - \ +beta^2]*r^4*R[r])), (C*(r^2 - y^2*(2 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])))/(Sqrt[1 - beta^2]*r^4*R[r]), + -((C*y*z*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))/(Sqrt[1 - \ +beta^2]*r^4*R[r]))}, {-((C*x*z*(2 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]))/(Sqrt[1 - beta^2]*r^4*R[r])), + -((C*y*z*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))/(Sqrt[1 - \ +beta^2]*r^4*R[r])), (C*(r^2 - z^2*(2 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])))/(Sqrt[1 - beta^2]*r^4*R[r])}}\ +\>", "\<\ + 2 + 2 2 2 2 2 C 2 M +{{(-2 beta C x (-r + x + y + z ) (1 + 2 Sqrt[1 + ----- - ----]) + + 4 R[r] + R[r] + + 2 \ + 2 2 + 2 2 2 C 2 M 2 \ + C 2 M 3 C 2 M 6 + (-((1 + beta ) C (r - x (2 + Sqrt[1 + ----- - ----]))) - 2 beta M r \ +x Sqrt[1 + ----- - ----]) R[r] + 2 beta x (-1 + Sqrt[1 + ----- - ----]) R[r] \ +) / + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + \ + 2 + 2 4 4 2 2 2 2 2 \ + C 2 M + ((-1 + beta ) r R[r] ), (-2 beta C y (-r + x + y + z ) (1 + 2 Sqrt[1 \ ++ ----- - ----]) + + \ + 4 R[r] + \ + R[r] + + 2 \ + 2 2 + 3 2 C 2 M 2 \ + C 2 M C 2 M 3 + y R[r] ((1 + beta ) C x (2 + Sqrt[1 + ----- - ----]) - 2 beta M r \ +Sqrt[1 + ----- - ----] + 2 beta (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + \ + 2 + 2 4 4 2 2 2 2 2 \ + C 2 M + ((-1 + beta ) r R[r] ), (-2 beta C z (-r + x + y + z ) (1 + 2 Sqrt[1 \ ++ ----- - ----]) + + \ + 4 R[r] + \ + R[r] + + 2 \ + 2 2 + 3 2 C 2 M 2 \ + C 2 M C 2 M 3 + z R[r] ((1 + beta ) C x (2 + Sqrt[1 + ----- - ----]) - 2 beta M r \ +Sqrt[1 + ----- - ----] + 2 beta (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 2 + C 2 M 2 \ + 2 C 2 M C 2 M + C x y (2 + Sqrt[1 + ----- - ----]) C (r - \ +y (2 + Sqrt[1 + ----- - ----])) C y z (2 + Sqrt[1 + ----- - ----]) + 4 R[r] \ + 4 R[r] 4 R[r] + 2 4 4 R[r] \ + R[r] R[r] + ((-1 + beta ) r R[r] )}, {-(----------------------------------), \ +----------------------------------------, \ +-(----------------------------------)}, + 2 4 \ + 2 4 2 4 + Sqrt[1 - beta ] r R[r] \ +Sqrt[1 - beta ] r R[r] Sqrt[1 - beta ] r R[r] + + 2 2 \ + 2 + C 2 M C 2 M \ + 2 2 C 2 M + C x z (2 + Sqrt[1 + ----- - ----]) C y z (2 + Sqrt[1 + ----- - \ +----]) C (r - z (2 + Sqrt[1 + ----- - ----])) + 4 R[r] 4 R[r] \ + 4 R[r] + R[r] R[r] \ + R[r] + {-(----------------------------------), \ +-(----------------------------------), \ +----------------------------------------}} + 2 4 2 4 \ + 2 4 + Sqrt[1 - beta ] r R[r] Sqrt[1 - beta ] r R[r] \ + Sqrt[1 - beta ] r R[r]\ +\>"], "Output", + CellChangeTimes->{3.5414913608543873`*^9, 3.5414917545772123`*^9, + 3.541522074603095*^9, 3.5415653581570053`*^9, 3.541565842812215*^9, + 3.541566164039706*^9, 3.5415668963680162`*^9, 3.541568770309145*^9, + 3.54157189097235*^9, 3.541573655439375*^9, 3.541576226550655*^9, + 3.541576525743513*^9, 3.541650670025827*^9, 3.5416507115342493`*^9, + 3.541653279624494*^9, 3.541659142523543*^9, 3.5416694933816967`*^9}] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"All", ",", "1"}], "]"}], "]"}], "=", + RowBox[{ + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"All", ",", "1"}], "]"}], "]"}], "*", "gamma"}]}]], "Input", + CellChangeTimes->{{3.541491364187389*^9, 3.541491382802622*^9}, + 3.541566902491549*^9}], + +Cell[OutputFormData["\<\ +{(-2*beta*C^2*x*(-r^2 + x^2 + y^2 + z^2)*(1 + 2*Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]) + + (-((1 + beta^2)*C*(r^2 - x^2*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))) - \ +2*beta*M*r^2*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + 2*beta*x*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6)/(Sqrt[1 - \ +beta^2]*(-1 + beta^2)*r^4*R[r]^4), + -((C*x*y*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))/((1 - beta^2)*r^4*R[r])), \ +-((C*x*z*(2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))/((1 - beta^2)*r^4*R[r]))}\ +\>", "\<\ + 2 + 2 2 2 2 2 C 2 M +{(-2 beta C x (-r + x + y + z ) (1 + 2 Sqrt[1 + ----- - ----]) + + 4 R[r] + R[r] + + 2 \ + 2 2 + 2 2 2 C 2 M 2 \ + C 2 M 3 C 2 M 6 + (-((1 + beta ) C (r - x (2 + Sqrt[1 + ----- - ----]))) - 2 beta M r x \ +Sqrt[1 + ----- - ----]) R[r] + 2 beta x (-1 + Sqrt[1 + ----- - ----]) R[r] ) \ +/ + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 + C 2 M \ + C 2 M + C x y (2 + Sqrt[1 + ----- - \ +----]) C x z (2 + Sqrt[1 + ----- - ----]) + 4 \ +R[r] 4 R[r] + 2 2 4 4 R[r] \ + R[r] + (Sqrt[1 - beta ] (-1 + beta ) r R[r] ), \ +-(----------------------------------), -(----------------------------------)} + 2 4 \ + 2 4 + (1 - beta ) r R[r] \ + (1 - beta ) r R[r]\ +\>"], "Output", + CellChangeTimes->{3.541491383696937*^9, 3.5415220781391897`*^9, + 3.541565846794992*^9, 3.541566176805335*^9, 3.541566906067164*^9, + 3.541568799285861*^9, 3.541571893799857*^9, 3.541573658120102*^9, + 3.541576228269662*^9, 3.541576534368664*^9, 3.5416507159632397`*^9, + 3.541653282067799*^9, 3.54165914794071*^9, 3.541669504568797*^9}] +}, Open ]], + +Cell[BoxData[ + RowBox[{"K", ":=", + RowBox[{"Table", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", + RowBox[{"(", + RowBox[{"2", "*", "aalpha"}], ")"}]}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{"beta", "*", + RowBox[{"g3dx", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], "+", + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], "-", + RowBox[{"G", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], "+", + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"j", ",", "i"}], "]"}], "]"}], "-", + RowBox[{"G", "[", + RowBox[{"[", + RowBox[{"j", ",", "i"}], "]"}], "]"}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.54141422395354*^9, 3.5414143223670197`*^9}, { + 3.541414361138858*^9, 3.541414403010789*^9}, {3.541491428085569*^9, + 3.541491430682425*^9}, {3.541491470357881*^9, 3.541491473762384*^9}, { + 3.5415669323933067`*^9, 3.5415669347687483`*^9}, 3.541650719124674*^9, + 3.541669536464787*^9}], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"KK", "=", + RowBox[{ + RowBox[{"FullSimplify", "[", "K", "]"}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{"r", "^", "2"}], "\[Rule]", + RowBox[{ + RowBox[{"x", "^", "2"}], "+", + RowBox[{"y", "^", "2"}], "+", + RowBox[{"z", "^", "2"}]}]}], "}"}]}]}]], "Input", + CellChangeTimes->{{3.5414914805661373`*^9, 3.541491535504921*^9}, { + 3.541491566316037*^9, 3.5414915746432867`*^9}, {3.54152312708692*^9, + 3.54152315393946*^9}, {3.541565859908424*^9, 3.5415659005325747`*^9}, { + 3.541573665884598*^9, 3.541573669055928*^9}, {3.54157599958922*^9, + 3.541576013023212*^9}}], + +Cell[OutputFormData["\<\ +{{-((Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + R[r]^4))/(beta^2*C^2*(r - \ +x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]* + (beta*(-(C^2*x*(3*(x^2 + y^2 + z^2) + (x^2 + y^2 + z^2)*(-2 - 5*Sqrt[1 + \ +C^2/R[r]^4 - (2*M)/R[r]]) + 3*(2*x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])) - + 2*beta^2*M^2*r^4*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] + beta*C*M*(x^2 \ ++ y^2 + z^2)*(3*x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + beta^3*M*r^4*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]*R[r]^4 + (C*(-x^2 - \ +y^2 - z^2 - (y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + + x^2*(1 + 2*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])) + 2*beta*M*x*(x^2 + \ +y^2 + z^2)*(1 - 2*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]))*R[r]^6 + + beta*x*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r]^7))/((1 - beta^2)^(3/2)*r^4*R[r]^3*(C^2 - 2*M*R[r]^3 + \ +R[r]^4))), + (y*(beta*C*(C*(y^2 + z^2 + x^2*(1 + 5*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + \ +2*(y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] - + 3*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +beta*M*x*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + (beta*M*(x^2 + y^2 + z^2)*(-2 + 3*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +3*C*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6 - + beta*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r]^7))/ + (r^4*R[r]*Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + \ +R[r]^4))/(beta^2*C^2*(r - x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + \ +z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]*(beta^2*C^2*(-y^2 - z^2) + \ +2*beta*(-(C*x) + beta*M*(x^2 + y^2 + z^2))*R[r]^3 - beta^2*(x^2 + y^2 + \ +z^2)*R[r]^4 + + R[r]^6)), + (z*(beta*C*(C*(y^2 + z^2 + x^2*(1 + 5*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + \ +2*(y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] - + 3*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +beta*M*x*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + (beta*M*(x^2 + y^2 + z^2)*(-2 + 3*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +3*C*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6 - + beta*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r]^7))/ + (r^4*R[r]*Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + \ +R[r]^4))/(beta^2*C^2*(r - x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + \ +z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]*(beta^2*C^2*(-y^2 - z^2) + \ +2*beta*(-(C*x) + beta*M*(x^2 + y^2 + z^2))*R[r]^3 - beta^2*(x^2 + y^2 + \ +z^2)*R[r]^4 + + R[r]^6))}, + {(y*(beta*C*(C*(y^2 + z^2 + x^2*(1 + 5*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + \ +2*(y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] - + 3*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +beta*M*x*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + (beta*M*(x^2 + y^2 + z^2)*(-2 + 3*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +3*C*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6 - + beta*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r]^7))/ + (r^4*R[r]*Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + \ +R[r]^4))/(beta^2*C^2*(r - x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + \ +z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]*(beta^2*C^2*(-y^2 - z^2) + \ +2*beta*(-(C*x) + beta*M*(x^2 + y^2 + z^2))*R[r]^3 - beta^2*(x^2 + y^2 + \ +z^2)*R[r]^4 + + R[r]^6)), (Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + \ +R[r]^4))/(beta^2*C^2*(r - x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + \ +z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]*(beta*C^2*x*(x^2 + y^2 + \ +z^2 + (x^2 + y^2 + z^2)*(-2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - + (x^2 - 2*y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + \ +R[r]^3*(-2*beta*M*x*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]) + + C*(x^2 + x^2*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + (-2*y^2 + \ +z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + + beta*x*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r])))/(Sqrt[1 - beta^2]*r^4*(C^2 - 2*M*R[r]^3 + R[r]^4)), + (-3*C*y*z*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]*(-(beta*C*x) + \ +R[r]^3)*Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + R[r]^4))/ + (beta^2*C^2*(r - x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + \ +z^2))*R[r]^3 + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)])/ + (Sqrt[1 - beta^2]*r^4*(C^2 - 2*M*R[r]^3 + R[r]^4))}, + {(z*(beta*C*(C*(y^2 + z^2 + x^2*(1 + 5*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + \ +2*(y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]] - + 3*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +beta*M*x*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + + (beta*M*(x^2 + y^2 + z^2)*(-2 + 3*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - \ +3*C*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6 - + beta*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r]^7))/ + (r^4*R[r]*Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + \ +R[r]^4))/(beta^2*C^2*(r - x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + \ +z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]*(beta^2*C^2*(-y^2 - z^2) + \ +2*beta*(-(C*x) + beta*M*(x^2 + y^2 + z^2))*R[r]^3 - beta^2*(x^2 + y^2 + \ +z^2)*R[r]^4 + + R[r]^6)), (-3*C*y*z*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]*(-(beta*C*x) + \ +R[r]^3)* + Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + R[r]^4))/(beta^2*C^2*(r - \ +x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)])/(Sqrt[1 - beta^2]*r^4*(C^2 \ +- 2*M*R[r]^3 + R[r]^4)), + (Sqrt[((-1 + beta^2)*R[r]^2*(C^2 - 2*M*R[r]^3 + R[r]^4))/(beta^2*C^2*(r - \ +x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + z^2))*R[r]^3 + + beta^2*(x^2 + y^2 + z^2)*R[r]^4 - R[r]^6)]*(beta*C^2*x*(x^2 + y^2 + \ +z^2 + (x^2 + y^2 + z^2)*(-2 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) - + (x^2 + y^2 - 2*z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + \ +R[r]^3*(-2*beta*M*x*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]]) + + C*(x^2 + x^2*(-1 + Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + (y^2 - \ +2*z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]]) + + beta*x*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \ +(2*M)/R[r]])*R[r])))/(Sqrt[1 - beta^2]*r^4*(C^2 - 2*M*R[r]^3 + R[r]^4))}}\ +\>", "\<\ + 2 2 2 3 \ + 4 + (-1 + beta ) R[r] (C - 2 M R[r] + \ +R[r] ) +{{-((Sqrt[--------------------------------------------------------------------\ +--------------------------------------] + 2 2 2 2 2 \ +3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + z )) \ +R[r] + beta (x + y + z ) R[r] - R[r] + + 2 \ + 2 + 2 2 2 2 2 2 2 C \ + 2 M 2 2 2 C 2 M + (beta (-(C x (3 (x + y + z ) + (x + y + z ) (-2 - 5 Sqrt[1 + \ +----- - ----]) + 3 (2 x + y + z ) Sqrt[1 + ----- - ----])) - + \ +4 R[r] 4 R[r] + R[r]\ + R[r] + + 2 \ + 2 + 2 2 4 C 2 M 2 2 2 \ + 2 2 2 C 2 M 3 + 2 beta M r x Sqrt[1 + ----- - ----] + beta C M (x + y + z ) \ +(3 x + y + z ) Sqrt[1 + ----- - ----]) R[r] + + 4 R[r] \ + 4 R[r] + R[r] \ + R[r] + + 2 \ + 2 2 + 3 4 C 2 M 4 2 2 2 2 \ + 2 C 2 M 2 C 2 M + beta M r x Sqrt[1 + ----- - ----] R[r] + (C (-x - y - z - (y \ ++ z ) (-1 + Sqrt[1 + ----- - ----]) + x (1 + 2 Sqrt[1 + ----- - ----])) + + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 + 2 2 2 C 2 M 6 \ + 2 2 2 C 2 M 7 + 2 beta M x (x + y + z ) (1 - 2 Sqrt[1 + ----- - ----])) R[r] \ ++ beta x (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] \ + 4 R[r] + R[r] \ + R[r] + + 2 3/2 4 3 2 3 4 + ((1 - beta ) r R[r] (C - 2 M R[r] + R[r] ))), (y (beta C + + 2 \ + 2 2 + 2 2 2 C 2 M 2 2 \ + C 2 M 2 2 2 C 2 M + (C (y + z + x (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y + z ) Sqrt[1 \ ++ ----- - ----] - 3 (x + y + z ) Sqrt[1 + ----- - ----]) - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 2 + 2 2 2 C 2 M 3 \ +2 2 2 C 2 M C 2 M \ + 6 + beta M x (x + y + z ) Sqrt[1 + ----- - ----]) R[r] + (beta M (x \ + + y + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \ +R[r] - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 + 2 2 2 C 2 M 7 + beta (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] + R[r] + + 2 2 2 \ + 3 4 + 4 (-1 + beta ) R[r] (C - 2 M \ +R[r] + R[r] ) + (r R[r] \ +Sqrt[-------------------------------------------------------------------------\ +---------------------------------] + 2 2 2 2 \ +2 3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + \ +z )) R[r] + beta (x + y + z ) R[r] - R[r] + + 2 2 2 2 2 2 2 3 \ + 2 2 2 2 4 6 + (beta C (-y - z ) + 2 beta (-(C x) + beta M (x + y + z )) R[r] - \ +beta (x + y + z ) R[r] + R[r] )), + + 2 \ + 2 2 + 2 2 2 C 2 M 2 2 \ + C 2 M 2 2 2 C 2 M + (z (beta C (C (y + z + x (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y + z ) \ +Sqrt[1 + ----- - ----] - 3 (x + y + z ) Sqrt[1 + ----- - ----]) - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 2 + 2 2 2 C 2 M 3 \ +2 2 2 C 2 M C 2 M \ + 6 + beta M x (x + y + z ) Sqrt[1 + ----- - ----]) R[r] + (beta M (x \ + + y + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \ +R[r] - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 + 2 2 2 C 2 M 7 + beta (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] + R[r] + + 2 2 2 \ + 3 4 + 4 (-1 + beta ) R[r] (C - 2 M \ +R[r] + R[r] ) + (r R[r] \ +Sqrt[-------------------------------------------------------------------------\ +---------------------------------] + 2 2 2 2 \ +2 3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + \ +z )) R[r] + beta (x + y + z ) R[r] - R[r] + + 2 2 2 2 2 2 2 3 \ + 2 2 2 2 4 6 + (beta C (-y - z ) + 2 beta (-(C x) + beta M (x + y + z )) R[r] - \ +beta (x + y + z ) R[r] + R[r] ))}, + + 2 \ + 2 2 + 2 2 2 C 2 M 2 2 \ + C 2 M 2 2 2 C 2 M + {(y (beta C (C (y + z + x (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y + z ) \ +Sqrt[1 + ----- - ----] - 3 (x + y + z ) Sqrt[1 + ----- - ----]) - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 2 + 2 2 2 C 2 M 3 \ +2 2 2 C 2 M C 2 M \ + 6 + beta M x (x + y + z ) Sqrt[1 + ----- - ----]) R[r] + (beta M (x \ + + y + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \ +R[r] - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 + 2 2 2 C 2 M 7 + beta (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] + R[r] + + 2 2 2 \ + 3 4 + 4 (-1 + beta ) R[r] (C - 2 M \ +R[r] + R[r] ) + (r R[r] \ +Sqrt[-------------------------------------------------------------------------\ +---------------------------------] + 2 2 2 2 \ +2 3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + \ +z )) R[r] + beta (x + y + z ) R[r] - R[r] + + 2 2 2 2 2 2 2 3 \ + 2 2 2 2 4 6 + (beta C (-y - z ) + 2 beta (-(C x) + beta M (x + y + z )) R[r] - \ +beta (x + y + z ) R[r] + R[r] )), + + 2 2 2 3 \ + 4 + (-1 + beta ) R[r] (C - 2 M R[r] + \ +R[r] ) + (Sqrt[---------------------------------------------------------------------\ +-------------------------------------] + 2 2 2 2 2 \ +3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + z )) R[r] \ + + beta (x + y + z ) R[r] - R[r] + + 2 \ + 2 + 2 2 2 2 2 2 2 C 2 M \ + 2 2 2 C 2 M + (beta C x (x + y + z + (x + y + z ) (-2 + Sqrt[1 + ----- - ----]) \ +- (x - 2 y + z ) Sqrt[1 + ----- - ----]) + + 4 R[r] \ + 4 R[r] + R[r] \ + R[r] + + 2 \ + 2 2 + 3 2 2 2 C 2 M \ +2 2 C 2 M 2 2 C 2 M + R[r] (-2 beta M x (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) + C \ +(x + x (-1 + Sqrt[1 + ----- - ----]) + (-2 y + z ) Sqrt[1 + ----- - ----]) \ ++ + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 + 2 2 2 C 2 M \ + 2 4 2 3 4 + beta x (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r]))) / \ +(Sqrt[1 - beta ] r (C - 2 M R[r] + R[r] )), + 4 R[r] + R[r] + + 2 + C 2 M 3 + -3 C y z Sqrt[1 + ----- - ----] (-(beta C x) + R[r] ) Sqrt[ + 4 R[r] + R[r] + + 2 2 2 3 \ + 4 + (-1 + beta ) R[r] (C - 2 M R[r] + \ +R[r] ) + -----------------------------------------------------------------------\ +-----------------------------------] + 2 2 2 2 2 3 \ + 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + z )) R[r] \ ++ beta (x + y + z ) R[r] - R[r] + ---------------------------------------------------------------------------\ +------------------------------------------------------------------------------\ +------------- + 2\ + 4 2 3 4 + Sqrt[1 - beta \ +] r (C - 2 M R[r] + R[r] ) + + 2 \ + 2 2 + 2 2 2 C 2 M 2 \ +2 C 2 M 2 2 2 C 2 M + }, {(z (beta C (C (y + z + x (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y + \ +z ) Sqrt[1 + ----- - ----] - 3 (x + y + z ) Sqrt[1 + ----- - ----]) - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 \ + 2 2 + 2 2 2 C 2 M 3 \ +2 2 2 C 2 M C 2 M \ + 6 + beta M x (x + y + z ) Sqrt[1 + ----- - ----]) R[r] + (beta M (x \ + + y + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \ +R[r] - + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 + 2 2 2 C 2 M 7 + beta (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / + 4 R[r] + R[r] + + 2 2 2 \ + 3 4 + 4 (-1 + beta ) R[r] (C - 2 M \ +R[r] + R[r] ) + (r R[r] \ +Sqrt[-------------------------------------------------------------------------\ +---------------------------------] + 2 2 2 2 \ +2 3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + \ +z )) R[r] + beta (x + y + z ) R[r] - R[r] + + 2 2 2 2 2 2 2 3 \ + 2 2 2 2 4 6 + (beta C (-y - z ) + 2 beta (-(C x) + beta M (x + y + z )) R[r] - \ +beta (x + y + z ) R[r] + R[r] )), + + 2 + C 2 M 3 + -3 C y z Sqrt[1 + ----- - ----] (-(beta C x) + R[r] ) Sqrt[ + 4 R[r] + R[r] + + 2 2 2 3 \ + 4 + (-1 + beta ) R[r] (C - 2 M R[r] + \ +R[r] ) + -----------------------------------------------------------------------\ +-----------------------------------] + 2 2 2 2 2 3 \ + 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + z )) R[r] \ ++ beta (x + y + z ) R[r] - R[r] + ---------------------------------------------------------------------------\ +------------------------------------------------------------------------------\ +------------- + 2\ + 4 2 3 4 + Sqrt[1 - beta \ +] r (C - 2 M R[r] + R[r] ) + + 2 2 2 3 \ + 4 + (-1 + beta ) R[r] (C - 2 M R[r] \ ++ R[r] ) + , (Sqrt[------------------------------------------------------------------\ +----------------------------------------] + 2 2 2 2 2 \ + 3 2 2 2 2 4 6 + beta C (r - x) (r + x) + 2 beta (C x - beta M (x + y + z )) \ +R[r] + beta (x + y + z ) R[r] - R[r] + + 2 \ + 2 + 2 2 2 2 2 2 2 C 2 M \ + 2 2 2 C 2 M + (beta C x (x + y + z + (x + y + z ) (-2 + Sqrt[1 + ----- - ----]) \ +- (x + y - 2 z ) Sqrt[1 + ----- - ----]) + + 4 R[r] \ + 4 R[r] + R[r] \ + R[r] + + 2 \ + 2 2 + 3 2 2 2 C 2 M \ +2 2 C 2 M 2 2 C 2 M + R[r] (-2 beta M x (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) + C \ +(x + x (-1 + Sqrt[1 + ----- - ----]) + (y - 2 z ) Sqrt[1 + ----- - ----]) + + 4 R[r] \ + 4 R[r] 4 R[r] + R[r] \ + R[r] R[r] + + 2 + 2 2 2 C 2 M \ + 2 4 2 3 4 + beta x (x + y + z ) (-1 + Sqrt[1 + ----- - ----]) R[r]))) / \ +(Sqrt[1 - beta ] r (C - 2 M R[r] + R[r] ))}} + 4 R[r] + R[r]\ +\>"], "Output", + CellChangeTimes->{3.5415659162551613`*^9, 3.5415670154406013`*^9, + 3.541568858667211*^9, 3.5415719548630667`*^9, 3.541573731918861*^9, + 3.541576030889586*^9, 3.541576277931704*^9, 3.541576595673015*^9, + 3.541650765091838*^9, 3.541653324871229*^9, 3.541659221800127*^9, + 3.54166957932972*^9}] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"KK", "[", + RowBox[{"[", + RowBox[{"1", ",", "3"}], "]"}], "]"}], "/.", + RowBox[{"{", + RowBox[{"beta", "\[Rule]", "0"}], "}"}]}], "]"}]], "Input", + CellChangeTimes->{{3.541575263897523*^9, 3.541575270303776*^9}, { + 3.541575301276211*^9, 3.541575304678358*^9}, {3.541575343858225*^9, + 3.541575357692504*^9}, {3.54157604137838*^9, 3.541576051460884*^9}, { + 3.54165209894792*^9, 3.541652100910768*^9}, {3.541652175159631*^9, + 3.541652175210434*^9}, {3.541652808490815*^9, 3.5416528107242107`*^9}, { + 3.541653334914645*^9, 3.541653335598914*^9}, {3.541653384003544*^9, + 3.541653384830377*^9}, {3.5416593421904993`*^9, 3.541659342312196*^9}}], + +Cell[OutputFormData["\<\ +(-3*C*x*z)/(r^4*R[r])\ +\>", "\<\ +-3 C x z +-------- + 4 +r R[r]\ +\>"], "Output", + CellChangeTimes->{3.54157605191965*^9, 3.541576301187193*^9, + 3.541576603679607*^9, 3.5416507867271214`*^9, 3.541652102035613*^9, + 3.541652175760277*^9, 3.541652812453547*^9, 3.5416533360248537`*^9, + 3.541653385183237*^9, 3.541659241340852*^9, 3.541659342728558*^9}] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{ + RowBox[{"CForm", "[", + RowBox[{"FullSimplify", "[", + RowBox[{"KK", "[", + RowBox[{"[", + RowBox[{"2", ",", "3"}], "]"}], "]"}], "]"}], "]"}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "\[Rule]", "R"}], ",", + RowBox[{"M", "\[Rule]", "MASS"}]}], "}"}]}]], "Input", + CellChangeTimes->CompressedData[" +1:eJxTTMoPSmViYGCQAGIQ7Z0h4sic/MqxuFHTCUQbhBl4g2iTE8b+IHr+pFt5 +IHpbybV8EP36Sm8diP5msgJMv/MqbgbRC/51g+kjeos6QPQxPfFOEH3iu9BE +EF3wXhRMHzKZNB1Es+ZNB9PWlz2/aQFpsfcn/oPoZaszWbWBtNTMZjCt+E2N +G0Q3PQgB0+c8vgmB6A5rXmEQ/evze0kQbbP/C5iOOFIpB6KD/teC6TOLa8/r +Aumt+TPBdESa8S0QPavVDEyv3xh2D0TrJHY/BNExVlufguibUy+A6ReNJ96B +6JyYB2D62N3CW0ZA2m/H+jsgmmmm0EMQXblhyRMQ7Ra86AWIFgi/8QZEy9iv +/QCiw3Q2gmkuW9avILpbnBNMAwDRxLZ+ + "]], + +Cell["\<\ +(-3*C*Sqrt(1 + Power(C,2)/Power(R,4) - (2*MASS)/R)*(Power(R,3) - beta*C*x)*y*z* + Sqrt(((-1 + Power(beta,2))*Power(R,2)*(Power(C,2) - 2*MASS*Power(R,3) + \ +Power(R,4)))/ + (-Power(R,6) + Power(beta,2)*Power(C,2)*(r - x)*(r + x) + \ +Power(beta,2)*Power(R,4)*(Power(x,2) + Power(y,2) + Power(z,2)) + + 2*beta*Power(R,3)*(C*x - beta*MASS*(Power(x,2) + Power(y,2) + \ +Power(z,2))))))/ + (Sqrt(1 - Power(beta,2))*Power(r,4)*(Power(C,2) - 2*MASS*Power(R,3) + \ +Power(R,4)))\ +\>", "Output", + CellChangeTimes->{ + 3.541572113644475*^9, {3.541572187630993*^9, 3.5415722179745502`*^9}, + 3.5415724981770363`*^9, 3.5415726073312407`*^9, 3.5415726447101507`*^9, + 3.5415726909153967`*^9, 3.541572746514616*^9, 3.541572798320188*^9, { + 3.541653453666379*^9, 3.5416535026673393`*^9}, 3.5416535510033216`*^9, + 3.541653596991078*^9, 3.54165367433136*^9, {3.541653722246106*^9, + 3.541653749013379*^9}, 3.5416592620393057`*^9, 3.541659306343055*^9, { + 3.541659351273757*^9, 3.5416594076604347`*^9}, 3.5416594419313107`*^9, + 3.541659529013665*^9, 3.541669608978157*^9, {3.541669643694899*^9, + 3.541669670981832*^9}, {3.541669720959494*^9, 3.541669735859336*^9}, { + 3.5416697765681667`*^9, 3.5416698011993017`*^9}}] +}, Open ]], + +Cell[BoxData[ + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"KK", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}], "-", + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"(", + RowBox[{"g3u", ".", "KK", ".", "g3u"}], ")"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], + RowBox[{"KK", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.541497080925271*^9, 3.541497142282139*^9}, { + 3.5414971792901707`*^9, 3.541497201004222*^9}, {3.541497232025593*^9, + 3.5414972603387737`*^9}, {3.541523114657205*^9, 3.541523114912692*^9}, { + 3.541556931292919*^9, 3.54155693146544*^9}}], + +Cell[BoxData[""], "Input", + CellChangeTimes->{{3.541567755294497*^9, 3.541567760710601*^9}}] +}, +WindowSize->{1472, 1200}, +WindowMargins->{{0, Automatic}, {Automatic, 0}}, +FrontEndVersion->"7.0 for Linux x86 (64-bit) (February 25, 2009)", +StyleDefinitions->"Default.nb" +] +(* End of Notebook Content *) + +(* Internal cache information *) +(*CellTagsOutline +CellTagsIndex->{} +*) +(*CellTagsIndex +CellTagsIndex->{} +*) +(*NotebookFileOutline +Notebook[{ +Cell[545, 20, 137, 4, 32, "Input"], +Cell[685, 26, 407, 9, 32, "Input"], +Cell[1095, 37, 444, 10, 32, "Input"], +Cell[1542, 49, 440, 10, 32, "Input"], +Cell[1985, 61, 379, 11, 32, "Input"], +Cell[2367, 74, 327, 8, 32, "Input"], +Cell[2697, 84, 976, 28, 32, "Input"], +Cell[3676, 114, 803, 21, 32, "Input"], +Cell[4482, 137, 258, 5, 32, "Input"], +Cell[4743, 144, 300, 8, 32, "Input"], +Cell[5046, 154, 196, 4, 32, "Input"], +Cell[5245, 160, 948, 25, 32, "Input"], +Cell[6196, 187, 849, 24, 32, "Input"], +Cell[7048, 213, 1036, 27, 32, "Input"], +Cell[8087, 242, 234, 5, 32, "Input"], +Cell[8324, 249, 445, 10, 32, "Input"], +Cell[8772, 261, 511, 12, 32, "Input"], +Cell[9286, 275, 1826, 46, 32, "Input"], +Cell[CellGroupData[{ +Cell[11137, 325, 1899, 50, 99, "Input"], +Cell[13039, 377, 7415, 132, 588, "Output"] +}, Open ]], +Cell[CellGroupData[{ +Cell[20491, 514, 337, 10, 32, "Input"], +Cell[20831, 526, 2882, 49, 238, "Output"] +}, Open ]], +Cell[23728, 578, 1232, 34, 32, "Input"], +Cell[CellGroupData[{ +Cell[24985, 616, 623, 15, 32, "Input"], +Cell[25611, 633, 27033, 490, 2337, "Output"] +}, Open ]], +Cell[CellGroupData[{ +Cell[52681, 1128, 728, 14, 32, "Input"], +Cell[53412, 1144, 377, 11, 69, "Output"] +}, Open ]], +Cell[CellGroupData[{ +Cell[53826, 1160, 791, 20, 32, "Input"], +Cell[54620, 1182, 1244, 21, 112, "Output"] +}, Open ]], +Cell[55879, 1206, 1101, 33, 32, "Input"], +Cell[56983, 1241, 92, 1, 32, "Input"] +} +] +*) + +(* End of internal cache information *) diff --git a/trumpet2.nb b/trumpet2.nb new file mode 100644 index 0000000..c14dfb5 --- /dev/null +++ b/trumpet2.nb @@ -0,0 +1,1441 @@ +(* Content-type: application/mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 7.0' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 145, 7] +NotebookDataLength[ 56179, 1432] +NotebookOptionsPosition[ 53828, 1360] +NotebookOutlinePosition[ 54165, 1375] +CellTagsIndexPosition[ 54122, 1372] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ +Cell[BoxData[ + RowBox[{"\[IndentingNewLine]", + RowBox[{ + RowBox[{"defderiv", "=", + RowBox[{"{", + RowBox[{ + RowBox[{"alpha", "\[Rule]", + RowBox[{"alpha", "[", + RowBox[{"R", "[", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}], "]"}]}], ",", + RowBox[{"beta1", "\[Rule]", + RowBox[{"beta1", "[", + RowBox[{"x", ",", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", + RowBox[{"R", "[", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}], "]"}]}], ",", + RowBox[{"beta2", "->", + RowBox[{"beta2", "[", + RowBox[{"y", ",", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", + RowBox[{"R", "[", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}], "]"}]}], ",", + RowBox[{"beta3", "->", + RowBox[{"beta3", "[", + RowBox[{"z", ",", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", + RowBox[{"R", "[", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}], "]"}]}], ",", + RowBox[{"R", "\[Rule]", + RowBox[{"R", "[", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], "]"}]}], ",", + RowBox[{"r", "\[Rule]", + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}]}]}], "}"}]}], + ";"}]}]], "Input", + CellChangeTimes->{{3.544057963240076*^9, 3.5440579895811253`*^9}, { + 3.544058055917317*^9, 3.544058056730595*^9}}], + +Cell[BoxData[{ + RowBox[{ + RowBox[{"undefderiv", "=", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"alpha", "[", "R_", "]"}], "\[Rule]", "alpha"}], ",", + RowBox[{ + RowBox[{"beta1", "[", + RowBox[{"x", ",", "r_", ",", "R_"}], "]"}], "\[Rule]", "beta1"}], ",", + + RowBox[{ + RowBox[{"beta2", "[", + RowBox[{"y", ",", "r_", ",", "R_"}], "]"}], "\[Rule]", "beta2"}], ",", + + RowBox[{ + RowBox[{"beta3", "[", + RowBox[{"z", ",", "r_", ",", "R_"}], "]"}], "\[Rule]", "beta3"}], ",", + + RowBox[{ + RowBox[{"R", "[", "r_", "]"}], "\[Rule]", "R"}], ",", + RowBox[{ + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], "\[Rule]", "r"}]}], "}"}]}], + ";"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"expderiv", "=", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"R", "'"}], "[", "r_", "]"}], "\[Rule]", + RowBox[{"R", "*", + RowBox[{"alpha", "/", "r"}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", "x"}], "]"}], + "\[Rule]", + RowBox[{"x", "/", "r"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", "y"}], "]"}], + "\[Rule]", + RowBox[{"y", "/", "r"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"r", "[", + RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", "z"}], "]"}], + "\[Rule]", + RowBox[{"z", "/", "r"}]}], ",", + RowBox[{ + RowBox[{ + RowBox[{"alpha", "'"}], "[", "R_", "]"}], "\[Rule]", + RowBox[{ + RowBox[{"(", + RowBox[{ + RowBox[{"2", "*", + RowBox[{"M", "/", + RowBox[{"R", "^", "2"}]}]}], " ", "-", " ", + RowBox[{"4", + RowBox[{ + RowBox[{"C", "^", "2"}], "/", + RowBox[{"R", "^", "5"}]}]}]}], ")"}], "/", + RowBox[{"(", + RowBox[{"2", "*", "alpha"}], ")"}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta1", "[", + RowBox[{"x", ",", "r_", ",", "R_"}], "]"}], ",", "x"}], "]"}], + "\[Rule]", + RowBox[{"beta1", "/", "x"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta1", "[", + RowBox[{"x", ",", "r_", ",", "R_"}], "]"}], ",", "r_"}], "]"}], + "\[Rule]", + RowBox[{ + RowBox[{"-", "2"}], "*", + RowBox[{"beta1", "/", "r"}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta1", "[", + RowBox[{"x", ",", "r_", ",", "R_"}], "]"}], ",", "R_"}], "]"}], + "\[Rule]", + RowBox[{ + RowBox[{"-", "beta1"}], "/", "R"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta2", "[", + RowBox[{"y", ",", "r_", ",", "R_"}], "]"}], ",", "y"}], "]"}], + "\[Rule]", + RowBox[{"beta2", "/", "y"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta2", "[", + RowBox[{"y", ",", "r_", ",", "R_"}], "]"}], ",", "r_"}], "]"}], + "\[Rule]", + RowBox[{ + RowBox[{"-", "2"}], "*", + RowBox[{"beta2", "/", "r"}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta2", "[", + RowBox[{"y", ",", "r_", ",", "R_"}], "]"}], ",", "R_"}], "]"}], + "\[Rule]", + RowBox[{ + RowBox[{"-", "beta2"}], "/", "R"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta3", "[", + RowBox[{"z", ",", "r_", ",", "R_"}], "]"}], ",", "z"}], "]"}], + "\[Rule]", + RowBox[{"beta3", "/", "z"}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta3", "[", + RowBox[{"z", ",", "r_", ",", "R_"}], "]"}], ",", "r_"}], "]"}], + "\[Rule]", + RowBox[{ + RowBox[{"-", "2"}], "*", + RowBox[{"beta3", "/", "r"}]}]}], ",", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"beta3", "[", + RowBox[{"z", ",", "r_", ",", "R_"}], "]"}], ",", "R_"}], "]"}], + "\[Rule]", + RowBox[{ + RowBox[{"-", "beta3"}], "/", "R"}]}]}], "}"}]}], + ";"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"expbeta", "=", + RowBox[{"{", + RowBox[{ + RowBox[{"beta1", "\[Rule]", + RowBox[{"C", "*", + RowBox[{"x", "/", + RowBox[{"(", + RowBox[{ + RowBox[{"r", "^", "2"}], "*", "R"}], ")"}]}]}]}], ",", + RowBox[{"beta2", "\[Rule]", + RowBox[{"C", "*", + RowBox[{"y", "/", + RowBox[{"(", + RowBox[{ + RowBox[{"r", "^", "2"}], "*", "R"}], ")"}]}]}]}], ",", + RowBox[{"beta3", "\[Rule]", + RowBox[{"C", "*", + RowBox[{"z", "/", + RowBox[{"(", + RowBox[{ + RowBox[{"r", "^", "2"}], "*", "R"}], ")"}]}]}]}]}], "}"}]}], + ";"}]}], "Input", + CellChangeTimes->{{3.5440579944883223`*^9, 3.544058014412415*^9}, { + 3.5440580588081303`*^9, 3.544058061055698*^9}, {3.544058181178363*^9, + 3.5440581953198423`*^9}, {3.5440582274336863`*^9, 3.5440582332787323`*^9}, { + 3.5440582935208063`*^9, 3.544058495273024*^9}, {3.544058574181799*^9, + 3.5440586234092607`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"psi", "=", + RowBox[{"Sqrt", "[", + RowBox[{"R", "/", "r"}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544016806249206*^9, 3.54401682306608*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"beta1", "=", + RowBox[{ + RowBox[{"psi", "^", "4"}], "*", " ", "x", "*", + RowBox[{"C", "/", + RowBox[{"R", "^", "3"}]}]}]}], ";"}]], "Input", + CellChangeTimes->{{3.541405061030883*^9, 3.541405081211104*^9}, { + 3.541568693038992*^9, 3.54156869591467*^9}, 3.541576353980927*^9, { + 3.541578112640421*^9, 3.541578113917652*^9}, 3.541650183093624*^9, { + 3.544016810104436*^9, 3.544016839992446*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"beta2", "=", + RowBox[{ + RowBox[{"psi", "^", "4"}], "*", " ", "y", "*", + RowBox[{"C", "/", + RowBox[{"R", "^", "3"}]}]}]}], ";"}]], "Input", + CellChangeTimes->{{3.541405087435377*^9, 3.541405089672179*^9}, { + 3.5415686986023483`*^9, 3.5415687083941183`*^9}, 3.5415763576834927`*^9, { + 3.5415781173360577`*^9, 3.541578119110003*^9}, {3.541650185131393*^9, + 3.541650185952448*^9}, {3.5440168142204943`*^9, 3.5440168435077477`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"beta3", "=", + RowBox[{ + RowBox[{"psi", "^", "4"}], " ", "*", "z", "*", + RowBox[{"C", "/", + RowBox[{"R", "^", "3"}]}]}]}], ";"}]], "Input", + CellChangeTimes->{{3.541405093909099*^9, 3.541405096428109*^9}, { + 3.541568711652349*^9, 3.5415687140903873`*^9}, 3.5415763611864567`*^9, { + 3.541578120891581*^9, 3.541578122699366*^9}, {3.541650188191567*^9, + 3.541650188911887*^9}, {3.544016816725215*^9, 3.54401684689876*^9}}], + +Cell[BoxData[ + RowBox[{"alpha", ":=", + RowBox[{"Sqrt", "[", + RowBox[{"1", "-", + RowBox[{"2", "*", + RowBox[{"M", "/", + RowBox[{"R", "[", "r", "]"}]}]}], "+", + RowBox[{ + RowBox[{"C", "^", "2"}], "/", + RowBox[{ + RowBox[{"R", "[", "r", "]"}], "^", "4"}]}]}], "]"}]}]], "Input", + CellChangeTimes->{{3.5414051009800997`*^9, 3.541405117339251*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"gamma", "=", + RowBox[{"1", "/", + RowBox[{"Sqrt", "[", + RowBox[{"1", "-", + RowBox[{"beta", "^", "2"}]}], "]"}]}]}], ";"}]], "Input", + CellChangeTimes->{{3.541405132291383*^9, 3.541405138684085*^9}, + 3.541568684612611*^9, 3.5415687173641*^9, {3.541573628926816*^9, + 3.5415736363091927`*^9}, {3.544016850032361*^9, 3.54401685080128*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"gorig", "=", + RowBox[{"{", + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"-", + RowBox[{"alpha", "^", "2"}]}], "+", + RowBox[{ + RowBox[{"psi", "^", + RowBox[{"(", + RowBox[{"-", "4"}], ")"}]}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{"beta1", "^", "2"}], "+", + RowBox[{"beta2", "^", "2"}], "+", + RowBox[{"beta3", "^", "2"}]}], ")"}]}]}], ",", " ", "beta1", ",", + " ", "beta2", ",", " ", "beta3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"beta1", ",", + RowBox[{"psi", "^", "4"}], ",", "0", ",", "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"beta2", ",", "0", ",", + RowBox[{"psi", "^", "4"}], ",", "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"beta3", ",", "0", ",", "0", ",", + RowBox[{"psi", "^", "4"}]}], "}"}]}], "}"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541405161936751*^9, 3.541405263881534*^9}, { + 3.544016854927958*^9, 3.5440168600784397`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"Lambda", "=", + RowBox[{"{", + RowBox[{ + RowBox[{"{", + RowBox[{"gamma", ",", + RowBox[{ + RowBox[{"-", "gamma"}], "*", "beta"}], ",", "0", ",", "0"}], "}"}], + ",", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"-", "gamma"}], "*", "beta"}], ",", "gamma", ",", "0", ",", + "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"0", ",", "0", ",", "1", ",", "0"}], "}"}], ",", + RowBox[{"{", + RowBox[{"0", ",", "0", ",", "0", ",", "1"}], "}"}]}], "}"}]}], + ";"}]], "Input", + CellChangeTimes->{{3.541405309979129*^9, 3.5414053406975393`*^9}, { + 3.541576211047379*^9, 3.541576214954173*^9}, {3.5415766134805937`*^9, + 3.541576615866742*^9}, {3.541577989494315*^9, 3.541577993214196*^9}, { + 3.541659117039507*^9, 3.541659120385696*^9}, {3.544016863413391*^9, + 3.5440168650751677`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g", "=", + RowBox[{"FullSimplify", "[", + RowBox[{"Lambda", ".", "gorig", ".", "Lambda"}], "]"}]}], ";"}]], "Input",\ + + CellChangeTimes->{{3.5416502169868383`*^9, 3.5416502287092876`*^9}, { + 3.5416506915108023`*^9, 3.541650691651325*^9}, {3.543728529472261*^9, + 3.5437285297106037`*^9}, {3.54389469154531*^9, 3.543894697383884*^9}, { + 3.5440168674066153`*^9, 3.544016868165395*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g3d", "=", + RowBox[{"g", "[", + RowBox[{"[", + RowBox[{ + RowBox[{"2", ";;", "4"}], ",", + RowBox[{"2", ";;", "4"}]}], "]"}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541405355823965*^9, 3.541405399740391*^9}, { + 3.5416502327337837`*^9, 3.541650240217704*^9}, {3.5440168703453913`*^9, + 3.5440168718342323`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g3u", "=", + RowBox[{"FullSimplify", "[", + RowBox[{"Inverse", "[", "g3d", "]"}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.5414054223252373`*^9, 3.541405432235853*^9}, { + 3.54401687376943*^9, 3.544016874555067*^9}, {3.5440881435466833`*^9, + 3.544088156846258*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g3dy", "=", + RowBox[{ + RowBox[{"FullSimplify", "[", + RowBox[{"D", "[", + RowBox[{ + RowBox[{"g3d", "/.", "defderiv"}], ",", "y"}], "]"}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541406634290625*^9, 3.541406800404162*^9}, { + 3.541406833684698*^9, 3.541406879015818*^9}, {3.5414069575741167`*^9, + 3.541406967124864*^9}, {3.5416532441838408`*^9, 3.541653248021184*^9}, { + 3.543677941065379*^9, 3.543677941977008*^9}, {3.5436782551946383`*^9, + 3.543678326932147*^9}, {3.543678397848588*^9, 3.5436784166317263`*^9}, { + 3.5436790498272038`*^9, 3.543679059902656*^9}, 3.543679289765324*^9, { + 3.543679351247592*^9, 3.543679352442617*^9}, {3.543680143285253*^9, + 3.5436801455670424`*^9}, {3.543714895986582*^9, 3.5437149543781424`*^9}, { + 3.5440168775572777`*^9, 3.5440169018251762`*^9}, {3.544016938370672*^9, + 3.544016939324675*^9}, 3.5440169859268417`*^9, {3.544057525032783*^9, + 3.5440576833886757`*^9}, 3.544057750052329*^9, {3.544057979664452*^9, + 3.5440580091736813`*^9}, {3.544058066678643*^9, 3.544058077555372*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g3dz", "=", + RowBox[{ + RowBox[{"FullSimplify", "[", + RowBox[{"D", "[", + RowBox[{ + RowBox[{"g3d", "/.", "defderiv"}], ",", "z"}], "]"}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541406979031295*^9, 3.541406987189321*^9}, { + 3.541653251676744*^9, 3.541653254754826*^9}, {3.543678332496558*^9, + 3.543678349142379*^9}, {3.54367844552579*^9, 3.5436784495093393`*^9}, { + 3.543679067977374*^9, 3.5436790758239098`*^9}, {3.543680152505425*^9, + 3.5436801543446617`*^9}, {3.54371498646307*^9, 3.543714996298427*^9}, { + 3.54401695830163*^9, 3.544016981007206*^9}, {3.544057715097053*^9, + 3.544057753592243*^9}, {3.5440580843461943`*^9, 3.544058096494072*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g3dx", "=", + RowBox[{ + RowBox[{"FullSimplify", "[", + RowBox[{"gamma", "*", + RowBox[{"D", "[", + RowBox[{ + RowBox[{"g3d", "/.", "defderiv"}], ",", "x"}], "]"}]}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541406991078109*^9, 3.5414069986545753`*^9}, { + 3.541491207633357*^9, 3.5414912099279613`*^9}, {3.5415665727617483`*^9, + 3.541566574896678*^9}, {3.5415668117623043`*^9, 3.541566812953938*^9}, { + 3.5416532580164423`*^9, 3.541653261549081*^9}, {3.543678356527137*^9, + 3.5436783634796352`*^9}, {3.543678457207299*^9, 3.5436784728819313`*^9}, { + 3.543679085541403*^9, 3.543679105802004*^9}, {3.543680167746161*^9, + 3.543680169937104*^9}, {3.543715007673709*^9, 3.543715030389812*^9}, { + 3.5440169932934837`*^9, 3.544017017415502*^9}, {3.544057769348524*^9, + 3.544057793540884*^9}, {3.5440581138803864`*^9, 3.544058123641137*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"g3diff", "=", + RowBox[{"{", + RowBox[{"g3dx", ",", "g3dy", ",", "g3dz"}], "}"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541407188389018*^9, 3.54140719780622*^9}, { + 3.54140730189473*^9, 3.541407302322214*^9}, {3.5440170215459623`*^9, + 3.544017022160943*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"bbeta", "=", + RowBox[{"g", "[", + RowBox[{"[", + RowBox[{"1", ",", + RowBox[{"2", ";;", "4"}]}], "]"}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{ + 3.541407735337234*^9, {3.5414077707003508`*^9, 3.541407781471999*^9}, { + 3.541408115610737*^9, 3.541408162402279*^9}, {3.541409085265256*^9, + 3.541409086933219*^9}, {3.54140912603513*^9, 3.54140913825906*^9}, { + 3.541650280572554*^9, 3.541650285777712*^9}, {3.544017024321209*^9, + 3.5440170252281237`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"aalpha", "=", + RowBox[{"FullSimplify", "[", + RowBox[{"Sqrt", "[", + RowBox[{ + RowBox[{"-", "1"}], "/", + RowBox[{ + RowBox[{"Inverse", "[", "g", "]"}], "[", + RowBox[{"[", + RowBox[{"1", ",", "1"}], "]"}], "]"}]}], "]"}], "]"}]}], + ";"}]], "Input", + CellChangeTimes->{{3.541650300536737*^9, 3.541650308665511*^9}, { + 3.5416503687152977`*^9, 3.5416503804793386`*^9}, {3.541650532412928*^9, + 3.541650564370451*^9}, {3.5416505986603622`*^9, 3.54165060601678*^9}, { + 3.544017027963932*^9, 3.54401702858636*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"G", " ", "=", " ", + RowBox[{"Table", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", "2"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"k", ",", "l"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "i", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"j", ",", "l"}], "]"}], "]"}], "+", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "j", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "l"}], "]"}], "]"}], "-", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "l", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ")"}], "*", + RowBox[{"bbeta", "[", + RowBox[{"[", "k", "]"}], "]"}]}], ",", + RowBox[{"{", + RowBox[{"k", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"l", ",", "3"}], "}"}]}], "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.541407259375214*^9, 3.541407310526156*^9}, + 3.541407527171105*^9, {3.541408223327443*^9, 3.541408315725686*^9}, { + 3.54140835840478*^9, 3.541408488907057*^9}, {3.541408591554949*^9, + 3.541408641405265*^9}, {3.541408706675763*^9, 3.541408716630164*^9}, { + 3.541408795207923*^9, 3.5414088035820436`*^9}, {3.5414088523415213`*^9, + 3.541408982237076*^9}, 3.5414090560364122`*^9, {3.54140915459509*^9, + 3.541409166780575*^9}, {3.54140921014863*^9, 3.541409244803606*^9}, { + 3.541411331885972*^9, 3.541411350959804*^9}, {3.541491196736491*^9, + 3.541491196874372*^9}, {3.544017032403316*^9, 3.544017033316012*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"Dbeta", " ", "=", " ", + RowBox[{ + RowBox[{"FullSimplify", "[", + RowBox[{"Table", "[", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{ + RowBox[{"bbeta", "[", + RowBox[{"[", "i", "]"}], "]"}], "/.", "defderiv"}], ",", "j"}], + "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", + RowBox[{"{", + RowBox[{"x", ",", "y", ",", "z"}], "}"}]}], "}"}]}], "]"}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.5414135046567793`*^9, 3.541413520732307*^9}, { + 3.541413560650477*^9, 3.541413627625822*^9}, {3.541413658200732*^9, + 3.5414136887418222`*^9}, {3.541413860975049*^9, 3.541413865563333*^9}, { + 3.541413895894105*^9, 3.541414042262004*^9}, 3.541491352776266*^9, { + 3.541653270790123*^9, 3.541653274140605*^9}, {3.543678568798662*^9, + 3.5436786478435793`*^9}, {3.543679126902885*^9, 3.543679142493442*^9}, { + 3.543680191732459*^9, 3.543680206934805*^9}, {3.544017054683242*^9, + 3.544017083270424*^9}, 3.544017113692617*^9, {3.544058141437265*^9, + 3.5440581696753807`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{ + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"All", ",", "1"}], "]"}], "]"}], "=", + RowBox[{ + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"All", ",", "1"}], "]"}], "]"}], "*", "gamma"}]}], + ";"}]], "Input", + CellChangeTimes->{{3.541491364187389*^9, 3.541491382802622*^9}, + 3.541566902491549*^9, 3.544017133923883*^9}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"K", "=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{ + RowBox[{"Table", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", + RowBox[{"(", + RowBox[{"2", "*", "aalpha"}], ")"}]}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{"beta", "*", + RowBox[{"g3dx", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], "+", + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], "-", + RowBox[{"G", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], "+", + RowBox[{"Dbeta", "[", + RowBox[{"[", + RowBox[{"j", ",", "i"}], "]"}], "]"}], "-", + RowBox[{"G", "[", + RowBox[{"[", + RowBox[{"j", ",", "i"}], "]"}], "]"}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}], "/.", "expderiv"}], "/.", + "expbeta"}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.54141422395354*^9, 3.5414143223670197`*^9}, { + 3.541414361138858*^9, 3.541414403010789*^9}, {3.541491428085569*^9, + 3.541491430682425*^9}, {3.541491470357881*^9, 3.541491473762384*^9}, { + 3.5415669323933067`*^9, 3.5415669347687483`*^9}, 3.541650719124674*^9, + 3.541669536464787*^9, {3.543683508858923*^9, 3.543683509963663*^9}, { + 3.5440171547763557`*^9, 3.544017157219846*^9}, {3.544017194631675*^9, + 3.5440172179441557`*^9}, {3.544019527438822*^9, 3.5440195294562674`*^9}, + 3.544058194323106*^9, {3.544058510276647*^9, 3.544058527744874*^9}, { + 3.54405862957839*^9, 3.544058632923155*^9}}], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"K", "[", + RowBox[{"[", + RowBox[{"1", ",", "3"}], "]"}], "]"}], ",", + RowBox[{"Assumptions", "\[Rule]", + RowBox[{"{", + RowBox[{ + RowBox[{"r", ">", "0"}], ",", + RowBox[{"R", ">", "0"}], ",", + RowBox[{"alpha", ">", "0"}], ",", + RowBox[{"beta", "<", "1"}], ",", + RowBox[{"beta", ">", "0"}]}], "}"}]}]}], "]"}]], "Input", + CellChangeTimes->CompressedData[" +1:eJwd0F1Ik3EYBfA3K+aGIzIpttbeuTBhbVqNDC8sbXYRq+i1XAUzxNLhGNkq +dLMPyC6SJnazKDeYA0e2DS3aWl4UfYwgFlkzK1dBBUVhJfSBzk1n//NcHH4X +51yd4qa2uuY8juO0LNCml1oklsmaqecHW+GGkM0sYzZEdzTAXZ9yTfBjXGOD +FZmlx+H2wGA7bByejsP7Z2fJW58VBauZpqdaMlo/1gc9M9/ISllhP/X21AB8 +59fdhNMXjaSnyhSBP53pKJR8zT2ASfXhJ1C54OcVzPLqkrXQEH5WBn2P5eXw +71GHAd6ritfC+RU/BFi0bVkz1GdWWuFosLANhlrO90AuJlyGk6rlV6CyW0NO +XAh7Yc/6ZD/sGh8JQE1slJS7OCfPtNXwZJHV021nvmr3kflf3G4oW9NHNhp9 +Xsi13CV3J3YOQPlVB3nj1MNB+FL8hjTlSo6dYJa+30Nq7b9XnWSqXf/IiFhI +dTAzc1ZyKlKXhoey++dhcOYP78CvhtsqeO7tvjJ4Jli7Ec6JFipg/tYjW6Ci +ozoCxwUzmYjqFjuZrtO9S6A7oLNCqcFIXpe9DkH/WJpMxEYmhpi/FhWkoKTL +Mgt788JZqI6ZRcPYizvJTiGph9+LP5B7paIDj5iqTTzpuaZsJb0CeSm3OfuC +ue5OJfkfx2VVOg== + "]], + +Cell[OutputFormData["\<\ +(Sqrt[(-1 + beta^2)/(alpha^2*beta^2*r^2*R^4 - (R^3 - \ +beta*C*x)^2)]*z*(3*alpha*C*R^6*x + beta^2*C*x*(alpha*M*r^2*R^3 - (1 + \ +2*alpha)*C^2*(r^2 - x^2 - y^2 - z^2)) + + beta*R^3*(alpha*r^2*R^3*(-M + (-1 + alpha)*alpha*R) + C^2*((1 + \ +2*alpha)*r^2 - (1 + 5*alpha)*x^2 - (1 + 2*alpha)*(y^2 + z^2)))))/(alpha*(-1 + \ +beta^2)*r^4*R^4)\ +\>", "\<\ + 2 + -1 + beta 6 2 \ + 2 3 2 2 2 2 2 +(Sqrt[-------------------------------------] z (3 alpha C R x + beta C x \ +(alpha M r R - (1 + 2 alpha) C (r - x - y - z )) + + 2 2 2 4 3 2 + alpha beta r R - (R - beta C x) + + 3 2 3 2 2 \ + 2 2 2 2 4 4 + beta R (alpha r R (-M + (-1 + alpha) alpha R) + C ((1 + 2 alpha) r \ +- (1 + 5 alpha) x - (1 + 2 alpha) (y + z ))))) / (alpha (-1 + beta ) r R )\ +\ +\>"], "Output", + CellChangeTimes->{{3.543683900162929*^9, 3.543683915088058*^9}, { + 3.5437146403530817`*^9, 3.5437146576437883`*^9}, {3.543714691785604*^9, + 3.543714741795236*^9}, 3.543715119550694*^9, 3.5437151656114902`*^9, + 3.543715361456665*^9, {3.543717981704191*^9, 3.5437179858770523`*^9}, + 3.543718365293414*^9, 3.543719120149531*^9, {3.543719162855845*^9, + 3.5437191996413193`*^9}, 3.543719290954267*^9, {3.5437193240853653`*^9, + 3.543719336451874*^9}, {3.543719395894435*^9, 3.5437194674442797`*^9}, { + 3.54372019053549*^9, 3.54372021773365*^9}, 3.543720313473928*^9, { + 3.54372047739721*^9, 3.543720487662917*^9}, {3.5437206073406754`*^9, + 3.543720744869968*^9}, 3.543720785632041*^9, {3.543720818872993*^9, + 3.543720825402874*^9}, 3.543720917537565*^9, 3.543720954445459*^9, { + 3.543721061604096*^9, 3.543721083666382*^9}, 3.543721130682632*^9, { + 3.543721183717269*^9, 3.543721215495632*^9}, 3.543721256997097*^9, + 3.543727049771803*^9, 3.543894804553688*^9, 3.543895111317523*^9, + 3.543895205593939*^9, 3.54389526354342*^9, 3.54389532474303*^9, + 3.543895360910643*^9, 3.543896947092414*^9, 3.5438983043246403`*^9, + 3.543908046333356*^9, 3.543908096578384*^9, 3.5439083400536413`*^9, { + 3.543908566347658*^9, 3.543908572584536*^9}, 3.543908616377408*^9, { + 3.5439087260706253`*^9, 3.5439087473407097`*^9}, 3.543908796692197*^9, + 3.543909788800337*^9, 3.543910212168428*^9, {3.543910426145859*^9, + 3.543910437941873*^9}, 3.543911250989028*^9, 3.543911771948946*^9, + 3.544018654022167*^9, 3.5440188928090887`*^9, 3.544018947339971*^9, { + 3.544019005226565*^9, 3.544019030754849*^9}, 3.544019362192422*^9, + 3.544019559037004*^9, 3.5440585565498343`*^9, 3.5440586463797007`*^9, + 3.544058699668764*^9, {3.544088532274184*^9, 3.544088539598835*^9}}] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"FullSimplify", "[", + RowBox[{"%", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{"z", "^", "2"}], "\[Rule]", + RowBox[{ + RowBox[{"r", "^", "2"}], "-", + RowBox[{"x", "^", "2"}], "-", + RowBox[{"y", "^", "2"}]}]}], "}"}]}], "]"}]], "Input", + CellChangeTimes->{{3.543720827486745*^9, 3.5437209153574343`*^9}, { + 3.5437209562880287`*^9, 3.543720963571994*^9}, {3.544018965251566*^9, + 3.544018974899355*^9}, {3.544019033199492*^9, 3.544019037652479*^9}}], + +Cell[OutputFormData["\<\ +(Sqrt[(-1 + beta^2)/((alpha*beta*r*R^2 + R^3 - beta*C*x)*(alpha*beta*r*R^2 - \ +R^3 + beta*C*x))]*(-(beta*r^2*R^3*(M - (-1 + alpha)*alpha*R)) + \ +beta^2*C*M*r^2*x + + 3*C*R^3*x - 3*beta*C^2*x^2)*z)/((-1 + beta^2)*r^4*R)\ +\>", "\<\ + 2 + -1 + beta \ + 2 3 2 2 3 2 \ + 2 +Sqrt[-------------------------------------------------------------------] \ +(-(beta r R (M - (-1 + alpha) alpha R)) + beta C M r x + 3 C R x - 3 \ +beta C x ) z + 2 3 2 3 + (alpha beta r R + R - beta C x) (alpha beta r R - R + beta C x) +------------------------------------------------------------------------------\ +------------------------------------------------------------------------------\ +----- + \ + 2 4 + (-1 + \ +beta ) r R\ +\>"], "Output", + CellChangeTimes->{{3.5437208434909897`*^9, 3.54372087769923*^9}, { + 3.5437209084203453`*^9, 3.543720920915968*^9}, 3.543720964536048*^9, { + 3.543721070114254*^9, 3.543721089600729*^9}, 3.543721138629195*^9, { + 3.543721189140917*^9, 3.543721216745717*^9}, 3.543721261353259*^9, + 3.54389512195747*^9, 3.543895162479344*^9, {3.543895206961536*^9, + 3.543895212983779*^9}, 3.5438952718573093`*^9, 3.543895326578341*^9, + 3.5438953649768744`*^9, {3.5438969490691433`*^9, 3.5438969540118723`*^9}, { + 3.543898305715353*^9, 3.54389831825931*^9}, 3.543908047362484*^9, + 3.5439080991955*^9, 3.543908383731992*^9, 3.543908568789239*^9, + 3.543908621074088*^9, {3.543908734211472*^9, 3.543908748795429*^9}, + 3.543908845263538*^9, 3.543909792605887*^9, 3.543910215480422*^9, { + 3.5439104271318083`*^9, 3.5439104388580627`*^9}, 3.5439112619737177`*^9, + 3.5439117785085697`*^9, 3.543911869808456*^9, 3.544018673341402*^9, + 3.544018976832711*^9, {3.544019024001992*^9, 3.544019038291746*^9}, + 3.5440193701479597`*^9, 3.544019570105242*^9, 3.544058664600779*^9, + 3.5440885453533573`*^9}] +}, Open ]], + +Cell[BoxData[ + RowBox[{"FullSimplify", "[", + RowBox[{"%", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"x", "^", "2"}], "+", + RowBox[{"y", "^", "2"}], "+", + RowBox[{"z", "^", "2"}]}], "\[Rule]", + RowBox[{"r", "^", "2"}]}], "}"}]}], "]"}]], "Input", + CellChangeTimes->{{3.543720973240287*^9, 3.543720988542207*^9}}], + +Cell[BoxData[ + RowBox[{"FullSimplify", "[", + RowBox[{"%", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{"y", "^", "2"}], "\[Rule]", + RowBox[{ + RowBox[{"r", "^", "2"}], "-", + RowBox[{"x", "^", "2"}], "-", + RowBox[{"z", "^", "2"}]}]}], "}"}]}], "]"}]], "Input", + CellChangeTimes->{{3.5439118820154533`*^9, 3.543911902976988*^9}}], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"CForm", "[", "%", "]"}]], "Input", + CellChangeTimes->{{3.543909367655545*^9, 3.543909371533186*^9}}], + +Cell["\<\ +(Sqrt(-(1/(Power(alpha,2)*Power(beta,2)*Power(r,2)*Power(R,4) - \ +Power(Power(R,3) - beta*C*x,2))))* + (Power(beta,3)*M*Power(r,4)*(-Power(C,2) + Power(alpha,2)*Power(R,4))*x \ +- C*Power(R,6)*(Power(r,2) - 3*Power(x,2)) + + Power(beta,2)*C*M*Power(r,2)*Power(R,3)*(Power(r,2) + 2*Power(x,2)) + + beta*Power(R,3)*x*(Power(r,2)*Power(R,3)*(-2*M + (-1 + alpha)*alpha*R) \ ++ Power(C,2)*(Power(r,2) - 3*Power(x,2)))))/((-1 + \ +Power(beta,2))*Power(r,4)*Power(R,4))\ +\>", "Output", + CellChangeTimes->{3.5439093731092978`*^9, 3.543910544073845*^9, + 3.543912041830412*^9}] +}, Open ]], + +Cell[BoxData[ + RowBox[{ + RowBox[{"trK", "=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"(", + RowBox[{"K", ".", "g3u"}], ")"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "i"}], "]"}], "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "1", ",", "3"}], "}"}]}], "]"}], "/.", "expbeta"}], + "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544006507112256*^9, 3.544006588831873*^9}, { + 3.544006694121695*^9, 3.544006724831628*^9}, {3.544007709623177*^9, + 3.5440077121119127`*^9}, {3.5440077972005672`*^9, 3.544007797976552*^9}, + 3.5440079197147903`*^9, {3.544017333931843*^9, 3.544017350521927*^9}, { + 3.544058720993672*^9, 3.544058721668118*^9}, {3.544064151693574*^9, + 3.544064181134458*^9}, {3.5440774586394777`*^9, 3.544077461558529*^9}, { + 3.5440775131753607`*^9, 3.544077515654381*^9}, {3.544077573342401*^9, + 3.5440775758629713`*^9}, {3.5440927897805*^9, 3.5440928305033197`*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"KijKij", "=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"K", "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"i", ",", "k"}], "]"}], "]"}], "*", + RowBox[{"K", "[", + RowBox[{"[", + RowBox[{"k", ",", "l"}], "]"}], "]"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"l", ",", "j"}], "]"}], "]"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "1", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "1", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"k", ",", "1", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"l", ",", "1", ",", "3"}], "}"}]}], "]"}], "/.", "expbeta"}], + "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544007008524529*^9, 3.54400709476969*^9}, { + 3.5440071267361507`*^9, 3.5440071351581783`*^9}, {3.544007808321087*^9, + 3.5440078189822397`*^9}, 3.5440079238254757`*^9, {3.5440173536906853`*^9, + 3.54401736521487*^9}, {3.544058757300873*^9, 3.54405876061347*^9}, { + 3.544077259196781*^9, 3.544077320969109*^9}, {3.544077377099451*^9, + 3.544077385426319*^9}, {3.544077464178179*^9, 3.544077522243578*^9}, { + 3.544077565576346*^9, 3.5440775704241858`*^9}, {3.544088193358122*^9, + 3.5440882806176367`*^9}, 3.544088947579814*^9}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chrx", " ", "=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"Table", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", "2"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"1", ",", "l"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "i", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"j", ",", "l"}], "]"}], "]"}], "+", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "j", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "l"}], "]"}], "]"}], "-", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "l", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"l", ",", "3"}], "}"}]}], "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}], "/.", "expderiv"}], "]"}]}], + ";"}]], "Input", + CellChangeTimes->{{3.544007905566985*^9, 3.5440079389346447`*^9}, { + 3.544008002960771*^9, 3.544008014468917*^9}, 3.544009905931024*^9, { + 3.544010015976405*^9, 3.5440100192081013`*^9}, 3.544010050024156*^9, + 3.544010101005965*^9, 3.544010185487176*^9, {3.544012950334074*^9, + 3.544012984685404*^9}, {3.54401749920787*^9, 3.544017505546061*^9}, { + 3.544017622151351*^9, 3.544017648952813*^9}, 3.54401767998133*^9, { + 3.544017730971414*^9, 3.544017737469659*^9}, 3.544017772371719*^9, { + 3.544018016295224*^9, 3.544018024001652*^9}, 3.54401999184079*^9, + 3.5440589360598383`*^9, {3.544059060834366*^9, 3.544059064057261*^9}, { + 3.544096205270871*^9, 3.544096229242694*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chry", "=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"Table", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", "2"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"2", ",", "l"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "i", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"j", ",", "l"}], "]"}], "]"}], "+", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "j", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "l"}], "]"}], "]"}], "-", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "l", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"l", ",", "3"}], "}"}]}], "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}], "/.", "expderiv"}], "]"}]}], + ";"}]], "Input", + CellChangeTimes->{{3.544008025458044*^9, 3.5440080342870407`*^9}, + 3.544010214195671*^9, {3.544012989094122*^9, 3.5440130088019323`*^9}, { + 3.544017786723536*^9, 3.5440178044782543`*^9}, {3.54401784778026*^9, + 3.544017851516535*^9}, {3.5440180328162193`*^9, 3.544018033257707*^9}, + 3.544019997831236*^9, 3.544058951415701*^9, {3.5440590747502003`*^9, + 3.544059076781601*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chrz", "=", + RowBox[{"FullSimplify", "[", + RowBox[{ + RowBox[{"Table", "[", + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"1", "/", "2"}], "*", + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"3", ",", "l"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "i", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"j", ",", "l"}], "]"}], "]"}], "+", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "j", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "l"}], "]"}], "]"}], "-", + RowBox[{ + RowBox[{"g3diff", "[", + RowBox[{"[", "l", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"i", ",", "j"}], "]"}], "]"}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"l", ",", "3"}], "}"}]}], "]"}], ",", + RowBox[{"{", + RowBox[{"i", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"j", ",", "3"}], "}"}]}], "]"}], "/.", "expderiv"}], "]"}]}], + ";"}]], "Input", + CellChangeTimes->{{3.544008036562434*^9, 3.54400804252567*^9}, + 3.5440102172871017`*^9, {3.54401301289886*^9, 3.54401301782524*^9}, + 3.544013369916349*^9, {3.544017814744141*^9, 3.54401782146946*^9}, { + 3.544018040107895*^9, 3.544018040453911*^9}, 3.544020005366929*^9, { + 3.544058890331402*^9, 3.544058891502262*^9}, 3.544058922446652*^9, { + 3.544059082787991*^9, 3.544059085112075*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chr", "=", + RowBox[{"{", + RowBox[{"chrx", ",", "chry", ",", "chrz"}], "}"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544008049376486*^9, 3.544008076289404*^9}, { + 3.5440083206106577`*^9, 3.544008321701839*^9}, {3.544010112889945*^9, + 3.544010117596229*^9}, 3.5440101486649237`*^9, 3.544010229916206*^9, { + 3.544013374704088*^9, 3.5440133755479927`*^9}, {3.544017826157701*^9, + 3.544017903029872*^9}, 3.5440589604546413`*^9}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chrdx", "=", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{ + RowBox[{"gamma", "*", "chr"}], "/.", "defderiv"}], ",", "x"}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544008346953102*^9, 3.5440084122972918`*^9}, { + 3.544009612246992*^9, 3.54400964212689*^9}, {3.5440097274760723`*^9, + 3.5440097501356153`*^9}, {3.5440102622427053`*^9, 3.5440102835084133`*^9}, { + 3.54401337980471*^9, 3.5440133835801153`*^9}, {3.544017921626112*^9, + 3.544017956055159*^9}, {3.544058986313904*^9, 3.544058994254533*^9}, { + 3.544059097958869*^9, 3.544059159850791*^9}, {3.544059228160652*^9, + 3.5440592355914183`*^9}, {3.544059271535349*^9, 3.544059273923382*^9}, { + 3.544059352238576*^9, 3.544059355254695*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chrdy", "=", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"chr", "/.", "defderiv"}], ",", "y"}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544008405876137*^9, 3.544008426278288*^9}, { + 3.544013394496241*^9, 3.544013396257222*^9}, {3.544017967934407*^9, + 3.544017977830834*^9}, {3.544059002531275*^9, 3.5440590351681023`*^9}, { + 3.544059100184265*^9, 3.54405910191995*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chrdz", "=", + RowBox[{ + RowBox[{"D", "[", + RowBox[{ + RowBox[{"chr", "/.", "defderiv"}], ",", "z"}], "]"}], "/.", + "undefderiv"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544008427704495*^9, 3.544008433179277*^9}, { + 3.5440133996295767`*^9, 3.544013401420314*^9}, {3.544018000953216*^9, + 3.544018006072641*^9}, {3.544059365160574*^9, 3.544059372944562*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"chrd", "=", + RowBox[{"{", + RowBox[{"chrdx", ",", "chrdy", ",", "chrdz"}], "}"}]}], ";"}]], "Input", + CellChangeTimes->{{3.544008440450982*^9, 3.5440084486434298`*^9}, { + 3.5440134118396893`*^9, 3.544013419446802*^9}, {3.544013744528404*^9, + 3.5440137565978622`*^9}, {3.5440158163425293`*^9, 3.5440158229294567`*^9}, + 3.544015963651965*^9, {3.54401820859608*^9, 3.544018247516447*^9}, { + 3.544059404536923*^9, 3.544059406591332*^9}, {3.544095689519257*^9, + 3.544095756532833*^9}}], + +Cell[BoxData[ + RowBox[{ + RowBox[{"Ric", "=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"a", ",", "b"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"chrd", "[", + RowBox[{"[", "c", "]"}], "]"}], "[", + RowBox[{"[", "c", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"a", ",", "b"}], "]"}], "]"}], "-", + RowBox[{ + RowBox[{ + RowBox[{"chrd", "[", + RowBox[{"[", "b", "]"}], "]"}], "[", + RowBox[{"[", "c", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"a", ",", "c"}], "]"}], "]"}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"a", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"b", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"c", ",", "3"}], "}"}]}], "]"}], "+", + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + RowBox[{"g3u", "[", + RowBox[{"[", + RowBox[{"a", ",", "b"}], "]"}], "]"}], "*", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"chr", "[", + RowBox[{"[", "d", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"a", ",", "b"}], "]"}], "]"}], "*", + RowBox[{ + RowBox[{"chr", "[", + RowBox[{"[", "c", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"c", ",", "d"}], "]"}], "]"}]}], "-", + RowBox[{ + RowBox[{ + RowBox[{"chr", "[", + RowBox[{"[", "d", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"a", ",", "c"}], "]"}], "]"}], "*", + RowBox[{ + RowBox[{"chr", "[", + RowBox[{"[", "c", "]"}], "]"}], "[", + RowBox[{"[", + RowBox[{"b", ",", "d"}], "]"}], "]"}]}]}], ")"}]}], ",", + RowBox[{"{", + RowBox[{"a", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"b", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"c", ",", "3"}], "}"}], ",", + RowBox[{"{", + RowBox[{"d", ",", "3"}], "}"}]}], "]"}]}], "/.", "expderiv"}], "/.", + "expbeta"}], "]"}]}], ";"}]], "Input", + CellChangeTimes->{{3.5440084639862757`*^9, 3.544008639919712*^9}, { + 3.544008694766852*^9, 3.544008703414303*^9}, {3.5440134307113037`*^9, + 3.544013432965466*^9}, 3.544015831668213*^9, {3.544015901551384*^9, + 3.544015904899539*^9}, {3.544015970945381*^9, 3.544015999056787*^9}, { + 3.544018261052456*^9, 3.544018325665082*^9}, {3.544020609977902*^9, + 3.54402061074506*^9}, {3.5440594214718924`*^9, 3.544059424304337*^9}, { + 3.544095760178644*^9, 3.54409576284205*^9}, {3.5440962330985537`*^9, + 3.5440962416387653`*^9}, {3.544096801674099*^9, 3.54409680652934*^9}, { + 3.544097104353647*^9, 3.544097108270903*^9}, {3.544112426664811*^9, + 3.5441124770622063`*^9}, 3.544144664324232*^9}], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"Simplify", "[", + RowBox[{"Ric", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"alpha", "^", "2"}], "\[Rule]", + RowBox[{"1", " ", "-", " ", + RowBox[{"2", + RowBox[{"M", "/", "R"}]}], "+", + RowBox[{ + RowBox[{"C", "^", "2"}], "/", + RowBox[{"R", "^", "4"}]}]}]}], ",", + RowBox[{ + RowBox[{"y", "^", "2"}], "\[Rule]", + RowBox[{ + RowBox[{"r", "^", "2"}], "-", + RowBox[{"z", "^", "2"}], "-", + RowBox[{"x", "^", "2"}]}]}]}], "}"}]}], "]"}]], "Input", + CellChangeTimes->{{3.544008726663698*^9, 3.5440087296046333`*^9}, { + 3.5440087620823517`*^9, 3.5440087639953327`*^9}, 3.54401034307869*^9, + 3.544013742198639*^9, {3.544015842447304*^9, 3.544015844681971*^9}, { + 3.54401606265217*^9, 3.544016063565484*^9}, {3.5440183931583757`*^9, + 3.5440184110183363`*^9}, {3.544020687917218*^9, 3.544020701753339*^9}, { + 3.544059480020286*^9, 3.544059491204368*^9}, {3.544064010787756*^9, + 3.54406401296695*^9}, {3.544094195662027*^9, 3.544094273812009*^9}, + 3.5441125253515797`*^9}], + +Cell[OutputFormData["\<\ +(-2*(-3*C^2*R^12 + 4*beta*C*R^12*(3*M + (-1 + alpha)*R)*x + \ +beta^3*C*R^3*x*(C^4*(-r^2 + x^2) - C^2*R^3*(-7*M*(r^2 - x^2) + R*((9 + \ +alpha)*r^2 + (-7 + 5*alpha)*x^2)) + + R^6*(-4*M^2*(r^2 - x^2) + M*R*((11 + 2*alpha)*r^2 + (1 + 4*alpha)*x^2) + \ +R^2*((-4 + 3*alpha^3 + alpha^4)*r^2 - (2 - 3*alpha^3 + alpha^4)*x^2))) + + beta^4*x^2*(2*alpha^3*r^2*R^11*(M + (-1 + alpha)*alpha*R) + 2*C^6*(r^2 - \ +x^2) + C^4*R^3*(-8*M*(r^2 - x^2) + R*(-((-6 + alpha)*r^2) + (-4 + \ +3*alpha)*x^2)) + + C^2*R^6*(5*M^2*(r^2 - x^2) + M*R*(-7*r^2 + (1 - 2*alpha)*x^2) + \ +R^2*(-((-2 + alpha^3 + 2*alpha^4)*r^2) + (1 - 3*alpha^3 + 2*alpha^4)*x^2))) + \ + + beta^2*R^6*(-(C^4*(r^2 - 4*x^2)) + C^2*R^3*(M*(r^2 - 13*x^2) + R*(r^2 + \ +2*alpha*r^2 + x^2 - 2*alpha*x^2)) - + R^6*(M^2*(r^2 - x^2) + 2*M*R*(x^2 + alpha*(r^2 + x^2)) + \ +R^2*(-2*alpha^3*r^2 - x^2 + alpha^4*(2*r^2 + x^2))))))/ + (R^6*(R^6 - 2*beta*C*R^3*x + beta^2*(r^2*(2*M - R)*R^3 + C^2*(-r^2 + \ +x^2)))^2)\ +\>", "\<\ + 2 12 12 3 3 +(-2 (-3 C R + 4 beta C R (3 M + (-1 + alpha) R) x + beta C R x + + 4 2 2 2 3 2 2 2 \ + 2 + (C (-r + x ) - C R (-7 M (r - x ) + R ((9 + alpha) r + (-7 + 5 \ +alpha) x )) + + + 6 2 2 2 2 2 \ +2 3 4 2 3 4 2 + R (-4 M (r - x ) + M R ((11 + 2 alpha) r + (1 + 4 alpha) x ) + R \ + ((-4 + 3 alpha + alpha ) r - (2 - 3 alpha + alpha ) x ))) + + + 4 2 3 2 11 6 2 2 \ + 4 3 2 2 2 2 + beta x (2 alpha r R (M + (-1 + alpha) alpha R) + 2 C (r - x ) + \ +C R (-8 M (r - x ) + R (-((-6 + alpha) r ) + (-4 + 3 alpha) x )) + + + 2 6 2 2 2 2 2 2 \ + 3 4 2 3 4 2 + C R (5 M (r - x ) + M R (-7 r + (1 - 2 alpha) x ) + R (-((-2 + \ +alpha + 2 alpha ) r ) + (1 - 3 alpha + 2 alpha ) x ))) + + + 2 6 4 2 2 2 3 2 2 2 \ +2 2 2 + beta R (-(C (r - 4 x )) + C R (M (r - 13 x ) + R (r + 2 alpha r \ + + x - 2 alpha x )) - + + 6 2 2 2 2 2 2 2 3 2 \ + 2 4 2 2 + R (M (r - x ) + 2 M R (x + alpha (r + x )) + R (-2 alpha r - \ +x + alpha (2 r + x )))))) / + + 6 6 3 2 2 3 2 2 2 2 + (R (R - 2 beta C R x + beta (r (2 M - R) R + C (-r + x ))) )\ +\>"], "Output", + CellChangeTimes->{ + 3.544015860064528*^9, 3.544016024926996*^9, 3.5440160808547077`*^9, + 3.544018542238614*^9, 3.5440198928043833`*^9, 3.544020234146203*^9, + 3.544020703889607*^9, 3.544059492118401*^9, 3.544064013644176*^9, + 3.544089154943667*^9, {3.5440942093330173`*^9, 3.544094274174481*^9}, + 3.544112526655836*^9, 3.544113148651258*^9}] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"FullSimplify", "[", "%", "]"}]], "Input", + CellChangeTimes->{{3.544113149839019*^9, 3.5441131528418283`*^9}}], + +Cell[OutputFormData["\<\ +(2*(3*C^2*R^12 + 4*beta*C*R^12*(-3*M + R - alpha*R)*x + + beta^4*x^2*(r^2*(-2*C^6 + C^4*R^3*(8*M + (-6 + alpha)*R) - \ +2*alpha^3*R^11*(M + (-1 + alpha)*alpha*R) + C^2*R^6*(-5*M^2 + 7*M*R + (-2 + \ +alpha^3 + 2*alpha^4)*R^2)) + + C^2*(2*C^4 + C^2*R^3*(-8*M + 4*R - 3*alpha*R) + R^6*(5*M^2 + (-1 + \ +2*alpha)*M*R + (-1 + (3 - 2*alpha)*alpha^3)*R^2))*x^2) + + beta^3*C*R^3*x*(r^2*(C^4 + C^2*R^3*(-7*M + (9 + alpha)*R) + R^6*(4*M^2 - \ +(11 + 2*alpha)*M*R - (-4 + alpha^3*(3 + alpha))*R^2)) - + (C^4 + C^2*R^3*(-7*M + 7*R - 5*alpha*R) + R^6*(4*M^2 + (1 + 4*alpha)*M*R \ +- (2 + (-3 + alpha)*alpha^3)*R^2))*x^2) + + beta^2*R^6*(r^2*(C^4 - C^2*R^3*(M + R + 2*alpha*R) + R^6*(M^2 + \ +2*alpha*M*R + 2*(-1 + alpha)*alpha^3*R^2)) - + (4*C^4 + C^2*R^3*(-13*M + R - 2*alpha*R) + R^6*(M^2 - 2*(1 + alpha)*M*R \ +- (-1 + alpha^4)*R^2))*x^2)))/ + (R^6*(R^6 - 2*beta*C*R^3*x + beta^2*(-(r^2*(C^2 + R^3*(-2*M + R))) + \ +C^2*x^2))^2)\ +\>", "\<\ + 2 12 12 4 2 2 6 4 \ + 3 3 11 +(2 (3 C R + 4 beta C R (-3 M + R - alpha R) x + beta x (r (-2 C + C \ +R (8 M + (-6 + alpha) R) - 2 alpha R (M + (-1 + alpha) alpha R) + + + 2 6 2 3 4 2 2 + C R (-5 M + 7 M R + (-2 + alpha + 2 alpha ) R )) + C + + 4 2 3 6 2 \ + 3 2 2 + (2 C + C R (-8 M + 4 R - 3 alpha R) + R (5 M + (-1 + 2 alpha) \ +M R + (-1 + (3 - 2 alpha) alpha ) R )) x ) + + + 3 3 2 4 2 3 6 2 \ + 3 2 + beta C R x (r (C + C R (-7 M + (9 + alpha) R) + R (4 M - (11 + \ +2 alpha) M R - (-4 + alpha (3 + alpha)) R )) - + + 4 2 3 6 2 \ + 3 2 2 + (C + C R (-7 M + 7 R - 5 alpha R) + R (4 M + (1 + 4 alpha) M R \ +- (2 + (-3 + alpha) alpha ) R )) x ) + + + 2 6 2 4 2 3 6 2 \ + 3 2 + beta R (r (C - C R (M + R + 2 alpha R) + R (M + 2 alpha M R + 2 \ +(-1 + alpha) alpha R )) - + + 4 2 3 6 2 \ + 4 2 2 + (4 C + C R (-13 M + R - 2 alpha R) + R (M - 2 (1 + alpha) M R - \ +(-1 + alpha ) R )) x ))) / + + 6 6 3 2 2 2 3 2 2 2 + (R (R - 2 beta C R x + beta (-(r (C + R (-2 M + R))) + C x )) )\ +\>"], "Output", + CellChangeTimes->{3.544113156016739*^9}] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"CForm", "[", "%", "]"}]], "Input", + CellChangeTimes->{{3.544089172004738*^9, 3.544089175432055*^9}}], + +Cell["\<\ +(-2*(-3*Power(C,2)*Power(R,12) + 4*beta*C*Power(R,12)*(3*M + (-1 + \ +alpha)*R)*x + + Power(beta,3)*C*Power(R,3)*x*(Power(C,4)*(-Power(r,2) + Power(x,2)) - + Power(C,2)*Power(R,3)*(-7*M*(Power(r,2) - Power(x,2)) + R*((9 + \ +alpha)*Power(r,2) + (-7 + 5*alpha)*Power(x,2))) + + Power(R,6)*(-4*Power(M,2)*(Power(r,2) - Power(x,2)) + M*R*((11 + \ +2*alpha)*Power(r,2) + (1 + 4*alpha)*Power(x,2)) + + Power(R,2)*((-4 + 3*Power(alpha,3) + Power(alpha,4))*Power(r,2) \ +- (2 - 3*Power(alpha,3) + Power(alpha,4))*Power(x,2)))) + + Power(beta,4)*Power(x,2)*(2*Power(alpha,3)*Power(r,2)*Power(R,11)*(M + \ +(-1 + alpha)*alpha*R) + 2*Power(C,6)*(Power(r,2) - Power(x,2)) + + Power(C,4)*Power(R,3)*(-8*M*(Power(r,2) - Power(x,2)) + R*(-((-6 + \ +alpha)*Power(r,2)) + (-4 + 3*alpha)*Power(x,2))) + + Power(C,2)*Power(R,6)*(5*Power(M,2)*(Power(r,2) - Power(x,2)) + \ +M*R*(-7*Power(r,2) + (1 - 2*alpha)*Power(x,2)) + + Power(R,2)*(-((-2 + Power(alpha,3) + \ +2*Power(alpha,4))*Power(r,2)) + (1 - 3*Power(alpha,3) + \ +2*Power(alpha,4))*Power(x,2)))) + + Power(beta,2)*Power(R,6)*(-(Power(C,4)*(Power(r,2) - 4*Power(x,2))) + + Power(C,2)*Power(R,3)*(M*(Power(r,2) - 13*Power(x,2)) + \ +R*(Power(r,2) + 2*alpha*Power(r,2) + Power(x,2) - 2*alpha*Power(x,2))) - + Power(R,6)*(Power(M,2)*(Power(r,2) - Power(x,2)) + \ +2*M*R*(Power(x,2) + alpha*(Power(r,2) + Power(x,2))) + + Power(R,2)*(-2*Power(alpha,3)*Power(r,2) - Power(x,2) + \ +Power(alpha,4)*(2*Power(r,2) + Power(x,2)))))))/ + (Power(R,6)*Power(Power(R,6) - 2*beta*C*Power(R,3)*x + \ +Power(beta,2)*(Power(r,2)*(2*M - R)*Power(R,3) + Power(C,2)*(-Power(r,2) + \ +Power(x,2))),2))\ +\>", "Output", + CellChangeTimes->{3.544089176410659*^9, 3.544112601281684*^9}] +}, Open ]] +}, +WindowSize->{1472, 1200}, +WindowMargins->{{0, Automatic}, {Automatic, 0}}, +FrontEndVersion->"7.0 for Linux x86 (64-bit) (February 25, 2009)", +StyleDefinitions->"Default.nb" +] +(* End of Notebook Content *) + +(* Internal cache information *) +(*CellTagsOutline +CellTagsIndex->{} +*) +(*CellTagsIndex +CellTagsIndex->{} +*) +(*NotebookFileOutline +Notebook[{ +Cell[545, 20, 1648, 44, 77, "Input"], +Cell[2196, 66, 5337, 170, 143, "Input"], +Cell[7536, 238, 191, 5, 32, "Input"], +Cell[7730, 245, 447, 10, 32, "Input"], +Cell[8180, 257, 484, 10, 32, "Input"], +Cell[8667, 269, 475, 10, 32, "Input"], +Cell[9145, 281, 379, 11, 32, "Input"], +Cell[9527, 294, 393, 9, 32, "Input"], +Cell[9923, 305, 1070, 30, 32, "Input"], +Cell[10996, 337, 890, 24, 32, "Input"], +Cell[11889, 363, 426, 9, 32, "Input"], +Cell[12318, 374, 375, 10, 32, "Input"], +Cell[12696, 386, 314, 7, 32, "Input"], +Cell[13013, 395, 1136, 20, 32, "Input"], +Cell[14152, 417, 749, 15, 32, "Input"], +Cell[14904, 434, 939, 18, 32, "Input"], +Cell[15846, 454, 304, 7, 32, "Input"], +Cell[16153, 463, 518, 12, 32, "Input"], +Cell[16674, 477, 587, 15, 32, "Input"], +Cell[17264, 494, 1925, 47, 32, "Input"], +Cell[19192, 543, 1209, 28, 32, "Input"], +Cell[20404, 573, 387, 12, 32, "Input"], +Cell[20794, 587, 1792, 43, 32, "Input"], +Cell[CellGroupData[{ +Cell[22611, 634, 1165, 27, 32, "Input"], +Cell[23779, 663, 2892, 46, 111, "Output"] +}, Open ]], +Cell[CellGroupData[{ +Cell[26708, 714, 501, 12, 32, "Input"], +Cell[27212, 728, 2237, 38, 111, "Output"] +}, Open ]], +Cell[29464, 769, 353, 10, 32, "Input"], +Cell[29820, 781, 355, 10, 32, "Input"], +Cell[CellGroupData[{ +Cell[30200, 795, 124, 2, 32, "Input"], +Cell[30327, 799, 591, 11, 82, "Output"] +}, Open ]], +Cell[30933, 813, 1010, 22, 32, "Input"], +Cell[31946, 837, 1498, 36, 32, "Input"], +Cell[33447, 875, 1962, 47, 32, "Input"], +Cell[35412, 924, 1653, 43, 32, "Input"], +Cell[37068, 969, 1664, 43, 32, "Input"], +Cell[38735, 1014, 477, 9, 32, "Input"], +Cell[39215, 1025, 794, 16, 32, "Input"], +Cell[40012, 1043, 463, 11, 32, "Input"], +Cell[40478, 1056, 415, 10, 32, "Input"], +Cell[40896, 1068, 534, 10, 32, "Input"], +Cell[41433, 1080, 3246, 83, 99, "Input"], +Cell[CellGroupData[{ +Cell[44704, 1167, 1101, 26, 32, "Input"], +Cell[45808, 1195, 3113, 60, 270, "Output"] +}, Open ]], +Cell[CellGroupData[{ +Cell[48958, 1260, 133, 2, 32, "Input"], +Cell[49094, 1264, 2749, 54, 270, "Output"] +}, Open ]], +Cell[CellGroupData[{ +Cell[51880, 1323, 124, 2, 32, "Input"], +Cell[52007, 1327, 1805, 30, 266, "Output"] +}, Open ]] +} +] +*) + +(* End of internal cache information *) -- cgit v1.2.3