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Solved, see comments.

Solved, see comments.

`show(version())

var('tpar','lpar')

tpar=4 lpar=4

Parallelism().set('tensor',tpar) Parallelism().set('linbox',lpar) show(Parallelism())

var('a')

%display latex

viewer3D = 'threejs' # must be 'threejs', 'jmol', 'tachyon' or None (default)

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')

MCT = M.open_subset('MCT') #cartesian CT.<t,x,y,z> = MCT.chart(r't:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo) z:(-oo,+oo)')

MBL = M.open_subset('MBL') #boyer-lindquist BL.<t,r,th,phi> = MBL.chart(r't:(-oo,+oo) r:(0,+oo) th:\theta:(0,+pi):periodic phi:\phi:(0,2*pi):periodic) ') M.declare_union(MBL,MCT)

g= MBL.lorentzian_metric('g')

g00=-e^(2r) g11=e^(-2r) g22=e^((th+r)) g33=0 g03=1

g[0,0]=g00;g[1,1]=g11;g[2,2]=g22;g[3,3]=g33;g[0,3]=1/2*g03

Nabla=g.connection()

http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf equations 4.52 4.53

rho=g.ricci_scalar() R=g.ricci()

v1 = - Nabla(Nabla(rho))
v2 = + (g(Nabla(Nabla(rho)))).up(g,3)['^a_bca']
v3 = + rhoR
v4 = - 1/4(grho*rho)

V=(v1+v2+v3+v4) V[:]

u1 = + Nabla(Nabla(R)).up(g,3)['^a_bca']
u2 = - 1/2v2 u3 = - 2((g.weyl().down(g))(g.ricci().up(g)))['^ad_abcd'] u4 = - 2(R(R.up(g,1)))['^a_abc']
u5 = + 1/2(gR(R.up(g)))['^ad_bcad'] u6 = + 1/3(rhoR) u7 = - 1/12(grho*rho)

U=(u1+u2+u3+u4+u5+u6+u7)

show(' U/2: ',1/2U[:],' V/6: ',1/6V[:])

B=(1/2U)['_bc']+(1/6V)['_bc'] show('B: ',B[:]) Bach=Nabla(g.cotton()).up(g,3)['^u_aub'] +((g.schouten().up(g))*(g.weyl().down(g)))['^uv_aubv'] show('Bach: ',Bach[:]) show(g.bach()[:])
click to hide/show revision 3
No.3 Revision

updated 2021-06-24 12:35:42 +0100

slelievre gravatar image

Solved, see comments.

`show(version())

var('tpar','lpar')

tpar=4 lpar=4

var('tpar','lpar')

tpar = 4
lpar = 4

Parallelism().set('tensor',tpar)
Parallelism().set('linbox',lpar)
show(Parallelism())
 
var('a')
 
show(Parallelism())

var('a')

%display latex
 
latex

viewer3D = 'threejs'  # must be 'threejs', 'jmol', 'tachyon' or None (default)
 
(default)

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
 
structure='Lorentzian')

MCT = M.open_subset('MCT') #cartesian
 # cartesian
CT.<t,x,y,z>   =    = MCT.chart(r't:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo) z:(-oo,+oo)')
 
z:(-oo,+oo)')

MBL = M.open_subset('MBL') #boyer-lindquist
 # Boyer-Lindquist
BL.<t,r,th,phi>  = MBL.chart(r't:(-oo,+oo) r:(0,+oo) th:\theta:(0,+pi):periodic phi:\phi:(0,2*pi):periodic) ') 
M.declare_union(MBL,MCT)
 
g= MBL.lorentzian_metric('g')
 
g00=-e^(2'
                            r'th:\theta:(0,+pi):periodic '
                            r'phi:\phi:(0,2*pi):periodic)')
M.declare_union(MBL, MCT)

g = MBL.lorentzian_metric('g')

g00 = -e^(2*r)
g11 = e^(-2*r)
g22 = e^(th + r)
g11=e^(-2r)
g22=e^((th+r))
g33=0
g03=1
 
g[0,0]=g00;g[1,1]=g11;g[2,2]=g22;g[3,3]=g33;g[0,3]=1/2*g03
 
Nabla=g.connection()
 
http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf g33 = 0
g03 = 1

g[0, 0] = g00
g[1, 1] = g11
g[2, 2] = g22
g[3, 3] = g33
g[0, 3] = 1/2*g03

Nabla = g.connection()

# equations 4.52 4.53
 
rho=g.ricci_scalar()
R=g.ricci()
 
and 4.53 in
# http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf 

rho = g.ricci_scalar()
R = g.ricci()

v1 = - Nabla(Nabla(rho)) 
 -Nabla(Nabla(rho))
v2 = + (g(Nabla(Nabla(rho)))).up(g,3)['^a_bca'] 
 +(g*(Nabla(Nabla(rho)))).up(g, 3)['^a_bca']
v3 = + rhoR 
 +rho*R
v4 = - 1/4(grho*rho)  
 
-1/4*(g*rho*rho)

V=(v1+v2+v3+v4)
V[:]
 
V[:]

u1 = + Nabla(Nabla(R)).up(g,3)['^a_bca'] 
 +Nabla(Nabla(R)).up(g, 3)['^a_bca']
u2 = - 1/2v2
-1/2*v2
u3 = - 2((g.weyl().down(g))(g.ricci().up(g)))['^ad_abcd']
-2*(g.weyl().down(g)*(g.ricci().up(g)))['^ad_abcd']
u4 = - 2(R(R.up(g,1)))['^a_abc'] 
 -2*(R*R.up(g, 1))['^a_abc']
u5 = + 1/2(gR(R.up(g)))['^ad_bcad']
+1/2*(g*R*R.up(g))['^ad_bcad']
u6 = + 1/3(rhoR)
+1/3*(rho*R)
u7 = - 1/12(grho*rho)
 
U=(u1+u2+u3+u4+u5+u6+u7)
 
show(' U/2: ',1/2U[:],' V/6: ',1/6V[:])
 
B=(1/2U)['_bc']+(1/6V)['_bc']
show('B: ',B[:])
Bach=Nabla(g.cotton()).up(g,3)['^u_aub']  +((g.schouten().up(g))*(g.weyl().down(g)))['^uv_aubv']
show('Bach: ',Bach[:])
show(g.bach()[:])
 -1/12*(g*rho*rho)

U = (u1 + u2 + u3 + u4 + u5 + u6 + u7)

show('U/2:', 1/2*U[:], 'V/6:', 1/6*V[:])

B = (1/2*U)['_bc'] + (1/6*V)['_bc']
show('B:', B[:])
Bach = (Nabla(g.cotton()).up(g, 3)['^u_aub']
        + ((g.schouten().up(g))*(g.weyl().down(g)))['^uv_aubv'])
show('Bach:', Bach[:])
show(g.bach()[:])

Solved, see comments.

var('tpar','lpar')

tpar = 4
lpar = 4

Parallelism().set('tensor',tpar)
Parallelism().set('linbox',lpar)
show(Parallelism())

var('a')

%display latex

viewer3D = 'threejs'  # must be 'threejs', 'jmol', 'tachyon' or None (default)

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')

MCT = M.open_subset('MCT')  # cartesian
CT.<t,x,y,z> = MCT.chart(r't:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo) z:(-oo,+oo)')

MBL = M.open_subset('MBL')  # Boyer-Lindquist
BL.<t,r,th,phi> = MBL.chart(r't:(-oo,+oo) r:(0,+oo) '
                            ' r'th:\theta:(0,+pi):periodic '
                            ' r'phi:\phi:(0,2*pi):periodic)')
M.declare_union(MBL, MCT)

g = MBL.lorentzian_metric('g')

g00 = -e^(2*r)
g11 = e^(-2*r)
g22 = e^(th + r)
g33 = 0
g03 = 1

g[0, 0] = g00
g[1, 1] = g11
g[2, 2] = g22
g[3, 3] = g33
g[0, 3] = 1/2*g03

Nabla = g.connection()

# equations 4.52 and 4.53 in
# http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf 

rho = g.ricci_scalar()
R = g.ricci()

v1 = -Nabla(Nabla(rho))
v2 = +(g*(Nabla(Nabla(rho)))).up(g, 3)['^a_bca']
v3 = +rho*R
v4 = -1/4*(g*rho*rho)

V=(v1+v2+v3+v4)
V[:]

u1 = +Nabla(Nabla(R)).up(g, 3)['^a_bca']
u2 = -1/2*v2
u3 = -2*(g.weyl().down(g)*(g.ricci().up(g)))['^ad_abcd']
u4 = -2*(R*R.up(g, 1))['^a_abc']
u5 = +1/2*(g*R*R.up(g))['^ad_bcad']
u6 = +1/3*(rho*R)
u7 = -1/12*(g*rho*rho)

U = (u1 + u2 + u3 + u4 + u5 + u6 + u7)

show('U/2:', 1/2*U[:], 'V/6:', 1/6*V[:])

B = (1/2*U)['_bc'] + (1/6*V)['_bc']
show('B:', B[:])
Bach = (Nabla(g.cotton()).up(g, 3)['^u_aub']
        + ((g.schouten().up(g))*(g.weyl().down(g)))['^uv_aubv'])
show('Bach:', Bach[:])
show(g.bach()[:])
click to hide/show revision 5
No.5 Revision

updated 2021-06-24 19:33:45 +0100

slelievre gravatar image

Solved, see comments.

var('tpar','lpar')

tpar = 4
lpar = 4

Parallelism().set('tensor',tpar)
Parallelism().set('linbox',lpar)
show(Parallelism())

var('a')

%display latex

viewer3D = 'threejs'  # must be 'threejs', 'jmol', 'tachyon' or None (default)

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')

MCT = M.open_subset('MCT')  # cartesian
CT.<t,x,y,z> = MCT.chart(r't:(-oo,+oo) x:(-oo,+oo) y:(-oo,+oo) z:(-oo,+oo)')

MBL = M.open_subset('MBL')  # Boyer-Lindquist
BL.<t,r,th,phi> = MBL.chart(r't:(-oo,+oo) t_r_theta_phi = (r't:(-oo,+oo) r:(0,+oo) ' r'th:\theta:(0,+pi):periodic ' th:\theta:(0,+pi):periodic '
                 r'phi:\phi:(0,2*pi):periodic)')
BL.<t, r, th, phi> = MBL.chart(t_r_theta_phi)
M.declare_union(MBL, MCT)

g = MBL.lorentzian_metric('g')

g00 = -e^(2*r)
g11 = e^(-2*r)
g22 = e^(th + r)
g33 = 0
g03 = 1

g[0, 0] = g00
g[1, 1] = g11
g[2, 2] = g22
g[3, 3] = g33
g[0, 3] = 1/2*g03

Nabla = g.connection()

# equations 4.52 and 4.53 in
# http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf 

rho = g.ricci_scalar()
R = g.ricci()

v1 = -Nabla(Nabla(rho))
v2 = +(g*(Nabla(Nabla(rho)))).up(g, 3)['^a_bca']
v3 = +rho*R
v4 = -1/4*(g*rho*rho)

V=(v1+v2+v3+v4)
V[:]

u1 = +Nabla(Nabla(R)).up(g, 3)['^a_bca']
u2 = -1/2*v2
u3 = -2*(g.weyl().down(g)*(g.ricci().up(g)))['^ad_abcd']
u4 = -2*(R*R.up(g, 1))['^a_abc']
u5 = +1/2*(g*R*R.up(g))['^ad_bcad']
u6 = +1/3*(rho*R)
u7 = -1/12*(g*rho*rho)

U = (u1 + u2 + u3 + u4 + u5 + u6 + u7)

show('U/2:', 1/2*U[:], 'V/6:', 1/6*V[:])

B = (1/2*U)['_bc'] + (1/6*V)['_bc']
show('B:', B[:])
Bach = (Nabla(g.cotton()).up(g, 3)['^u_aub']
        + ((g.schouten().up(g))*(g.weyl().down(g)))['^uv_aubv'])
show('Bach:', Bach[:])
show(g.bach()[:])
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