Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

How to calculate some Ricci tensor derived quantities.

I am unsure of the correct way to calculate some Ricci tensor derived quantities, I have tried many different ways but can't get the correct value for the two independent Bach tensor parts U and V (W is zero in four dimensions).

The quantities I am unsure of are.

R_bc ;a^a

g_bc*R;a^a

R;bc

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') var('r','th') 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()

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

v1 = - Nabla(Nabla(g.ricci_scalar())) #Unsure v2 = + (g(Nabla(Nabla(g.ricci_scalar())))).up(g,3)['^a_bca'] #Unsure v3 = - g.ricci_scalar()g.ricci()
v4 = + 1/4(gg.ricci_scalar()*g.ricci_scalar())

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

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

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[:]) The correct Bach tensor is: (I have altered the cotton tensor 'convention') [ 1163/32e^(6r) 0 0 25/64e^(4r)] [ 0 -5/32e^(2r) 0 0] [ 0 0 -45/32e^(5r + th) 0] [ 25/64e^(4*r) 0 0 0]

How to calculate some Ricci tensor derived quantities.

I am unsure of the correct way to calculate some Ricci tensor derived quantities, I have tried many different ways but can't get the correct value for the two independent Bach tensor parts U and V (W is zero in four dimensions).

The quantities I am unsure of are.

R_bc ;a^a

g_bc*R;a^a

R;bc

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')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') 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); M.declare_union(MBL,MCT);

g= MBL.lorentzian_metric('g') var('r','th') MBL.lorentzian_metric('g'); var('r','th'); g00=-e^(2r) 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*g03r); 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()

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

v1 = - Nabla(Nabla(g.ricci_scalar())) ; #Unsure v2 = + (g(Nabla(Nabla(g.ricci_scalar())))).up(g,3)['^a_bca'] (Nabla(Nabla(g.ricci_scalar())))).up(g,3)['^a_bca']; #Unsure v3 = - g.ricci_scalar()g.ricci() ;
v4 = + 1/4(gg.ricci_scalar()*g.ricci_scalar()) g.ricci_scalar()*g.ricci_scalar());

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

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

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

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

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

The correct Bach tensor is: (I have altered the cotton tensor 'convention') 'convention.')

[ 1163/32e^(6r) 0 0 25/64e^(4r)] [ 0 -5/32e^(2r) 0 0] [ 0 0 -45/32e^(5r + th) 0] [ 25/64e^(4*r) e^(4r) 0 0 0]

How to calculate some Ricci tensor derived quantities.

I am unsure of the correct way to calculate some Ricci tensor derived quantities, I have tried many different ways but can't get the correct value for the two independent Bach tensor parts U and V (W is zero in four dimensions).

The quantities I am unsure of are.

R_bc ;a^a

g_bc*R;a^a

R;bc

show(version())`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');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'); 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); M.declare_union(MBL,MCT)

g= MBL.lorentzian_metric('g'); var('r','th'); MBL.lorentzian_metric('g') var('r','th') g00=-e^(2r); 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;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()

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

v1 = - Nabla(Nabla(g.ricci_scalar())) ; #Unsure v2 = + (g(Nabla(Nabla(g.ricci_scalar())))).up(g,3)['^a_bca']; (Nabla(Nabla(g.ricci_scalar())))).up(g,3)['^a_bca'] #Unsure v3 = - g.ricci_scalar()g.ricci() ;
v4 = + 1/4(gg.ricci_scalar()*g.ricci_scalar()); g.ricci_scalar()*g.ricci_scalar())

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

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

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

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

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

The correct Bach tensor is: (I have altered the cotton tensor 'convention.')

[ 1163/32e^(6r) 0 0 25/64e^(4r)] [ 0 -5/32e^(2r) 0 0] [ 0 0 -45/32e^(5r + th) 0] [ 25/64e^(4r) 0 0 0]

How to calculate some Ricci tensor derived quantities.

I am unsure of the correct way to calculate some Ricci tensor tensor derived quantities, I have tried many different ways ways but can't get the correct value for the two independent independent Bach tensor parts U and V (W is zero in four dimensions).

The quantities I am unsure of are.are labeled "Unsure" in the code below.

R_bc ;a^a

g_bc*R;a^a

R;bc

`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= M.declare_union(MBL,MCT) g = MBL.lorentzian_metric('g') var('r','th') 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()

var('r', 'th') 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() # http://liu.diva-portal.org/smash/get/diva2:244008/FULLTEXT01.pdf equations 4.52 4.53

4.53 v1 = - Nabla(Nabla(g.ricci_scalar())) #Unsure -Nabla(Nabla(g.ricci_scalar())) # Unsure v2 = + (g(Nabla(Nabla(g.ricci_scalar())))).up(g,3)['^a_bca'] #Unsure +(g*(Nabla(Nabla(g.ricci_scalar())))).up(g, 3)['^a_bca'] # Unsure v3 = - g.ricci_scalar()g.ricci()
-g.ricci_scalar()*g.ricci() v4 = + 1/4(gg.ricci_scalar()*g.ricci_scalar())

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

+1/4*(g*g.ricci_scalar()*g.ricci_scalar()) V = v1 + v2 + v3 + v4 V[:] u1 = + Nabla(Nabla(g.ricci())).up(g,3)['^a_bca'] #Unsure +Nabla(Nabla(g.ricci())).up(g, 3)['^a_bca'] # Unsure 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(g.ricci()(g.ricci())).up(g,3)['^a_abc']
+2*(g.ricci()*(g.ricci())).up(g, 3)['^a_abc'] u5 = - 1/2
(gg.ricci()(g.ricci().up(g)))['^ad_bcad'] -1/2*(g*g.ricci()*(g.ricci().up(g)))['^ad_bcad'] u6 = - 1/3(g.ricci_scalar()g.ricci()) -1/3*(g.ricci_scalar()*g.ricci()) u7 = + 1/12(gg.ricci_scalar()*g.ricci_scalar())

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

+1/12*(g*g.ricci_scalar()*g.ricci_scalar()) U = u1 + u2 + u3 + u4 + u5 + u6 + u7 show(' U/2: ',1/2U[:],' ', 1/2*U[:], ' V/6: ',1/6V[:])

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

', Bach[:])

The correct Bach tensor is: (I have altered the cotton tensor 'convention.')'convention') is:

[    1163/32e^(6r)                   0                   0       25/64e^(4r)]
1163/32*e^(6*r)                   0                   0       25/64*e^(4*r)]
[                  0       -5/32e^(2r) -5/32*e^(2*r)                   0                   0]
[                  0                   0 -45/32e^(5r -45/32*e^(5*r + th)                   0]
[      25/64e^(4r)                   0                   0                   0]

25/64*e^(4*r) 0 0 0]