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asked 2021-06-22 11:38:15 +0200

AlanTStafford gravatar image

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]

click to hide/show revision 4
None

updated 2021-06-22 12:04:01 +0200

slelievre gravatar image

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]
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