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]