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 labeled "Unsure" in the code below.
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^(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
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*(g*g.ricci_scalar()*g.ricci_scalar())
V = v1 + v2 + v3 + v4
V[:]
u1 = +Nabla(Nabla(g.ricci())).up(g, 3)['^a_bca'] # Unsure
u2 = -1/2*v2
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*(g*g.ricci()*(g.ricci().up(g)))['^ad_bcad']
u6 = -1/3*(g.ricci_scalar()*g.ricci())
u7 = +1/12*(g*g.ricci_scalar()*g.ricci_scalar())
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[:])
The correct Bach tensor (I have altered the cotton tensor 'convention') is:
[ 1163/32*e^(6*r) 0 0 25/64*e^(4*r)]
[ 0 -5/32*e^(2*r) 0 0]
[ 0 0 -45/32*e^(5*r + th) 0]
[ 25/64*e^(4*r) 0 0 0]
The quantities I am unsure of are: v1: R;_bc v2: g_bc * R;_a^a u1: R_bc;_a^a
I have changed my definition of the cotton tensor in metric.py to be:
if self._cotton is None: nabla = self.connection() s = self.schouten() cot = -(n-2)*nabla(s).antisymmetrize(1,2)I am not sure what your question is. Maybe it is a matter of convention about the Cotton and Bach tensors. Your "unsure" quantities
v1,v2andu1seem correctly evaluated (i.e. the formed expressions agree with those of your comment above).The cotton convention will mean that the above is not portable to machines where the cotton tensor is given a convention which agrees with maple. It is that v1, v2 and u1 agree with the quantities. I am asking which you think are right. I will keep trying alternatives.
Got this now. With the convention used in the paper I need to reverse the sign of any terms that have an even number of the Ricci scalar plus Ricci tensor terms.. That is v3, v4, u4, u5, u6 and u7. Then negate all of them.