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initial version

Try this version:

reset()
Man=Manifold(4, 'Man', r'\mathcal{M}')
BL.<t,r,th,ph> = Man.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
H=BL.function('H')(t,r,th,ph)  
assume (H<0)
term1=(1/2)*sqrt(2)*sqrt(I*H)
term2=(1/2)*sqrt(2)*sqrt(-I*H)
T1=Man.tensor_field(0,1,[0,0,term1,0])
T2=Man.tensor_field(0,1,[0,0,term2,0])
res=T1[2]*T2[2]
res

-1/2*H