The linearization of the metric of curved space-time

[maica]

Why Y^ab=X^ab-kH^ab+k²H^acH_c^b-... and not Y^ab=X^ab-kH^ab+(1/2)k²H^ac_c^b-...?
Y is the curved space-time metric
X is the planespace-time metric

 

[maica]

The second order term in k is H^acH_c^b and not H^ac_c^b

 

[Bill_K]

Because of the matrix identity (I + H)^-1 = I - H + HH - ...

 

[maica]

but this expansion is not an expansion matrix. these objects are components, numbers

 

[Nugatory]

but this expansion is not an expansion matrix. these objects are components, numbers

How do you calculate the value of (HH)^ab, given the components of H?

 

[maica]

I'm a little lost. The expression (I+H)^-1 = I - H + HH - ... is the Taylor series in powers of "H", but g^ij = η^ij + κh^ij + (1/2)κ²hⁱ_lh^lj, is a series in powers of "κ", and doesn't have the expoent "-1", which is responsible for the absence of the factor 1/2 in the Taylor series of "H". I can't see why the factor 1/2 is absent, in Taylor series of "κ" powers. I think that the expansion of (HH)^ab is identical of the g^ij expansion, because of de tensor rank.

 

[maica]

I think that I understood. If we consider the equation g^ikg_kj = δⁱ_j and if g_kj = η_kj + κ h_kj then g^ik is the inverse and its expansion is g^ik = η^ik - κ h^ik + κ² hⁱ_l h^lk - ... identical to the expression (I + H)^-1.