Raising and lowering indices in Linearized theory

[zn5252]

Hi there,
in Linearized theory we know that :
g_αβ = η_αβ + h_αβ
If I multiply out both terms by η^θαη^λβ, wouldn't one get :
g^θλ = η^θλ + h^θλ ? EQ1
But we already know g^θλ = η^θλ - h^θλ + O(h²) EQ2
How can we reconciliate EQ1 and EQ2 ? Was it an error to have raised the perturbation h with the η ?
Can we also say that g^θλ = g_θλ ?
I'm a bit confused.
Thanks,

 

[Bill_K]

zn5252, There are covariant tensors and contravariant tensors, and it's just a convention that we use the metric tensor to associate one with the other. That is, if B_μ = g_μνA^ν, its a convention to say that A and B are the 'same' tensor. For this to work you must be able to reverse the process and say A^ν = g^μνB_μ, and this requires g^μν to be the inverse of g_μν, matrixwise, that is, g^μνg_νσ = δ^μ_σ.

In linearized theory, g_μν = η_μν + h_μν. Writing this as a matrix equation, g = η + h, the inverse is g^-1 = (η + h)^-1 = η - ηhη + ... So again it's a helpful convention to raise and lower indices in the flat space using the flat space metric η_μν and its inverse η^μν, but the original definition of g^μν as the inverse matrix of g_μν must be kept.

 

[zn5252]

Remark : The product of the covariant g with its contravariant counterpart is assured to equal the unit tensor to first order In this case. See footnote of Landau & Lifgarbagez page 350.

 

[zn5252]

zn5252, There are covariant tensors and contravariant tensors, and it's just a convention that we use the metric tensor to associate one with the other. That is, if B_μ = g_μνA^ν, its a convention to say that A and B are the 'same' tensor. For this to work you must be able to reverse the process and say A^ν = g^μνB_μ, and this requires g^μν to be the inverse of g_μν, matrixwise, that is, g^μνg_νσ = δ^μ_σ.

In linearized theory, g_μν = η_μν + h_μν. Writing this as a matrix equation, g = η + h, the inverse is g^-1 = (η + h)^-1 = η - ηhη + ... So again it's a helpful convention to raise and lower indices in the flat space using the flat space metric η_μν and its inverse η^μν, but the original definition of g^μν as the inverse matrix of g_μν must be kept.

Thanks Bill . I'm just trying to convince the little mathematician in my head ...