Nearly Lorentz Coordinate Systems: Is h a Tensor?

[Silviu]

Hello! I am reading Schutz A first course in GR and he introduces the Nearly Lorentz coordinate systems as ones having a metric such that $g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}$, with h a small deviation from the normal Minkowski metric. Then he introduces the Background Lorentz transformations (this is section 8.3 in the second edition) in which all the points are transformed as $x^{\bar\alpha}=\Lambda_\beta^{\bar\alpha}x^\beta$. Applying this transformation to g he gets in the end that h transforms as $h_{\bar\alpha\bar\beta}=\Lambda_\mu^{\bar\alpha}\Lambda_\nu^{\bar\beta} h_{\mu\nu}$ and from here he says that we can treat h as if it was a tensor in SR and this simplify the calculations a lot. Can someone explain to me why ones need all these calculations for this? I am sure I am missing something but h is the difference between g and $\eta$ so isn't it a tensor, just because it is the difference between 2 tensors? Why do you need a proof for it? Moreover Schutz says that h "it is, of course, not a tensor, but just a piece of $g_{\alpha\beta}$". So can someone explain to me why isn't h a tensor and why my logic is flawed? Thank you!

 

[martinbn]

Why is $\eta_{\alpha\beta}$ a tensor?

 

[Silviu]

Why is $\eta_{\alpha\beta}$ a tensor?

Well a tensor (in this case a $(0,2)$ tensor) is a function that turns 2 vectors into a real number. $\eta$ is a 4x4 matrix so it behaves like a $(0,2)$ tensor, when applied to a 4D vector (which is our case).