Relativity: Gravity pseudotensor, Gibbons-Hawking-York term

[Bartolius]

In our course on General Relativity the professor explained the linearized version of Einstein's Equation and the emission of gavitational waves, then made a little digression on the conservation of energy in a gravitational system. He showed that the search for a conserved object (Momentum tensor of matter is covriantly conserved, so it doesn't fit our search) leads to the contruction of a pseudotensor from the sum of the Momentum Tensor of matter and the non-linear part of Einstein's Tensor. Then he constructed conserved quantities from the integral over all our volume and showed that all the information on conserved quantities lives on the boundary (a holographic property).
My questions are:
1 did I understand the logical process or did i mess up something? I ask this because I borrowed some notes from a friend and he seems to have done a little mess with covariant and classical derivatives of tensors.

2 How the covariant conservation of Einstein's tensor implies the conservation of it's linear part? on my friends notes is written that it can be shown by series, I assume that the covariant conservation equation for Einstein's tensor must hold at every order, and the linear term only receives contribution from the divergence (non covariant for the Christoffels are first order) of the linear part of Einsteins tensor.

3 this is about another topic: I studied Hilbert-Einstein's action and the necessity of a correction in the form of the Gibbons-Hawking-York term, but I lack any example in which that term plays a relevant role: I read around the internet that it is important for the entropy of horizons, and that it reproduces the famous A/4 proportionality relation for the entropy of a Schwarzschild Black-hole, but nothing more. Where can I find more informations on the importance of this term?

Thanks

 

[_coyote_]

in advance!1) Your understanding of the logical process is mostly correct, but there may be some details that you are missing. It is possible that your friend's notes contain some mistakes, so it would be best to double check any information against other sources. 2) The covariant conservation of Einstein's tensor implies the conservation of its linear part because the conservation equation for Einstein's tensor holds at every order. This means that the linear term only receives contribution from the divergence (non-covariant for the Christoffels are first order) of the linear part of Einsteins tensor. 3) The Gibbons-Hawking-York term is important for calculating the entropy of horizons. It can be used to calculate the entropy of a Schwarzschild Black-Hole, and has been used to calculate the entropy of other kinds of horizons as well. Additionally, it can be used to study the behavior of black holes in different spacetime dimensions. You can find more information on the importance of this term by searching online for academic papers on the subject.