I'm reading Winitzki's Introduction to Quantum Effects in Gravity and I get some questions:
In the remark of section 8.3, Page 105. When we calculate the energy-momentum tensor in Rindler space, because $$ <0_R|(\partial_u \hat{\phi})^2|0_R> $$ is singular on the future horizon at $u=0$ so the Rindler vacuum is a singular state. Here why should the energy-momentum contain the term $\partial_u \hat{\phi}$? $u$ is the coordinate in Minkovski spacetime and I think the EMT in Rindler spacetime should be in terms of $\partial_{\tilde{u}} \hat{\phi}$
In equation (8.45) the delta function $\delta (0)$ should the volume of the V. But it comes from $$ \int d\tilde{u} $$ in the equation (8.38) and I think it doesn't make sense to interpret it as volume which should be $$ \int dx $$
3.About the Hawking effect in section 9.1, page 117. It is said that a non-external black hole formed as a result of collapse need not to consider the left-moving(outgoing) modes $u$ on the past light-like infinity. Also other lectures about Hawking effect also expand the field on the past light-like infinity only in terms of ingoing modes $v$, and on the future light-like infinity in terms of outgoing modes $v$. I get confused about that. They are modes and they are not particles. If I expand the field both in terms of $v$ and $u$ then what is going wrong?
They may be some naive questions. Thank you! :)
