Saturday, 23 November 2019

vectors - Metric tensor and imaginary time


I just started a re-reading of the Conformal Field Theory yellow book by Di Francesco et al. In chapter two, after defining imaginary time $\tau$ as $t=-i\tau$, the authors state that the metric tensor, being in real time, e.g., $g_R=\mbox{diag}(1,-1,-1,-1)$, becomes in imaginary time $g_I=\mbox{diag}(1,1,1,1)$.


How can this statement be proven? I know it is a maybe stupid question, but I can't figure it out. For instance, if I consider the covariant four-vector $x^\mu_R=(t,x_0,x_1,x_2)$, the contravariant vector becomes, by application of $g_R$, $x_{R,\mu}=(t,-x_0,-x_1,-x_2)$. Performing Wick rotation, they become $x^\mu_I=(-i\tau,x_0,x_1,x_2)$ and $x_{I,\mu}=(-i\tau,-x_0,-x_1,-x_2)$, that are not transformed into each other by $g_I$! Where does my argument breaks?




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...