In the Arora-Barak book, in the Cook-Levin reduction, the resulting SAT formula is of size $T(n)\log(T(n))$, where $T(n)$ is the running time of the given Turing machine.
Is there a method to get a $O(T(n))$ length SAT formula?
The original TM is made oblivious (a TM is oblivious if the position of the head at the $i$th step depends only on the length of the input string). This step brings the $O(\log(n))$ factor. Once we have an oblivious TM that takes time $T'(n)$, the SAT formula is of length $O(T'(n))$.
Can we somehow bypass the step of making the TM oblivious?