On the Unruh effect and the thermofield double state

Abstract: The purpose of this review is to provide a pedagogical development of the Unruh effect and the thermofield double state. In Section 2, we construct Rindler spacetime and analyze the perspective of an observer undergoing constant acceleration in Minkowski spacetime, which motivates the establishment of the relationship between the Fourier modes in both geometries using the Bogoliubov-Valatin transformation. In Section 3, we explore the underlying physics leading to the Unruh effect, its analogy with the thermal radiation observed around a Schwarzschild black hole, and its manifestation through the coupling of a particle detector to the scalar field. Finally, in Section 4, we derive the thermofield double state by conducting a Euclidean analysis of the field and geometry.