Question

As far as I understand Wick rotation, it means the mathematical transformation

ct→jct
Where j is imaginary unit. While reading on CTC (closed timelike curves) in the Gödel metric I came across this particular diagram of rotating light cone (the orthogonal x−t axes are essentially rotating beyond a specific radius). Does this mean that after every 90 degree rotation space and time periodically switch their roles?
(x,t)→(−ct,x/c)→(−x,−t)→(ct,−x/c)→(x,t)
Secondly, is this role-switching anyhow related to that inside a black hole? And thirdly, can CTCs inside a rotating black hole be explained on similar lines?

Answer 1

Wick rotation involves using imaginary time to simplify mathematical models, and its relation to CTCs in the Gödel metric suggests a periodic switching of space and time which is an oversimplification. Lorentz transformations depict axes rotations in spacetime, affecting simultaneity and time measurements, and are essential in understanding relativistic phenomena like those in black holes.

Step-by-step explanation:

Understanding Wick rotation and its implications on closed timelike curves (CTCs) in the context of the Gödel metric involves a mix of theoretical physics and complex mathematics. The process of Wick rotation involves replacing the time coordinate with an imaginary time coordinate (ct → jct), which can help transition from a physical spacetime metric to an Euclidean metric for easier mathematical treatment. The rotation of the light cone in the Gödel metric suggests that beyond a specific radius, one can perceive the periodic switching of space and time coordinates as you described, but this interpretation is a simplification of more complex relativistic behavior. In terms of black holes, such as rotating black holes (Kerr black holes), there are regions where CTCs can theoretically exist. However, the presence of CTCs within a black hole's structure presents additional complexities, and while there are similarities in the mathematical descriptions, their physical interpretations may differ significantly from the Gödel model.

Relativity and Lorentz transformations play critical roles in understanding these phenomena, as these transformations can be visualized as a sort of 'rotation' in space-time where time and space coordinates are adjusted. However, unlike in normal spatial rotations, the Lorentz transformation alters scales along the axes and does not preserve perpendicularity. The Lorentz transformation incorporates the invariant nature of light's speed across all reference frames, which ensures that light cones maintain their structure under such transformations. Thus, aspects of simultaneity and the relative measurements of time between different observers can differ dramatically.