Can Different Vacuum States in Curved Spacetime Appear Thermal to Each Other?

[grammophon]

Recently I attend a course on quantum field theory in curved space and find it's very difficult for me to understand vacuum state in curved spacetime properly.

For example, as I try to recover the Gibbons-Hawking temperature in dS space, I was told to solve Klein-Gordon equation in both planar and static coordinates (I use the textbook by Birrell and Davies (5.54) and (5.76)), then the spectrum of radiation detected by static observer can be deduced from Bogoliubov coefficient which are used to related the distinct vacuum states (Bunch-Davies vacuum and static vacuum) in two coordinates.

It is very confuse for me that, do the comoving observer in planar coordinates should define BD vacuum as a no-particle state ? If a comoving observer (w.r.t conformal time) in BD vacuum could see nothing, is this correct to say that he would detect a thermal spectrum when he turns to be static (comoving w.r.t cosmic time) since BD vacuum appears thermal then from the view of new static vacuum?

 

[Naty1]

As you are likely aware, in general different observers do see different vacuum states.
For example, Rindler coordinates and Schwarzschild coordinates reveal different vacuum states for inertial and accelerating observers.

This discussion may or may not answer the precise questions you ask, but will I think at least provide some useful insights... https://www.physicsforums.com/showthread.php?t=574548

 

[bapowell]

It is very confuse for me that, do the comoving observer in planar coordinates should define BD vacuum as a no-particle state ? If a comoving observer (w.r.t conformal time) in BD vacuum could see nothing, is this correct to say that he would detect a thermal spectrum when he turns to be static (comoving w.r.t cosmic time) since BD vacuum appears thermal then from the view of new static vacuum?

Yes, each of their respective vacua appear thermal to the other.

 

[grammophon]

Yes, each of their respective vacua appear thermal to the other.

Do you mean Bunch-Davies vacuum also appears thermal to a comoving observer w.r.t conformal time? Is this conflict with the fact that Bunch-Davies vacuum is defined for this comoving observer as a no-particle state?

I thought about such a process: at first, for a comoving observer w.r.t conformal time Bob, he can define a no-particle state as Bunch-Davies vacuum and detect no particle creation. Then, Bob turns to be static and a new no-particle state can be defined as static vacuum. Of course, no particle creation can be detected in new static vacuum. However, Bob would find the BD vacuum he got before appears thermal w.r.t to new vacuum, then a thermal spectrum can be detected.

Am I right on above picture? Hope for more help!

 

[sardar]

I can provide a response to this question by explaining the concept of vacuum states in curved spacetime and how they may appear thermal to each other.

In quantum field theory, the vacuum state is defined as the lowest energy state of a quantum field. In flat spacetime, this vacuum state is unique and there is no ambiguity in its definition. However, in curved spacetime, the concept of vacuum state becomes more complex.

One of the key concepts in curved spacetime is the concept of observer-dependent vacuum states. This means that different observers, depending on their motion and position, may perceive the vacuum state differently. In other words, the vacuum state is not unique for all observers in curved spacetime.

In the example you provided, the Bunch-Davies (BD) vacuum and the static vacuum are two different vacuum states that are related to each other through a transformation known as the Bogoliubov coefficient. This transformation describes how particles in one vacuum state appear as particles in the other vacuum state.

Now, to answer the question of whether different vacuum states in curved spacetime can appear thermal to each other, the answer is yes. This is because, for a given observer, the vacuum state they perceive may contain particles and therefore have a non-zero temperature. However, for another observer, this same vacuum state may appear as a no-particle state and therefore have a temperature of absolute zero.

In the case of the comoving observer in the BD vacuum, they would perceive it as a no-particle state. However, when they switch to the static vacuum, they would perceive it as a thermal state with a non-zero temperature. This is because the transformation between the two vacuum states would result in a non-zero Bogoliubov coefficient, indicating a non-zero particle content in the static vacuum.

In conclusion, the concept of vacuum states in curved spacetime can be confusing, but it is important to understand that they are observer-dependent and may appear thermal to each other. The concept of Bogoliubov coefficients helps us understand how different vacuum states are related to each other and how particles in one vacuum state may appear as particles in another.