# Minkowski space and Rindler spaces, What's relation?

To describe the dynamic of a qubit one can use the Minkowski cordinates, But when that qubit is accelerated with Unruh acceleration we need Rindler cordinate .

So, what is UNRUH acceleration? and what's the relation between the two spaces?

• In Minkowski Space coordinate system S‘ move relative to coordinate system S with constant velocity, you can use die Lorentz Transformation. If die coordinate systems S‘ ans S are move relatively with constant acceleration you use the Rindler metric.The Unruh effect is the phenomenon that observer will measure a density of particles following a Bose-Einstein distribution, where the temperature is identified with the acceleration.thphys.uni-heidelberg.de/~witkowski/curved/talk8.pdf – Eli Jul 15 at 19:46
• The Unruh effect has not been observed and has been challenged in publications as non-existent. I don't know what your source is, but my guess is that "Unruh acceleration" refers to Rindler acceleration, which is simply a constant acceleration. – safesphere Jul 15 at 21:00

## 1 Answer

There is no such thing as Unruh acceleration! What you are looking for is probably Unruh effect or Unruh temperature!

If you compare the Hilbert space of a Minkowski observer to the Hilbert space of a Rindler observer (an accelerated observer), you can easily notice that the latter is actually half of the former.

It is also easily seen by comparing the Penrose diagram associated with both metrics. On the Penrose diagrams, you can see that the accelerated observer with $$\rho>0$$ doesn't have access to $$x<0$$ events and vice versa.

Therefore, on a constant time surface where you can define the Hilbert space, a Rindler observer doesn't even see half of the space, so its Hilbert space is half of the Minkowski observer, which can see the whole space.

On the other hand, we have Equivalence Principle, meaning that we require physical phenomena to be observer independent. As a result, both Rindler and Minkowski observers should have the same interpretation of physical phenomena and calculate equal values for physical properties such as n-point correlation functions.

This additional requirement that we impose on the theory will lead to the fact that the vacuum states of Rindler observer are different from vacuum states of Minkoski observer.

Now if you compare these vacuum states, you will find that the Rindler observer would associate a non-trivial density matrix which is in fact a thermal one, leading to the observation that this observer would experience a thermal distribution and therefore associate a non-trivial temperature to its spacetime. This is called the Unruh temperature.

This phenomena can also be interpreted as the pair production of virtual particles exactly on the horizon of the Rindler observer. This temperature is actually very small ($$T=\frac{ℏa}{2πK}$$), which is very hard to measure experimentally. Here $$a$$ is the value of the acceleration and $$K$$ is Boltzmann's constant.

However, there have been some experimental evidence for this effect that you can check, for example in Experimental evidence for the Unruh effect.