The Unruh Effect: Acceleration and Thermalization in Minkowski Vacuum

1. A Coordinate System for Accelerated Observers: Rindler Spacetime

Before we can quantize a field for an accelerating observer, we must first describe their motion and the spacetime they perceive. We are not dealing with a curved spacetime here—this entire analysis takes place in flat Minkowski space. We are simply describing it from the perspective of a non-inertial reference frame.

An observer with constant proper acceleration $a$ in the $x$-direction follows a hyperbolic trajectory in Minkowski spacetime, given by $x^2-c^2{t}^2=(c^4/a^2)$. This motivates a change of coordinates from the inertial Minkowski coordinates $(t,x)$ to a new set that is better adapted to the accelerating observer. These are called Rindler coordinates.

The video provides a great visual foundation. For our derivation, we'll use a slightly different but equivalent parameterization of Rindler coordinates, $(\eta ,\xi )$, which is standard in the literature on the Unruh effect. For an observer with a given acceleration, these coordinates provide a local notion of time ($\eta$) and space ($\xi$).

To summarize, the transformation from Rindler coordinates $(\eta ,\xi )$ to Minkowski coordinates $(t,x)$ (in natural units, $c=1$) is:

A crucial result is the form of the Minkowski line element in these new coordinates:
$d{s}^2=-d{t}^2+d{x}^2=e^{2a\xi }(-d{\eta }^2+d{\xi }^2)$
This metric is conformally flat, meaning it's the flat Minkowski metric multiplied by a position-dependent factor $e^{2a\xi }$. This property is key.

The Rindler coordinates $(\eta ,\xi )$ only cover the region of Minkowski space where $x>|t|$. This region is called the Rindler wedge. The boundaries of this wedge, $x=\pm t$, act as causal horizons for any observer confined to the wedge.

2. Field Quantization: Minkowski vs. Rindler

Now we consider a massless scalar field $\varphi$ propagating in this spacetime. We will quantize it from two different points of view.

A. The Inertial (Minkowski) Observer
This is the standard quantization you've seen before.

• The field equation is $\square \varphi =(-{\partial }_t^2+{\partial }_x^2)\varphi =0$.
• The observer defines positive-frequency modes with respect to their time coordinate $t$, using the timelike Killing vector ${\partial }_t$. A mode $u_k$ is positive frequency if ${\partial }_t{u}_k=-i{\omega }_k{u}_k$ with ${\omega }_k>0$. These modes are plane waves, $u_k\propto e^{-i{\omega }_{k}t+ikx}$.
• The field operator is expanded as $\hat{\varphi }=\int dk\left({\hat{a}}_k{u}_k+{\hat{a}}_k^{†}{u}_k^{\ast }\right)$.
• The Minkowski vacuum is defined as the state annihilated by all ${\hat{a}}_k$: ${\hat{a}}_k|0_M⟩=0$ for all $k$.

B. The Accelerating (Rindler) Observer
This observer lives entirely within the Rindler wedge.

• In Rindler coordinates, the Klein-Gordon equation $\square \varphi =0$ becomes $e^{-2a\xi }(-{\partial }_{\eta }^2+{\partial }_{\xi }^2)\varphi =0$, which simplifies to $(-{\partial }_{\eta }^2+{\partial }_{\xi }^2)\varphi =0$. This is just the wave equation in $(\eta ,\xi )$ coordinates!
• The Rindler observer has their own timelike Killing vector within the wedge, ${\partial }_{\eta }$. They use this to define their positive-frequency modes. A Rindler mode $f_k$ is positive frequency if ${\partial }_{\eta }{f}_k=-i{\Omega }_k{f}_k$ with ${\Omega }_k>0$. The solutions are plane waves in Rindler coordinates, $f_k\propto e^{-i{\Omega }_k\eta +iK\xi }$.
• The field operator is expanded using these modes: $\hat{\varphi }=\int dK\left({\hat{b}}_K{f}_K+{\hat{b}}_K^{†}{f}_K^{\ast }\right)$.
• The Rindler vacuum is defined as ${\hat{b}}_K|0_R⟩=0$ for all $K$.

We have two different definitions of the vacuum, $|0_M⟩$ and $|0_R⟩$, based on two different notions of time evolution. Our goal is to find out what an accelerating observer sees when the field is in the Minkowski vacuum. That is, we must compute the number of Rindler particles in the state $|0_M⟩$: $⟨0_M|{\hat{b}}_K^{†}{\hat{b}}_K|0_M⟩$.

3. The Derivation: Bogoliubov Coefficients via Analytic Continuation

To find the number of Rindler particles, we need the Bogoliubov coefficients relating the Minkowski and Rindler operators. We will follow a particularly elegant method pioneered by Unruh, which relies on analytic continuation in the complex plane. This avoids a direct, and more cumbersome, calculation of the overlap integrals.

Let's recap the stunning result of that derivation. The Bogoliubov transformation relating the Rindler annihilation operator ${\hat{b}}_k$ to the operators ${\hat{c}}_k^{(1)}$ and ${\hat{c}}_k^{(2)}$ (which define the Minkowski vacuum) is found to be:
${\hat{b}}_k=\frac{1}{\sqrt{2\sinh (\pi {\omega }_k/a)}}\left(e^{\pi {\omega }_k/2a}{\hat{c}}_k^{(1)}+e^{-\pi {\omega }_k/2a}{\hat{c}}_{-k}^{(2)†}\right)$
Here, ${\omega }_k$ is the frequency of the Rindler mode. Now, we compute the expectation value of the Rindler number operator ${\hat{n}}_{b_k}={\hat{b}}_k^{†}{\hat{b}}_k$ in the Minkowski vacuum $|0_M⟩$. Since ${\hat{c}}_k|0_M⟩=0$, the only term that survives is the one containing ${\hat{c}}^{†}\hat{c}$ after commuting the operators. The calculation yields:
$⟨{\hat{n}}_{b_k}⟩=⟨0_M|{\hat{b}}_k^{†}{\hat{b}}_k|0_M⟩=\frac{1}{e^{2\pi {\omega }_k/a}-1}$
This is a momentous result. The spectrum of particles seen by the accelerating observer is not just some random distribution; it is a perfect Planckian thermal spectrum.

4. The Unruh Temperature

By comparing this result to the Bose-Einstein distribution for thermal radiation, $⟨n⟩=\frac{1}{e^{E/k_{B}T}-1}$, we can identify the energy of the Rindler particle as $E={\omega }_k$ (setting $\hslash =1$) and the temperature of the thermal bath.

This gives the Unruh Temperature:
$k_{B}T=\frac{\hslash a}{2\pi c}$
Or in natural units ($\hslash =c=k_B=1$):
$T=\frac{a}{2\pi }$
The temperature measured by the accelerating observer is directly proportional to their proper acceleration.

The accelerations required to produce a measurable temperature are enormous. To see a temperature of just 1 Kelvin, you would need an acceleration of about $10^{20}{\text{ m/s}}^2$. Nevertheless, the Unruh effect is a fundamental prediction of quantum field theory and a crucial piece of the puzzle connecting gravity, quantum mechanics, and thermodynamics. It tells us that the vacuum state is more complex than it appears, and its particle content is in the eye of the beholder.

Conclusion

This lesson has been a deep dive into one of the most remarkable predictions of theoretical physics. By combining the formalism of Bogoliubov transformations with the kinematics of accelerated reference frames, we have derived the Unruh effect from first principles.

Key Takeaways:

• A uniformly accelerating observer is best described using Rindler coordinates, which cover a 'wedge' of Minkowski spacetime and feature a causal horizon.
• The definition of a vacuum state is observer-dependent. The quantization of a scalar field by an inertial (Minkowski) observer leads to a different vacuum state, $|0_M⟩$, than the one defined by an accelerating (Rindler) observer, $|0_R⟩$.
• By computing the Bogoliubov transformation between the two frames, we found that the Minkowski vacuum $|0_M⟩$ contains a thermal distribution of Rindler particles.
• The temperature of this thermal bath, the Unruh temperature, is directly proportional to the observer's proper acceleration: $T=a/(2\pi )$.

Preview of the Next Lesson:

The structure we've analyzed—an observer near a causal horizon—has a profound parallel in general relativity: an observer hovering just outside the event horizon of a black hole. According to the equivalence principle, the physics for this stationary observer in a gravitational field should be locally indistinguishable from the physics for our accelerating Rindler observer. This powerful analogy is the key to our next topic: formulating quantum field theory near a black hole's event horizon and, from there, deriving the famous Hawking radiation.