Moving with Constant Acceleration

Consider an observer, such as an astronaut in a spaceship, traveling through Minkowski spacetime with a steady acceleration. The metric tensor η can be represented in two dimensions as follows:

The line element corresponding to this scenario is expressed as:

By using the proper time τ to parametrize the observer's motion, we arrive at two key conditions (the second derived from the first):

The first condition indicates that the 2-velocity is normalized, while the second indicates the orthogonality of the 2-acceleration and velocity.

The Horizon of the Accelerated Observer

An observer experiencing constant acceleration follows a hyperbolic trajectory. As illustrated in the diagram below, an accelerated observer can surpass a light ray given an adequate head start, thus creating a hidden region delineated by a horizon. This phenomenon mirrors that of black holes, which also possess an unobservable region bounded by a horizon.

The spacetime diagram below depicts the interaction of two photons with the observer. Photon A, emitted from the origin before t=0, eventually catches up with the observer, while Photon B, which crosses the origin after t=0, does not.

In an instantaneously comoving inertial frame, where the observer remains stationary, we have:

From this, it follows that:

This condition holds true across all inertial frames, leading us to:

Our aim is to demonstrate that an accelerating observer detects particles, unlike a non-accelerated observer. To achieve this, we will employ a new set of coordinates known as lightcone coordinates.

Lightcone Coordinates

These coordinates are defined in relation to the original (t,x) coordinates as follows:

The corresponding metric tensor for Minkowski spacetime in these coordinates is given by:

Substituting η with the above expression in the previous equations yields:

After some algebraic manipulation, including rescaling and shifting the origin, we find:

This describes the trajectory of an accelerated observer.

From the definition of lightcone coordinates, we can derive:

In this coordinate system, the trajectory of the accelerated observer takes the shape of a hyperbola.

Thus, we conclude that in the t-x coordinate system, the worldline of an accelerated observer is indeed a hyperbola.

Comoving Frames

Next, we will establish a comoving frame for our accelerated observer. Following the insights of Mukhanov and Winitzki, we seek a frame where:

• The observer is at rest when the spatial component ξ¹=0
• The temporal coordinate ξ⁰=τ, representing the observer's proper time

It is advantageous for the metric in this frame to be conformally flat, a characteristic that will become significant when we integrate quantum mechanics.

In conformally flat manifolds, each point can be mapped to flat space via a conformal transformation. Consequently, the line element in the comoving frame takes the form:

where Ω remains undefined. To determine Ω(ξ⁰, ξ¹), we first define the lightcone coordinates of the comoving frame:

One can show that to avoid the emergence of quadratic differentials of the comoving lightcone coordinates on ds², we must have the following relations:

Following a few more straightforward steps, we can derive the explicit forms of these functions:

Now we can explicitly express the line element in the comoving frame:

This describes Rindler spacetime, which is equivalent to Minkowski spacetime but only encompasses a quarter of it, thus rendering it incomplete.

The original coordinates x and t can be expressed in terms of the ξ variables:

Introducing Quantum Fields

Let’s delve into the dynamics of a massless scalar field within a 1+1 dimensional spacetime. The action can be expressed as:

This action is conformally invariant:

This invariance elucidates the similarities between the actions in inertial and accelerating frames:

By employing lightcone coordinates, we can readily ascertain the field equations and their solutions, which consist of sums of right and left-moving modes:

This characteristic, coupled with the earlier equations, indicates that oppositely moving modes do not interfere with one another and can therefore be treated independently. Henceforth, we will focus exclusively on the right-moving modes for clarity.

Having conducted a classical analysis, we will now transition to quantization.

Within the Rindler wedge, the coordinate frames overlap, allowing us to apply the standard canonical quantization method to the quantum field operator ϕ:

where (LM) denotes left-moving modes. The operators defined by:

adhere to standard commutation relations, which will be omitted for brevity. It is important to note that there exist two distinct vacuum states:

The appropriate vacuum state is contingent upon the nature of the experiment. For instance, from the perspective of the accelerated observer, the Minkowski vacuum appears as a state containing particles. Conversely, if the quantum fields are in the Rindler vacuum, no particles will be detected.

Relationship Between a and b Operators

The transformations connecting these operators are termed Bogolyubov transformations, named after the renowned Soviet physicist Nikolay Bogolyubov.

Substituting the Bogolyubov transformations into the mode expansion and performing some manipulations leads us to:

The final results yield insights into the mean density of particles with frequency Ω as perceived by the accelerated observer and introduce the concept of Unruh temperature, which reflects the temperature of the Bose-Einstein distribution for massless particles detected by the accelerated observer in the Minkowski vacuum.

This temperature is defined as the so-called Unruh temperature:

Physical Interpretation

The Unruh effect can be interpreted as a consequence of quantum vacuum fluctuations interacting with the detector carried by the accelerated observer. This interaction results in the detector behaving as if it were in a thermal bath, with the temperature specified by the derived equations. The energy responsible for such fluctuations originates from the mechanism driving the acceleration, such as a propulsion system in the observer's spaceship.