Moving With Constant Acceleration

Consider an astronaut aboard a spaceship moving with uniform acceleration through Minkowski spacetime. The two-dimensional matrix representation of the metric tensor is given as follows:

The line element corresponding to this scenario is:

By parameterizing the observer's motion using proper time, we derive two conditions (the second being the derivative of the first):

The Horizon of the Accelerated Observer

An observer undergoing constant force moves in a hyperbolic manner. As illustrated in the following diagram, such an observer can overtake a light beam if given a sufficient head start, creating a hidden region delineated by a horizon—akin to the phenomenon in black holes.

In the spacetime diagram below, photon A, which departs from the origin at x=0 when t < 0, eventually catches up with the observer, while photon B, which crosses the origin when t > 0, does not.

In the comoving inertial frame, where the observer is at rest, we find:

Notably, from the previous equations, we establish:

Our objective is to demonstrate that an accelerating observer will perceive particles, unlike an observer at rest. To facilitate this, we will utilize new coordinates known as lightcone coordinates to describe Minkowski spacetime.

Lightcone Coordinates

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

The matrix representation of the corresponding metric tensor in Minkowski spacetime is:

By substituting the metric tensor into the earlier equations, we arrive at:

These can be readily solved, leading us to:

From the lightcone coordinates, we can derive:

It is essential to note that:

Thus, we can conclude that the worldline of an accelerated observer in the (t,x) coordinate system is represented as a hyperbola:

• The observer, initially at x ? ??, comes to rest at x = 1/a and then accelerates back towards x ? +?.
• As x approaches infinity, the trajectory nears the lightcone.

Comoving Frames

Next, we will identify a comoving frame for our accelerated observer. Following Mukhanov and Winitzki, we seek a reference frame where:

• The observer remains at rest when the spatial component x¹=0
• The time coordinate is defined as ?=t, the proper time of the observer along their worldline

For the metric in this comoving frame, it is advantageous for it to be conformally flat, a property that becomes relevant when integrating quantum mechanics into our analysis. A conformal map retains angles locally, although it does not necessarily preserve lengths.

In conformally flat manifolds, each point can be mapped to flat space using a conformal transformation. Therefore, the line element in the comoving frame can be expressed as:

Here, the function remains undetermined. To ascertain the expression for f(?,x¹), we first define the lightcone coordinates of the comoving reference frame:

One can demonstrate that, to avoid quadratic differentials of the comoving lightcone coordinates in ds², the following variable dependencies must hold:

After a few additional straightforward steps, we quickly derive the forms of these functions:

We can now utilize the results obtained to express the line element in the comoving frame:

This is referred to as Rindler spacetime, which is similar to Minkowski spacetime (devoid of curvature) but only encompasses a quarter of it (thus it is incomplete). The diagram below illustrates examples of Rindler accelerated observers.

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

Introducing Quantum Fields

Next, we explore a massless scalar field within a 1+1 dimensional spacetime. The action can be represented as:

This action exhibits conformal invariance:

Here, g signifies the determinant of the metric tensor, illustrating the similarity between S in both inertial and accelerated frames:

In lightcone coordinates, we can readily determine the field equations and their solutions (detailed in Mukhanov and Winitzki). The solutions consist of sums of right and left-moving modes:

This characteristic of the solutions, along with the earlier equations, indicates that opposite moving modes do not interfere with each other and can thus be treated independently. Moving forward, only the right-moving modes will be considered for clarity.

To this point, the analysis has remained classical, lacking quantum mechanics. We will now transition into quantizing the theory.

Within the Rindler wedge, the coordinate systems overlap, allowing us to employ the standard canonical procedure to quantize the theory and expand the quantum field operator ?:

In this expression, (LM) indicates left-moving modes. The operators

adhere to standard commutation relations, which will be omitted for brevity. It is crucial to note the existence of two vacuum states:

The appropriate vacuum state is contingent on the specific experiment conducted. For instance, from the viewpoint of the Rindler observer, the Minkowski vacuum appears to contain particles. In other words, if the quantum fields are in the Minkowski vacuum state, a detector operated by the Rindler observer will register massless particles. Conversely, if the quantum fields reside in the Rindler vacuum, no particles will be detected.

Relation Between a and b Operators

The transformations linking the operators are referred to as Bogolyubov transformations, named after the Soviet mathematician and physicist Nikolay Bogolyubov.

By substituting the Bogolyubov transformations into the quantum field operator expansion and performing straightforward manipulations, we can derive the coefficients:

The Unruh Temperature

We now arrive at significant findings:

• The average density of particles with frequency ? as perceived by the accelerating observer.
• The Unruh temperature, which corresponds to the temperature of the Bose-Einstein distribution governing the massless particles detected by an accelerated observer in the Minkowski vacuum.

The mean density is given by:

This corresponds to a temperature known as the Unruh temperature:

Physical Explanation

The Unruh effect can be understood as follows: fluctuations in the quantum vacuum interact with the detector carried by the accelerating observer. This interaction excites the detector, resembling a thermal environment at the temperature specified by the Unruh temperature. The energy resulting from these fluctuations is produced by the mechanism responsible for the acceleration, such as a propulsion system that accelerates the spaceship carrying the observer.

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