Unruh’s Effect Savan Kharel

Historical Introduction Unruh Effect shows us that accelerating observer in Minkowski vacuum state will observe a thermal spectrum of particles. Historically, this was discovered to understand the physics of Hawking effect. Hawking effect is similar but slightly different because it’s harder to solve wave equation in the curved spacetime.

Inertial Observers A Minkowski vacuum is something that all inertial (not accelerating) observers agree. If the stationary observer sees nothing, then the observer moving with constant v w.rt to the observer sees nothing. (as one is related to the other by the boost)

Non inertial Observer If the velocity is not constant, then moving observer is no longer inertial. So what happens when the observers are not inertial is the premise of Unruh’s effect. For simplicity, lets consider the Minkowski metric with just one spatial coordinate for simplicity:

Constant accelerating path asymptotic to the null path Consider an observer moving at a uniform acceleration in the x direction. We can parametrize: Its easy to see that by taking the magnitude . The trajectory is a hyperboloid which are asymptotic to the null path x=+t and x=-t.

Light Cone Coordinates The trajectory is a hyperbolae which is asymptotic to x= +t, x=-t. Also define a special convenient coordinate( L.C Coordinate) The Light Cone gives 4 wedges. Line element,

Rindler’s Coordinates Also, we will introduce new coordinates on two dimensional Minkowski space.Lets define: The metric would look like this:

Some definitions Lets simplify this further and write down: Now the metric in these coordinates take the following form. The metric in region I defines what is known as R.S. The metric in terms of the Rindler light cone coordinate:

Rindler Space So the region I with this metric is known as Rindler observer moves along a constant acceleration path. Here, one sees how equal-time lines evolve

A little pause We defined the inertial versus the accelerating observer. We introduced light cone coordinates and then define the Rindler Space. The goal here is to relate the inertial vacuum with the Rindler (accelerating) vacuum.

Klein Gordon Equation Consider a Klein Gordon Equation for some massless field Equations of motions are given by: As we well know this admits a positive plane wave solutions with the standard normalization factor and the negative energy solution is given by

Right Moving and Left Moving When we have the right moving positive energy wave When we have the left moving negative energy wave.

General Solution to the Wave Equation The general solution to the wave equation is given by: Now we can define the vacuum:

(Almost ) same Story for Rindler Observer However the crucial difference is that only covers the quarter of the Minkowski space. The wave equation now is: This leads to the positive energy solutions:

(Almost) Same Story When we have the right moving positive energy wave When we have the left moving negative energy wave.

Analytic continuation beyond Region I We can analytically continue these plane waves beyond region I by expressing Rindler’s light cone coordinates in terms of Minkowski light cone coordinate:

Rindler Wave Solution After analytic continuation we obtain two sets of modes using analytic continuation: Also we can define the Rindler Vacuum

Lets think a little about the inertial observer again Inertial Observers may be expanded in more general solution: where, and the vacuum of the inertial observer is defined as,

Relation of Modes Using the above definitions, we can relate the inertial modes to the Rindler modes: And we can find the commutation relation of the Minkowski mode w.rt Rindler’s mode:

Now lets relate these modes to Rindler Modes Here is the Bogoliubov Transformation that relates the Rindler modes with the creation and annihilation operators of the inertial observer:

Number Operator Assume that system is in Minkowski vacuum The number operator for the Rindler observer: So using Bogoliubov transformation and definition of inertial vacuum, the expected number of such particles of some frequency will be given by (the expectation value):

Emergence of Black Body Spectrum The above is the black body spectrum: The delta function is an artifact of plane wave basis modes. If we had constructed normalized wave packet then we would have obtained a finite result with the identical spectrum. This gives the blackbody spectrum with

Summary of what has happened so far Inertial versus accelerating observer We discussed light Cone Coordinates and Rindler Space Related the modes of Minkowski and the Rindler modes Showed the expectation value which lead us remarkably to the Planck Factor and the black body spectrum.

Conclusion An observer moving with uniform acceleration through the Minkowski vacuum observes a thermal spectrum of particles. This is called the Unruh Effect. This shows how different set of observers will describe the same state in very different ways. In some sense this reveals thermal nature of vacuum in quantum field theory.