FOURIER SPECTRUM OF SIGNAL CHIRPED BY ACCELERATION

Consider a plane wave

$$\psi=A\cos (\mathbf{k\cdot x})\equiv A\cos(k_xx+k_yy+k_z z-\omega t)$$ (1)

characterized by its wave propagation four-vector

$$\mathbf{k}:~(\omega_e,k_z,k_y,k_z)=(\omega_e,\omega_e,0,0),~~\omega_e>0~.$$

One finds that relative to a uniformly linearly accelerated rocket whose four-velocity is

$$\mathbf{u}:~(\cosh g\tau,\sinh g\tau,0,0),~~-\infty<\tau<\infty~,$$

the observed frequency of that wave is

$$\begin{eqnarray*} \omega_{obs}(\tau)&=&-\mathbf{k\cdot u}\\ &=&\omega_e\cosh g\tau-\omega_e\sinh g\tau\\ &=&\omega_e e^{-g\tau} ~. \end{eqnarray*}$$

This frequency depends on the rocket time $\tau$ ( $-\infty<\tau<\infty$), and thus refers to an exponentially ``chirped'' signal. It starts highly blue-shifted and becomes ultimately highly red-shifted.

a)

FIND the Fourier power spectrum of this plane wave as observed in the accelerated rocket frame.

Discussion:

The space and time coordinates of the accelerated frame are related to those of the inertial lab frame by

$$\left. \begin{array}{c} t=(\xi+g^{-1})\sinh g\tau\\ x=(\xi+... ...} -g^{-1}<\xi<\infty\\ -\infty<\tau<\infty \end{array}\right.$$ (2)

The rocket is located at $\xi=0$, and it traces out the hyperbolic world line

$$x^2-t^2=g^{-2}~.$$

To streamline the mathematical analysis, replace Eq.(2) with

$$\left. \begin{array}{c} t=\xi\sinh \tau\\ x=\xi\cosh \tau \... ...ray}{c} 0<\xi<\infty\\ -\infty<\tau<\infty \end{array}\right.$$ (3)

Thus, in this streamlined formulation the rocket is located at $\xi=g^{-1}$. Relative to these coordinates the phase

$$\mathbf{k\cdot x}=k_xx+k_yy+k_z-\omega_e t$$

of the plane wave becomes

$$\begin{eqnarray*} \mathbf{k\cdot x}&=&\omega_e \xi \cosh \tau-\omega_e \xi \sinh \tau\\ &=&\omega_e \xi e^{-\tau} \end{eqnarray*}$$

and the (chirped) plane wave becomes

$$\begin{eqnarray*} \psi&=&A\cos(\mathbf{k\cdot x})\\ &=&A\cos(\omega_e \xi e^{-... ...frac{A}{2} \exp(i\omega_e \xi e^{-\tau})+\textrm{compl. conj.}~. \end{eqnarray*}$$

The Fourier amplitude of this chiped wave train in the accelerated frame is

$$\begin{eqnarray*} \hat\psi(\Omega)&=&\int_{-\infty}^\infty \psi(\tau)e^{i\Omega\... ...au}}+e^{-i\omega_e \xi e^{-\tau}}\right] e^{i\Omega\tau}d\tau ~. \end{eqnarray*}$$

This Fourier integral is readily evaluated in terms of the familiar gamma function

$$\Gamma(z)=\int_0^\infty e^{-t}t^{z-1}dt ~,$$

or equivalently

$$\Gamma(z)=e^{\mp iz\pi/2}\int_0^\infty e^{\pm iw}w^{z-1}dw ~.$$

The evaluation is achieved by letting

$$\begin{eqnarray*} w&=&\omega_e \xi e^{-\tau}\\ d\tau&=&-\frac{dw}{w}~. \end{eqnarray*}$$

The Fourier integral becomes

$$\hat\psi(\Omega)=\frac{A}{2} \left(\omega_e\xi \right)^{-i\O... ...}\int_0^\infty \left[ e^{iw}+e^{-iw}\right]w^{i\Omega-1}dw ~.$$ (4)

Each of these two integrals is proportional to the gamma function $\Gamma(i\Omega)$. In fact, one has

$$\hat\psi(\Omega)=\frac{A}{2} \left[(\omega_e\xi \right)^{-i\... ...mma(i\Omega)\left[ e^{\pi\Omega/2}+e^{-\pi\Omega/2} \right] ~.$$

This is the Fourier amplitude of the chirped signal observed in the rocket frame at $\xi=const$.

The Fourier power spectrum is obtained by taking advantage of the identity

$$\Gamma(i\Omega)\Gamma(-i\Omega)=\frac{\pi}{\Omega\sinh\pi\Omega} ~.$$

End of discussion

b)

FIND the Fourier power spectrum of the phase shifted plane wave

$$\psi=A\cos (\mathbf{k\cdot x})\equiv A\cos(k_xx+k_yy+k_z-\omega t+\delta_{\omega_e}) ~,$$ (5)

where

$$\delta_{\omega_e}=\textrm{phase shift}$$

associated with the plane wave having frequency $\omega_e$ in the inertial lab frame.

c)

Does this power spectrum depend on the frequency $\omega_e$? If so, how?

d)

BRING that power spectrum into the form

$$\vert \hat\psi(\Omega)\vert^2=B\times\left[ \frac{1}{2}+ N(\... ...omega_e}\times \sqrt{N\left(\Omega)(N(\Omega)+1\right)}\right]$$ (6)

WHAT is the value of the constant $B$? EXHIBIT the function $N(\Omega)$.

CONGRATULATIONS! You have established a very deep connection between the theory of special functions (the ``gamma function'') and statistical physics! If you are familiar (or want to become familiar) with the statistical physics of thermal (black body) radiation, compare this power spectrum with that of black body radiation. You may do this by assuming that one has an ensemble of plane waves having frequencies

$$\omega_e=\omega_1,\omega_2,\cdots,\omega_n,\cdots,etc.~,$$

and respective random phase shifts

$$\delta_{\omega_e}=\delta_1,\delta_2,\cdots,\delta_n,\cdots,etc.~,$$

and answer the following questions:

e)

The function $N(\Omega)$ is named after a famous physicist who first introduced it into physics. WHO is he?

f)

WHAT is the value of the temperature? REEXPRESS this temperature in terms of the acceleration $g$, the speed of light $c$, all in conventional physical units.

CONGRATULATIONS! You have derived and identified the ``acceleration temperature'' for an accelerated frame. Its physical significance is a mystery to this day, twenty-seven years after its identification by William Unruh and Paul Davies in 1976.

g)

Discuss the meaning of the three terms in the square bracket in Eq.(6).