KLT of radio signals from relativistic spaceships in arbitrary motion

Abstract

In three papers [1–3] the present author applied the concept of time-rescaled Brownian motion to aspects of relativistic interstellar flight ranging from communication theory to genetics. The content of the present chapter was fully published in paper form in [11]. In particular, the Gaussian noise (Brownian motion) X(t), emitted by a relativistic spaceship traveling at a constant acceleration g in its own reference frame, was shown to be—see Equation (12.10) or [1, eq. (53)]

$$ X\left( t \right) = B\left( {\frac{c} {g}\arcsin h\left( {\frac{g} {c}t} \right)} \right) = B\left( {\frac{c} {g}\ln \left[ {\frac{g} {c}t + \sqrt {1 + \left( {\frac{g} {c}t} \right)^2 } } \right]} \right), $$

(13.1)

where c is the speed of light, B(t) denotes standard Brownian motion with mean zero, variance t, and initial condition B(0) = 0, and time ranges within the finite interval 0 ≤ t ≤ T.