An intuition on the Rindler modes

When we are solving the Klein-Gordon equation for the quantization of a massive scalar field on the Minkowski spacetime, we may use the global coordinates and obtain the usual quantization with plane waves of the form (first using the $t$-time translation invariance of the metric, then Fourier transform to solve the resulting equation): $$u(x,t) \sim \exp(-i\omega t + i k.x).$$ When one does the same thing in a different coordinate frame, corresponding to a constantly accelerating observer, restricted to a quarter of the Minkowski spacetime, one uses the $\eta$-time translation invariance, where $$t=\xi \sinh \eta, x= \xi \cosh \eta, \xi \in \mathbb{R}^+, \eta \in \mathbb{R},$$ and the solution to the resulting equation is this time a modified Bessel function (MacDonald) (we can see that in the Rindler-frame, there is a potential that goes to infinity when $\xi \rightarrow \infty$, so any wave packet coming from the left should return to the left). So those correspond to modes that diagonalize the boost generator, and we now write: $$v(\xi, \eta) \sim \exp (-i\omega \eta) K_{i\omega}(\xi).$$ In the Rindler-spacetime, the absolute value of this is independent on $\eta$, the wave follows the observer.

Now looking at what those modes look like by decomposing them in plane waves of $u$-type (I used a particular quantization based on operators that diagonalize the boost operator, still annihilating the same vacuum as the operators associated to $u$-modes), one gets something like this: $$\exp(-i\Omega \eta) K_{i\Omega}(\xi) = \sqrt{\frac{m}{2\hat\omega_k}}\int \frac{dk_1}{\omega_k} \exp\left(-i\omega_k t+ ik_1 + ix \cosh \eta - i \frac{\Omega}{2} \log\left(\frac{\omega_k + k_1}{\omega_k - k_1}\right)\right) \qquad (1)$$

$$\omega_k = \sqrt{m^2 + k_ik^i} , i =1, 2, 3$$ $$\hat\omega_k = \sqrt{m^2 + k_ik^i} , i =2, 3$$

where the x-axis represents $\eta$ and the y-axis is for $\log \xi$ (the wiggling part is coming from the numerical cutoff needed to evaluate the integral I think). The curve represents the maximum of the absolute value of a propagating mode through the Rindler spacetime (it thus comes from the left and returns to the left as said above). They thus seem to follow inertial trajectories (ie straight lines) in the Minkowski spacetime (as one can readily see that hyperbolic trajectories in one correspond to straight line in the other). But we only decomposed one eigenfunction of the boost generator, what happened?

I would greatly appreciate any help regarding this seemingly strange paradox, or indication towards what I have wrongly understood.

• Sorry, but it's not clear to me what exactly this paradox is that you're talking about? Can you reword what's confusing you? – Danu Mar 19 '16 at 9:13
• The left hand side of the equation (1) doesn't depend on $\eta$ when the absolute value is taken, but the right hand side does, as appears on the plot. I'll add details regarding to how I obtained this equation (1). – faero Mar 19 '16 at 10:25
• I would be interested in seeing those details. This is an interesting question, but it might be more transparent if an extra step or so were included – IntuitivePhysics Mar 24 '16 at 15:44