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I have seen a lot of posts asking about the difference between the rapidity $y$ and the pseudorapidity $\eta$, and I understand it well enough (at least in the context of heavy-ion collisions). They're defined, respectively, as $$ y=\frac{1}{2}\ln\left(\frac{E+p_z}{E-p_z}\right),\ \eta=\frac{1}{2}\ln\left(\frac{|p|+p_z}{|p|-p_z}\right). $$

However, sometimes another one of those appears: a space-time rapidity $$ \eta_s = \frac{1}{2}\ln\left(\frac{t+z}{t-z}\right). $$ I believe that this is a spatial coordinate, as the name suggests, accompanying the proper time $\tau=\sqrt{t^2-z^2}$. In some texts it's said to be approximate in value to $\eta$ or $y$, which seems to simplify some calculations. However, I can't see clearly the relation between these quantities, other than them being boost-additive. Can someone clarify this for me?

I can hand-wave an understanding that they are similar in a symmetric HIC, since at the center of the collision ($z=\eta_s=0$) the fluid has almost no longitudinal expansion ($y=\eta=0$), and the same for forward rapidities; but I wanted a more satisfactory answer.

Urb
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Renan Nobuyuki Hirayama
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1 Answers1

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In the case of a particle traveling at a constant velocity, $t$ and $z$ would be related by that velocity. In the relativistic limit ($p \gg mc \to E \approx |p|c$), in which rapidity $y$ is approximately equal to pseudorapidity $\eta = \frac{1}{2} \ln \left( \frac{1 + \cos \theta}{1 - \cos \theta} \right)$, the space-time rapidity (spatial rapidity) is:

$$\begin{equation} \eta_s = \frac{1}{2} \ln \left( \frac{c t + z}{c t - z} \right) = \frac{1}{2} \ln \left( \frac{c t + v_z t}{c t - v_z t} \right) = \frac{1}{2} \ln \left( \frac{1 + v_z / c}{1 - v_z / c} \right) = \frac{1}{2} \ln \left( \frac{1 + \cos \theta}{1 - \cos \theta} \right) \end{equation}$$

as well. In other words, they're all equivalent expressions of the angle $\theta$ of the particle's travel relative to the beamline.

Here is a nice reference for anyone looking for a brief introduction to space-time rapidity, rapidity, and pseudorapidity: https://doi.org/10.1088/bk978-0-750-31060-4ch2 Another is https://arxiv.org/abs/1804.06469 pp. 8, 12-3. (Note that both use natural units, i.e., $c = 1$.)

Urb
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dssx
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