 # Formula for the Bekenstein bound S &#x2264;<!-- ≤ --> 2 &#x03C0;<!-- π --> Ashley Fritz 2022-05-09 Answered
Formula for the Bekenstein bound
$S\le \frac{2\pi kRE}{\hslash c}$
where $E$ is the total mass-energy. That seems to imply that the presence of a black hole in the region is dependent on an observer's frame of reference. Yet, my understanding is that the Bekenstein bound is the maximum entropy that any area can withstand before collapsing into a black hole.
Does this mean that the existence of black holes is observer dependent? Or that even if an observer does not report a black hole in their frame, one is guaranteed to form there in the future?
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Individually, $E$ and $R$ are both frame-dependent, but at least in inertial reference frames, the product $ER$ is actually invariant.
Suppose we have a system with rest energy ${E}_{0}$ and proper length ${R}_{0}$. When we boost to a frame with Lorentz factor $\gamma$, in that frame the system has $E=\gamma {E}_{0}$ and $R=\frac{{R}_{0}}{\gamma }$, so $ER={E}_{0}{R}_{0}$.