What is the definition of density in a relativistic context? 
In this question, there seems to be a clear presence of ambiguity, which raises the question: what is density in relativity?
First of all, the question asks for the relativistic mass, "the apparent mass to people on Earth." I did find that.
Using length contraction formula, one can easily "plug-and-chug" and find that the cube will look like it is $43.6m * 100m * 100m$.
However, when it comes down to density, must one use relativistic mass, or the inertial mass?
Whether I use both masses and divide by the above volume of $43.6m * 100m * 100m$, I get the wrong answer. The answer is supposedly $4.39*10^{3} kg/m^{3} $.
 A: One should always specify whether one is talking about rest mass per unit rest frame volume, $\rho_0 = m_0/V_0$, rest mass per unit observer-frame volume, $D = m_0/(V_0/\gamma) = \gamma\rho_0$, or relativistic mass per unit observer-frame volume, $(\gamma m_0)/(V_0/\gamma) = \gamma^2\rho_0$.1 (I can't imagine the fourth case, relativistic mass per unit rest frame volume, $(\gamma m_0)/V_0$, ever being implied, but it's the same as $D$ so it doesn't matter anyway.) Thus I would say the problem isn't particularly well defined.
Now, most anyone working in relativity these days never uses "relativistic mass" $m = \gamma m_0$, precisely because of all the confusion it causes. Thus "mass" is always rest mass, and so "density" is often rest mass per unit something, which is still ambiguous between what I've called $\rho_0$ and $D$. That said, the way part (a) is worded makes me think the problem thinks of "mass" as the frame-dependent quantity, in which case the third option I give makes most sense. But you'd have to clarify what the instructor's expectations are.
Finally, I can't see how the answer of $4.39\times10^3\ \mathrm{kg/m^3}$ makes any sense. Even if we agree the $6$ should be a $9$, that is $10/\gamma$ times $\rho_0$.

1Note my choice of symbols $\rho_0$ and $D$ are commonly used in my field of relativistic fluid dynamics, but these are by no means universal.
A: I don't think we should use the term "relativist mass" for the reason mentioned above. Relativistic density is equal to relativistic energy (gamma times mass times C-square) divided by (C-square times Volume).
Note: assume we use relativistic mass for an electron that passes close to the planet earth, them if one uses Newton's gravitational law (Gm1 m2/r2) there will be very large force between earth and that electron. However, in practice that is not the case. This is one reason of not using relativistic mass. In the past physicist used it to explain why an object can't move as fast as C; in this case the mass will get so large to an extent one can't push it further to get to speed of light. Although, in this case relativistic mass, was  a good "explanation" but it is not correct concept and caused many other confusions.
