Q:

Density of a football at 0.9c

A rectangular block moving at $0.9c$ will have a length of $0.435l_0$ and mass of $2.298m_0$for a stationary observer. The observer will record a density of $5.28\rho_0$ where $\rho_0$  is the density of the body for the stationary observer when the body was at rest.
Would the density have changed if the body was a sphere of same volume?
According to $SR$ , the length along the direction of velocity contracts. So, the body should be sort of $3D$ $ellipsoid$, although it still seems to be a sphere, which is actually due to a different reason. Can the density be known by calculating volume of $3D$ $ellipsoid$? 

A:

Volume definition:
$$V=\int dV=\iiint dxdydz$$
Assume the ball moves along the $x$ direction, so
$$V_1=\iiint \frac{dx}{\gamma}dydz=\frac{V_0}{\gamma}$$
Mass is
$$m_1=\gamma m_0$$
So, the density is
$$\rho_1=\frac{m_1}{V_1}=\gamma^2\rho_0$$
If the speed is $\beta=0.9$, then $\rho_1=5.26\rho_0$. Clearly, density is independent of the shape of the ball.

