Special relativity and tensile stress If an observer studies a cubic structure crystal from a moving frame of reference while speeding towards the crystal, he would expect to measure the atoms in the crystal closer together in the direction of his travel compared to distance of atoms in a perpendicular direction.
How would this observer explain the tensile stress force he observes on the crystal which someone standing next to the crystal will not detect? 
 A: 
How would this observer explain the tensile stress force he observes on the crystal which someone standing next to the crystal will not detect?

Stress is the space-space components of the stress energy tensor. For the stationary observer the stress energy tensor is 
$$\left( \begin{array}{cccc}
\rho & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right)$$
For the moving observer the stress energy tensor is
$$\left( \begin{array}{cccc}
\gamma^2 \rho & v \gamma^2 \rho & 0 & 0 \\
v \gamma^2 \rho & v^2 \gamma^2 \rho & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right)$$
So there is a nonzero component of stress in the moving frame in the $xx$ direction. In an engineering stress tensor this would represent a compressive stress rather than a tensile stress. However, in relativity this term includes a momentum convection term. So according to the moving observer, the reason that the term is zero for the stationary observer is that because he is comoving with the crystal there is no momentum convection past him. 
A: The moving observer won't see any stress.
This is essentially Bell's spaceship paradox.
http://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html
From the point of view of the moving observer it is the cube that is moving.
In detail, at the atomic level, from the point of view of the observer, if you solve for the required equilibrium position of say two atoms (given appropriate semi classical assumptions) you find that two atoms moving at high speed have an equilibrium position that is closer together. At this point the observer does not have to think in relativistic terms at all. They just solve the electromagnetic equations and say - hey, whatever, atoms move closer together when they travel at high speed (along the axis of the molecule). 
For example, this one is easy to do with few assumptions and only basic vector calculus, solve for the field around a point charge that is stationary, and then solve for the field around a point charge moving at constant velocity. The field around the moving charge is not a moving version of the field around the stationary charge. It is contracted in the direction of motion. This is the Fitzgerald contraction in its original context.
This exercise required zero relativity. 
The contraction comes simply from solving Maxwell's equations. 
This is the historical context of the development of the theory of relativity. Einstein and Lorentz knew and liked each other and Lorentz even wrote 100-page book saying how great Einstein's theory was. 
