Q:

Paradox regarding phase transitions in relativistic systems

The main question I would like to ask is whether quantities such as density are dependent on the frame of reference. I have searched several forums and the answer is somewhat controversial. Some answers use the concept of relativistic mass to justify that it is invariant. Some of the answers say that relativistic mass is not a correct concept (given in Classical Mechanics by John R. Taylor pg 633) and that mass is invariant and hence the density must be an observer dependent quantity. This is a little odd because of the following thought experiment:
Imagine that a container filled with liquid is made to travel at relativistic speeds. In the frame of the container, the density is d at temperature T and pressure P. To a person in the ground frame the volume of the liquid will decrease because of length contraction. At the critical density there must be a phase transition from liquid to solid. In the moving frame, the container has a liquid but in the rest frame, the container is filled with a solid. 
So based on the above I have the following questions:

Is there anything wrong in the above argument and why is it incorrect? (eg. Phase diagram changes depending on the velocity of the object)
If the density is observer-dependent, could it mean that thermodynamics is observer-dependent?

A:

Every time someone comes up with an apparent "paradox" in Special Relativity, it is always related to some application of the "Lorentz contraction" or "time dilation" concepts in some questionable contexts. Over the years of many similar discussions,  I've worked out the recipe for dissolution of such "paradoxes": just reduce your "paradox" down to the Lorentz transformations. They are fundamental to the Special Relativity -- both time dilation and the Lorentz contraction are derived from the Lorentz transformations. So, whatever you get from Lorentz transformation is what Special Relativity actually predicts. Any contradiction with the result you get from using Lorentz transformation is a mistake somewhere in your reasoning.    
In order to reduce your "paradox" to these fundamental equations of Special Relativity - you first have to formulate your problem in terms of space-time events and world lines. When dealing with thermodynamic systems you'll have to get down to individual particles making up your bodies. Extract their worldlines and Lorentz-transform them to get your trajectories in moving reference frames. 
To illustrate this, I've randomly generated a 1000 particles bouncing in a 3x1 box with Maxwell-distributed velocities:

I've saved the worldlines of all the particles and applied the Lorentz transform with $\gamma=2$ to each of the particles. Plotting the resulting coordinates, I've got this:

Above I've kept the scale size of both scales and made sure that the center of the scale is at the center of the resulting particle cloud. You can see that the density has, indeed, increased. Notice that the velocities of the particles along the x-axis are changed in respect to the bounding volume -- it is not a Maxwell distribution anymore.

In the moving frame, the container has a liquid but in the rest frame, the container is filled with a solid.

As for your thought experiment, you can apply this reduction to Lorentz transformations to realize that there is no way you can get dynamic motions of individual particles of a liquid and then Lorentz-transform them to get static (relative to each other) positions of the same particles in a solid body.

A:

Is there anything wrong in the above argument and why is it incorrect?

You have implicitly assumed that the pressure remains constant, and therefore that since density is increasing, at some point as the speed of the medium increases you will hit a "critical point" and undergo a phase transition.  This does not follow, since the pressure of the matter will increase along with the density. 
Specifically, the mass/energy density, pressure, and momentum density of a bulk solid all form part of the stress-energy tensor of the material, where
$$
T_{00} = \text{mass/energy density} \\
T_{0i} = \text{momentum density vector} \\
T_{ij} = \text{stress tensor}
$$
(working in units where $c = 1$.) In particular, let's suppose that we have an isotropic fluid at rest, so that $T_{00} = \rho$, $T_{ij} = P \delta_{ij}$, and all other components of the stress-energy tensor vanish.  When we transform into another frame moving at speed $\beta$ in the $x$-direction relative to the rest, we find that 
$$
T'_{00} = \gamma^2 (\rho + \beta^2 P) \\
T'_{11} = \gamma^2 (P + \beta^2 \rho) 
$$
The key realization here is while the observed density of the medium ($T_{00}$) increases as the speed of the medium increases, the pressure ($T_{11}$) will also increase.  The argument that "since the density increases, you'll eventually get a phase transition" is therefore not necessarily correct;  the density at which a phase transition occurs depend on the pressure, which is also changing.

A:

A phase transition should occur when the attracting forces between particles becomes high relative to their relative motion (broadly speaking).
Thought experiment: We have two objects with a certain opposite electrical charge. Now we move with respect to the objects, so that the distance between them becomes smaller (and charge won't). Does this change the attractive forces between them? The laws of nature don't change for a moving observer (the fundamental assumption in relativity), so I would say yes. With a factor $\gamma^2$, according to Coulomb's law, the attractive force would increase. However, would it be in disagreement with what an observer that doesn't move (w.r.t. the objects) would see?
Let's say the objects have equal mass and the acceleration between the objects in the 'standing still'-frame is $g/r^2$ (relative acceleration), and they start with a velocity such that they will make a circular motion with radius $R$. For the moving observer this should Lorentz-contract to an ellipse. When the objects align according to your line of motion, the attraction would be bigger, but the velocity of the objects change with a factor $\gamma$ too (addition of perpendicular velocities. So the centripetal force should change with a factor $\gamma^2$, which we have seen it does, so there it tempts to keep a circular motion. When the objects align perpendicular to your line of motion, the force doesn't change, but the velocity decreases, so there they would fall a bit to each other. If someone is very skeptic whether their movement will be elliptic, you can do it explicitly by finding an expression for their attraction as a function of their angle and solve the differential equation with the right starting conditions. But this broadly explains why there would be no issue with the attracting forces being bigger.
Now a liquid in a container is the same problem with more bodies. So generalizing this result, they should move in a way that would give same liquid when Lorentz-transformed. So if my reasoning is correct, for a moving observer it is possible for a liquid to have a higher density than would be possible in a resting frame and still be a liquid, i.e.: the phase transition shifts.

