Relative black hole? When some object is moving, it appears smaller and more massive. What if it's moving so fast it should become a black hole? Some calculations:

Let observed radius $R_0\gamma$ ($R_0$ - radius at rest) be equal Schwarzschild radius for object with observed mass $M_0\gamma$ ($M_0$ - mass at rest):

$$R_0 / \gamma = \frac{2GM_0\gamma}{c^2}$$

$$\gamma^2=\frac{R_0c^2}{2GM_0}$$

$$\frac{v}{c}=\sqrt{1-\frac{2GM_0}{R_0c^2}}$$

For Earth it gives $v=0.999999999304c$ and for Sun $v=0.999997877903c$. Is my reasoning correct? How would we see such objects?
To answer quickly, the answer is that no, an object cannot be made to collapse as such. 

There are multiple reasons as to why. Pela brings up a great argument here, reminding us that if a black hole could be formed this way, then I could be made to collapse into a black hole just by something moving by me at extreme velocities. Clearly, this cannot occur.

To understand why this is, we must first completely throw away the idea you have of relativistic mass. The entire concept of relativistic mass was created to allow for Newtonian mechanics to still hold in relativistic situations. By modifying the mass of an object at extreme velocities, equations such as $F=ma$ can be made to still hold. To paraphrase Philip Gibbs, gravity couples not only to mass but also to momentum and itself. Therefore, a description of spacetime in the vicinity of an object will be incomplete without momentum considerations. From this relativistic momentum, we can compute the relativistic energy.

While these quantities are useful for giving detailed descriptions of the body, the answer is actually somewhat simpler. We must remember that an object must behave the same in all inertial reference frames. If the object does not become a black hole in its own frame (as it clearly will not), then it cannot appear to be one in any other. 

If you are wondering how to reconcile this with a relativistic mass that approaches infinity, the answer is simply that relativistic mass is not real mass. The metrics which describe spacetime in the vicinity of these objects refer to the invariant mass, which is the rest mass of the body.