Explanation of Supplee's paradox I was reading about Supplee's paradox, which is about whether a relativistic projectile, subject to uniform gravitational acceleration, would float or sink underwater. However the solution of the paradox seems a little unclear to me, as given in the link.

Given certain assumptions about how to treat the gravitational force, he argued that the bullet sinks with acceleration $g (\gamma^2-1)$, where g is the acceleration due to gravity (assumed to be uniform over the scale of the thought experiment) and $\gamma^2$ is the factor mentioned above.

I understand that in the frame of the bullet, the shape of the container of fluid is altered ,i.e, the sea floor is curved upwards.
But what are the certain assumptions? How do we get the acceleration of the bullet? And how do we conclude that the bullet sinks?
 A: I'm a bit unsure where you are unsure, but let me try. The "real" behaviour is best seen in the frame of the liquid. In this frame the bullet will shrink compared to it's stationary size, and so it will have a higher density than the water (supposing they have the same density when stationary).
The acceleration downwards will be given by the difference in density ($\rho$) between the bullet and water as 
$$ a_{\mathrm{b}} = g\frac{\rho_b-\rho_w}{\rho_b} $$
using that $\rho_b=\rho_w \gamma^2$ then 
$$ a_{\mathrm{b}} = g\frac{\gamma^2-1}{\gamma^2} $$
This is not exactly like in the wikipedia article (but makes more sense as the acceleration will reach free fall $a=g$ when $\gamma>>1$.
Anyway: the bullet sinks.
Now, from the point of view of the bullet the water gets denser by a factor of $\gamma^2$ so by the same argument the bullet should rise with acceleration
$$ a_{\mathrm{b}} = g(1-\gamma^2). $$
This time however there is an extra effect. Not only will the water get denser, the floor of the container will also begin to rise. The upward acceleration of the container floor is faster than the acceleration upward for the bullet.
So, even though the bullet is rising through the fluid, the floor rises even faster and eventually the bullet will hit the bottom of the tank. 
Paradox resolved ;)
To understand why the floor rises one can think as follows. To model the existence of gravity you may use the equivalence principle and see it as if the sea floor is accelerating upward with constant acceleration $g$. This you will see even if you are stationary. What happens then you start to move is that the floor will length-contract (just as the water). As a result it will look as if the upward acceleration of the floor gets a boost (by some factor of $\gamma$) 
This boosts is enough to make the floor catch up with the bullet.
In the language of a gravitational force this would correspond to that the bullet feels the force of gravity stronger than the surrounding water (as it is moving) and therefore gets pulled down.
A: Since buoyancy is a contact force and the motion is fast enough to be relativistic, it's ridiculous to neglect drag.  In fact, I think it's crazy to neglect the non-uniformities that will arise if the bullet is going that fast in water.  Its path will bend, but not because of gravity.  It'll bend in an unpredictable direction.  That being said...


*

*Supplee assumed he could treat gravity using the equivalence principle.  He calculated only the force of buoyancy, and allowed the gravitational force to be accounted for by an acceleration.  


*

*An object under free fall would have a zero acceleration (instead of what we think of as accelerating downward at $g$).

*An object "at rest" such as the water is not at zero acceleration but at an acceleration of $g$ upward.

*The bullet moving quickly through the water under neutral buoyancy) is not at zero acceleration (like we would say in Newtonian mechanics), but it is accelerating upward because of the buoyant force.


*Calculating the acceleration of the bullet (in the inertial frame with no gravity) is just a matter of $F_B = \gamma m_0 a$ with the buoyant force decreased by a factor of $\gamma$ because of length contraction.  Thus the acceleration is $g/\gamma^2$ upward.

*In the lab frame (water initially at rest with no gravity), the bullet sinks because its upward acceleration is less than the upward acceleration of the water and lake bed ($g$).  This is transforming the acceleration into the lab frame where the water actually is at rest.

*In the bullet frame (bullet initially at rest with no gravity), the bullet accelerates upwards at $g$ and the water/seabed accelerates upward at $g/\gamma^2$ (the reverse of the previous frame).  This can come from Lorentz transforming the lake's acceleration and the buoyant force.

*The sea floor is "curved upward" even in Newtonian mechanics.  An object "at rest" using the equivalence principle has a non-relativistic position of $y=\tfrac12 g t^2$.  The relativistic difference is that because of time dilation, the curve is different.  This ends up resolving the apparent paradox.
For more detail, see Supplee's article (referenced in the Wikipedia article).  You can probably find a copy online.  Just search for the text of the abstract.
A: 
I was reading about Supplee's paradox, which is about whether a relativistic projectile, subject to uniform gravitational acceleration, would float or sink underwater. However the solution of the paradox seems a little unclear to me, as given in the link.

It isn't very clear, is it?  

Given certain assumptions about how to treat the gravitational force, he argued that the bullet sinks with acceleration $g (\gamma^2-1)$, where g is the acceleration due to gravity (assumed to be uniform over the scale of the thought experiment) and $\gamma^2$ is the factor mentioned above.

I think that's broadly correct, but that there's a simpler way to understand why.  

I understand that in the frame of the bullet, the shape of the container of fluid is altered ,i.e, the sea floor is curved upwards. But what are the certain assumptions? How do we get the acceleration of the bullet? And how do we conclude that the bullet sinks?

A moving body doesn't change the things around it. The body changes, and the way it interacts with other things changes, but those other things don't. Anyway: how do we conclude that the bullet sinks? We replace it with a very long rod. An "infinite" rod. When it's motionless there's a given mass-energy density or energy per unit length. Let's say that this is equal to that of the water. So the rod doesn't sink, or rise. But now let's add energy to it by pushing it lengthways, faster and faster. We have increased the energy, and the mass-energy density per unit length, so the rod sinks. 
