The conclusion derives from looking at how tachyon signals would behave as seen in slower-than-light inertial frames, not from trying to consider a tachyon's own "time" (if you can call it that, since a tachyon's worldline would have to be space-like, not time-like)--basically, it's a consequence of the relativity of simultaneity. It can be shown that any signal that moves even slightly faster than light in one frame would move instantaneously in some other inertial frame, and if the first postulate of SR applies to tachyons, then if it's possible to send a message instantaneously in one frame, this must be possible in all inertial frames. Likewise, if a signal travels instantaneously in one frame, there must be some other inertial frame where it actually travels backwards in time (i.e. the event of the signal being received occurs before the event of it being sent), and if that's possible in any frame, it must be possible in all frames too.
If you're familiar with the basics of SR spacetime diagrams and how a surface of simultaneity in one frame looks tilted in other frames, then you could take a look at the helpful explanation with diagrams on this page, which shows how the ability of two different slower-than-light observers to send signals that move instantaneously as seen in their own frames implies that they can bounce a message back and forth, and in each one's frame the other one's signal is going backwards in time, so that the message gets returned to the original sender at an earlier point on his worldline than the point where he sent the message, a clear causality violation in all frames. You could also take a look at the tachyonic antitelephone article which goes into more detail, with equations and a numerical example.
Note that this answer is really about why the ability to send FTL signals would violate causality--as other answers have noted, in quantum field theories there could be tachyons that would be impossible to use for transmitting information, in which case no causality violation would occur (the last section of this article has a helpful discussion about why tachyons in QFT wouldn't be usable for information transmission).
"Suppose two objects collide and combine into a single object, will the total relativistic momentum and relativistic mass stay the same?"
The answer is "yes", or rather their sums over the system of bodies will stay the same, but I would counsel you to stop using the term relativistic mass. It's going out of use for a number of good reasons that I won't bore you with now.
$m \gamma c^2$ in which m is the body's mass (formerly called 'rest mass') represents the sum of the body's internal energy, $mc^2$ and its kinetic energy ($\gamma m - m)c^2$. So for a closed system its sum over the bodies of the system is conserved in collisions, elastic or inelastic. Think of $\gamma m$ as the body's total energy, expressed in mass units.
The beauty of it is that a body's total energy (divided by the mere constant, c) and the three components of its momentum $(m \gamma u_x, m \gamma u_y, m \gamma u_z)$ make up a 4-component vector (or 4-vector): $(m \gamma c, m \gamma u_x, m \gamma u_y, m \gamma u_z)$. So for a closed system, despite collisions elastic or inelastic, the vector sum of these vectors is conserved, that is the sums of each component separately. One conserved 4-vector deals with conservation of energy and conservation of momentum.
Note also that the modulus of the 4-vector, defined as $\sqrt{(m \gamma c)^2 - (m \gamma u_x)^2 - (m \gamma u_y)^2 - (m \gamma u_z)^2}$, is simply $mc$, the body's mass multiplied by the mere constant, c.
$m$ is a constant for the body (provided we don't tamper with the body, e.g. by changing its internal energy!) and doesn't vary from frame to frame. It is a Lorentz invariant. [Beware: the sum of the masses (rest masses) of bodies in a system has no obvious significance; it is certainly not the mass (rest mass) of the system!]
I've gone on longer than I should have done. It's all so wonderful. A classic introduction to Special Relativity, first rate on concepts, is Spacetime Physics by Taylor and Wheeler.
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Physically, you math guys aren't allowed to
crossnear the boundary $c$ (speed of light). Special Relativity does that. SR says that it would be impossible for a particle to be accelerated to $c$ because the speed of light (maximum possible measured velocity) is constant in vacuum for all inertial observers (i.e.) Observers in all inertial frames would measure the same value for $c$. Not only the fact that infinite energies are required to accelerate objects to speed of light, (but) an observer would see things going crazy around the guy (or an object) traveling at $c$ such as length contraction (length would be contracted to zero), time dilation (time would freeze around him) & infinite mass. You can't enjoy anything when you travel at $c$. But, the stationary observer who's measuring your speed (relative to his frame) would definitely suffer..!Note: But, there are some quantum mechanical solutions that allow negative masses like the expression for relativistic energy-momentum. Let's try not to make the subject more complicated. $$E^2=p^2c^2+m^2c^4$$
There are hypothetical particles (having negative mass squared (or) imaginary mass) always traveling faster than the speed of light called Tachyon. This was assumed by Physicists in order to investigate the faster than light case. So When $v>c$, the denominator becomes a imaginary. But, Energy is an observable. It should be some integer. A consistent theory could be made if their mass is made to be imaginary and Energy to be negative. Using these data in the E-p relation, we would arrive at a point $p^2-E^2=m^2$, where $m$ is real. This makes Tachyons behave a kind of opposite to that of ordinary particles. When they gain energy, their momentum decreases (which strongly disproves all our assumptions).
The first reason that this investigation blown off is Cherenkov radiation where particles traveling faster than light emit this kind of radiation. As far as now, No such radiation has been observed in vacuum proving the existence of these..! It's like making a pencil to stand at its graphite tip. If it would stand, physicists would've to blow up their heads :-)
There are tougher stories on the topic when you Google it out...