I'm about as qualified to discuss SR as I am to play power forward in the NBA, however:
I have seen Cavediver (I think) describe travel through space-time like a vector: we are all "moving" at a constant rate through space-time; but the ratio of our rate of travel through space to our rate of travel through time can change.
Picture a meter stick pointing straight up. This represents an object "standing still" in space in a particular FOR. Because it is not moving in space, it is moving as fast as possible through time. Now tip the meter stick over 5 degrees. The overall rate of travel through space-time (the lenght of the meter stick) has not changed, but now the object is moving slowly through space, and is moving somewhat slower through time than it was before. As the object speeds up through space, the meter stick tips more and more.
If I understand it correctly, when the meter stick reaches the horizontal, that represents an object moving at LS. You can't go faster than this because you can't get the meter stick to deflect horizontally any more than this.
Don't ask me what it means to continue tipping the stick so it starts to point down, or if this is even possible.
Edited by Aware Wolf, : Clarity, I hope.