Let us suppose we have an object travelling at faster than the
speed of light; the familiar
tachyon. So we have an object with v(
velocity)>c(
speed of light). Taking
Einstein's ratio
gamma from the equation E=gamma*m*c^2 to be 1/(1-v^2/c^2)
1/2 we see that any object with v>c gamma is an
imaginary number:
v>c
v2>c2
v2/c2>1
1-v2/c2 <0
1/(1-v2/c2)1/2 is in the set of imaginary numbers
thus gamma is imaginary. Let us call the modulus of gamma "k" because my keyboard has no gamma. so we can say gamma=ik E=ikmc^2. Now in order for E to be real m or c^2 has to be imaginary, and c^2 can't very well be, so let's say m is. thus we have defined a tachyon as an object with imginary mass. Seeing as the curvature of spacetime around an object is proportional o the energy of the object, we can say that if an said is somehow brought to a standstill, it will have imaginary Energy and thus imaginary curvature. If the curvature between two such objects is dependent on the products of their curvature, then logically the curvature will be negative, or repulsive; i2=-1
im1*im2=i2m1*m2
-1*m1*m2
-m1*m2.
This is just my uneducated ramblings, but similarly if you could accelerate two objects of normal mass above the speed of light then they too should become gravitationally "repulsive", as their energy would be imaginary as well. Still decelerating tachyons would require the application of imaginary energy , and though I have heard of negative energy I have yet to hear of imaginary energy. Still if we could harness this negative energy we might just be able to travel faster than the speed of light.
...Just ramblings...