Tachyons – What Are They?

Some people are curious about tachyons. So what the hell are they? I will endeavour to explain.
Tachyons are particles with spacelike four-momentum and imaginary mass. What does this mean? Well it is probably best to explain this using a space time diagram.
Here the ordinary spatial dimensions are in the x-y plane and the z axis represents time since an event. The event is at the origin. The -z axis is thus the past and the +z axis the future.
A particle travelling at the speed of light is represented as a line with slope c [the slope is drawn as slope=1. This is because some people, especially particle physicist like setting everything =1. Other people actually define the z axis as ct instead of t as i said]. This “world-line” defines the “light cone” shown in the diagram. Any world-line represents a path of a particle moving at a constant speed. As we’ve said the slope of the line gives the speed in units of the speed of light c. So particles travelling at v>c will have world-lines outside the light cone. The lines inside, along and outside the light cone are called “time-like”, “light-like” and “space-like” in the jargon of relativity.
Tachyons are the space-like particles. Now one important aspect of special relativity is that everything is relative – you see particles move differently based on your reference frame. If you know what a world-line looks like in one reference frame you can see what it looks like in a reference frame by applying a Lorentz transform. These are just equations that relate position and velocity between two reference frames (discovered by Joseph Larmor and Hendrik Lorentz with Lorentz getting the credit as he had a knack of doing as Ludvig Lorenz would have attested to).
An important result is that Lorentz transforms transform time-like to time-like and space-like to space-like, i.e. if you are in the cone you will be seen differently by different observers but you will always be seen to be moving along a line inside the cone. Likewise if you are outside the cone you will be seen differently by different people but you will always be outside the cone. Now this is very important! As z>0 is the future, z=0 is now and z<0 is the past then this means that if you are moving along a space-like world-line you can simply move to a refrence frame in which your world-line points backwards in time!! You can then send messages back to yourself and others through time!
There are many examples of why this is bad (not in the sense of bad for book-makers, lottery operators, etc. but bad in a logical sense! google the grandfather paradox to see what i mean) and this is why tachyons are held by most as not existing. They are generally viewed as hypothetical objects.
The second thing i mentioned is that they have imaginery rest mass. The energy of a particle is :

But if v>c then the denominator is imaginery (it contains the square root of a negative number). So as energy is not imaginery it means that the mass must be to cancel the imaginery denominator.
It is worth noting that this problem can easily be circumvented by defining the mass* to be:

So the energy equation of relativity becomes:

which basically says that while ordinary particles cannot accelerate to the speed of light (it would require infinite energy) tachyons cannot slow down to the speed of light!
If i think of other basic properties of tachyons i will add them here.

* note that this is perfectly fine to do mathematically speaking