# What Are Tachyons?

During the 1960s, it was realized that general relativity said a lot about traveling at speeds near c but never mentioned anything about something moving faster than that speed outside of a reference frame. Gerald Feinberg and George Sudarshan were able to show that if such a particle existed then it *couldn’t* move any slower than c – that is, it was always faster than the speed of light. Now called the tachyon, this hypothetical particle would have many weird propertied, such as having tis energy decrease as its velocity increased. Therefore, as it approached infinite speed, the energy would approach zero! It and its antimatter counterpart would pop in and out of the quantum vacuum as virtual particles (Morris 214-5, Arianrhod).

However, no experimental evidence for their existence has been found. Either tachyons interact with matter weakly or they interact not at all. More than likely, they are just an interesting idea. Even Feinberg doesn’t think they truly exist. But what if they do exist and we just can’t find them…what then? (Morris 215)

## Einstein Talk

When scientists talk about tachyons, they use the theory of relativity that Einstein developed in the early 20^{th} century. This means we got to talk about Lorentz transformations and reference frames, but where relativity shows the means of travelling at less than c, tachyons would require the opposite (and, as it turns out, backwards in space-time on some occasions). And how can they achieve their FTL speeds if relativity says nothing moves faster than c? Well, it actually states nothing can *speed up* to c, but if it was already going at that speed from, say the Big Bang, then nothing is violated. The quantum theory of virtual particles is also valid, because it comes into existence and has no speeding up. Possibilities are numerous here (Vieria 1-2).

Does relativity predict tachyons? It sure does. Remember that E^{2}=p^{2}c^{2 }+ m^{2}c^{4} where E is energy, p is momentum, c is speed of light, and m is rest mass. If one were to solve for E, a positive and negative root arise and relativity currently concerns itself with the positive one. But what about the negative one? That would arise from backward motion through time, the counter to the positive solution. To interpret this, we call upon the switching principle, which shows that a forward particle will look the same as a backward one with its properties reversed, and such. But the moment a backward or forward particle encounters a photon, *that* is the transition to its compliment. But to us, we see the photon only and know that something must have hit our particle, which in particle physics is the anti-particle. *That* is why the two have opposite properties, and is an interesting non-quantum approach to proving antiparticles and in this case a tachyon-like particle (3-4).

Alright, now let’s look at some math here. After all, that is a rigorous and universal way to describe what is happening as we transition with tachyons. In relativity, we talk about reference frames and the motion *of* them and *through* them. So, if I move from one reference frame to another but limit my travel to one direction, then with a backward moving particle in reference frame R we can describe the distance travelled as x=ct, or x^{2} – c^{2}t^{2}=0. In a different reference frame R^{’}, we can say we moved x^{’}=ct^{’} or x^{’2}-c^{2}t^{’2}=0. Why squared? Because it takes care of signs. Now, if I wanted to relate the two motions between the frames R and R^{’}, we need an Eigenvalue to relate the two motions together. This can be written as x^{’2}-c^{2}t^{’2}=λ(v)( x^{2} – c^{2}t^{2}). What if I went backwards from R^{’} to R with –v? We would have x^{2}-c^{2}t^{2}=λ(-v)( x’^{2} – c^{2}t’^{2}). Using algebra, we can rework the two systems and arrive at λ(v) λ(-v)=1. Because physics works the same no matter the direction of the velocity, λ(v) λ(-v)= λ(v)^{2} so λ(v) =±1 (4).

For the case λ(v) =1, we arrive at the familiar Lorentz transformations. But for λ(v) =-1, we get x^{’2}-c^{2}t^{’2}=(-1)( x^{2} – c^{2}t^{2}) = c^{2}t^{2}-x^{2}. We don’t have the same format now! But if we made x=iX and ct=icT, we would have instead X^{2}-c^{2}T^{2} and so we have our familiar Lorentz transformations ct^{’}=(cT-Xv/c)/(1-v^{2}/c^{2})^{1/2} and x^{’}=(X-vT)/( 1-v^{2}/c^{2})^{1/2}. Plugging back in for x and t and rationalizing gives us ct^{’}=±(ct-xv/c)/(v^{2}/c^{2}-1)^{1/2} and x^{’}=±(x-vt)/(v^{2}/c^{2}-1)^{1/2}. This should look familiar, but with a twist. Notice the root: if v is less than c, we get non-real answers. We have our tachyons represented here! As for the sign in the front, that is just relative to the direction of travel (5).

## Mechanics

In physics, it is convenient to talk about action, denoted by S, which is either a max or a min for any movement we make. Without any forces acting on something, Newton’s Third Law states that the tachyon will move in a straight line, so we can say that the differential dS=a*ds where a is a coefficient relating the infinitesimal differential of action to that of a line segment. For a tachyon, that differential dS = a*c*(v^{2}/c^{2}-1)^{1/2}dt. That inner component is our action, and from physics we know that momentum is the change in action with respect to velocity, or p(v) = (a*c*(v^{2}/c^{2}-1)^{1/2}). Also, since energy is the change in momentum with respect to time, E(v)=v*p(v) + a*c*(v^{2}/c^{2}-1)^{1/2} (which arises from the Product Rule). Simplifying this gives us p(v)= (a*v/c)/(v^{2}/c^{2}-1)^{1/2} and E(v)= (a*c)/(v^{2}/c^{2}-1)^{1/2}. Notice that as we limit these as velocity gets larger and larger, p(v)=a and E(v)=0. How *weird*! Energy goes to zero the faster and faster we go, and the momentum converges upon our constant of proportionality! Do note that this was a heavily simplified version of what the possible reality of tachyons are, but nonetheless is a useful tool in gaining intuition (10-1).

## Huge Event

Now, what can generate tachyons? According to Herb Fried and Yves Gabellini, some huge event that dumps a ton of energy into the quantum vacuum could cause those virtual particles to fly apart and enter the real vacuum. These tachyons and their antimatter particles interact with electrons and positrons (which themselves pop into existence from virtual particles), for the math that Fried and Gabellini uncovered implied imaginary masses being in existence. What has mass with an imaginary coefficient? Tachyons. And the interactions between these particles can explain inflation, dark matter, and dark energy (Arianrhod).

So the huge event that generated them was likely the Big Bang, but how does it explain dark matter? Turns out, tachyons can exhibit a gravitational force and also absorb photons, rendering them invisible to our instruments. And speaking of the Big Bang, it could have been generated by a tachyon meeting its antimatter counterpart and causes a tear into the quantum vacuum dumping a lot of energy into the real vacuum, starting a new Universe. It all fits well, but like much cosmological theories it remains to be tested, if it ever can be (Ibid).

## Works Cited

Arianrhod, Robyn. “Can faster-than-light particles explain dark matter, dark energy, and the Big Bang?” *cosmosmagazine.com*. 30 June 2017. Web. 25 Sept. 2017.

Morris, Richard. The Universe, The Eleventh Dimension, and Everything Else. Four Walls Eight Undous, New York, 1999: 214-5. Print.

Vieria, Ricardo S. “An Introduction to the Theory of Tachyons.” arXiv:1112.4187v2.

**© 2018 Leonard Kelley**

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