Tachyons and Anti-Telephones

Tachyons are faster than light particles. Are such things possible and if so, are there odd effects which may arise because of them? Here, we consider the tachyonic anti-telephone as proposed by Benford, Book, and Newcomb (1970).

Before we get on to the tachyonic anti-telephone, I want to remind you about some things about simultaneity.

    SIMULTANEITY

    Suppose the train is stationary and a light turns on in the center of the train at time t = 0. The light travels to each end of the train and arrives simultaneously according to observers both in the train and on the ground (in the lab frame). This is illustrated on the plot to the right. The upward slanting lines represent two beams of light which connect the two events; the turning on of the light and the arrival of the light at the front (right hand line) and the back (left hand line) of the train.

Next, suppose the train is moving and a light again turns on at its center at t' = 0. The light again travels to each end of the train and, according to the observer riding in the train, the signal arrives simultaneously at the front and back of the train. This is illustrated in the frame to the left where we show the worldlines for the front and back of the train and the times as measured by a stationary observer. Note that the ends of the train move to larger x as t increases (the lines slant upward to the right). It is clear that the signal arrives at the back of the train before it reaches the front according to the stationary observer (the times as measured on the vertical time axis).

In the panel to the right, we show three cases: a stationary train where the line of simultaneity is horizontal, a frame moving to the right with v < c for which the line of simultaneity slants upward to the right, and finally for a frame moving to the left with v < c for for which the line of simultaneity slants upward to the left.

    The Light Cone, the Future, and the Past

    Consider a two-dimensional space + time (see the left panel). If we plot (x,y,ct) so that a trajectory with v = c has slope of 1. In our 2+1 space, this curve sweeps out a cone, the Light Cone. For motion with v < c, the world lines fall inside the cone either pointing upward into the future or backward into the past. If v > c, then the world lines fall outside the light cone.

    Causality

    Suppose a signal is emitted at Point A and then received at point B. This trajectory falls within the light cone and so the two events are connected by v < c. For a stationary observer, the event occurs at the marked time and we see that A occurs before B. Further, for any frame moving with v < c, A precedes B and causality is preserved. This situation is referred to as time-like because a frame exists with v < c where A and B occur at the same point in space separated only in time.

    A signal sent from A to C however, is another matter. This signal propagates with v > c (slope v/c > 1, but a worldline with slope < 1). What happens in this case? Well, for a stationary observer, A precedes C and causality is preserved, but the signal seems to arrive faster than it should. For a frame which moves sufficiently fast, we can get a more peculiar result in that C may precede A, its cause. In this case, we may construct what is referred to as an anti-telephone. We discuss this possibility in the next section.

For the signal which travels from P to Q, there may be issues for an observer who moves fast enough, even if the observer moves with v < c. The black line marks the curve vt = ct' and the magenta line marks the line of simultaneity (the x'-axis). Note that the signal is sent at time ct = ct' = 0, but is received at time ct' <0 ===> it is received before it is sent.

This odd result can arise if the observer moves with speed

v > 2 a /(1+a2),

where the tachyonic signal travels with speed ac. The possibility of sending a message to the past leads to issues embodied in the what is called the The Tolman Paradox. Benford, Book, and Newcomb (1970) coined one such device an Anti-Telephone ( 1970, Phys. Rev. D, 2 # 2, 263).

Such anti-telephones are theoretically possible, but can people travel backwards in time?

    Tachyons

    In Special Relativity, because we require that the laws of pages are of the same form and that the speed of light is the same for all observers in inertial frames, the energy and momentum of particles must have different forms than which you are familiar. To the left is shown the energy of a particle in free space in Special Relativity. You are more familar with the form

    E = 0.5 mv2 + mc2
    Actually we don't usually add the rest mass energy because in classical physics we can't tap this energy and so we just usually redefine the energy as E' = E + mc2.

Comments:
  • If v < c initially, then as v ===> c, 1 - (v/c)2 ===> 0 and the energy blows-up; E ===> infinity. This says that if we wish to accelerate a particle whose initial v is < c to v = c, we would need to invest an infinite amount of energy! For tardyons (normal matter), the upper limit to their speed is then the speed of light.
  • If v > c iniitally, we don't run into this problem and we can have particles with super-light (superluminal) speed. We refer to these particles as Tachyons.
  • If v > c then 1 - (v/c)2 < 0 and we have issues. To make the energy real (as it must be because it is something which we can measure) then this implies that the rest mass of the tachyon must be imaginary. This is okay, however, as tachyons can never be stopped and their rest masses cannot be measured.
  • Why can't tachyons be stopped? Well, for tachyons, v > c so that if a tachyon slows down, 1 - (v/c)2 ===> 0 and the energy blows-up that is the energy of a tachyon become infinite as it slows to v = c. For tachyons, the speed of light is a lower limit to its speed.
  • If tachyons are charged, then because they have speeds > c, they will emit something known as Cerenkov radiation (see figure in the top panel). As they radiate, they lose energy and so speed up which causes them to radiate more energy which causes them to speed up. This instability drives tachyons to zero energy.

    If tachyons exist, then signals can be carried backward in time. But note that we cannot travel back in time as our speeds are always restricted to v < c and we can never become superluminal.