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Impact on Special Relativity

Tachyon in Physics

Neutrino as Tachyon or Sterilized

Tachyon is any object moving with greater than light speed. By this definition the object reaches the observer (at point P in Figure 01) before being seen. After such object passed over, there are two images coming to view. There is first a bluish one with an elongated shape created by the Doppler shift and the aberration of light as it approaches to the observer. The second image is in red color and shrunk by the same kind of effects for a receding object. Figure 01 also shows the Cerenkov radiation, which is the bluish light produced by charged particles moving through a | |

## Figure 01 Appearance of a Tachyon [view large image] |
transparent medium at a speed greater than the speed of light in the medium. This is the optical equivalent of a sonic boom. In theory even though the object is not charged, it would induce gravitational radiation via the interaction with the mass in the medium. |

Physics books before 2011 invariably consider tachyon as non-existing. If such object appears in a theory, then there must be something wrong with the analysis, or the system is unstable and would quickly decay to a state without such annoyance. This kind of view has changed somewhat as OPERA (Oscillation Project with Emulsion-tracking Apparatus) announced in March 2011 that they have detected neutrino moving slightly faster than the velocity of light. See Update.

In Einstein's first paper on Special Relativity, he pointed out that as the velocity of an object or frame of reference v approaches the velocity of light c, many formulas in the theory become infinity. He then went on to declare that nothing can move pass the speed of light, and moving just at the speed of light is also impossible as it requires infinite amount of energy to attain that. This statement has been accepted by all text books in physics as self-evident. What had been neglected to mention is the fact that photon moves with the speed of light at the moment of its creation, | ||

## Figure 02 Mass and Velocity [view large image] |
## Figure 03 Minkowski Space-time |
since it doesn't need to go through the barrier. Similarly, a tachyon can be created conceivably with a speed exceeding the velocity of light - again there is no barrier to cross (the barrier is shown in Figure 02). |

Since there is some sort of symmetry between the under and over velocity of light objects, Special Relativity for tachyon can be better illustrated by referring to those properties for under the barrier objects (see appendix on Relativity). Special attention has been given to distinguish the velocity of the object v and the velocity of the frame of reference V. In order to simplify the formulation, the followings deal with only the case where velocity of the frame V < c and velocity of the object v > c.

- For the above-mentioned special case, the formulas for the Lorentz transformation, the Lorentz contraction, time dilation, and simultaneity remain unchanged because only V is involved.
- The space-time interval ds
^{2}= -(v^{2}/c^{2}-1)c^{2}dt^{2}becomes negative for v > c, i.e., it is space-like as shown by event 3 in Figure 3. In conventional physics (i.e., in the absence of tachyon) all events in the time-like region can be linked to event 1 (at the center) via signal with velocity c. The signal is sent to event 1 in the past light cone, and sent from event 1 in the future cone. The tachyon in the past light cone could connect with event 1 and its light signal arrives at event 1a a bit later (see event 4 in Figure 3, and illustration in Figure 1), but event 1 and 3 would not be able to connect by conventional mean. - According to the Lorentz transformation, the space and time intervals between 2 frames are transformed by the formulas:

dx' = (1 - V/v)dx

dt' = (1 - vV/c^{2})dt

where =(1-V^{2}/c^{2})^{-1/2}. Thus, by suitably choosing a frame of reference such that vV > c^{2}, then dt' becomes negative which means time points backward to the past - creating the possibility of running time machine (Figure 04) and contradictions such as the grandfather paradox, which asks what happens if a time-traveler kills his grandmother before she ever meets his grandfather. Fortunately, it is noticed that the energy E also transforms similarly, i.e., E' = (1 - vV/c^{2})E

For example, for the case of dt' > 0, a process would emits +|E| particle earlier at t'_{1}and it is absorbed later at t'_{2}, i.e., X^{-}@t'_{1}X^{-}@t'_{2}.

To resolve the problem with backward time flow for dt' < 0, when the observer would see the process as emission of -|E| particle at t'_{2}and its subsequent absorption at t'_{1}, i.e., X^{-}<-E>@t'_{1}X^{-}<-E>@t'_{2}; such process can be re-interpreted as the emission of an anti-particle at t'_{1}and its subsequent absorption at t'_{2}, i.e., X^{+}@t'_{1}X^{+}@t'_{2}. - Using the proper time ds as the parameter independent of different inertial frames, the velocity in the 4 dimensional space-time can be defined as:

u_{}= dx_{}/ds = v_{}/c

where =(1-v^{2}/c^{2})^{-1/2}, the Greek indices run from 1 to 3 representing x, y, z, and the 4th component

u_{4}= icdt/ds = i.

By definition the square of the 4-velocity u^{2}= -1 is invariant under the Lorentz transformation.

In special relativity the equation of motion for the tachyon is:

f_{i}= m_{t}c^{2}(du_{i}/ds)

where the space components of the 4-vector f_{i}form a 3-vector**f**/(1-v^{2}/c^{2})^{1/2}, the time component is f_{4}= i**f**(**v**/c)/(1-v^{2}/c^{2})^{1/2}. This equation is invariant under the Lorentz transformation. - Similarly the 4-momentum in term of the tachyon mass m
_{t}is defined as:

p_{}= m_{t}v_{},

and p_{4}= im_{t}c.

The square of the 4-momentum p^{2}= -m_{t}^{2}c^{2}is another 4-scalar quantity invariant under the Lorentz transformation.

The 4th component is linked to the total energy E by the formula:

E = -ip_{4}c = m_{t}c^{2}= mc^{2}

where m = m_{t}. It seems that the most celebrated formula_{}is also valid for tachyon. However, there is a problem, because =(1-v^{2}/c^{2})^{-1/2}is an imaginary quantity. In order to preserve E as a positive quantity, then the tachyon mass has to be an imaginary quantity as well, i.e., m_{t}= im_{0}, where m_{0}is a real and positive quantity. Then the energy can be recast into the form : E = m_{0}c^{2}/(v^{2}/c^{2}-1)^{1/2}. It shows that E 0 as v , and E as v c. - Thus, the mass of a particle moving at velocity v > c can be re-written as m = m
_{o}/ ( v^{2}/c^{2}-1)^{1/2}. The components of the 3-momentum**p**also becomes real, i.e., p_{}= m_{0}v_{}/(v^{2}/c^{2}-1)^{1/2}, and p_{4}= im_{0}c/(v^{2}/c^{2}-1)^{1/2}. Similarly, the imaginary sign in the equation of motion disappears under such re-definition. - Now in term of the 3-momentum
**p**, E = (**p**^{2}c^{2}- m_{o}^{2}c^{4})^{1/2}. The three-momentum and energy together form a 4-vector (**p**_{x},**p**_{y},**p**_{z}, iE/c) in the Minkowski space. Its magnitude m_{o}^{2}c^{4}is invariant under the Lorentz transformation. This formula reveals that for a tachyon |**p**|c > E, and the range of each is given by : m_{0}c < |**p**| < , 0 < E < . - From the relations between E, m, and
**p**, the velocity v can be expressed in term of the 3-momentum**p**:

(v/c)^{2}= 1/[1 - (m_{o}c/**p**)^{2}] or**p**^{2}= m_{o}^{2}c^{2}{(v/c)^{2}/[(v/c)^{2}- 1]}

which shows that v c as**p**, and v as**p**m_{o}c (see Figure 02). - The consideration above demonstrates that with slight alteration for some of the formulas, tachyon is not in contradiction with Special Relativity. Usually, it is maintained that the doctrine on "no signal can excess the velocity of light" is still valid because there is no rest frame to assemble a message with the tachyon. However as long as the source can be secured in a rest frame, there is no problem of sending off signals with all kinds of velocities including tachyons (that is, if we can find such source).

## Figure 04 Time Machine |
In this way, the problem with causality is avoided. Such interpretation is consistent with quantum theory, the force-free solution of which always involves a factor of e^{-iEt}. It remains unchanged with the simultaneous reversal of sign for E and t. |

- The followings describe the appearance of tachyon in some theories of physics.
- Virtual Particle - In quantum field theory, interaction is mediated by virtual particles. These particles exist for only a
*very short time interval*but can move with all kinds of speeds including those beyond the velocity of light. Computation of the S-matrix (probability for a certain process) depends on summing over interactions via the virtual particles. The classical process such as scattering happens to involve only virtual particles with velocity up to the speed of light (no tachyon), while other novel processes such as - Spontaneous Symmetry Breaking - In the Standard Model for elementary particles, The mass for all the particles was obtained through a process of Spontaneous Symmetry Breaking. All the particles were massless at the time of Big Bang. The Higgs field was in an unstable but symmetrical false vacuum
*with m*as shown in Figure 06 (diagram on the left). Spontaneous symmetry breaking occurred at a temperature corresponding to about 250 Gev in the electroweak era soon after the end of inflation^{2}< 0 (i.e., tachyonic) - Entanglement - The idea can be traced back to the 1920s and 1930s in the form of EPR (Einstein-Podolsky-Rosen) paradox. These scientists had a lot of problems with quantum mechanics in general and entanglement in particular. Entanglement occurs for example, in the decay of the pi meson into an electron-positron pair (Figure 09). Since the spin for the pi meson is 0, the spin for the electron-positron pair must be opposite according to the conservation of angular momentum. Therefore, no matter how far apart are the members of this pair, if the spin is flipped for one of the member, the spin for the other member

## Figure 05 Feynman Diagrams [view large image] |
self-energy involve the "off mass-shell" virtual particles as well (Figure 05). "Off mass-shell" means that the relation E^{2} = p^{2}c^{2} + m_{0}^{2}c^{4} is violated - turning into tachyon. |

## Figure 06 Spontaneous Symmetry Breaking |
~ 10^{-32} sec after the Big Bang. The process turns m^{2} > 0, and endowed mass to the particles. |

Bosonic Strings - It was found in the 1960s that the spin (or angular momentum) of a family of resonances (short-lived elementary particles) is related to the square of mass by a simple line on a graph, which is known as Regge trajectory. Figures 07 and 08 show the theoretical derivation of such relationship for | ||

## Figure 07 Regge Trajectories, closed string [view large image] |
## Figure 08 Regge Trajectories, open string [view large image] |
a few low-lying string states of the closed and open bosonic strings respectively (from the string theory). |

will also be flipped to the opposite at precisely the same moment. This non-local influence (non-locality) occur instantaneously, as if some form of communication, which Einstein called a "spooky action at a distance", operates not just faster than the speed of light, but infinitely fast. An experiment in the 1980s confirmed that entanglement does occur. Nobody knows how information of one particle can be transmitted to its partner instantaneously, but | |

## Figure 09 Entanglement [view large image] |
whatever involved would likely be a tachyon. However, it has also been shown that the nonlocal correlations seen in entanglement cannot actually be used to transmit classical information faster than light, so that relativistic causality is preserved. |

which generates the neutrinos. Experimenters determined the flight time by comparing the shape of the proton signal at CERN to that of the neutrino signal received at Gran Sasso (Figure 10). After considering details like the curvature of the Earth, the tidal effects of the moon, the general relativistic effects on the GPS synchronization, and a lot of checking for experimental errors (Figure 11), they still left with 60 nanoseconds arrival time earlier | ||

## Figure 10 Velocity of Neutrino [view large image] |
## Figure 11 Explanation of Superluminality |
than if the neutrinos were traveling at the speed of light. |

Although superluminal neutrino is apparently a tachyon (if confirmed by later experiments), another equally exotic explanation involves the extra-dimension invoked in the superstring theory. According to the theory our world is on a 3+d_{11} dimensional brane, which floats in a higher dimensional space-time called the bulk (Figure 12). Only the gravitons and possibly the sterile neutrinos can jump over to the bulk (since they are closed string). The shorten arrival time of the neutrinos is attributed to the short-cut taking by the sterile neutrinos when they appear in portions of the flight path due to neutrino oscillation. It is suggested that confirmation of the OPERAs result
| |

## Figure 12 3+d |
would provide support for the existence of sterile neutrino, the first glimpse of extra-dimensions, and a step forward to vindicate the superstring theory. |

A study in September, 2011 shows that the superluminal neutrinos would lost energy (at the high end) through the process of bremsstrahlung (braking radiation) producing electron-positron pairs : + e ^{-} + e^{+}The ICARUS team (in the same location of OPERA) follows up the idea with their data but for a different process in which the neutrino turns into a muon. Figure 13 compares the data (blue points) with a theoretical model (in red) for non-superluminal neutrinos. The result indicates that there is no superluminal neutrinos, otherwise the data would deviate a lot more from the theoretical curve. | |

## Figure 13 Test for Super-luminal Neutrinos |

Another report in "Nature News" on February 2012 by the OPERA team indicates that there are possibly two problems involving the use of GPS. The first is the incorrect calculation of the time interval from the GPS to the synchronize atomic clocks at each end of its neutrino beam. The other involves a possible faulty connection between the GPS signal and the OPERA master clock. The two errors together can account for exactly the 60-nanosecond difference. | |

## Figure 14 Detection Errors |

According to April 2, 2012 Nature Online : OPERA's physics coordinator Autiero resigned from the collaboration on 30 March, a day after OPERA spokesman Antonio Ereditato of the University of Bern. The moves followed months of internal tension and media leaks and, last week, votes of no confidence in Ereditato and Autiero by OPERA's collaboration board, which consists of representatives from its member institutions. After the motions won a majority - although not the two-thirds of votes required to force the leadership out - the two men stepped down voluntarily.