Tachyons in relativistic theory Tachyon

1 tachyons in relativistic theory

1.1 mass
1.2 speed
1.3 neutrinos
1.4 cherenkov radiation
1.5 causality

1.5.1 reinterpretation principle

tachyons in relativistic theory

in special relativity, faster-than-light particle have space-like four-momentum, in contrast ordinary particles have time-like four-momentum. although in theories mass of tachyons regarded imaginary, in modern formulations mass considered real, formulas momentum , energy being redefined end. moreover, since tachyons constrained spacelike portion of energy–momentum graph, not slow down subluminal speeds.

mass

in lorentz invariant theory, same formulas apply ordinary slower-than-light particles (sometimes called bradyons in discussions of tachyons) must apply tachyons. in particular energy–momentum relation:

e

2


=
(
p
c

)

2


+
(
m

c

2



)

2





{\displaystyle e^{2}=(pc)^{2}+(mc^{2})^{2}\;}

(where p relativistic momentum of bradyon , m rest mass) should still apply, along formula total energy of particle:

e
=



m

c

2




1
−



v

2



c

2







.


{\displaystyle e={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.}

this equation shows total energy of particle (bradyon or tachyon) contains contribution rest mass (the rest mass–energy ) , contribution motion, kinetic energy. when v larger c, denominator in equation energy imaginary , value under radical negative. because total energy must real, numerator must imaginary: i.e. rest mass m must imaginary, pure imaginary number divided pure imaginary number real number.

in modern formulations of theory, mass of tachyons regarded real.

speed

one curious effect that, unlike ordinary particles, speed of tachyon increases energy decreases. in particular,



e


{\displaystyle e}

approaches 0 when



v


{\displaystyle v}

approaches infinity. (for ordinary bradyonic matter, e increases increasing speed, becoming arbitrarily large v approaches c, speed of light). therefore, bradyons forbidden break light-speed barrier, tachyons forbidden slowing down below c, because infinite energy required reach barrier either above or below.

as noted einstein, tolman, , others, special relativity implies faster-than-light particles, if existed, used communicate backwards in time.

neutrinos

in 1985, chodos proposed neutrinos can have tachyonic nature. possibility of standard model particles moving @ superluminal speeds can modeled using lorentz invariance violating terms, example in standard-model extension. in framework, neutrinos experience lorentz-violating oscillations , can travel faster light @ high energies. proposal criticized.

cherenkov radiation

a tachyon electric charge lose energy cherenkov radiation—just ordinary charged particles when exceed local speed of light in medium (other hard vacuum). charged tachyon traveling in vacuum, therefore, undergoes constant proper time acceleration and, necessity, worldline forms hyperbola in space-time. reducing tachyon s energy increases speed, single hyperbola formed of 2 oppositely charged tachyons opposite momenta (same magnitude, opposite sign) annihilate each other when simultaneously reach infinite speed @ same place in space. (at infinite speed, 2 tachyons have no energy each , finite momentum of opposite direction, no conservation laws violated in mutual annihilation. time of annihilation frame dependent.)

even electrically neutral tachyon expected lose energy via gravitational cherenkov radiation, because has gravitational mass, , therefore increase in speed travels, described above. if tachyon interacts other particles, can radiate cherenkov energy particles. neutrinos interact other particles of standard model, , andrew cohen , sheldon glashow used argue faster-than-light neutrino anomaly cannot explained making neutrinos propagate faster light, , must instead due error in experiment.

causality

causality fundamental principle of physics. if tachyons can transmit information faster light, according relativity violate causality, leading logical paradoxes of kill own grandfather type. illustrated thought experiments such tachyon telephone paradox or logically pernicious self-inhibitor.

the problem can understood in terms of relativity of simultaneity in special relativity, says different inertial reference frames disagree on whether 2 events @ different locations happened @ same time or not, , can disagree on order of 2 events (technically, these disagreements occur when spacetime interval between events space-like , meaning neither event lies in future light cone of other).

if 1 of 2 events represents sending of signal 1 location , second event represents reception of same signal @ location, long signal moving @ speed of light or slower, mathematics of simultaneity ensures reference frames agree transmission-event happened before reception-event. however, in case of hypothetical signal moving faster light, there frames in signal received before sent signal said have moved backward in time. because 1 of 2 fundamental postulates of special relativity says laws of physics should work same way in every inertial frame, if possible signals move backward in time in 1 frame, must possible in frames. means if observer sends signal observer b moves faster light in s frame backwards in time in b s frame, , b sends reply moves faster light in b s frame backwards in time in s frame, work out receives reply before sending original signal, challenging causality in every frame , opening door severe logical paradoxes. mathematical details can found in tachyonic antitelephone article, , illustration of such scenario using spacetime diagrams can found in baker, r. (2003)

reinterpretation principle

the reinterpretation principle asserts tachyon sent in time can reinterpreted tachyon traveling forward in time, because observers cannot distinguish between emission , absorption of tachyons. attempt detect tachyon future (and violate causality) create same tachyon , send forward in time (which causal).

however, principle not accepted resolving paradoxes. instead, required avoid paradoxes unlike known particle, tachyons not interact in way , can never detected or observed, because otherwise tachyon beam modulated , used create anti-telephone or logically pernicious self-inhibitor . forms of energy believed interact @ least gravitationally, , many authors state superluminal propagation in lorentz invariant theories leads causal paradoxes.