Tachyonic antitelephone
Encyclopedia
A tachyonic antitelephone is a hypothetical device in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

 that could be used to send signals into one's own past
Past
Most generally, the past is a term used to indicate the totality of events which occurred before a given point in time. The past is contrasted with and defined by the present and the future. The concept of the past is derived from the linear fashion in which human observers experience time, and is...

. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 in 1907
presented a thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

 of how faster-than-light
Faster-than-light
Faster-than-light communications and travel refer to the propagation of information or matter faster than the speed of light....

 signals can lead to a paradox of causality, which was described by Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...

 in 1910 as a means "to telegraph into the past". The same thought experiment was described by Richard Chace Tolman in 1917, therefore it is also known as Tolman's paradox.

A device capable of "telegraphing into the past" was later also called "tachyonic antitelephone" by Gregory Benford
Gregory Benford
Gregory Benford is an American science fiction author and astrophysicist who is on the faculty of the Department of Physics and Astronomy at the University of California, Irvine...

 et al. According to the current understanding of physics, no such faster-than-light transfer of information is actually possible. For instance, the hypothetical tachyon
Tachyon
A tachyon is a hypothetical subatomic particle that always moves faster than light. In the language of special relativity, a tachyon would be a particle with space-like four-momentum and imaginary proper time. A tachyon would be constrained to the space-like portion of the energy-momentum graph...

 particles which give the device its name do not exist even theoretically in the standard model of particle physics, due to tachyon condensation
Tachyon condensation
In particle physics, theoretical processes that eliminate or resolve particles or fields into better understood phenomena are called, by extension and metaphor with the macroscopic process, "condensation"...

, and there is no experimental evidence that suggests that they might exist. The problem of detecting tachyons via causal contradictions was treated scientifically.

One-way path

Tolman used the following variation of Einstein's thought experiment: Imagine a distance with endpoints and . Let a signal be sent from A propagating with velocity a towards B. All of this is measured in an inertial frame where the endpoints are at rest. The arrival at B is given by:


Here, the event at A is the cause of the event at B. However, in the inertial frame moving with relative velocity v, the time of arrival at B is given according to the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...

:


It can be easily shown that if a > c, then certain values of v can make t' negative. In other words, the effect arises before the cause in this frame. Einstein and Tolman concluded that this result contains in his view no logical contradiction; however, it contradicts the totality of our experience so that the impossibility of a > c is sufficiently proven.

Two-way path

A more common variation of this thought experiment is to send back the signal to the sender (a similar one was given by David Bohm
David Bohm
David Joseph Bohm FRS was an American-born British quantum physicist who contributed to theoretical physics, philosophy, neuropsychology, and the Manhattan Project.-Youth and college:...

). Suppose Alice
Alice and Bob
The names Alice and Bob are commonly used placeholder names for archetypal characters in fields such as cryptography and physics. The names are used for convenience; for example, "Alice sends a message to Bob encrypted with his public key" is easier to follow than "Party A sends a message to Party...

 (A) is on a spacecraft
Spacecraft
A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....

 moving away from the Earth in the positive x-direction with a speed , and she wants to communicate with Bob (B) back home. Assume both of them have a device that is capable of transmitting and receiving faster-than-light signals at a speed of
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

 with . Alice uses this device to send a message to Bob, who sends a reply back. Let us choose the origin of the coordinates of Bob's reference frame, , to coincide with the reception of Alice's message to him. If Bob immediately sends a message back to Alice, then in his rest frame the coordinates of the reply signal (in natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

 so that c=1) are given by:


To find out when the reply is received by Alice, we perform a Lorentz transformation to Alice's frame moving in the positive x-direction with velocity with respect to the Earth. In this frame Alice is at rest at position , where is the distance that the signal Alice sent to Earth traversed in her rest frame. The coordinates of the reply signal are given by:


The reply is received by Alice when . This means that and thus:


Since the message Alice sent to Bob took a time of to reach him, the message she receives back from him will reach her at time:


later than she sent her message. However, if then and Alice will receive the message back from Bob before she sends her message to him in the first place.

Benford et al. wrote about such paradoxes in general:
They concluded that superluminal particles such as Tachyon
Tachyon
A tachyon is a hypothetical subatomic particle that always moves faster than light. In the language of special relativity, a tachyon would be a particle with space-like four-momentum and imaginary proper time. A tachyon would be constrained to the space-like portion of the energy-momentum graph...

s are therefore not allowed to convey signals.
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