Editing Special relativity (section)

=== Causality and prohibition of motion faster than light ===
{{See also|Causality (physics)|Tachyonic antitelephone}}
[[File:Simple light cone diagram.svg|thumb|Figure 4–6. [[Light cone]]]]
In Fig.&nbsp;4-6, the time interval between the events A (the "cause") and B (the "effect") is 'timelike'; that is, there is a frame of reference in which events A and B occur at the ''same location in space'', separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect).

The interval AC in the diagram is 'spacelike'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. But no frames are accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, paradoxes of causality would result.

For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).<ref>{{cite book | first = Richard C.|last = Tolman|title =The Theory of the Relativity of Motion |location=Berkeley|publisher = University of California Press|date = 1917|page = 54|url = https://books.google.com/books?id=8yodAAAAMAAJ&q=54}}</ref><ref group=p>{{cite journal|author1=G. A. Benford |author2=D. L. Book |author3=W. A. Newcomb |name-list-style=amp |doi=10.1103/PhysRevD.2.263|title=The Tachyonic Antitelephone|date=1970|journal=Physical Review D|volume=2|issue=2|pages=263–265|bibcode = 1970PhRvD...2..263B |s2cid=121124132 }}</ref> A variety of causal paradoxes could then be constructed.

{{multiple image
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 | image1 = Causality violation 1.svg
 | image2 = Causality violation 2.svg
 | alt1 = Causality violation: Beginning of scenario resulting from use of a fictitious instantaneous communicator   
 | alt2 = Causality violation: B receives the message before having sent it. 
 | footer_align = center
 | footer = Figure 4-7. Causality violation by the use of fictitious<br/>"instantaneous communicators"
}}
Consider the spacetime diagrams in Fig.&nbsp;4-7. A and B stand alongside a railroad track, when a high-speed train passes by, with C riding in the last car of the train and D riding in the leading car. The [[world lines]] of A and B are vertical (''ct''), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (''{{prime|ct}}''), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.
# Fig.&nbsp;4-7a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the <math>-x'</math> axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives ''earlier'' than it was sent.
# Fig.&nbsp;4-7b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the <math>+x</math> axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, in the primed frames of C and D, B will receive the message before it was sent out, a violation of causality.<ref name="Takeuchi">{{cite web |last1=Takeuchi |first1=Tatsu |title=Special Relativity Lecture Notes – Section 10 |url=https://www1.phys.vt.edu/~takeuchi/relativity/notes/section10.html |publisher=Virginia Tech |access-date=31 October 2018}}</ref>

It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the <math>x'</math> axis (and the signal from A to B slightly steeper than the <math>x</math> axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the <math>ct'</math> and <math>x'</math> axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only ''slightly'' faster than the speed of light will result in causality violation.<ref name="Morin2017">{{cite book|last1=Morin|first1=David|title=Special Relativity for the Enthusiastic Beginner|date=2017|publisher=CreateSpace Independent Publishing Platform|pages=90–92|isbn=9781542323512}}</ref>

Therefore, '''if''' [[causality]] is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel [[faster than light]] in vacuum.

This is not to say that ''all'' faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light.<ref>{{cite web |last1=Gibbs |first1=Philip |title=Is Faster-Than-Light Travel or Communication Possible? |url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html |website=Physics FAQ |publisher=Department of Mathematics, University of California, Riverside |access-date=31 October 2018}}</ref> For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).<ref>{{cite book |title=Applications of Electrodynamics in Theoretical Physics and Astrophysics |edition=illustrated |first1=David |last1=Ginsburg |publisher=CRC Press |year=1989 |isbn=978-2-88124-719-4 |page=206 |url=https://books.google.com/books?id=Lh0tjaBNzg0C|bibcode=1989aetp.book.....G }} [https://books.google.com/books?id=Lh0tjaBNzg0C&pg=PA206 Extract of page 206]</ref><ref>{{cite book |title=Four Decades of Scientific Explanation |author=Wesley C. Salmon |publisher=University of Pittsburgh |date=2006 |isbn=978-0-8229-5926-7 |page=107 |url=https://books.google.com/books?id=FHqOXCd06e8C}}, [https://books.google.com/books?id=FHqOXCd06e8C&pg=PA107 Section 3.7 page 107]</ref><!-- a pair of diagrams, with x–t and x'–t' coordinates would help here -->