A CONCEPTUAL ANAL YSIS OF THE RELA TIVISTIC CLOCK P ARADO - - PowerPoint PPT Presentation

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A CONCEPTUAL ANAL YSIS OF THE RELA TIVISTIC CLOCK P ARADO - - PowerPoint PPT Presentation

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A CONCEPTUAL ANAL YSIS OF THE RELA TIVISTIC CLOCK P ARADO X Gergely Szk ely h ttp://www.ren yi.h u/turms lok p arado x PSfrag replaemen ts ClkP : time b ( e a , e c ) > time a ( e a , e )+ time ( e, e c ) b e c

slide-1
SLIDE 1 A CONCEPTUAL ANAL YSIS OF THE RELA TIVISTIC CLOCK P ARADO X Gergely Szk ely h ttp://www.ren yi.h u/turms
slide-2
SLIDE 2 lo k p arado x PSfrag repla emen ts

e ea ec

a b

ClkP: time

b(ea, ec) > time a(ea, e)+time (e, ec)
slide-3
SLIDE 3 v ariants
  • f
lo k p arado x PSfrag repla emen ts

e ea ec

a b

NoClkP

: time b(ea, ec) = time a(ea, e) + time (e, ec)

AntiClkP : time

b(ea, ec) < time a(ea, e) + time (e, ec)
slide-4
SLIDE 4 first-order logi framew
  • rk
f
  • r
sp a e-times Language: Q :

; B : ;

  • PSfrag
repla emen ts

Q

  • quan
tities

B

  • b
  • dies
. . .
slide-5
SLIDE 5 Dimension
  • f
spa e-time: d ≥ 2. Language: Q :

; B : Ob ; W

PSfrag repla emen ts

p1 p2 pd Q

  • quan
tities

W Ob B

  • b
  • dies

m b

. . . W
  • rld-view
relation: W(m, b,

p )

  • Observ
er m
  • rdinatizes
b
  • dy

b

at spa e-time lo ation

p

  • (at
time p1 and spa e p2 . . . pd)
slide-6
SLIDE 6 str u ture
  • f
quantities Language: Q : <, +, ·, 0, 1; B : Ob

; W

PSfrag repla emen ts

1 p1 p2 pd Q

  • quan
tities

W Ob B

  • b
  • dies

m b

. . .

AxEOF:

The quan tit y part Q; +, ·, <, 0, 1 is a Eu lidean
  • rdered
eld. (P
  • sitiv
e elemen ts ha v e square ro
  • ts.)
slide-7
SLIDE 7 axioms
  • f
kinema ti s PSfrag repla emen ts

m Qd AxSelf

: The
  • bserv
er are in rest a ording to themselv es.
slide-8
SLIDE 8 axioms
  • f
kinema ti s PSfrag repla emen ts

m k 1 2 Qd AxLinTime :

The life-lines
  • f
  • bserv
ers are lines and time is passing uniformly
  • n
the life-lines.
slide-9
SLIDE 9 axioms
  • f
kinema ti s
  • PSfrag
repla emen ts

m m k k e Ev

  • p
  • q

Qd Qd AxEv

: Ev ery
  • bserv
er
  • rdinatize
the same ev en ts.

Kinem0 := {AxEOF, AxSelf, AxLinTime, AxEv}

slide-10
SLIDE 10 mink
  • wski
sphere PSfrag repla emen ts at
  • n
v ex
  • n a
v e

m k 1 Qd MSm

is the set
  • f
time-unit v e tors.
slide-11
SLIDE 11 geometri al hara teriza tion Theorem: Assume Kinem0 . Then

∀m ∈ Ob MSm

is
  • n
v ex

= ⇒ ClkP, ∀m ∈ Ob MSm

is at

= ⇒ NoClkP, ∀m ∈ Ob MSm

is
  • n a
v e

= ⇒ AntiClkP.

slide-12
SLIDE 12 geometri al hara teriza tion Theorem: Assume Kinem0 +AxDispl . Then

∀m ∈ Ob MSm

is
  • n
v ex

⇐ ⇒ ClkP, ∀m ∈ Ob MSm

is at

⇐ ⇒ NoClkP, ∀m ∈ Ob MSm

is
  • n a
v e

⇐ ⇒ AntiClkP. AxDispl

is te hni al axiom. It is used to displa e
  • bserv
ers in
  • rder
to reate t win parado x situations.
slide-13
SLIDE 13
  • nsequen es
newtonian kinema ti s

AxUnivTime

: The
  • bserv
ers measure the same time b et w een ev en ts. Theorem: AxEOF + AxUnivTime |

= NoClkP

slide-14
SLIDE 14
  • nsequen es
newtonian kinema ti s

AxUnivTime

: The
  • bserv
ers measure the same time b et w een ev en ts. Theorem: AxEOF + AxUnivTime |

= NoClkP AxOb+

: Observ ers an mo v e in an y dire tion with an y nite sp eed. Theorem: Kinem0 + AxOb+ + NoClkP |

= AxUnivTime

slide-15
SLIDE 15 spe ial rela tivity Language: Q : <, +, ·, 0, 1; B : Ob, Ph; W PSfrag repla emen ts

1 p1 p2 pd Q

  • quan
tities

W Ob Ph B

  • b
  • dies

m b

. . .
slide-16
SLIDE 16 spe ial rela tivity

AxPh:

F
  • r
ev ery
  • bserv
er, the sp eed
  • f
ligh t is 1.

SpecReld

0 := {AxEOF, AxSelf, AxPh, AxEv}

slide-17
SLIDE 17 spe ial rela tivity

AxPh:

F
  • r
ev ery
  • bserv
er, the sp eed
  • f
ligh t is 1.

SpecReld

0 := {AxEOF, AxSelf, AxPh, AxEv}

W e ha v e to w eak en AxOb+ sin e SpecReld implies the imp
  • ssibilit
y
  • f
faster than ligh t motions for
  • bserv
ers (if d ≥ 3).

AxOb:

Observ ers an mo v e in an y dire tion with an y sp eed less than 1 (less that the sp eed
  • f
ligh t).
slide-18
SLIDE 18
  • nsequen es
  • n
spe ial rela tivity

SlowTime

: Relativ ely mo ving
  • bserv
ers' lo ks slo w do wn. Thm(d ≥ 3): SpecReld

0 + AxLinTime + SlowTime |

= ClkP

slide-19
SLIDE 19
  • nsequen es
  • n
spe ial rela tivity

SlowTime

: Relativ ely mo ving
  • bserv
ers' lo ks slo w do wn. Thm(d ≥ 3): SpecReld

0 + AxLinTime + SlowTime |

= ClkP

Thm(d ≥ 3): SpecReld

0 + AxLinTime + AxOb + ClkP |

= SlowTime

slide-20
SLIDE 20
  • nsequen es
  • n
spe ial rela tivity

AxSimDist:

If ev en ts e1 and e2 are sim ultaneous for b
  • th
  • bserv
ers

m

and k , then m and k agree
  • n
the spatial distan e b et w een

e1

and e2 . Thm(d ≥ 3): SpecReld

0 + AxSimDist |

= ClkP

slide-21
SLIDE 21
  • nsequen es
  • n
spe ial rela tivity

AxSimDist:

If ev en ts e1 and e2 are sim ultaneous for b
  • th
  • bserv
ers

m

and k , then m and k agree
  • n
the spatial distan e b et w een

e1

and e2 . Thm(d ≥ 3): SpecReld

0 + AxSimDist |

= ClkP

Thm(d ≥ 3): SpecReld

0 + AxLinTime + AxOb + ClkP |

= AxSimDist

slide-22
SLIDE 22 a question f
  • r
fur ther resear h Question: What is the logi al
  • nne tion
b et w een AxSimDist and

SlowTime

? Remark: If Q = R , then AxSimDist and SlowTime are equiv alen t in the mo dels
  • f SpecReld

0 + AxLinTime + AxDispl + AxOb.