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

A CONCEPTUAL ANAL YSIS OF THE RELA TIVISTIC CLOCK P ARADO X Gergely Sz�k ely h ttp://www.ren yi.h u/�turms

lo k p arado x PSfrag repla emen ts ClkP : time b ( e a , e c ) > time a ( e a , e )+ time ( e, e c ) b e c a e e a

v ariants of lo k p arado x PSfrag repla emen ts : time b ( e a , e c ) = time a ( e a , e ) + time ( e, e c ) b e c a AntiClkP : time b ( e a , e c ) < time a ( e a , e ) + time ( e, e c ) NoClkP e e a

PSfrag repla emen ts first-order logi framew ork f or sp a e-times Language: � Q : � quan tities � b o dies ; B : ; � Q B . . .

PSfrag repla emen ts Dimension of spa e-time: d ≥ 2 . Language: � Q : � quan tities � b o dies ; B : Ob ; W � W Q B . . . p 1 Ob p 2 m b p d W orld-view relation: W ( m, b, � � � Observ er m o ordinatizes b o dy at spa e-time lo ation � � (at time p 1 and spa e � p 2 . . . p d � ) p ) b p

PSfrag repla emen ts str u ture of quantities Language: � Q : <, + , · , 0 , 1; B : Ob � quan tities � b o dies ; W � W Q B . . . p 1 Ob 1 m p 2 0 b p d AxEOF : The quan tit y part � Q ; + , · , <, 0 , 1 � is a Eu lidean ordered �eld. (P ositiv e elemen ts ha v e square ro ots.)

axioms of kinema ti s m PSfrag repla emen ts : The observ er are in rest a ording to themselv es. Q d AxSelf

axioms of kinema ti s PSfrag repla emen ts m k 2 1 0 AxLinTime : The life-lines of observ ers are lines and time is passing uniformly on the life-lines. Q d

axioms of kinema ti s PSfrag repla emen ts k m k m � p q � � � Q d Q d : Ev ery observ er o ordinatize the same ev en ts. Ev e AxEv Kinem 0 := { AxEOF , AxSelf , AxLinTime , AxEv }

PSfrag repla emen ts mink o wski sphere on v ex �at k m 1 on a v e 0 Q d is the set of time-unit v e tors. MS m

geometri al hara teriza tion Theorem: Assume Kinem 0 . Then is on v ex is �at is on a v e ∀ m ∈ Ob MS m = ⇒ ClkP , ∀ m ∈ Ob MS m = ⇒ NoClkP , ∀ m ∈ Ob MS m = ⇒ AntiClkP .

geometri al hara teriza tion Theorem: Assume Kinem 0 + AxDispl . Then is on v ex is �at is on a v e ∀ m ∈ Ob MS m ⇐ ⇒ ClkP , ∀ m ∈ Ob MS m ⇐ ⇒ NoClkP , ∀ m ∈ Ob MS m ⇐ ⇒ AntiClkP . is te hni al axiom. It is used to displa e observ ers in order to reate t win parado x situations. AxDispl

onsequen es newtonian kinema ti s : The observ ers measure the same time b et w een ev en ts. Theorem: AxEOF + AxUnivTime | AxUnivTime = NoClkP

onsequen es newtonian kinema ti s : The observ ers measure the same time b et w een ev en ts. Theorem: AxEOF + AxUnivTime | AxUnivTime : Observ ers an mo v e in an y dire tion with an y �nite sp eed. = NoClkP Theorem: Kinem 0 + AxOb + + NoClkP �| AxOb + = AxUnivTime

PSfrag repla emen ts spe ial rela tivity Language: � Q : <, + , · , 0 , 1; B : Ob , Ph ; W � � quan tities � b o dies W Q B . . . p 1 Ob 1 m p 2 0 b p d Ph

spe ial rela tivity AxPh : F or ev ery observ er, the sp eed of ligh t is 1 . SpecRel d 0 := { AxEOF , AxSelf , AxPh , AxEv }

spe ial rela tivity AxPh : F or ev ery observ er, the sp eed of ligh t is 1 . W e ha v e to w eak en AxOb + sin e SpecRel d implies the imp ossibilit y SpecRel d 0 := { AxEOF , AxSelf , AxPh , AxEv } of faster than ligh t motions for observ ers (if d ≥ 3 ). AxOb : Observ ers an mo v e in an y dire tion with an y sp eed less 0 than 1 (less that the sp eed of ligh t).

onsequen es on spe ial rela tivity : Relativ ely mo ving observ ers' lo ks slo w do wn. Thm( d ≥ 3 ): SpecRel d SlowTime 0 + AxLinTime + SlowTime | = ClkP

onsequen es on spe ial rela tivity : Relativ ely mo ving observ ers' lo ks slo w do wn. Thm( d ≥ 3 ): SpecRel d SlowTime Thm( d ≥ 3 ): SpecRel d 0 + AxLinTime + SlowTime | = ClkP 0 + AxLinTime + AxOb + ClkP �| = SlowTime

onsequen es on spe ial rela tivity AxSimDist : If ev en ts e 1 and e 2 are sim ultaneous for b oth observ ers and k , then m and k agree on the spatial distan e b et w een and e 2 . Thm( d ≥ 3 ): SpecRel d m e 1 0 + AxSimDist | = ClkP

onsequen es on spe ial rela tivity AxSimDist : If ev en ts e 1 and e 2 are sim ultaneous for b oth observ ers and k , then m and k agree on the spatial distan e b et w een and e 2 . Thm( d ≥ 3 ): SpecRel d m e 1 Thm( d ≥ 3 ): SpecRel d 0 + AxSimDist | = ClkP 0 + AxLinTime + AxOb + ClkP �| = AxSimDist

a question f or fur ther resear h Question: What is the logi al onne tion b et w een AxSimDist and ? Remark: If Q = R , then AxSimDist and SlowTime are equiv alen t in the mo dels of SpecRel d 0 + AxLinTime + AxDispl + AxOb . SlowTime