Relativity Theory by H . Andréka , J. X. Madarász, I . & P . Németi , G. Székely

1. A Logic Based Foundation and Analysis of Relativity Theoryby H.Andréka, J. X. Madarász, I. & P. Németi, G. Székely Relativity Theory and Logic

2. Special Relativity Part i Relativity Theory and Logic

3. Language for specrel Bodies (test particles), Inertial Observers, Photons, Quantities, usual operations on it, Worldview B Q= number-line IOb Ph W 0 Relativity Theory and Logic

4. Language for specrel W(m, t x y z, b) body “b” is present at coordinates “t x y z” for observer “m” m t b (worldline) x y Relativity Theory and Logic

5. Axioms for specrel • AxField Usual properties of addition and multiplication on Q : Q is an ordered Euclidean field. 1. ( Q , + , ∙ ) is a field in the sense of abstract algebra (with 0 , −, 1 , / as derived operations) 2. 3. Ordering derived: Relativity Theory and Logic

6. Axiomsforspecrel • AxPh For all inertial observers the speed of light is the same in all directions and is finite. In any direction it is possible to send out a photon. t Formalization: ph1 ph2 ph3 x y Relativity Theory and Logic

7. Axiomsforspecrel • AxEv All inertial observers coordinatize the same events. m k t t b1 b1 b2 b2 Wk Wm x x y Formalization: y Relativity Theory and Logic

8. Axioms for specrel • AxSelf An inertial observer sees himself as standing still at the origin. t m t = worldline of the observer Wm Formalization: x y Relativity Theory and Logic

9. Axioms for Specrel • AxSymd Any two observers agree on the spatial distance between two events, if these two events are simultaneous for both of them, and |vm(ph)|=1. t m ph t k x x e1 Formalization: e2 y y is the event occurring at p in m’s worldview Relativity Theory and Logic

10. Axiomsforspecrel What is speed? m b pt p 1 qt q vm(b) qs ps Relativity Theory and Logic

11. Specrel SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd} Theorems SpecRel Thm1 Thm2 Thm3 Thm4 Thm5 AxField AxPh AxEv AxSelf AxSymd Proofs … Relativity Theory and Logic

12. Specrel • Thm1 • Thm2 Relativity Theory and Logic

13. Specrel • Proof of Thm2 (NoFTL): k - AxField Tangent plane ph t - AxPh ph1 ph m k - AxSelf e1 e2 t - AxEv e1 q e3 e2 p ph1 x x y y k asks m : where did ph and ph1 meet? Relativity Theory and Logic

14. Specrel Relativity Theory and Logic • Conceptual analysis • Which axioms are needed and why • Project: • Find out the limits of NoFTL • How can we weaken the axioms to make NoFTL go away

15. Specrel It is an importantresearchtodaytostudyspacetimeswith more thanonetimedimensions Relativity Theory and Logic Igor D. Novikov, September 3 Budapest:

16. Specrel • Thm4 • No axioms of SpecRel is provable from the rest • Thm5 • SpecRel is complete with respect to Minkowski geometries • (e.g. implies all the basic paradigmatic effects of Special Relativity - even quantitatively!) Relativity Theory and Logic • Thm3 • SpecRel is consistent

17. Specrel • Thm6 • SpecRel generates an undecidable theory. Moreover, it enjoys both of Gödel’s incompleteness properties • Thm7 • SpecRel has a decidable extension, and it also has a hereditarily undecidable extension. Both extensions are physically natural. Relativity Theory and Logic

18. Relativistic effects Thm8 vk(m) • Moving clocks get out of synchronism. • Captainclaimsthatthetwoclocks show thesametime. Relativity Theory and Logic

19. Moving clocks get out of synchronism • Thm8 (formalization of clock asynchronism) Assume SpecRel. Assume m,kϵIOband events e, e’ are simultaneous for m, (1) Assume e, e’ are separated in the direction of motion of m in k’s worldview, k m m v 1t xm e’ |v| e e e’ xk 1x xm yk Relativity Theory and Logic

20. Moving clocks get out of synchronism k m (2) e, e’ are simultaneous for k, too e, e’ are separated orthogonally to vk(m) in k’s worldview xm e xk e’ ym yk Relativity Theory and Logic

21. Moving clocks get out of syncronism Thought-experiment for proving relativity of simultaneity. Relativity Theory and Logic

22. Moving clocks get out of syncronism Thought-experiment for proving relativity of simultaneity. Relativity Theory and Logic

23. Moving clocks get out of syncronism Relativity Theory and Logic

24. Moving clocks get out of syncronism Black Hole Wormhole Timewarp-theory Relativity Theory and Logic

25. Moving clocks tick slowly Moving spaceships shrink Relativistic effects Thm9 Relativity Theory and Logic

26. Moving clocks Tickslowly • Thm9 (formalization of time-dilation) Assume SpecRel. Let m,kϵIOband events e, e’ are on k’s lifeline. k k m e’ v e’ e e 1 xm xk Relativity Theory and Logic

27. Moving clocks tick slowly Relativity Theory and Logic

28. Moving spaceshipsshrink • Thm10 (formalization of spaceship shrinking) Assume SpecRel. Let m,k,k’ ϵIOband assume k’ k k k’ m e e’ xm x Relativity Theory and Logic

29. Moving spaceshipsshrink • Experiment for measuring distance (for m) by radar: m’ m’’ m Distm(e,e’) e’ e Relativity Theory and Logic

30. Moving spaceshipsshrink Relativity Theory and Logic

31. Relativistic effects v = speed of spaceship Relativity Theory and Logic

32. Worldview transformation wmk q p m k t t b1 b1 b2 evm evk b2 Wk Wm x x y y Relativity Theory and Logic

33. specrel • Thm11 • The worldview transformations wmk are Lorentz transformations (composed perhaps with a translation). Relativity Theory and Logic

34. Otherformalisations of specrel Minkowskian Geometry: Top-down approach to SR, observer-free. Robert Goldblatt: complete FOL axiomsystemMinkGeo Relativity Theory and Logic

35. hierarchy of theoriesondifferentlanguages Definitional equivalence SpecRel + simplifying axioms MinkGeo+ “meterrod” Interpretations SpecRel MinkGeo Relativity Theory and Logic

36. Otherformalisations of specrel Minkowskian Geometry James Ax: Signals Alfred Robb: causality Patrick Suppes: worldview transformations … Connections between theories. Dynamics of theories. Interpretations between them. Theory morphisms. Definitional equivalence. Tamás Füzessy, Judit Madarász and Gergely Székely began joint work in this Definability theory of logic! (Tarski, Makkai) Contribution of relativity to logic: definability theory with new objects definable (and not only with new relations definable). J. Madarász’ dissertation. Relativity Theory and Logic

37. Specrel SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd} Theorems SpecRel Thm1 Thm2 Thm3 Thm4 Thm5 AxField AxPh AxEv AxSelf AxSymd Proofs … Relativity Theory and Logic

38. specrel • New theory is coming: Relativity Theory and Logic Conceptual analysis of SR goes on … on our homepage