Relativity Theory by Hajnal Andréka , István & Péter Németi

1. A Logic Based Foundation and Analysis of Relativity Theoryby HajnalAndréka, István & Péter Németi Relativity Theory and Logic

2. intro Relativity Theory and Logic

3. Welcome Relativity Theory and Logic • Our Research Group: H. Andréka, J. X. Madarász, I. & P. Németi, G. Székely, R. Tordai. Cooperation with: L. E. Szabó, M. Rédei.

4. Aims of our school Relativity Theory and Logic • Analysis of the logical structure of R. Theory • Base the theory on simple, unambiguous axioms with clear meanings • Make Relativity Theory: • More transparent logically • Easier to understand and teach • Easier to change • Modular • Demystify Relativity Theory

5. PLAN of OUR presentations SpecRel AccRel GenRel Relativity Theory and Logic Logical analysis of: • Special relativity theory Transition to: • Accelerated observers and Einstein's EP Transition to: • General relativity • Exotic space-times, black holes, wormholes • Application of general relativity to logic • Visualizations of Relativistic Effects • Relativistic dynamics, Einstein’s E=mc2 • SpecRel → AccRel → GenRel transition in detail

6. Structure of our theories SpecRel RelDyn (E=mc2) AccRel GenRel Relativity Theory and Logic

7. Aims of our school • R.T.’s as theories of First Order Logic S. R. G. R. Relativity Theory and Logic

8. Logic axiomatization of R.T. Theorems Thm … Thm … Thm … Thm … Thm … Thm … Axioms Proofs Ax…. Ax…. Ax…. Ax…. … Streamlined Economical Transparent Small Rich Relativity Theory and Logic

9. Special Relativity Part i Relativity Theory and Logic

10. 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

11. 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

12. 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

13. 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

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

15. 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

16. 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

17. 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

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

19. Specrel • Thm1 • Thm2 Relativity Theory and Logic

20. 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

21. 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

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

23. Specrel • Thm6 • SpecRelgenerates 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

24. Relativistic effects Thm8 vk(m) • Moving clocks get out of synchronism. Relativity Theory and Logic

25. 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

26. 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

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

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

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

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

31. Moving clocks slow down Moving spaceships shrink Relativistic effects Thm9 Relativity Theory and Logic

32. Moving clocks slow down • 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

33. Moving clocks slow down Relativity Theory and Logic

34. 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

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

36. Moving spaceshipsshrink Relativity Theory and Logic

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

38. 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

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

40. Otherformalisations of specrel MinkowskianGeometry: Hierarchy of theories. Interpretations between them. Definitional equivalence. 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

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

42. AcceleratedObservers Part ii Relativity Theory and Logic

43. Theory of Accelerated observers AccRel = SpecRel + AccObs. AccRel is stepping stone to GenRel via Einstein’s EP AccRel Accelerated Observers AccRel SpecRel GenRel S. R. G. R. Relativity Theory and Logic

44. Language for accrel The same as for SpecRel. Recallthat Relativity Theory and Logic

45. Language for accrel Cd(m) Cd(k) wmk m k t t b1 b1 evm evk b2 b2 Wk Wm x x y y World-view transformation Relativity Theory and Logic

46. Axioms for accelerated observers kϵOb p mϵIOb Formalization: Let F be an affine mapping (definable). Relativity Theory and Logic AxCmv At each moment p of his worldline, the accelerated observer k “sees” (=coordinatizes) the nearby world for a short while as an inertial observer m does, i.e. “the linear approximation of wmk at p is the identity”.

47. Thought experiment The ViennaCircle and Hungary, Vienna, May 19-20, 2008 Relativity Theory and Logic

48. Thought experiment Relativity Theory and Logic

49. Axioms for accelerated observers t m k b2 t k b2 Wk Wm x x y Formalization: y Relativity Theory and Logic AxEv- If m “sees” k participate in an event, then k cannot deny it.

50. Axioms for accelerated observers Relativity Theory and Logic • AxSelf- • The world-line of any observer is an open interval of the time-axis, in his own world-view • AxDif • The worldview transformations have linear approximations at each point of their domain (i.e. they are differentiable). • AxCont • Bounded definable nonempty subsets of Q have suprema. Here “definable” means “definable in the language of AccRel, parametrically”.