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A Logic Based Foundation and Analysis of Relativity Theory and LogicPage: 1

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Relativity Theory and LogicPage: 2

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

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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 Relativity Theory and LogicPage: 4

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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=mc 2 SpecRel → AccRel → GenRel transition in detail Relativity Theory and LogicPage: 5 SpecRel AccRel GenRel

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Relativity Theory and LogicPage: 6

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R.T.’s as theories of First Order Logic Relativity Theory and LogicPage: 7

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Relativity Theory and LogicPage: 8 Streamlined Economical Transparent Small Rich Theorems Axioms Proofs …

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Special Relativity Relativity Theory and LogicPage: 9

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Relativity Theory and LogicPage: 10 Bodies (test particles), Inertial Observers, Photons, Quantities, usual operations on it, Worldview B IOb Ph 0 Q = number-line W

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Relativity Theory and LogicPage: 11 W(m, t x y z, b) body “b” is present at coordinates “t x y z” for observer “m” t x y b (worldline) m

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1. ( Q, +, ∙ ) is a field in the sense of abstract algebra (with 0, +, 1, / as derived operations) 2. 3. Ordering derived: Relativity Theory and LogicPage: 12 AxField Usual properties of addition and multiplication on Q : Q is an ordered Euclidean field.

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.Relativity Theory and LogicPage: 13 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 x y ph 1 ph 2 ph 3 Formalization:

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v m (b) Relativity Theory and LogicPage: 14 What is speed? m b p ptpt psps qsqs qtqt q 1

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Relativity Theory and LogicPage: 15 AxEv All inertial observers coordinatize the same events. Formalization: t x y WmWm b1b1 b2b2 m

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Relativity Theory and LogicPage: 16 Formalization: AxSelf An inertial observer sees himself as standing still at the origin. t x y t = worldline of the observer m WmWm

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Relativity Theory and LogicPage: 17 AxSymd Any two observers agree on the spatial distance between two events, if these two events are simultaneous for both of them, and |v m (ph)|=1. Formalization: k t x y m x y t e1 e2 ph is the event occurring at p in m’s worldview

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Relativity Theory and LogicPage: 18 SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd} SpecRel Theorems Proofs …

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Relativity Theory and LogicPage: 19 Thm2 Thm1

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t x y m Relativity Theory and LogicPage: 20 Proof of Thm2 (NoFTL): e1 k ph - AxSelf - AxPh e2 ph1 e3 e2 ph1 - AxEv t x y k e1 - AxField ph p q Tangent plane k asks m : where did ph and ph1 meet?

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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 Relativity Theory and LogicPage: 21

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Thm3 SpecRel is consistent Relativity Theory and LogicPage: 22 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!)

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Relativity Theory and LogicPage: 23 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.

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Relativity Theory and LogicPage: 24 Moving clocks get out of synchronism. Thm8 v k (m)

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Relativity Theory and LogicPage: 25 Thm8 (formalization of clock asynchronism) (1) Assume e, e’ are separated in the direction of motion of m in k’s worldview, m ykyk k xkxk 1t1t v 1x1x xmxm e e’ |v| Assume SpecRel. Assume m,k IOb and events e, e’ are simultaneous for m, m xmxm e’ e

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Relativity Theory and LogicPage: 26 (2) e, e’ are simultaneous for k, too e, e’ are separated orthogonally to v k (m) in k’s worldview m xmxm ykyk k xkxk ymym e e’

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Relativity Theory and LogicPage: 27 Thought-experiment for proving relativity of simultaneity.

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Relativity Theory and LogicPage: 28 Thought-experiment for proving relativity of simultaneity.

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Relativity Theory and LogicPage: 31 Moving clocks slow down Moving spaceships shrink Thm9

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Relativity Theory and LogicPage: 32 Thm9 (formalization of time-dilation) Assume SpecRel. Let m,k IOb and events e, e’ are on k’s lifeline. m xmxm e e’ k 1 v k xkxk e

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Relativity Theory and LogicPage: 34 Thm10 (formalization of spaceship shrinking) Assume SpecRel. Let m,k,k’ IOb and assume m xmxm k k’ e’ e k x k’

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Relativity Theory and LogicPage: 35 Experiment for measuring distance (for m) by radar: m’ m’’ m e e’ Dist m (e,e’)

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Relativity Theory and LogicPage: 37 v = speed of spaceship

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Relativity Theory and LogicPage: 38 t x y WmWm b1b1 b2b2 m w mk ev m ev k p q

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Relativity Theory and LogicPage: 39 Thm11 The worldview transformations w mk are Lorentz transformations (composed perhaps with a translation).

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Relativity Theory and LogicPage: 40 Minkowskian Geometry: 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.

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Conceptual analysis of SR goes on … on our homepage Relativity Theory and LogicPage: 41 New theory is coming:

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Accelerated Observers Relativity Theory and LogicPage: 42

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Relativity Theory and LogicPage: 43 AccRel = SpecRel + AccObs. AccRel is stepping stone to GenRel via Einstein’s EP AccRel

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Relativity Theory and LogicPage: 44 The same as for SpecRel. Recall that

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Relativity Theory and LogicPage: 45 World-view transformation t x y WmWm b1b1 b2b2 m w mk Cd(k) ev m ev k Cd(m)

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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 w mk at p is the identity”. Relativity Theory and LogicPage: 46 kOb mIOb p Formalization: Let F be an affine mapping (definable).

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Relativity Theory and LogicPage: 47 The Vienna Circle and Hungary, Vienna, May 19-20, 2008

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Relativity Theory and LogicPage: 48

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AxEv - If m “sees” k participate in an event, then k cannot deny it. Relativity Theory and LogicPage: 49 t x y WmWm k b2b2 m t x y WkWk b2b2 k Formalization:

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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”. Relativity Theory and LogicPage: 50

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Relativity Theory and LogicPage: 51 AccRel = SpecRel + AxCmv + AxEv - + AxSelf - + AxDif + AxCont AccRel Theorems Proofs …

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Thm101 Relativity Theory and LogicPage: 52 Appeared: Found. Phys. 2006, Madarász, J. X., Németi, I., Székely, G. IOb Ob t1t1 t1’t1’ t t’

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Relativity Theory and LogicPage: 53 Acceleration causes slow time. Thm102 a k (m)

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Relativity Theory and LogicPage: 54 Thm102 I.e., clocks at the bottom of spaceship run slower than the ones at the nose of the spaceship, both according to nose and bottom (watching each other by radar). Appeared: Logic for XXIth Century. 2007, Madarász, J. X., Németi, I., Székely, G.

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ACCELERATING SPACEFLEET Relativity Theory and LogicPage: 55

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ACCELERATING SPACEFLEET Relativity Theory and LogicPage: 56

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ACCELERATING SPACEFLEET Relativity Theory and LogicPage: 57

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Relativity Theory and LogicPage: 58 + - - - - + - - - - + - - - - - - - - + - - - - + - - - - +

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h 0 Relativity Theory and LogicPage: 59 Thm103 Proof: We will present a recipe for constructing a world-view for any accelerated observer living on a differentiable “time-like” curve (in a vertical plane): r 0 m wline m h q d

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UNIFORMLY ACCELERATED OBSERVERS Relativity Theory and LogicPage: 60 World-lines are Minkowski-circles (hyperbolas)

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UNIFORMLY ACCELERATED OBSERVERS Relativity Theory and LogicPage: 61 Result of the general recipe

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UNIFORMLY ACCELERATED OBSERVERS Relativity Theory and LogicPage: 62

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UNIFORMLY ACCELERATING SPACESHIP Relativity Theory and LogicPage: 63 outside viewinside view

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UNIFORMLY ACCELERATING SPACESHIP Relativity Theory and LogicPage: 64 1. Length of the ship does not change seen from inside. Captain measures length of his ship by radar:

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UNIFORMLY ACCELERATING SPACESHIP Relativity Theory and LogicPage: 65 2. Time runs slower at the rear of the ship, and faster at the nose of ship. Measured by radar:

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Relativity Theory and LogicPage: 66 via Einstein’s Equivalence Principle

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Relativity Theory and LogicPage: 67 -7 μs/day because of the speed of sat. approx. 4 km/s (14400 km/h) +45 μs/day because of gravity Relativistic Correction: 38 μs/day approx. 10 km/day Global Positioning System

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Relativity Theory and LogicPage: 68 via Einstein’s Equivalence Principle

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Relativity Theory and LogicPage: 69 via Einstein’s Equivalence Principle

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Relativity Theory and LogicPage: 70 via Einstein’s Equivalence Principle

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Relativity Theory and LogicPage: 71 Breaking the Turing-barrier via GR Relativistic Hyper Computing

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Conceptual analysis of AccRel goes on … on our homepage Relativity Theory and LogicPage: 72 New theory is coming:

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General Relativity Relativity Theory and LogicPage: 73

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Relativity Theory and LogicPage: 74 Einstein’s strong Principle of Relativity: “All observers are equivalent” (same laws of nature) Abolish different treatment of inertial and accelerated observers in the axioms

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Language of GenRel: the same as that of SpecRel. Recipe to obtain GenRel from AccRel: delete all axioms from AccRel mentioning IOb. But retain their IOb-free logical consequences. Relativity Theory and LogicPage: 75 AxSelf AxPh AxSymt AxEv AxSelf- AxPh- AxSymt- AxEv- AxCmv AxDif

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AxPh- The velocity of photons an observer “meets” is 1 when they meet, and it is possible to send out a photon in each direction where the observer stands AxSym- Meeting observers see each other’s clocks slow down with the same rate Relativity Theory and LogicPage: 76 mOb 1t1t 1 t’

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Relativity Theory and LogicPage: 77 GenRel = AxField +AxPh - +AxEv - + AxSelf - +AxSymt - +AxDif+AxCont GenRel Theorems Proofs …

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Thm1001 Relativity Theory and LogicPage: 78 AccRel was stepping stone from SpecRel to GenRel:

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Manifold Thm1002 GenRel is complete wrt Lorentz manifolds. Relativity Theory and LogicPage: 79 k’ p k LpLp M=Events ev m ev k w km wkwk wmwm Metric:

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UNIFORMLY ACCELERATED OBSERVERS Relativity Theory and LogicPage: 80

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How to use GenRel? Recover IOb’s by their properties In AccRel by the “twin paradox theorem” IOb’s are those observers who maximize wristwatch-time locally Recover LIFs Relativity Theory and Logic Page: 81

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Relativity Theory and LogicPage: 82 Timelike means: exist comoving observer

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Thm1003 Relativity Theory and LogicPage: 83

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Relativity Theory and LogicPage: 90 More in our papers in General Relativity and Gravitation 2008 & in arXiv.org 2008 Lightcones open up.

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Relativity Theory and LogicPage: 91 Timetravel in Gödel’s univers

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Relativity Theory and LogicPage: 92 More concrete material available from our group: (1) Logic of Spacetime http://ftp.math-inst.hu/pub/algebraic-logic/Logicofspacetime.pdf (2) in Foundation of Physics http://ftp.math-inst.hu/pub/algebraic-logic/twp.pdf (3) First-order logic foundation of relativity theories http://ftp.math-inst.hu/pub/algebraic-logic/springer.2006-04-10.pdf (4) FOL 75 papers http://www.math-inst.hu/pub/algebraic-logic/foundrel03nov.html http://www.math-inst.hu/pub/algebraic-logic/loc-mnt04.html (5) our e-book on conceptual analysis of SpecRel http://www.math-inst.hu/pub/algebraic-logic/olsort.html (6) More on István Németi's homepage http://www.renyi.hu/~nemeti/http://www.renyi.hu/~nemeti/ (Some papers available, and some recent work) Thank you!

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