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

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

Relativity Theory and LogicPage: 2

 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

 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

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

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

Special Relativity Relativity Theory and LogicPage: 9

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

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

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.

.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:

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

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

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

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

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

Relativity Theory and LogicPage: 19  Thm2  Thm1

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?

 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

 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!)

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.

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

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

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’

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

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

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’

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

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

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

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.

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

Accelerated Observers Relativity Theory and LogicPage: 42

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

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

 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)

 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).

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

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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:

 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

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

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

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

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.

ACCELERATING SPACEFLEET Relativity Theory and LogicPage: 55

ACCELERATING SPACEFLEET Relativity Theory and LogicPage: 56

ACCELERATING SPACEFLEET Relativity Theory and LogicPage: 57

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

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

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

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

UNIFORMLY ACCELERATED OBSERVERS Relativity Theory and LogicPage: 62

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

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:

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:

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

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

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

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

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

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

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

General Relativity Relativity Theory and LogicPage: 73

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

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

 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’

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

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

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:

UNIFORMLY ACCELERATED OBSERVERS Relativity Theory and LogicPage: 80

 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

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

 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.

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

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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