1. 1 Emergence of Relativistic Space-Time in a Computational Universe Michael Nicolaidis TIMA Laboratory

2. 2 Outline - Motivation: paradoxes and fracture of physics - Special relativity versus computation rules (interaction laws) - Space-time structure in special relativity - Implications of the relativistic space-time structure - Treatment of Special Relativity: the approach - Condition for the emergence of relativistic space time - Illustration examples - Proof Generalization - Perspectives - Motivation: paradoxes and fracture of physics - Special relativity versus computation rules (interaction laws) - Space-time structure in special relativity - Implications of the relativistic space-time structure - Treatment of Special Relativity: the approach - Condition for the emergence of relativistic space time - Illustration examples - Proof Generalization - Perspectives

3. 3 Paradoxes, Metaphysics, Fracture of Physics A dilemma: - In a 3D world it exists a division of events into past, present, and future. Which is not consistent with relativity. - In the alternative view – a 4D space-time world – there is (i) no objective time flow (all events of spacetime are equally existent), (ii) absolute determinism (at the macro scale), and (iii) no free will. These consequences make most physicists and philosophers agree that this world view is undoubtedly wrong. But so far, no one has succeeded in formulating a view that avoids the above dilemma and is compatible with relativity. From the foundation text of International Conference on the Nature and Ontology of Space-time

4. 4 Paradoxes, Metaphysics, Fracture of Physics Metaphysics: structure of space-time  length contraction of objects (by which metaphysical mean space constraints objects to contract). Idem for time dilatation effects on processes. Fracture of physics: since space, time and elementary objects are primary ingredients of the universe, two theories are necessary to describe the world (structure of space-time, behavior of particles)  Is there a vision which eliminates these paradoxes and unifies the physics ?

5. 5 Space-time versus computation (interaction) laws Space and time are engendered by the evolution of the states of the elementary particles composing the universe. The rules of computation (interaction laws) determine the evolution of these states. The interaction laws should determine the structure of space-time. But how ? Unifying approach since a complete theory of interactions could describe everything (relativity becomes a by-product). Space and time are engendered by the evolution of the states of the elementary particles composing the universe. The rules of computation (interaction laws) determine the evolution of these states.  The interaction laws should determine the structure of space-time. But how ?  Unifying approach since a complete theory of interactions could describe everything (relativity becomes a by-product).

6. 6 Space-time structure in special relativity Space-time structure defined by Lorentz transformations S0S0 S : u = v//x x x’ t’ t

7. 7 Implications of the relativistic space- time structure Implications : S0S0 S Desynchronization: Desynchronization: t2 - t1 =  x  v v/c 2  t’ xx tt  x’ xx t’ t2 t1 Length contraction: Length contraction:  x =  x’/  v Time dilatation: Time dilatation:  t =  v  t’ t’ t

8. 8 Treatment of Special Relativity: the approach Space and time are engendered by the evolution of the states of elementary particles. Space and time are engendered by the evolution of the states of elementary particles. Their structure (e.g. length contraction, time dilation), reflects the impact of the laws governing the evolution of particles on the dimensions of the objects, their distances, their movement, and the pace of their evolution. Their structure (e.g. length contraction, time dilation), reflects the impact of the laws governing the evolution of particles on the dimensions of the objects, their distances, their movement, and the pace of their evolution.  The computation laws governing the evolution of particles determinestructure.  The computation laws governing the evolution of particles determine the space-time structure.

9. 9 Treatment of Special Relativity: the approach Goal: Determine the conditions which imply that the space-time engendered by the computation of the states of a set of particles (a computational universe) is in conformity with special relativity. Idea 1: The intensity of the interactions (computation rules)change with the speed of the particles Idea 1: The intensity of the interactions (computation rules) change with the speed of the particles  modification of the distances of equilibrium of the particles composing an object (length contraction) and of the pace of evolution of a set of particles engendering a process (time dilation).  it is not a veritable structure of space-time that forces the objects to modify their dimensions and the processes to modify their pace of evolution (by which means?)  it is not a veritable structure of space-time that forces the objects to modify their dimensions and the processes to modify their pace of evolution (by which means?) Idea 2:The accelerations induced by the particle interactions must be in conformity with special relativity. Idea 2: The accelerations induced by the particle interactions must be in conformity with special relativity.  Condition which engenders a space-time conform to Lorentz transformations. u 1 = 0 u 2 = 0 u 1 = vu 2 = v

10. 10 Treatment of Special Relativity: "privileged" reference frame Start with apparent relativistic “no-sense”: A particle determines only one value for its position variable and for its speed variable A particle determines only one value for its position variable and for its speed variable  the variables of position computed by the meta-objects define by default a (privileged) reference frame: S 0 But relativity does not admit privileged reference frames. But relativity does not admit privileged reference frames.  Admit such a reference frame in order to develop the approach.  Then, show that for any observer being part of the universe, S 0 is not distinguishable from any other inertial frame.

11. 11 S0S0 r=(r x, r y, r z ) u 2 =(u x2, u y2, u z2 ) a 2 =(a x2, a y2, a z2 ) u 1 =(u x1, u y1, u z1 ) Special Relativity : the condition (Relativistic Constraint of Accelerations) Introduce an inertial frame: the values of position variables of particles determined by the computation are considered as positions in an inertial frame S 0 The accelerations induced by the interactions* must change when the speed of particles changes. RCA describes the impact of speed change on the accelerations *interactions propagated at speed c are considered: interaction takes its source at a past position of particle 1 (  t = c.r)

12. 12 Special Relativity : the condition (Relativistic Constraint of Accelerations) S0S0 r’=(r x /  v, r y, r z )

13. 13 Special Relativity : the condition (RCA) r=(r x /  v, r y, r z )

14. 14 Emergence of relativistic space-time Show that RCA implies: 1. In S 0, the dimensions of objects, the distances between objects and the pace of evolution of processes that are in movement, undergo the Lorentz contraction of lengths time dilation. 2. The relations between the measurements of the dimensions of objects, of the distances between objects and of the paces of evolution of processes, carried out in various inertial reference frames, conform the Lorentz transformations.

15. 15 S0S0 u 1 = 0 Length Contraction Example u 2 = 0 u 1 = v//xu 2 = v//x  

16. 16 S0S0 u 1 = 0 u 2 = 0 u 1 = v//x u 2 = v//x u 2 = 0 u 2 = v//x  For the above object and laws, length contraction conforms special relativity on all directions Length Contraction Example

17. 17 Frame S u = v//x Frame S 0 u = 0 Motionless clock: P1 interacts with P2 at rest in S 0 (law g 1 g 2 /m 2 l 0 2 ) it carries out a circular trajectory of ray l 0 around P2, as checked by the length reference l 0. Moving clock : P1 under the same interaction performs a trajectory around particle P2 which is in rest in S (S: u=v). The clock is the same, since the new trajectory is also circular and of ray l 0 for the observer of S, as checked by his length reference l 0. Time Dilatation: we find T’ = T  v. Time Dilatation Example P1 P2

18. 18 Synchronization of distant clocks in S From M launchtowards H1 and H2 two objects having speed = c (photons) From M launch towards H1 and H2 two objects having speed = c (photons)  clocks synchronized at S have a desynchronization inS 0 equal to  t =  v vl/c 2.  clocks synchronized at S have a desynchronization in S 0 equal to  t =  v vl/c 2. Lengths at rest inS  contraction in S 0 Lengths at rest in S  contraction in S 0 Time intervals measured in the same position of S  dilatation in S 0 Time intervals measured in the same position of S  dilatation in S 0 Relation of desynchronization Relation of desynchronization  Lorentz Transformations S : u = v//x M l H1H2

19. 19 Synchronization using moving clocks What about if we don’t know in advance particles with invariant speed?  Launch towards H1 and H2 two objects having equal speeds in S. But how to measure their speeds in S without synchronized clocks? But how to measure their speeds in S without synchronized clocks? Idea: launch two identical clocks towards H1 and H2.  Check if they indicate identical times when they cross equidistantly distributed positions.  Same desynchronization relation  Important equations to seek laws that could produce non-Lorentzian “relativistic” space-time H1H2 lll lllll

20. 20 Generalization Illustrations for simple objects and processes and simple interaction laws are given. But we need a general proof.  Prove length contraction for any object (any # and type of particles, in any repartition in space, at rest or moving around their positions of equilibrium...)  Prove time dilation for any process.  General proof for all laws obeying the RCA, in order to cover all possible interactions (existing or imagined). Such demonstration is impossible by means of analytical expression of the evolution of all possible objects and processes (complexity + inapplicable if we don’t know the exact expression of interaction laws). Such demonstration is impossible by means of analytical expression of the evolution of all possible objects and processes (complexity + inapplicable if we don’t know the exact expression of interaction laws).

21. 21 Generalization Approach Approach  We consider that the observers of S and S 0 decide arbitrarily to: synchronize their clocks in a way that verify the relation  t =  v vl/c 2. synchronize their clocks in a way that verify the relation  t =  v vl/c 2. use as length references objects which verify the relation of length contraction use as length references objects which verify the relation of length contraction use as time references, processes which verify the relation of time dilation use as time references, processes which verify the relation of time dilation  We show that all measurements conform Lorentz transformations Arbitrary choices of measurement means imply Lorentz transformations !? Arbitrary choices of measurement means imply Lorentz transformations !?

22. 22 Generalization  These arbitrary choices will not engender by miracle a relativistic space-time. The results of measurements will not be coherent if the laws of interactions are arbitrary: the objects selected as length references n S and S0 will not be identical. Idem for processes used as time reference the objects selected as length references n S and S0 will not be identical. Idem for processes used as time reference measurements of the same object or processes carried out in S and S 0 will give different results measurements of the same object or processes carried out in S and S 0 will give different results  But we now dispose in S and S 0 a tool (Lorentz transformations) which we will use to show that these choices are coherent.

23. 23 Generalization  We show that if the laws of interactions verify RCA, then, these arbitrary choices imply identical results when we measure the same objects and processes in S and S 0.  the results of measurements are coherent.  the arbitrary choice of length and time references becomes coherent: use of the same object and of the same process as length reference and time reference in S and in S 0.  space-time verifies the transformations of Lorentz.  all laws of interaction which verify RCA engender a space-time conform to special relativity.

24. 24 Generalization  We show that: RCA defines the largest family of laws that engender a space-time conform to special relativity. RCA defines the largest family of laws that engender a space-time conform to special relativity. All laws compatible with relativity verify RCA. All laws compatible with relativity verify RCA.  For stochastic laws (quantum laws of interactions): RCA applies to the average values on the macroscopic scale. RCA applies to the average values on the macroscopic scale. Develop stochastic form of RCA on the quantum scale. Develop stochastic form of RCA on the quantum scale.  Fusion of special relativity with the quantum laws of interactions that are compatible with relativity (electromagnetic, weak and strong).  Idem for the gravitation, if current efforts lead in quantum law of gravitation.

25. 25 Perspectives  Pieces of physics already treated under the vision of Computational Universes (special relativity, quantum systems)  Further treatments needed for: General Relativity General Relativity Quantum theories of interactions Quantum theories of interactions Gravitational interaction Gravitational interaction  Discreet space and time.  Meaning and emergence of conscience in Computational Universes …….