### Time ‘betwins’

Thierry Grandou and Jacques L. RUBIN

Institut du Non Lineaire de Nice

U.M.R. 6618, C.N.R.S. Universite de Nice Sophia-Antipolis

**Abstract**

Discussions on the Langevin Twins ‘paradox’ are most often based on a “triangular ”diagram which outlines the twins spacetime travels. It won’t be our way,avoiding what we think to be a problem at the basis of numerous controversies, i.e., the “time” dynamics at the apex of the triangle at which the space traveling twin comes backward to the on-earth twin. Sometimes,in order to smooth the apex of the “triangle ”diagram,acceleration is considered and arguments are then based on what we call a “smooth triangle ”diagram. We do not consider such “smooth ”triangular approaches,but really a full non-triangular one,smoothed or not. Our approach relies on a fundamentally different Equivalence Principle, namely the so-called “Punctual Equivalence Principle” [GB01 ], from which we think that the very conformal aspect of our framework proceeds. We focus on the following crucial questions: 1.How to physically compare distances on different spacetime paths ? 2.Is acceleration relevant to the Twins Paradox ? In fact,we suggest that spacetime physical paths can ’t be conceived in special relativity and that this point could be at the origin of the main sources of misconceptions.Hence,our proposal concerning the Twins Paradox deals with a di .erent notion of a spacetime path. We present a resolution of this paradox in the framework of the so-called “scale gravity”. This resolution hinges on a clear determinism of the Twins proper times,in some de .nite situations,and a fundamental under- determinism in some other particular ones,the physical discrimination of which being achieved out of a precise mathematical description of conformal geometry. As a consequence,the concept of spacetime as a given geometrical man-ifold,providing it with the status of a genuine “telos”,can ’t be maintained any longer,in order to de .ne,in an empirical and realistic way,time and duration. Moreover,we .nd that the time discrepancy between the twins,could somehow be at the root expression of the second fundamental law of thermo-dynamics.Other conceptions and consequences will be presented,in partic-ular related to the 3D/4D controversy and the question of pseudo-Euclidean versus Euclidean geometry.
References

[GB01] Ghins, M., Budden,T.: “The Equivalence Principle”. Stud. Hist. Phil. Mod. Phys. 32 , pp. 33–51 (2001).