Unifying relativity and classical dynamics 1, 1,2, Mozafar Karamian,∗ Mahdi Atiq, † Fatemeh Najdat,‡ and Mehdi Golshani § 1Foundations of Physics Group, School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran 2Department of Physics, Sharif University of Technology, Tehran 11365-8639, Iran Relativity and classical dynamics, as defined so far, form distinct parts of classical physics and are formulated based on independent principles. We propose that the formalism of classical dynamics can be considered as the theoretical foundation of the current theory of relativity and may be employed for exploring possibilities beyond the current theory. We show that special-relativistic kinematics, including universality of the speed of massless particles relative to inertial frames, is a consequence of the formalism of classical dynamics, with no assumptions other than spacetime point transformations and Euclidean geometry of space in inertial frames. We discuss that energy- independent velocity is a general concept in classical dynamics, applicable even to massive objects, in appropriate canonical coordinates. The derivation of Lorentz symmetry is inherently local and allows the speed of massless particles (relative to local inertial frames) to vary with space and time globally, which may provide a theoretical foundation for variable speed of light cosmology. We obtain no kinematical scales other than the light-speed, specially no scale of energy or momentum as has been suggested in some quantum gravity investigations. We argue that this is a consequence of spacetime point transformations making the momentum space linear, and a possible second scale must require non-point transformations as a necessary condition, which seems compatible with the notion of relative locality in curved momentum space. PACS numbers: 03.30.+p, 45.20.Jj, 98.80.-k, 04.60.-m Keywords: Lorentz Symmetry, Classical Dynamics, Energy-independent Velocity, Variable Speed of Light Cosmology, Quantum Gravity, Doubly Special Relativity, Relative Locality 1. INTRODUCTION although the kinematics itself has a dynamical character (in the presence of gravity). Classical physics, as has been developed and formulated Therefore, according to the current definition of the so far, consists of two basic parts, relativity and classi- theory, while spacetime interacts with matter, no theo- cal dynamics, which are defined as distinct theoretical retical relationship between properties of spacetime and frameworks and are formulated using independent prin- the classical or quantum nature of the dynamics of matter ciples. is assumed. Accordingly, although the theory deals with Relativity, as the more fundamental theory, is defined motions of particles and reference frames in the classical as the theory of properties and dynamics of spacetime, sense and using spacetime geodesics whose equations are distinct from theories that describe properties and dy- mathematically equivalent to some Lagrange equations, namics of matter in spacetime, and spacetime is defined the pseudo-Riemannian structure of the underlying man- in the theory as an arena on which in principle any theory ifold is not theoretically related to Lagrangian formalism. of matter, classical or quantum, can be defined. Apart It is assumed as a characteristic or defining property of from the Einstein field equations describing the dynam- gravitational field. ics of spacetime, the theory is based on the principles of This theoretical independence between relativity and equivalence of inertial frames, universality of the speed classical dynamics, however helpful in developing differ- of massless particles relative to inertial frames, inertia- ent dynamical frameworks, including quantum ones, does gravity equivalence, and general covariance. not seem theoretically satisfactory in some aspects. Rela- tivity in its current form is a classical (non-quantum) the- arXiv:1807.10607v2 [physics.gen-ph] 9 Aug 2018 Classical dynamics, on the other hand, is considered as a restricted (non-quantum) framework of dynamics, ory in every aspects. Inertial frames, which have a cen- which by itself can even be non-relativistic, and is for- tral role in the theory, are indeed rest frames of objects mulated based on the action principle. It, therefore, is which move according to the laws of classical dynamics, considered as one of the possible dynamical frameworks with definite trajectory, and time, length, velocity, mass, that can be developed on the relativistic spacetime. energy, momentum and so on in those frames are classical In other words, relativity defines the fundamental kine- concepts developed in terms of the classical behavior of matics upon which theories of dynamics are developed, objects. The theory fundamentally stands upon concepts taken from classical dynamics, and the relativistic kine- matics which is used as a foundation even in quantum field theory gets its meaning originally from the classical ∗ [email protected] † mma [email protected] behavior of objects. ‡ fa [email protected] But, apart from the classical nature of the theory, the § [email protected] mathematical formalism of the theory has essential fea- 2 tures in common with the formalism of classical dynam- dently explained as incompatibility of absolute time with ics, which should be considered as indicating the require- the intrinsic invariance of action and even with the Eu- ment of a theoretical unification between them. As men- clidean geometry of space in inertial frames. The impos- tioned above, geodesic equations can be expressed in the sibility of a well-defined inertial rest frame for massless form of Lagrange equations, and the action principle be- particles is also concluded, very straightforwardly, from comes equivalent to the condition of shortest path (min- the invariance of action. imal proper time) in this case. Even without presum- We present a detailed derivation of Lorentz symme- ing anything about Einsteinian relativity with its spe- try and universality of the speed of massless particles as cific principles mentioned above, Lagrange equations are consequences of the invariance of action and Euclidean intrinsically invariant under arbitrary changes of coordi- geometry of space. nate system in the configuration space, which is practi- The equivalence between inertial frames is not required cally the requirement of general covariance in relativity. to be assumed as an independent or fundamental princi- More importantly, any Lagrangian which is homogeneous ple besides the Lagrangian formalism. Instead, it can be of degree one in its velocity arguments provides a mea- considered as a systematic consequence of the formalism. sure of length and a metric tensor on the configuration The connection between inertiality of the frame and the space [1]. This means that Lagrangian formalism, by Euclidean geometry of space (more accurately speaking, its intrinsic properties, can be employed for defining a equivalence between them) is also explained using La- Finsler or Riemann metric on the configuration space, grangian formalism.1 and therefore the concept of metric can be introduced As a supporting theoretical evidence for the fundamen- very naturally into the theory starting from Lagrangian tal role of classical dynamics in the foundations of relativ- formalism. Such intrinsic properties and capabilities of ity, we show that energy-independent velocity, which is at- the Lagrangian formalism can be considered as theoret- tributed only to massless particles in the Einsteinian for- ical evidences for a unified point of view regarding the mulation of relativity, is actually a more general concept current theory of relativity and classical dynamics. In in classical dynamics for the generalized meaning of the other words, the mathematical formalism of classical dy- term velocity, and is applicable to massive and massless namics is so powerful that one may consider this formal- systems, in appropriate canonical coordinates. An exam- ism not just as a framework of dynamics (a restricted ple of such a velocity in a massive system is the energy- one), but even as the foundation of the theory of relativ- independent angular velocity of harmonic oscillator. We ity, and specifically and importantly, as the theoretical show that this similarity with the energy-independent foundation of relativistic kinematics. speed of light is not accidental, but is a consequence of In the following, we treat classical dynamics not just the vanishing of Lagrangian, and appears in any system as a theoretical framework governing classical motions of in which the Lagrangian vanishes during the evolution of particles and reference frames, but also as the theoretical the system. The concept of energy-independent velocity origin of relativistic kinematics. Our basic intention is to is not canonically-invariant in general, as is observed from demonstrate that one can start completely and funda- the example of harmonic oscillator which has an energy- mentally from the intrinsic properties of Lagrangian for- dependent linear velocity, but an energy-independent an- malism (without making any assumption about the func- gular velocity. The linear energy-independent velocity tionality of the Lagrangian) instead of the above men- of massless particles in the ordinary space is, however, tioned principles of Einsteinian relativity and still end up observer-independent, as far as the transformations be- with relativistic kinematics, and those principles should tween observers are