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Register a new userIntroduction to Special Relativity
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Eliana Gianfelice-Wendt
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The goal of this lecture is to introduce the student to the theory of Special Relativity. Not to overload the content with mathematics, the author will stick to the simplest cases; in particular only reference frames using Cartesian coordinates and translating along the common x-axis as in Fig. 1 will be used. The general expressions will be quoted or may be found in the cited literature.
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We present a comprehensive introduction to the kinematics of special relativity based on Minkowski diagrams and provide a graphical alternative to each and every topic covered in a standard introductory sequence. Compared to existing literature on the subject, our introduction of Minkowski diagrams follows a more structured and contemporary approach. This work also demonstrates new ways in which Minkowski diagrams can be used and draws several new insights from the diagrams constructed. In this regard, the sections that stand out are: 1. the derivation of Lorentz transformations (section IIIA through IIID), 2. the discussion of spacetime (section III F), 3. the derivation of velocity addition rules (section IV C), and 4. the discussion of relativistic paradoxes (section V). Throughout the development, special attention has been placed on the needs and strengths of current undergraduate audiences.
This Letter, i.e. for the first time, proves that a general invariant velocity is originated from the principle of special relativity, namely, discovers the origin of the general invariant velocity, and when the general invariant velocity is taken as the invariant light velocity in current theories, we get the corresponding special theory of relativity. Further, this Letter deduces triple special theories of relativity in cosmology, and cancels the invariant presumption of light velocity, it is proved that there exists a general constant velocity K determined by the experiments in cosmology, for K > 0, = 0 and < 0, they correspond to three kinds of possible relativistic theories in which the special theory of relativity is naturally contained for the special case of K > 0, and this Letter gives a prediction that, for K < 0, there is another likely case satisfying the principle of special relativity for some special physical systems in cosmology, in which the relativistic effects observed would be that the moving body would be lengthened, moving clock would be quickened. And the point of K = 0 is a bifurcation point, through which it gives out three types of possible universes in the cosmology (or multiverse). When a kind of matter with the maximally invariant velocity that may be superluminal or equal to light velocity is determined by experiments, then the invariant velocity can be taken as one of the general invariant velocity achieved in this Letter, then all results of current physical theories are consistent by utilizing this Letters theory.
We introduce the special issue on the Statistical Mechanics of Climate published on the Journal of Statistical Physics by presenting an informal discussion of some theoretical aspects of climate dynamics that make it a topic of great interest for mathematicians and theoretical physicists. In particular, we briefly discuss its nonequilibrium and multiscale properties, the relationship between natural climate variability and climate change, the different regimes of climate response to perturbations, and critical transitions.
This pair of CAS lectures gives an introduction for accelerator physics students to the framework and terminology of machine learning (ML). We start by introducing the language of ML through a simple example of linear regression, including a probabilistic perspective to introduce the concepts of maximum likelihood estimation (MLE) and maximum a priori (MAP) estimation. We then apply the concepts to examples of neural networks and logistic regression. Next we introduce non-parametric models and the kernel method and give a brief introduction to two other machine learning paradigms, unsupervised and reinforcement learning. Finally we close with example applications of ML at a free-electron laser.
It is shown how a Doubly-Special Relativity model can emerge from a quantum cellular automaton description of the evolution of countably many interacting quantum systems. We consider a one-dimensional automaton that spawns the Dirac evolution in the relativistic limit of small wave-vectors and masses (in Planck units). The assumption of invariance of dispersion relations for boosted observers leads to a non-linear representation of the Lorentz group on the $(omega,k)$ space, with an additional invariant given by the wave-vector $k=pi /2$. The space-time reconstructed from the $(omega,k)$ space is intrinsically quantum, and exhibits the phenomenon of relative locality.
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