Einstein and philosophy: A new definition of 'simultaneous'

Modification of special relativity and local structures of gravity-free space and time

Training deep reinforcement learning (RL) agents necessitates overcoming the highly unstable nonconvex stochastic optimization inherent in the trial-and-error mechanism. To tackle this challenge, we propose a physics-inspired optimization algorithm called relativistic adaptive gradient descent (RAD), which enhances long-term training stability. By conceptualizing neural network (NN) training as the evolution of a conformal Hamiltonian system, we present a universal framework for transferring long-term stability from conformal symplectic integrators to iterative NN updating rules, where the choice of kinetic energy governs the dynamical properties of resulting optimization algorithms. By utilizing relativistic kinetic energy, RAD incorporates principles from special relativity and limits parameter updates below a finite speed, effectively mitigating abnormal gradient influences. In addition, RAD models NN optimization as the evolution of a multiparticle system where each trainable parameter acts as an independent particle with an individual adaptive learning rate. We prove RAD's sublinear convergence under general nonconvex settings, where smaller gradient variance and larger batch sizes contribute to tighter convergence. Notably, RAD degrades to the well-known adaptive moment estimation (ADAM) algorithm when its speed coefficient is chosen as one and symplectic factor as a small positive value. Experimental results show RAD outperforming nine baseline optimizers with five RL algorithms across twelve environments, including standard benchmarks and challenging scenarios. Notably, RAD achieves up to a 155.1% performance improvement over ADAM in Atari games, showcasing its efficacy in stabilizing and accelerating RL training.

We have developed an innovative ray-tracing simulation algorithm to describe Relativistic Effects in SpaceTime ("REST"). Our algorithm, called REST-frame, models light rays that have assumed infinite speed in conventional ray-tracing to have a finite speed in spacetime, and uses the non-Newtonian Lorentz Transformation to relate measurements of a single event in different inertial coordinate systems (inertial frames). Our earlier work [5][6][7] explored the power of REST-frame as an experimentation tool to study the rich visual properties in natural world modeled by Special Relativity. Non-intuitive images of the anisotropic deformation ("warping") of space, the intensity concentration/spreading of light sources in spacetime, and the relativistic Doppler shift were visualized from our simulations.

This book has some good points and was clearly a labour of love for the author. Unfortunately, I am not able to recommend it to readers of Physics Education. The author sets out to discuss redshift, with particular reference to its importance in astronomy and cosmology. The main text is fairly descriptive, with `mathematical notes' giving a more quantitative treatment. There are separate chapters dealing with the nature of light, the Doppler effect, gravitational redshift, cosmological redshift and unconventional interpretations of redshift. Quite rightly, the author is very firm about the fact that cosmological redshift is not a Doppler shift. So what are the problems? First, and least important, is the production and presentation of the book. The text is set in a large sans-serif typeface which I found uninviting to read, and the diagrams are highly stylized, often with broad black lines and sometimes with sharp corners where there should be smooth curves, e.g. in graphs of redshift as a function of velocity. The book would benefit from the attentions of a professional editor - there are numerous minor, but irritating, misspellings, inconsistent symbols and proof-reading errors (for example, `Niel' Bohr and `Kirchoff' for Niels Bohr and Kirchhoff; a θ mysteriously becomes a J in mid-derivation; some figure labels are incomplete or misplaced; a vital `exp' is missing from a formula for a Gaussian line profile; megaparsec is abbreviated throughout as mpc (milliparsec) rather than Mpc). Far more serious, though, are the problems with the science. Despite the author's laudable aim of presenting the `laws of light' and explaining the various distinct ways in which redshifts might arise, the treatment given here has many shortcomings, exemplified by one particularly serious error that occurs in the first chapter. Following some discussion of the origins of special relativity, the author provides an example using regular pulses of light being sent from a moving to a stationary car in which he shows that the time interval between successive pulses being received at the stationary car is longer than the interval between successive pulses being sent from the receding car. However, this has nothing to do with relativity. It is merely a consequence of light travelling at a finite speed and the fact that the observers have measured the time intervals between two different pairs of events. The author seems to have completely missed the crucial relativistic effect, namely that two observers in relative motion will disagree about the time interval between the same two events (e.g. the sending of two light pulses) and that the disagreement depends only on the relative speed and not on whether the two are approaching or receding. The author extends his initial derivation to arrive at what he quite correctly describes as the non-relativistic Doppler effect, but then goes on to reiterate that it illustrates special relativity. Similiar confusions are to be found elsewhere in the book. By no means everything is incorrect, but clarity and physical insight are sadly lacking. At the beginning, I did refer to `good points', and in fairness to the author I should mention some of them. In the first chapter, there are some good descriptions of early attempts to measure the speed of light (I liked Galileo's, which involved two people some miles apart using buckets to cover and uncover candles). There is a good account of the discovery and interpretation of `nebulae' which led to Hubble's model of the universe and the birth of modern cosmology. The transverse Doppler shift is included. There are some nice diagrams that illustrate the distinction between cosmological and Doppler shifts. The final chapter (on 'unconventional interpretations') makes interesting reading, though it does mix the respectable with the unorthodox. The overall plan of the book is well thought out, and (despite needing an editor) it is written in a readable style. However, these redeeming features do not alter the fact that anyone using this book to study `the laws of light' is likely to end up as confused as the author seems to be.

BEI~GSTR(iM (1) has demonstrated that the magnetic force in a moving system is a Coriolis force resulting from the Thomas rotation caused by the electric force. TOLMAN (3), many years ago, considered the apparent force between two charged particles in vacuo as measured in an inertial frame S in which both particles appear to move at general velocities. In Tholman's paper it is asserted that in the frame S', in which the particle of charge q'l appears to be at rest and the particle of charge q~ has an instantaneous position vector r ' and velocity u ' , the force F ' on q~ due to q~ is electrostatic. The transforms of special relativity are applied to find the force F on q2, due to ql, in the frame S in which ql appears to have velocity v. These two papers, separated by a span of sixty years, represent the closest approach to date to an explanation of the magnetic field. In the earlier paper (2), allowance was not made for the effect of retardation, i.e. for the fact that, in S, the force field due to ql (moving) travels at a finite speed c towards q2. If allowance is made for this (~ retardation effect ,>, the result includes terms of order v/c, (v/c) 2 . . . . independent of u. To avoid this complication it is only necessary to regard ql as the charge instantaneously in a small volume ~V~, fixed in S, containing charge density ~ (q~ = Q~ 3V~) and with the charge flowing through it at a velocity v, both ~1 and v remaining sensibly constant during a short period of order r/c and constituting a current density J~. Then the charges in 8V~ are continually being replaced by identical charges following each other in succession. The result of this is that retardation complications may be ignored. With the ((succession effect ~> in force, for both particles for the sake of symmetry, Tolman's (3) result can be adjusted to give, in MKS units,

The appearance of fast moving objects can be calculated according to the Theory of Special Relativity. In addition to the Lorentz contraction the effects of finite light speed and aberration play an important role. These phenomena were first discovered and described correctly in 1959 by Penrose and Terrel. Concerning the visualization of the phenomena there already exist systems with relativistic ray tracing and polygon rendering. Investigating these approaches in detail we found a reformulation of the problem which allows the treatment of acceleration in real-time. Therefore, user interaction could be integrated and a virtual reality for special relativity was possible.

We present an analytic study of light propagation in a simple Michelson-Morley interferometer as observed by inertial observers to understand any connections among classical physics, optical physics, and special relativity. To that end, we develop coordinate transformations of wave propagation as observed in inertial frames (i.e., non-accelerating reference frames). We find that relativistic and other effects appear naturally as a result of finite light speed. Such effects include wavefront tilt relative to the normal of energy flow with the wavefront normal tilting relative to the Poynting vector.

A phenomenological, general relativistic theory of dissipative elastic solids whose equations form a hyperbolic system is proposed. The non-stationary transport equations for dissipative fluxes containing new cross-effect terms, as required by compatibility with irreversible thermodynamics, have been adopted. As opposed to some conventional theories which are parabolic and predict instantaneous propagation of wavefronts, the theory formulated, consisting of 14 partial differential equations (in the case of special relativity), of total order 17, is hyperbolic and predicts, for all existing propagation modes, finite front speeds. The complete system of special relativistic propagation modes of an elastic solid is determined from the linearised equations. There are four mutually distinct non-trivial propagation modes, two for longitudinal waves and two for transverse waves. If the rigidity modulus decreases to zero one obtains as a special case the normal modes for fluid according to the theory of Muller (1972) and Israel (1976). Weber's equation is also briefly discussed.

Context. Understanding dark energy and measuring the topology of the Universe are two of the biggest open questions in physical cosmology. It was previously shown that multiple connectedness, via the twin paradox of special relativity, provides a novel physical justification for an assumption of the standard FLRW model: it implies a favoured space-time splitting (comoving coordinates). Aims. Could cosmic topology also imply dark energy? Methods. We use a weak field (Newtonian) approximation of gravity and consider the gravitational effect from distant, multiple copies of a large, collapsed (virialised) object today (i.e. a massive galaxy cluster), taking into account the finite propagation speed of gravity, in a flat, multiply connected universe, and assume that due to a prior epoch of fast expansion (e.g. inflation), the gravitational effect of the distant copies is felt locally, from beyond the naively calculated horizon. Results. We find that for a universe with a T' x R 2 spatial section, the residual Newtonian gravitational force (to first order) provides an anisotropic effect that repels test particles from the cluster in the compact direction, in a way algebraically similar to that of dark energy. For a typical test object at comoving distance X from the nearest dense nodes of the cosmic web of density perturbations, the pressure-to-density ratio w of the equation of state in an FLRW universe, is ω ∼ -(X/L) 3 , where L is the size of the fundamental domain, i.e. of the Universe. Clearly, |ω| « 1. For a T 3 spatial section of exactly equal fundamental lengths, the effect cancels to zero. For a T3 spatial section of unequal fundamental lengths, the acceleration effect is anisotropic in the sense that it will tend to equalise the three fundamental lengths. Conclusions. Provided that at least a modest amount of inflation occurred in the early Universe, and given some other conditions, multiple connectedness does generate an effect similar to that of dark energy, but the amplitude of the effect at the present epoch is too small to explain the observed dark energy density and its anisotropy makes it an unrealistic candidate for the observed dark energy.

Information transmittal, time uncertainty and special relativity

In this paper, we are interested in a general equation that has finite speed of propagation compatible with Einstein's theory of special relativity. This equation without external force fields has been derived recently by means of optimal transportation theory. We first provide an argument to incorporate the external force fields. Then, we are concerned with comparison and maximum principles for this equation. We consider both stationary and evolutionary problems. We show that the former satisfies a comparison principle and a strong maximum principle while the latter fulfils weaker ones. The key technique is a transformation that matches well with the gradient flow structure of the equation.

We have developed an innovative ray-tracing algorithm to describe Relativistic Effects in SpaceTime (“REST”). Our algorithm, called REST-frame, simulates a generalized world in Spacetime and gives the fine details implicit in the Special Theory of Relativity that have not yet been made apparent. These novel simulations disclose the non-intuitive realm of Special Relativity and, by visualization means, advance beyond the findings of past revelations concerning relativistic effects. Through the application of state-of-the-art computation technology and simulation techniques to earlier quests in Physics, REST-frame offers a flexible visualization tool to study some of the most exciting aspects of the natural world; particularly, the rich visual properties associated with the finite speed of light.

3 - Surprising Properties of Our Universe

Special Relativity: The Newton - Maxwell Conflict Einstein's Light and Relativity Postulates How Time Slows and Space Shrinks with Motion Simultaneity The Equivalence of Mass and Energy Per E=mc2 The Unification of Space and Time in the Spacetime Interval The Twins Paradox General Relativity: The Equivalence Principle Gravitational Time Dilation Ehrenfest's Paradox The Warping of Time and Space Time Travel Local and Global Gravity Spacetime Curvature The Einstein Equation Black Holes Gravitational Lensing Gravity Waves Frame Dragging Wormholes The Expansion of the Universe The Big Bang Cosmic Inflation Dark Matter Dark Energy The Future of the Universe.

In the past few years, the perturbation due to the finite speed of light was among the most inconsistent in corner-cube absolute gravimeters. For the relativistic treatment of the perturbation based on Lorentz transformation, the relation between Doppler shift in special relativity and time delay in classical domain is easily misunderstood, leading to spurious conclusions. To avoid these issues, we apply a ‘relativistic geometrical method’ based on the motion of photons to calculate the frequency shift in corner-cube absolute gravimeters, where the misunderstood relation has been corrected. Additionally, we find that the modern corner-cube absolute gravimeters cannot sense effects of the special relativity.