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Numerical solution of ODE problems
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
Notice that, in our reasoning, we have assumed that the observers agree on the same way to describe motion based on the space coordinate, the velocity, and the time, and other experiments may include the mass and forces of different type. This agreement requires to formulate physical laws using those quantities that can be unambiguously correlated through transformation (position, velocity, force, … ) or are invariant (mass, time, …), which is the general principle of covariance. In most cases, this principle, as first formulated by Galileo Galilei, results in equations that are form-invariant under the given transformation.
Application of Tensors in General Theory of Relativity
Published in Bhaben Chandra Kalita, Tensor Calculus and Applications, 2019
For the “principle of covariance,”‡ an essence of general relativity, all natural laws must be expressed in tensor forms (Covariant) for their validity in all coordinate systems including non-inertial frames of curved space required for general theory of relativity. Hence, Equation (7.6.1) needs to be expressed completely in tensor form.
Maxwell’s theory of electromagnetism
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
The essence of special relativity is that the mathematical forms of Maxwell’s equations are identical in all inertial reference frames: frames moving with uniform velocities relative to the laboratory frame of reference in which we perform our measurements. This form invariance of Maxwell’s equations is a specific example of the general physical principle of covariance. In the laboratory frame we write the differential equations of Maxwell’s theory as ∇×E(r,t)=−∂B(r,t)∂t,∇×H(r,t)=J(r,t)+∂D(r,t)∂t,∇⋅D(r,t)=ρ(r,t),∇⋅B(r,t)=0,∇⋅J(r,t)=−∂ρ(r,t)∂t.
Travelling light
Published in Journal of Modern Optics, 2021
The following argument in the book Relativity in Our Time: From Physics to Human Relations by Mendel Sachs resembles Einstein’s argument: The basic idea of the theory of relativity that convinced Einstein of its extreme simplicity is his principle of covariance, also referred to as the principle of relativity.This principle asserts that the laws of nature must have expressions independent of the frame of reference in which they are represented—from any particular observer’s view. This is equivalent to saying that the laws of nature are totally objective.We see, then, that the theory of relativity is based on a premise that is a law about laws, rather than a law that deals directly with physical phenomena. The idea about the objectivity of the laws of nature is, however, not really that new! For how could a law be a law, by definition of the word ‘law’, if it were not totally objective? [23]If the principle of relativity were equivalent to saying that the laws of nature are totally objective, that would be a powerful argument for it. It is hard to be against objectivity. However, this is a false equivalence, for two reasons. First, a law of nature, or any statement about nature, can be totally objective even if it has nothing to do with inertial coordinate systems. Objectivity is no more dependent on using inertial coordinate systems than on using Morse code. Second, Sachs confuses objectivity with uniformity. A coordinate-system-involving statement that is true in all inertial coordinate systems would have a certain kind of uniformity. It might also be objective, but in that case, it would not be objective because it was uniform. A statement that something is true in some inertial coordinate systems but not in all of them can also be objective. The principle of relativity imposes a uniformity condition that is compatible with objectivity but incidental to it.