On the equations of motion in Minkowski space (original) (raw)
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Invariants and conservation laws of physical quantities in Minkowski space
2020
It is shown that invariants and relativistically invariant laws of conservation of physical quantities in Minkowski space follow from 4-tensors of the second rank, which are four-dimensional derivatives of 4-vectors, tensor products of 4-vectors and inner products of 4-tensors of the second rank. Two forms of the system of equations of conservation laws for a number of physical quantities in Minkowski space are obtained. The four-dimensional law of conservation of energy-momentum combines the three-dimensional laws of conservation of energy, momentum and angular momentum. The equations of the four-dimensional laws of conservation of physical quantities in explicit or implicit form contain the wave part Based on a system of four-dimensional kinematic conservation equations, the reason for the stability of vortex rings in liquids and gases is explained.
Generalized relativistic velocity addition with spacetime algebra
Arxiv preprint physics/0511247, 2005
Using spacetime algebra-the geometric algebra of spacetime-the general problem of relativistic addition of velocities is addressed. The successive application of noncollinear Lorentz boosts is then studied in Minkowski spacetime. Even spatial vectors, such as the relative velocity of two reference frames, are analyzed in their proper setting-as vectors belonging to a four-dimensional spacetime manifold and not as vectors in ordinary three-dimensional (Euclidean) space. This is a clear and physical result that illustrates the fact that Lorentz boosts do not form a group. The entire derivation is carried through without using Einstein's second postulate on the speed of light, thereby stressing that the framework of special relativity does not depend on electromagnetism.
2021
Zbigniew Oziewicz was a pioneer of the 4D space-time approach to covariant relative velocities. In the 1980s (according to private correspondence) he discovered two types of 4D relative velocities: binary and ternary, along with the rules for adding them. They were first published in conference materials in 2004, and the second time in a peer-reviewed journal in 2007. These physically logical and mathematically precise concepts are so subtle that Oziewicz’s numerous preprints have yet to receive the recognition they deserve. Part I of the work was planned to be a more review, but a thorough review of the little-known results was made in an original synthetic manner with numerous generalizations. The Part I presents the Oziewicz–Bolós (and Bini–Carini– Jantzen) binary relative velocity and the Oziewicz–Dragan ternary relative velocity. The Einstein–Oziewicz and Einstein–Minkowski velocities, which are a four-dimensional generalization of Einstein velocities addition, also have a tern...
Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime
Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension ict, with the unit imaginary producing the correct spacetime distance x 2 {c 2 t 2 , and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary i~ffi ffiffiffiffiffiffi ffi {1 p , with the Clifford bivector i~e 1 e 2 for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis e 1 and e 2 . We find that with this model of planar spacetime, using a twodimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.
Minkowski Geometry and Space-Time Manifold in Relativity
Space-time manifold plays an important role to express the concepts of Relativity properly. Causality and space-time topology make easier the geometrical explanation of Minkowski space-time manifold. The Minkowski metric is the simplest empty spacetime manifold in General Relativity, and is in fact the space-time of the Special Relativity. Hence it is the entrance of the General Relativity and Relativistic Cosmology. No material particle can travel faster than light. So that null space is the boundary of the space-time manifold. Einstein equation plays an important role in Relativity. Some related definitions and related discussions are given before explaining the Minkowski geometry. In this paper an attempt has been taken to elucidate the Minkowski geometry in some details with easier mathematical calculations and diagrams where necessary.
International Journal of Modern Physics A, 2009
We perform a Noether analysis for a description of translation transformations in 4D κ-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. The techniques that some of us had previously developed (hep-th/0607221) for a description of translation transformations based on a 4D differential calculus turn out to be applicable without any modification, and they allow us to show that the basis usually adopted for the 5D calculus does not take into account certain aspects of the structure of time translations in κ-Minkowski. We propose a change of basis for the 5D calculus which leads to a more intuitive description of time translations. * Electronic address: giovanni.amelino-camelia@roma1.infn.it † Electronic address: antonino.marciano@roma1.infn.it ‡ Electronic address: danielepra@libero.it
Generalised Minkowski spacetime
arXiv (Cornell University), 2015
The four dimensional spacetime continuum, as originally conceived by Minkowski, has become the default framework for describing physical laws. Due to its fundamental importance, there have been various attempts to find the origin of this structure from more elementary principles. In this paper, we show how the Minkowski spacetime structure arises naturally from the geometrical properties of three dimensional space when modeled by Clifford geometric algebra of three dimensions Cℓ(ℜ 3). We find that a time-like dimension along with the three spatial dimensions, arise naturally, as well as four additional degrees of freedom that we identify with spin. Within this expanded eight-dimensional arena of spacetime, we find a generalisation of the invariant interval and the Lorentz transformations, with standard results returned as special cases. The value of this geometric approach is shown by the emergence of a fixed speed for light, the laws of special relativity and the form of Maxwell's equations, without recourse to any physical arguments.
International Journal of Modern Physics A, 2009
We perform a Noether analysis for a description of translation transformations in 4D κ-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. The techniques that some of us had previously developed (hep-th/0607221) for a description of translation transformations based on a 4D differential calculus turn out to be applicable without any modification, and they allow us to show that the basis usually adopted for the 5D calculus does not take into account certain aspects of the structure of time translations in κ-Minkowski. We propose a change of basis for the 5D calculus which leads to a more intuitive description of time translations. * Electronic address: giovanni.amelino-camelia@roma1.infn.it † Electronic address: antonino.marciano@roma1.infn.it ‡ Electronic address: danielepra@libero.it
We begin with a lightening review of the relevant concepts of special relativity. The basic postulate of relativity is that the laws of physics are the same in all inertial reference frames. The theory of special relativity tells us how things look to observers who are moving relative to each other. Consider the first observer who sits in an inertial frame S with space-time coordinates (ct, x, y, z) while the second observer sits in an inertial frame S with space-time coordinates (ct , x , y , z). If S is taken to be moving with speed v in the x-direction relative to S then the coordinate systems are related by the Lorentz boost x = γ x − v c ct andct = γ ct − v c x while y = y and z = z with c being the speed of light and γ being the ubiquitous factor γ = 1 1 − v 2 /c 2. A. Four-Vectors To make things easier and useful, the space-time coordinates can be packaged in 4-vectors with indices running from µ = 0 to µ = 3 X µ = (ct, x, y, z), µ = 0, 1, 2, 3. Note that the index is a superscript rather than a subscript. This will bear great importance in the discussion which follows. A general Lorentz transformation is a linear map from X to X of the form (X) µ = Λ µ ν X ν. Here Λ is a 4 × 4 which obeys the matrix equation Λ T ηΛ = η ⇐⇒ Λ ρ µ η ρσ Λ σ ν = η µν where η µν = diag(+1, −1, −1, −1). The solutions to Λ T ηΛ = η ⇐⇒ Λ ρ µ η ρσ Λ σ ν = η µν fall into two classes. The first class is simply rotations. Given a 3×3 rotation matrix R obeying R T R = 1, we can construct a Lorentz transformation Λ by embedding R in the spatial part 1 0 0 0 0 0 R 0 These transformations describe how to relate the coordinates of two observers who are rotated with respect to each other. The second class is that of Lorentz boosts. These are transformations appropriate for observers moving relative to each other. The Lorentz transformation x = γ x − v c ct andct = γ ct − v c x