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A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations: [3] = (,,,) = + + + = + = where A α is the magnitude component and E α is the basis vector component; note that both are necessary to make a vector, and that when A α is seen alone, it refers strictly to the components of the vector.
In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions x μ (τ), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates.
The 4-wavevector is the 4-gradient of the negative phase (or the negative 4-gradient of the phase) of a wave in Minkowski Space: [6]: 387 = = (,) = [] = [] This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave ): K ⋅ X = ω t − k → ⋅ x → = − Φ {\displaystyle \mathbf {K} \cdot ...
The 4-divergence of this current is: = + where ∂ μ is the 4-gradient and μ is an index labeling the spacetime dimension. Then the continuity equation is: ∂ μ J μ = 0 {\displaystyle \partial _{\mu }J^{\mu }=0} in the usual case where there are no sources or sinks, that is, for perfectly conserved quantities like energy or charge.
Also known as vector current, it is used in the geometric context of four-dimensional spacetime, rather than separating time from three-dimensional space. Mathematically it is a four-vector and is Lorentz covariant. This article uses the summation convention for indices.
The 3-space momentum = (,,) is conserved (not to be confused with the classic non-relativistic momentum ). Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame and potential energy from forces between the particles contribute to the ...
Family and friends are back at home and work following the holiday weekend, but fridges may still be packed full of Thanksgiving leftovers.For folks wondering how long the turkey, potatoes and ...
of a point-like particle are combined into a four-dimensional bivector in terms of the 4-position X and the 4-momentum P of the particle: [8] [9] = where ∧ denotes the exterior product. This tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each constituent of the system.