The inertial mass of a system is defined as the value connecting the four-momentum and four-velocity of the center of momentum of the system, and can be calculated using the square of the four-momentum. It is shown that the canonical representation of the angular momentum pseudotensor is ...
The resulting gyrovector spaces enable Euclidean trigonometry to be extended to hyperbolic trigonometry. In particular, we present the hyperbolic law of cosines and sines and the Hyperbolic Pythagorean Theorem emerges when the common vector addition is replaced by the Einstein velocity addition....
Looking at the results reported in Tables 1–12, it appears that the aberrated velocity modulus values obtained from the input value β″ = 0.1 are all very close to this value, confirming the reliability of the algorithm. In addition, the variations of the aberrated velocity modulus decreased...
velocity given by v m = g 4 + g 2 u 0 v 0 , u 0 = 1 + | u | 2 , g 2 = s − 4 , s = 2 ( u 0 v 0 − u ⋅ v + 1 ) . the function σ ( g , θ ) is called the differential cross section or the scattering kernel, and it measures the interactions ...
(2.4). We can see that in the non-adiabatic limit, γw > Mϕ2 vTnuc , (2.7) the exponential goes to one and the density becomes independent of the velocity of the wall vw, as opposed to particle production via dimension five and six operators, as we will see below. The final ...
We present the state-of-the-art concerning the relativistic constraints, which describe the geometry of hypersurfaces in a spacetime subject to the Einstei
velocity of the particles) to counter the debunching effects caused by space-charge and rf-induced energy spread. In this scheme the particle bunch lags behind the decelerating crest of the wave, with the particles at the bunch tail losing ...
To properly understand these ambiguities, one must realize that the fluid four-velocity uμ, energy density e, fluid pressure p, the heat current qμ, and shear tensor πμν may be considered unobservable figments of the theorists’ imagination, since what is actually observed is the entire ...
Here ρ is the macroscopic fluid density, and u is the fluid 3-velocity. To close this system, one must prescribe an equation of state p = P(ρ). A well-investigated choice is again the polytropic one, i.e., P(ρ) = ργ , γ > 1. When γ > 6 5 there exist spherically ...
Additionally, in the ideal MHD framework, the electric field E is a function of the fluid velocity v and the magnetic field B(5)cE=−v×B, where c is the speed of light. Analytically, the commutativity of spatial derivatives endows the induction equation with the solenoidal condition for ...