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Free Fall and Harmonic Oscillators
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The left side contains the curvature of spacetime as determined by the metric gµν. The Einstein tensor, Gμv=Rμv−12Rgμv, is determined from the curvature tensor Rµν and the scalar curvature R. These in turn are obtained from the metric tensor. Λ is the famous cosmological constant, which Einstein originally introduced to maintain a static universe, that has since taken on a different role. The right-hand side of Einstein’s equation involves the familiar gravitational constant, the speed of light, and the stress-energy tensor Tµν. Georges Lemaître (1894–1966) had actually predicted the expansion of the universe in 1927 and proposed what later became known as the Big Bang Theory.
The limits of Riemann solutions to the relativistic van der Waals fluid
Published in Applicable Analysis, 2021
The system (1) owns a very strong physical background in special relativity fluids. It is usually used to describe the dynamics of plane waves in a two-dimensional Minkowski space–time where is the stress-energy tensor, and all indices run from 0 to 1 with , denotes the flat Minkowski metric, u the 2-velocity of the fluid particle, and ρ the mass-energy density of the fluid as measured in units of mass in a frame moving with the fluid particle.
Implications of causality for quantum biology – I: topology change
Published in Molecular Physics, 2018
The Hodge norm of a quantity is invariant with respect to coordinate transformations because of the Hodge-⋆ and integration. It is invariant with respect to matrix-valued gauge transformations because of the trace. The trace converts the gauge group-valued quantity to a real scalar, the * above includes complex conjugation. This Hodge norm can be used to define a least action principle. In fact, for a Abelian gauge field with 1D representations K = F, we have for an action principle of electrodynamics: . The scalar term within the integral is called a Lagrangian density. For non-Abelian fields, using the Hodge norm of the gauge curvature, we have an action principle = extremum; in terms of the gauge curvature scalar where: Here ΓR is the Rth representation of the gauge symmetry group. The condition = extremum states the curvature of the QMST is to be an extremum. If one solves for and sets , it is possible to associate the non-Abelian gauge curvature with a matrix-valued Abelianised gauge field and the contraction of a matrix-valued torsion tensor . This shows the gauge curvature can be replaced by a torsion, so the dynamics can be described over a geometrically flat space (i.e. with vanishing Riemann curvature but non-vanishing torsion). The action is solely a functional of the currents . From such an action, one can use the proof of a theorem of Montesinos and Flores [28] to verify the following result for a symmetric stress-energy tensor:
The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
Alberto Roper Pol, Axel Brandenburg, Tina Kahniashvili, Arthur Kosowsky, Sayan Mandal
There is a striking analogy between the normalised radiation energy density in the present context of an ultrarelativistic plasma and the mass density in the usual MHD equations and in the Lighthill equation (2). Note that here “normalised” just refers to division by , not the normalisation presented in appendix 1. This analogy was employed by Christensson et al. (2001) and Banerjee and Jedamzik (2004) to argue that the equations for the early universe could be solved using just ordinary MHD codes. Here and below, we expand γ in , i.e. , including only terms up to second order. The term enters in the stress–energy tensor because in a relativistic plasma, the gas pressure is equal to one third of the radiation energy density, . Using the ultrarelativistic equation of state, we have , so the prefactor of in (5) reduces to . Hence, similar 4/3 factors appear in the MHD equations for an ultrarelativistic gas in a flat expanding universe (Brandenburg et al.1996, 2017; Kahniashvili et al.2017), which are given by where are the components of the rate-of-strain tensor with commas denoting partial derivatives, is the current density, ν is the viscosity, and η is the magnetic diffusivity. We assume constant η and ν in all our simulations. We emphasise that all variables have been scaled appropriately so that terms proportional to appear neither in (25)–(27) nor in (3), i.e. we use comoving variables that already take into account the effect of the expansion of the universe; see appendix 1. Nevertheless, there remains the term on the right-hand side of the GW equation (3), which means that the source of GWs gradually declines during the radiation-dominated epoch of the universe.