Stress Energy Tensor Study in Fluid Mechanics.pdf (original) (raw)
The energy-momentum tensor for a dissipative fluid in general relativity
General Relativity and Gravitation, 2016
Considering the growing interest of the astrophysicist community in the study of dissipative fluids with the aim of getting a more realistic description of the universe, we present in this paper a physical analysis of the energy-momentum tensor of a viscous fluid with heat flux. We introduce the general form of this tensor and, using the approximation of small velocity gradients, we relate the stresses of the fluid with the viscosity coefficients, the shear tensor and the expansion factor. Exploiting these relations, we can write the stresses in terms of the extrinsic curvature of the normal surface to the 4-velocity vector of the fluid, and we can also establish a connection between the perfect fluid and the symmetries of the spacetime. On the other hand, we calculate the energy conditions for a dissipative fluid through contractions of the energy-momentum tensor with the 4-velocity vector of an arbitrary observer. This method is interesting because it allows us to compute the conditions in a reasonably easy way and without considering any approximation or restriction on the energy-momentum tensor.
2021
We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically and intuitively motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the source matter fields’ energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter fields must be divergence free, when spacetime is 4 dimensional. Moreover, if the total source matter fields are assumed to be divergence free, then either the spacetime is of dimension 4 or the spacetime has constant scalar curvature. Mathematics Subject Classification (2000) : 83C05, 83C40, 83C99.
Inverse approach to Einstein’s equations for fluids with vanishing anisotropic stress tensor
Physical Review D, 2008
We expand previous work on an inverse approach to Einstein Field Equations where we include fluids with energy flux and consider the vanishing of the anisotropic stress tensor. We consider the approach using warped product spacetimes of class B1. Although restricted, these spacetimes include many exact solutions of interest to compact object studies and to cosmological models studies. The question explored here is as follows: given a spacetime metric, what fluid flow (timelike congruence), if any, could generate the spacetime via Einstein's equations. We calculate the flow from the condition of a vanishing anisotropic stress tensor and give results in terms of the metric functions in the three canonical types of coordinates. A condition for perfect fluid sources is also provided. The framework developed is algorithmic and suited for the study and validation of exact solutions using computer algebra systems. The framework can be applied to solutions in comoving and non-comoving frames of reference, and examples in different types of coordinates are worked out.
On the equivalence among stress tensors in a gauge-fluid system
International Journal of Modern Physics A, 2017
In this paper, we bring out the subtleties involved in the study of a first-order relativistic field theory with auxiliary field variables playing an essential role. In particular, we discuss the nonisentropic Eulerian (or Hamiltonian) fluid model. Interactions are introduced by coupling the fluid to a dynamical Maxwell (U(1)) gauge field. This dynamical nature of the gauge field is crucial in showing the equivalence, on the physical subspace, of the stress tensor derived from two definitions, i.e. the canonical (Noether) one and the symmetric one. In the conventional equal-time formalism, we have shown that the generators of the space–time transformations obtained from these two definitions agree modulo the Gauss constraint. This equivalence in the physical sector has been achieved only because of the dynamical nature of the gauge fields. Subsequently, we have explicitly demonstrated the validity of the Schwinger condition. A detailed analysis of the model in lightcone formalism ha...
Viscous Liquid Spacetime and Its Consequences
Recent experimental measurements performed by American satel-Iite Gravity Probe B showed fhat Einstein spacetime warped and is pulled by earth's movemenL Thk important experimental result could signify that Einstein spacetime exhibits as a viscous behavior. Moreover, in a recent publication,l T. Padmanabhan deman-stt'ated that general relativity and Navier-Stokes equations can be identical. In a previous paper,2 we showed how equatiotts estab-lislrcd for large scale physics could be applied at atomic scale. Baseil on this important result ûble to unify physics laws, we calculated in this papu the stress produced by each element of periodic classîfication in liquid spacetime. A correlation between the stress produced by each atom and their density was founel showing that a purely mechqnical description could be available at atomic scale. From this analysis, several aspects of physics*Iike the effect of hish intensity force ftelds on aether or the effect of highly asymmetric force ftelds*are discassed.
Variational aspects of relativistic field theories, with application to perfect fluids
Annals of Physics, 1977
By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully cova.riant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor Py. By perturbing the Noether operator's definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwell's equations for FPV and Euler's equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) + (c)l. General relativity is a theory which does have an unconstrained variational principle but the definition of Noether operator is more ambiguous than for other fields. We define a pseudotensorial operator which includes the Einstein and Landau-Lifschitz complexes as special cases and satisfies a certain criterion on the asymptotic behavior. Then our extremal theorem leads to a proof of the uniqueness of Minkowski space: It is the only asymptotically flat, stationary, vacuum solution to Einstein's equations having Iw4 global topology and a maximal spacelike hypersurface. We next consider perfect fluid dynamics. The failure of the extremal-energy theorem elucidates why constraints have always been used in variational principles that lead to Euler's equations. We discuss their meaning and give what we consider to be the "minimally constrained" principle. A discussion of one constraint, "preservation of particle identity," from the point of view of path-integral quantum mechanics leads to the conclusion that it is inapplicable to degenerate Bose fluids, and this gives immediately the well-known irrotational flow of such fluids. Finally, we develop a restricted extremal theorem for the case of perfect fluids with self-gravitation, which has the same form as before except that certain perturbations are forbidden in (c). We show that it is a generalization of the Bardeen-Hartle-Sharp variational principle for relativistic stellar structure. It may be useful in constructing nonaxisymmetric stellar models (generalized Dedekind ellipsoids). We also give the Newtonian versions of the main results here, and we show to what extent the extremal theorems extend to fields that may not even have a variational principle.
On Tensorial Concomitants and the Non-Existence of a Gravitational Stress-Energy Tensor
The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bun...
On a viable first-order formulation of relativistic viscous fluids and its applications to cosmology
International Journal of Modern Physics D, 2017
We consider a first-order formulation of relativistic fluids with bulk viscosity based on a stress-energy tensor introduced by Lichnerowicz. Choosing a barotropic equation-of-state, we show that this theory satisfies basic physical requirements and, under the further assumption of vanishing vorticity, that the equations of motion are causal, both in the case of a fixed background and when the equations are coupled to Einstein's equations. Furthermore, Lichnerowicz's proposal does not fit into the general framework of first-order theories studied by Hiscock and Lindblom, and hence their instability results do not apply. These conclusions apply to the full-fledged nonlinear theory, without any equilibrium or near equilibrium assumptions. Similarities and differences between the approach explored here and other theories of relativistic viscosity, including the Mueller–Israel–Stewart formulation, are addressed. Cosmological models based on the Lichnerowicz stress-energy tensor a...
Shear-free Perfect Fluids in General Relativity: Gravito-magnetic Spacetimes
General Relativity and Gravitation, 2000
We investigate shear-free, perfect fluid solutions of Einstein's field equations in which the perfect fluid satisfies a barotropic equation of state p = p(w) such that w + p = 0. We find that if the electric part of the Weyl tensor (with respect to the fluid flow) vanishes and the spacetime is not conformally flat then the fluid volume expansion is zero but the vorticity is necessarily nonzero. In addition, we show that if p = −w/3 then necessarily either the fluid expansion is zero or the fluid vorticity is zero.