scholarly journals Dimensional analysis in relativity and in differential geometry

2020 ◽  
Author(s):  
PierGianLuca Porta Mana
Keyword(s):  
Differential Geometry ◽  
General Relativity ◽  
Dimensional Analysis ◽  
Absolute Dimension ◽  
Intrinsic Dimension ◽  
Stress Energy ◽  
Curvature Tensors

This note provides a short guide to dimensional analysis in Lorentzian and general relativity and in differential geometry. It tries to revive Dorgelo and Schouten's notion of 'intrinsic' or 'absolute' dimension of a tensorial quantity. The intrinsic dimension is independent of the dimensions of the coordinates and expresses the physical and operational meaning of a tensor. The dimensional analysis of several important tensors and tensor operations is summarized. In particular it is shown that the components of a tensor need not have all the same dimension, and that the Riemann (once contravariant and thrice covariant), Ricci (twice covariant), and Einstein (twice covariant) curvature tensors are dimensionless. The relation between dimension and operational meaning for the metric and stress-energy-momentum tensors is discussed; and the possible conventions for the dimensions of these two tensors and of Einstein's constant $\kappa$, including the curious possibility $\kappa = 8\pi G$ without $c$ factors, are reviewed.

scholarly journals Stress Energy Tensor Study in Fluid Mechanics

2019 ◽  
Author(s):  
Roman Baudrimont
Keyword(s):  
General Relativity ◽  
Fluid Mechanics ◽  
Space Time ◽  
Newtonian Fluids ◽  
Third Party ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Perfect Fluids ◽  
Stress Energy

This paper is to summarize the involvement of the stress energy tensor in the study of fluid mechanics. In the first part we will see the implication that carries the stress energy tensor in the framework of general relativity. In the second part, we will study the stress energy tensor under the mechanics of perfect fluids, allowing us to lead third party in the case of Newtonian fluids, and in the last part we will see that it is possible to define space-time as a no-Newtonian fluids.

scholarly journals Counterexamples to the Maximum Force Conjecture

Universe ◽  
2021 ◽  
Vol 7 (11) ◽  
pp. 403
Author(s):  
Aden Jowsey ◽  
Matt Visser
Keyword(s):  
General Relativity ◽  
Dimensional Analysis ◽  
Maximum Force ◽  
Explicit Calculation ◽  
General Notion ◽  
Physical Situation ◽  
Speed Of Light ◽  
Maximum Tension ◽  
Dimensionless Function ◽  
Fluid Spheres

Dimensional analysis shows that the speed of light and Newton’s constant of gravitation can be combined to define a quantity F*=c4/GN with the dimensions of force (equivalently, tension). Then in any physical situation we must have Fphysical=fF*, where the quantity f is some dimensionless function of dimensionless parameters. In many physical situations explicit calculation yields f=O(1), and quite often f≤1/4. This has led multiple authors to suggest a (weak or strong) maximum force/maximum tension conjecture. Working within the framework of standard general relativity, we will instead focus on idealized counter-examples to this conjecture, paying particular attention to the extent to which the counter-examples are physically reasonable. The various idealized counter-examples we shall explore strongly suggest that one should not put too much credence into any truly universal maximum force/maximum tension conjecture. Specifically, idealized fluid spheres on the verge of gravitational collapse will generically violate the weak (and strong) maximum force conjectures. If one wishes to retain any truly general notion of “maximum force” then one will have to very carefully specify precisely which forces are to be allowed within the domain of discourse.

Precision and Dimensional Analysis about Parallel Mechanism Having Less Degree of Freedom Based on Differential Geometry

2014 ◽  
Vol 940 ◽  
pp. 509-512 ◽  
Author(s):  
De Xing Zheng ◽  
Bin Wang
Keyword(s):  
Differential Geometry ◽  
Dimensional Analysis ◽  
Parallel Mechanism ◽  
Structural Parameters ◽  
Maximum Error ◽  
Error Model ◽  
Degree Of Freedom ◽  
Vector Model ◽  
Parallel Mechanisms ◽  
Output Error

A 3-PUU parallel mechanism was studied, which can perform one-dimensional translation about z-axis and two-dimensional rotations about y-axis and x-axis. This paper shows the study about the precision analysis, the dimensional analysis and the synthesis problem of the parallel mechanism having less degree of freedom. First, the closed-loop vector model was built based on analyzing the branched-chains of 3-PUU, the pose error model was built through the differential geometry, and the error model of positive solutions was obtained which contains all the structural parameters errors. For the error of structural parameters given, the pose output error can be solved out by application of this model, the area of maximum error is found out. And the influence of pose output error was analyzed with postures changed. The dimensional analysis of 3-PUU was also done by this error model, and the reasonable selection of mechanism parameters was discussed. And then the rationality of structure dimensional design was discussed in this paper. Finally, this method also may be used in the dimensional analysis of other less freedom parallel mechanisms.

scholarly journals Stress Energy Tensor Study in Fluid Mechanics

2019 ◽  
Author(s):  
Roman Baudrimont
Keyword(s):  
General Relativity ◽  
Fluid Mechanics ◽  
Space Time ◽  
Newtonian Fluids ◽  
Third Party ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Perfect Fluids ◽  
Stress Energy

This paper is to summarize the involvement of the stress energy tensor in the study of fluid mechanics. In the first part we will see the implication that carries the stress energy tensor in the framework of general relativity. In the second part, we will study the stress energy tensor under the mechanics of perfect fluids, allowing us to lead third party in the case of Newtonian fluids, and in the last part we will see that it is possible to define space-time as a no-Newtonian fluids.

scholarly journals Quasilocal conservation laws in cosmology: A first look

2020 ◽  
Vol 29 (14) ◽  
pp. 2043029
Author(s):  
Marius Oltean ◽  
Hossein Bazrafshan Moghaddam ◽  
Richard J. Epp
Keyword(s):  
General Relativity ◽  
Cosmological Constant ◽  
Conservation Laws ◽  
Perfect Fluid ◽  
Vacuum Energy ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Stress Energy ◽  
Fluid Matter ◽  
Stress Energy Momentum Tensor

Quasilocal definitions of stress-energy–momentum—that is, in the form of boundary densities (in lieu of local volume densities) — have proven generally very useful in formulating and applying conservation laws in general relativity. In this Essay, we take a basic look into applying these to cosmology, specifically using the Brown–York quasilocal stress-energy–momentum tensor for matter and gravity combined. We compute this tensor and present some simple results for a flat FLRW spacetime with a perfect fluid matter source. We emphasize the importance of the vacuum energy, which is almost universally underappreciated (and usually “subtracted”), and discuss the quasilocal interpretation of the cosmological constant.

Hodge Theory on Compact Two-Dimensional Spacetimes and the Uniqueness ofgijwith a SpecifiedRij

1987 ◽  
Vol 39 (6) ◽  
pp. 1459-1474
Author(s):  
Edwin Ihrig
Keyword(s):  
Ricci Tensor ◽  
Energy Tensor ◽  
Main Question ◽  
Two Dimensional ◽  
Stress Energy Tensor ◽  
Riemann Curvature ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Stress Energy ◽  
Curvature Tensors

The main question we wish to address in this paper is to what extent does the Ricci curvature of a spacetime determine the metric of that spacetime. Although it is relatively easy to see that the full Riemann curvature uniquely determines the metric for a generic choice of curvature tensors (see[4], [10], [11], [14] and [15], and the references contained therein), very little has been discovered about whether, if ever, Ric (or the stress energy tensor in Einstein's equations for that matter) determinesg. Most exact solution techniques for Einstein's equations look only for solutions that have the same symmetries as Ric. It is not true in general thatgmust inherit the symmetries of Ric. It is not even clear that there is a Ric such that everygwith this Ricci tensor is known.

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