On Lynden-Bell and Katz's definition of gravitational field energy

1986 ◽  
Vol 18 (9) ◽  
pp. 889-897 ◽  
Author(s):  
Ø. Grøn
Keyword(s):  
Gravitational Field ◽  
Field Energy ◽  
Definition Of

Variational thermodynamic derivation of the formula for pressure difference across a charged conducting liquid surface and its relation to the thermodynamics of electrical capacitance

1995 ◽  
Vol 450 (1938) ◽  
pp. 177-192 ◽  
Keyword(s):  
Free Energy ◽  
Pressure Difference ◽  
Liquid Surface ◽  
Field Energy ◽  
Conducting Liquid ◽  
Difference Formula ◽  
Direct Energy ◽  
Free Case ◽  
Definition Of ◽  
Work Done

The formula for pressure difference across a charged conducting liquid surface has conventionally been derived by adding a Maxwell stress term to the pressure-difference formula for the field-free case. As far as can be established, no derivation applying direct energy-based methods to the charged-surface case has ever been clearly formulated. This paper presents a first-principles variational derivation, starting from the laws of thermodynamics and modelled on Gibbs’s (1875) approach to the field-free case. The derivation applies to the static equilibrium situation. The method is to treat the charged liquid and its environment as a heterogeneous system in thermodynamic equilibrium, and consider the effects of a small virtual variation in the shape of the conducting-liquid surface. Expressions can be obtained for virtual changes in the free energies of relevant system components and for the virtual electrical work done on the system. By converting the space integral of the variation in electrostatic field energy to an integral over the surface of the liquid electrode, the usual pressure-difference formula is retrieved. It is also shown how the problem can be formulated, in various ways, as a free-energy problem in a situation involving electric stresses and capacitance. The most satisfactory approach involves the definition of an unfamiliar form of free energy, that can be seen as the electrical analogue of the Gibbs free energy and may have use in other contexts.

scholarly journals WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

2007 ◽  
Vol 22 (35) ◽  
pp. 2675-2687 ◽  
Author(s):  
LUIS F. BARRAGÁN-GIL ◽  
ABEL CAMACHO
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Field ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Approximation Method ◽  
Bound States ◽  
The Other ◽  
Homogeneous Gravitational Field ◽  
Definition Of ◽  
Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

Formation of an expanded criteria base for managing the effectiveness of the most important elements of advanced space systems for monitoring the Earth's gravitational field

2021 ◽  
pp. 17-25
Author(s):  
Л.Г. Азаренко
Keyword(s):  
Gravitational Field ◽  
Space Systems ◽  
Earth’S Gravitational Field ◽  
Expanded Criteria ◽  
Definition Of

В статье рассматриваются подходы к формированию критериальной базы оценки эффективности при проектировании перспективных космических систем на примере космических систем мониторинга гравитационного поля Земли. Сформулированы основные требования к созданию критериальной базы. Дано определение критерия и обобщенного критерия эффективности применительно к элементам перспективных космических систем. Рассмотрены общие и частные критерии эффективности перспективных космических систем в зависимости от назначения конкретной системы (оборонного применения, гражданские, многоцелевые). The article considers approaches to the formation of a criteria base for evaluating the effectiveness of the design of advanced space systems on the example of space systems for monitoring the Earth's gravitational field. The main requirements for creating a criteria base are formulated. The definition of the criterion and the generalized criterion of efficiency in relation to the elements of advanced space systems is given. General and specific criteria for the effectiveness of advanced space systems are considered, depending on the purpose of a particular system (defense, civil, multi-purpose).

scholarly journals Gravitational energy in the framework of embedding and splitting theories

2018 ◽  
Vol 27 (02) ◽  
pp. 1750188 ◽  
Author(s):  
D. A. Grad ◽  
R. V. Ilin ◽  
S. A. Paston ◽  
A. A. Sheykin
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Gravitational Energy ◽  
Einstein Equations ◽  
Field Energy ◽  
Energy Localization ◽  
Embedding Theory ◽  
Isometric Embeddings ◽  
Definition Of ◽  
Noether Current

We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge–Teitelboim approach. For the embedding theory, we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case, the independent definition of gravitational energy–momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory), we consider Noether procedure and the alternative method of energy–momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result, we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed.

On the attractive force of the gravitational field in static space times

1970 ◽  
Vol 68 (1) ◽  
pp. 187-197 ◽  
Author(s):  
H. Müller zum Hagen
Keyword(s):  
Gravitational Field ◽  
Test Particle ◽  
Gravitational Force ◽  
Killing Vector ◽  
Attractive Force ◽  
Potential Surfaces ◽  
Asymptotically Flat ◽  
Definition Of ◽  
Meaningful Definition ◽  
Static Space

AbstractA static metric is considered. A meaningful definition of gravitational force is given and the potential, which is the norm of the Killing vector ξa, is studied. For the case that the metric is asymptotically flat, the following is shown: The equi-potential surfaces are closed 2-dimensional surfacesSlying in the rest space V3, which is the hypersurface orthogonal to ξa. All the surfacesSenclose matter, and the gravitational force points intoStowards the enclosed matter. A test particle starting atSwill be pulled into the domain bounded bySand will never leave this domain.

scholarly journals Unification of Gravity and Electromagnetism

2021 ◽  
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Mathematical Structure ◽  
Momentum Tensor ◽  
Dirac Field ◽  
Field Equations ◽  
Mathematical Structures ◽  
Gravitational Field Equations ◽  
Definition Of ◽  
Two Sides

Gravity and electromagnetism are two sides of the same coin, which is the clue of this unification. Gravity and electromagnetism are representing by two mathematical structures, symmetric and antisymmetric respectively. Einstein gravitational field equation is the symmetric mathematical structure. Electrodynamics Lagrangian is three parts, for electromagnetic field, Dirac field and interaction term. The definition of canonical energy momentum tensor was used for each term in Electrodynamics Lagrangian to construct the antisymmetric mathematical structure. Symmetric and antisymmetric gravitational field equations are two sides of the same Lagrangian

Sign in / Sign up

Export Citation Format

Share Document