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The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles: setting the coupling constant of interaction in the Gross–Pitaevskii equation to zero (see the following section) recovers the single-particle Schrödinger equation describing a particle inside a trapping potential.
It can be adapted to similar equations e.g. F = ma, v = fλ, E = mcΔT, V = π r 2 h and τ = rF sinθ. When a variable with an exponent or in a function is covered, the corresponding inverse is applied to the remainder, i.e. = and = .
The equation of motion for the particle derived above = + + can be rewritten using the definition of the Schwarzschild radius r s as = [] + + (+) which is equivalent to a particle moving in a one-dimensional effective potential = + (+) The first two terms are well-known classical energies, the first being the attractive Newtonian gravitational ...
In the context of general relativity, it means the problem of finding solutions to Einstein's field equations — a system of hyperbolic partial differential equations — given some initial data on a hypersurface. Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising ...
Diagram regarding the confirmation of gravitomagnetism by Gravity Probe B. Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity.
The "slide" analogy is a reference to the slide projector, a device that has become somewhat obsolete due to the use of presentation software. Slides can be printed, or (more usually) displayed on-screen and navigated through at the command of the presenter. An entire presentation can be saved in video format. [6]
The gravitational potential (V) at a location is the gravitational potential energy (U) at that location per unit mass: =, where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.
For two pairwise interacting point particles, the gravitational potential energy is the work that an outside agent must do in order to quasi-statically bring the masses together (which is therefore, exactly opposite the work done by the gravitational field on the masses): = = where is the displacement vector of the mass, is gravitational force acting on it and denotes scalar product.