Gravitational field

In classical mechanics, a gravitational field is a physical quantity. A gravitational field can be defined using Newton's law of universal gravitation. Determined in this way, the gravitational field around a single particle of mass is a vector field consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated by applying the universal law, and represents the force per unit mass on any object at that point in space. Because the force field is conservative, there is a scalar potential energy per unit mass, , at each point in space associated with the force fields; this is called gravitational potential. The gravitational field equation is \mathbf{g}=\frac{\mathbf{F}}{m}=\frac{d^2\mathbf{R}}{dt^2}=-GM\frac{\mathbf{R}}{\left|\mathbf{R}\right|^3} = -\nabla\Phi , where is the gravitational force, is the mass of the test particle, is the radial vector of the test particle relative to the mass (or for Newton's second law of motion which is a time dependent function, a set of positions of test particles each occupying a particular point in space for the start of testing), is time, is the gravitational constant, and is the del operator. This includes Newton's law of universal gravitation, and the relation between gravitational potential and field acceleration. and are both equal to the gravitational acceleration (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass density of the attracting mass is: \nabla\cdot\mathbf{g}=-\nabla^2\Phi=-4\pi G\rho which contains Gauss's law for gravity, and Poisson's equation for gravity. Newton's law implies Gauss's law, but not vice versa; see ''Relation between Gauss's and Newton's laws''. These classical equations are differential equations of motion for a test particle in the presence of a gravitational field, i.e. setting up and solving these equations allows the motion of a test mass to be determined and described. The field around multiple particles is simply the vector sum of the fields around each individual particle. A test particle in such a field will experience a force that equals the vector sum of the forces that it would experience in these individual fields. This is \mathbf{g} = \sum_{i}\mathbf{g}_i = \frac{1}{m}\sum_{i}\mathbf{F}_i = - G\sum_{i}m_i\frac{\mathbf{R}-\mathbf{R}_i}{\left|\mathbf{R}-\mathbf{R}_i\right|^3} = - \sum_{i}\nabla\Phi_i , i.e. the gravitational field on mass is the sum of all gravitational fields due to all other masses mi, except the mass itself. is the position vector of the gravitating particle , and is that of the test particle. == General relativity ==

A freely moving particle in gravitational field has the equations of motion: \frac{d^2x^\lambda}{d\tau^2} + \Gamma^\lambda_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0 where \tau is the proper time for the particle, \Gamma^\lambda_{\mu\nu} are the Christoffel symbols and repeated indices are summed over. \mathbf{G} = \kappa \mathbf{T} , where is the stress–energy tensor, is the Einstein tensor, and is the Einstein gravitational constant. The latter is defined as , where is the Newtonian constant of gravitation and is the speed of light. These equations are dependent on the distribution of matter, stress and momentum in a region of space, unlike Newtonian gravity, which is depends on only the distribution of matter. The fields themselves in general relativity represent the curvature of spacetime. General relativity states that being in a region of curved space is equivalent to accelerating up the gradient of the field. By Newton's second law, this will cause an object to experience a fictitious force if it is held still with respect to the field. This is why a person will feel himself pulled down by the force of gravity while standing still on the Earth's surface. In general the gravitational fields predicted by general relativity differ in their effects only slightly from those predicted by classical mechanics, but there are a number of easily verifiable differences, one of the most well known being the deflection of light in such fields. == Embedding diagram ==

Embedding diagrams are three dimensional graphs commonly used to educationally illustrate gravitational potential by drawing gravitational potential fields as a gravitational topography, depicting the potentials as so-called gravitational wells, sphere of influence. == See also ==