Gravitational field

 ( Theories Of Gravity ) 

In physics, a gravitational field or gravitational acceleration field is a vector field field used to explain the influences that a body extends into the space around itself.

A gravitational field is used to explain Gravity phenomena, such as the gravitational force field exerted on another massive body. It has dimension of acceleration (L/T²) and it is measured in units of newtons per kilogram (N/kg) or, equivalently, in per second squared (m/s²).

In its original concept, gravity was a force between point . Following Isaac Newton, Pierre-Simon Laplace attempted to model gravity as some kind of radiation field or fluid, and since the 19th century, explanations for gravity in classical mechanics have usually been taught in terms of a field model, rather than a point attraction. It results from the spatial gradient of the gravitational potential field.

In general relativity, rather than two particles attracting each other, the particles distort spacetime via their mass, and this distortion is what is perceived and measured as a "force". In such a model one states that matter moves in certain ways in response to the curvature of spacetime,

and that there is either no gravitational force,

(2025). 9780387691992, Springer Japan. . ISBN 9780387691992

or that gravity is a fictitious force.

(2025). 9780387260785, Springer Science & Business. . ISBN 9780387260785

Gravity is distinguished from other forces by its obedience to the equivalence principle.

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Classical mechanics In classical mechanics, a gravitational field is a physical quantity.

Richard Feynman (1970). 9780201021158, Addison Wesley Longman. . ISBN 9780201021158

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.

(2025). 9780470014608, Wiley. ISBN 9780470014608

The gravitational field equation is

(1991). 9780895737526, Wiley-VCH. . ISBN 9780895737526

p. 451 $$\backslash mathbf\{g\}=\backslash frac\{\backslash mathbf\{F\}\}\{m\}=\backslash frac\{d^2\backslash mathbf\{R\}\}\{dt^2\}=-GM\backslash frac\{\backslash mathbf\{R\}\}\{\backslash left|\backslash mathbf\{R\}\backslash right|^3\}\; =\; -\backslash nabla\backslash Phi\; ,$$ where is the gravitational force, is the mass of the test mass, 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: $$\backslash nabla\backslash cdot\backslash mathbf\{g\}=-\backslash nabla^2\backslash Phi=-4\backslash pi\; G\backslash rho$$ 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

$$\backslash mathbf\{g\}\; =\; \backslash sum\_\{i\}\backslash mathbf\{g\}\_i\; =\; \backslash frac\{1\}\{m\}\backslash sum\_\{i\}\backslash mathbf\{F\}\_i\; =\; -\; G\backslash sum\_\{i\}m\_i\backslash frac\{\backslash mathbf\{R\}-\backslash mathbf\{R\}\_i\}\{\backslash left|\backslash mathbf\{R\}-\backslash mathbf\{R\}\_i\backslash right|^3\}\; =\; -\; \backslash sum\_\{i\}\backslash nabla\backslash Phi\_i\; ,$$ i.e. the gravitational field on mass is the sum of all gravitational fields due to all other masses m _i, except the mass itself. is the position vector of the gravitating particle , and is that of the test particle.

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General relativity A freely moving particle in gravitational field has the equations of motion: $$\backslash frac\{d^2x^\backslash lambda\}\{d\backslash tau^2\}\; +\; \backslash Gamma^\backslash lambda\_\{\backslash mu\backslash nu\}\backslash frac\{dx^\backslash mu\}\{d\backslash tau\}\backslash frac\{dx^\backslash nu\}\{d\backslash tau\}\; =\; 0$$ where $\backslash tau$ is the proper time for the particle, $\backslash Gamma^\backslash lambda\_\{\backslash mu\backslash nu\}$ are the Christoffel symbols and repeated indices are summed over. The proper time can be expressed in terms of the metric tensor: $$d\backslash tau^2\; =\; -g\_\{\backslash mu\backslash nu\}\; dx^\backslash mu\; dx^\backslash nu.$$ The field that determines the gravitational force is the Christoffel symbols and its derivatives, the metric tensor plays the role of the gravitational potential.

In general relativity, the gravitational field is determined by solving the Einstein field equations

Steven Weinberg (1972). 9780471925675, John Wiley & Sons. . ISBN 9780471925675

$$\backslash mathbf\{G\}\; =\; \backslash kappa\; \backslash mathbf\{T\}\; ,$$ 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 acceleration 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.

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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.

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See also

• Classical mechanics
• Entropic gravity
• Gravitation
• Gravitational energy
• Gravitational potential
• Gravitational wave
• Gravity map
• Newton's law of universal gravitation
• Newton's laws of motion
• Potential energy
• Specific force
• Speed of gravity
• Tests of general relativity

Categories: Theories Of Gravity, Geodesy, General Relativity

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