Gravitational field

In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself.[1] A gravitational field is used to explain gravitational phenomena, such as the gravitational force field excerted 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 meters per second squared (m/s²).

Representation of the gravitational field of Earth and Moon combined (not to scale). Vector field (blue) and its associated scalar potential field (red). Point P between earth and moon is the point of equilibrium.

In its original concept, gravity was a force between point masses. 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,[2] and that there is either no gravitational force,[3] or that gravity is a fictitious force.[4]

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

Classical mechanics

In classical mechanics, a gravitational field is a physical quantity.[5] A gravitational field can be defined using Newton's law of universal gravitation. Determined in this way, the gravitational field $g$ around a single particle of mass $M$ 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, $Φ$ , at each point in space associated with the force fields; this is called gravitational potential.[6] The gravitational field equation is[7]

\mathbf {g} ={\frac {\mathbf {F} }{m}}={\frac {d^{2}\mathbf {R} }{dt^{2}}}=-GM{\frac {\mathbf {\hat {R}} }{\left|\mathbf {R} \right|^{2}}}=-\nabla \Phi

where $F$ is the gravitational force, $m$ is the mass of the test particle, $R$ is the position of the test particle (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), $R̂$ is a unit vector in the radial direction of $R$ , $t$ is time, $G$ is the gravitational constant, and $\nabla$ is the del operator.

This includes Newton's law of universal gravitation, and the relation between gravitational potential and field acceleration. Note that $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}d2R/dt2$ and $F / m$ are both equal to the gravitational acceleration $g$ (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass[8]). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass density $ρ$ 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. An object in such a field will experience a force that equals the vector sum of the forces it would experience in these individual fields. This is mathematically[9]

\mathbf {g} _{j}^{\text{(net)}}=\sum _{i\neq j}\mathbf {g} _{i}={\frac {1}{m_{j}}}\sum _{i\neq j}\mathbf {F} _{i}=-G\sum _{i\neq j}m_{i}{\frac {\mathbf {\hat {R}} _{ij}}{\left|\mathbf {R} _{i}-\mathbf {R} _{j}\right|^{2}}}=-\sum _{i\neq j}\nabla \Phi _{i}

i.e. the gravitational field on mass $m j$ is the sum of all gravitational fields due to all other masses m_i, except the mass $m j$ itself. The unit vector $R̂ ij$ is in the direction of $R j - R i$ (pointing from particle i to particle j).

General relativity

In general relativity, the Christoffel symbols play the role of the gravitational force field and the metric tensor plays the role of the gravitational potential.

In general relativity, the gravitational field is determined by solving the Einstein field equations[10]

\mathbf {G} =\kappa \mathbf {T} ,

where $T$ is the stress–energy tensor, $G$ is the Einstein tensor, and $κ$ is the Einstein gravitational constant. The latter is defined as $κ = 8 πG / c 4$ , where $G$ is the Newtonian constant of gravitation and $c$ is the speed of light.

These equations are dependent on the distribution of matter and energy in a region of space, unlike Newtonian gravity, which is dependent only on 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.

Notes

Feynman, Richard (1970). The Feynman Lectures on Physics. Vol. I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
Geroch, Robert (1981). General Relativity from A to B. University of Chicago Press. p. 181. ISBN 978-0-226-28864-2.
Grøn, Øyvind; Hervik, Sigbjørn (2007). Einstein's General Theory of Relativity: with Modern Applications in Cosmology. Springer Japan. p. 256. ISBN 978-0-387-69199-2.
Foster, J.; Nightingale, J. D. (2006). A Short Course in General Relativity (3 ed.). Springer Science & Business. p. 55. ISBN 978-0-387-26078-5.
Feynman, Richard (1970). The Feynman Lectures on Physics. Vol. II. Addison Wesley Longman. ISBN 978-0-201-02115-8. A 'field' is any physical quantity which takes on different values at different points in space.
Forshaw, J. R.; Smith, A. G. (2009). Dynamics and Relativity. Wiley. ISBN 978-0-470-01460-8.
Lerner, R. G.; Trigg, G. L., eds. (1991). Encyclopaedia of Physics (2nd ed.). Wiley-VCH. ISBN 978-0-89573-752-6.
Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 978-0-7195-3382-2.
Kibble, T. W. B. (1973). Classical Mechanics. European Physics Series (2nd ed.). UK: McGraw Hill. ISBN 978-0-07-084018-8.
Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973). Gravitation. W. H. Freeman & Co. p. 404. ISBN 978-0-7167-0344-0.

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[1] Feynman, Richard (1970). The Feynman Lectures on Physics. Vol. I. Addison Wesley Longman. ISBN 978-0-201-02115-8.

[2] Geroch, Robert (1981). General Relativity from A to B. University of Chicago Press. p. 181. ISBN 978-0-226-28864-2.

[3] Grøn, Øyvind; Hervik, Sigbjørn (2007). Einstein's General Theory of Relativity: with Modern Applications in Cosmology. Springer Japan. p. 256. ISBN 978-0-387-69199-2.

[4] Foster, J.; Nightingale, J. D. (2006). A Short Course in General Relativity (3 ed.). Springer Science & Business. p. 55. ISBN 978-0-387-26078-5.

[5] Feynman, Richard (1970). The Feynman Lectures on Physics. Vol. II. Addison Wesley Longman. ISBN 978-0-201-02115-8. A 'field' is any physical quantity which takes on different values at different points in space.

[6] Forshaw, J. R.; Smith, A. G. (2009). Dynamics and Relativity. Wiley. ISBN 978-0-470-01460-8.

[7] Lerner, R. G.; Trigg, G. L., eds. (1991). Encyclopaedia of Physics (2nd ed.). Wiley-VCH. ISBN 978-0-89573-752-6.

[8] Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 978-0-7195-3382-2.

[9] Kibble, T. W. B. (1973). Classical Mechanics. European Physics Series (2nd ed.). UK: McGraw Hill. ISBN 978-0-07-084018-8.

[10] Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973). Gravitation. W. H. Freeman & Co. p. 404. ISBN 978-0-7167-0344-0.

Gravitational field

Classical mechanics

General relativity

See also

Notes