Gravitational field

Last updated March 26, 2024

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 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 meters per second squared (m/s²).

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".^{[ citation needed ]} 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 {R} }{\left|\mathbf {R} \right|^{3}}}=-\nabla \Phi ,

where $F$ is the gravitational force, $m$ is the mass of the test particle, $R$ 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), $t$ 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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}d2R/dt2$ 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. 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^[9]

\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 $m j$ is the sum of all gravitational fields due to all other masses m_i, except the mass $m j$ itself. $R i$ is the position vector of the gravitating particle $i$ , and $R$ is that of the test particle.

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, 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.

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In physics, a force is an influence that can cause an object to change its velocity, i.e., to accelerate, meaning a change in speed or direction, unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol $F$ .

<span class="mw-page-title-main">Momentum</span> Property of a mass in motion

In Newtonian mechanics, momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If $m$ is an object's mass and $v$ is its velocity, then the object's momentum $p$ is:

In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of potentiality.

An electric field is the physical field that surrounds electrically charged particles. Charged particles exert attractive forces on each other when their charges are opposite, and repulse each other when their charges are the same. Because these forces are exerted mutually, two charges must be present for the forces to take place. The electric field of a single charge describes their capacity to exert such forces on another charged object. These forces are described by Coulomb's law, which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Thus, we may informally say that the greater the charge of an object, the stronger its electric field. Similarly, an electric field is stronger nearer charged objects and weaker further away. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, Electromagnetism is one of the four fundamental interactions of nature.

Newton's laws of motion are three laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:

A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force.
The net force on a body is equal to the body's instantaneous acceleration multiplied by its instantaneous mass or, equivalently, the rate at which the body's momentum changes with time.
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In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

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In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's law for gravity is often more convenient to work from than Newton's law.

<span class="mw-page-title-main">Classical mechanics</span> Description of large objects physics

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<span class="mw-page-title-main">Lagrangian mechanics</span> Formulation of classical mechanics

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique.

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. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.

In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, while electrodynamics can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field.

References

↑ 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.^{[ page needed ]}
↑ Lerner, R. G.; Trigg, G. L., eds. (1991). Encyclopaedia of Physics (2nd ed.). Wiley-VCH. ISBN 978-0-89573-752-6. p. 451
↑ Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 978-0-7195-3382-2.^{[ page needed ]}
↑ Kibble, T. W. B. (1973). Classical Mechanics. European Physics Series (2nd ed.). UK: McGraw Hill. ISBN 978-0-07-084018-8.^{[ page needed ]}
↑ 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.^{[ page needed ]}

[7] Lerner, R. G.; Trigg, G. L., eds. (1991). Encyclopaedia of Physics (2nd ed.). Wiley-VCH. ISBN 978-0-89573-752-6. p. 451

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

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

[10] 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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