Gravitational field

In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenomena, and is measured in newtons per kilogram (N/kg). 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 have usually been taught in terms of a field model, rather than a point attraction.

In a field model, 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, or that gravity is a fictitious force.

Classical mechanics

In classical mechanics as in physics, 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 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 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. The gravitational field equation is

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

where F is the gravitational force, m is the mass of the test particle, R is the position of the test particle, R̂ is a unit vector in the direction of R, t is time, G 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. Note that 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). 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 and Gauss's law are mathematically equivalent, and are related by the divergence theorem. Poisson's equation is obtained by taking the divergence of both sides of the previous equation. 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

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

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_i − R_j.

General relativity

See also: Gravitational acceleration § General relativity, and Gravitational potential § 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

$\mathbf {G} ={\frac {8\pi G}{c^{4}}}\mathbf {T}$

Here T is the stress–energy tensor, G is the Einstein tensor, 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 bending of light in such fields.

See also

• Classical mechanics
• Gravitation
• Gravitational potential
• Newton's law of universal gravitation
• Newton's laws of motion
• Potential energy
• Speed of gravity
• Tests of general relativity
• Defining equation (physics)