Field (physics)

In classical gravitation, mass is the source of an attractive gravitational field g.

A classical field theory describing gravity is Newtonian gravitation, which describes the gravitational force as a mutual interaction between two masses.

Any body with mass M is associated with a gravitational field g which describes its influence on other bodies with mass. The gravitational field of M at a point r in space corresponds to the ratio between force F that M exerts on a small or negligible test mass m located at r and the test mass itself:[12]

${\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}.}$

Stipulating that m is much smaller than M ensures that the presence of m has a negligible influence on the behavior of M.

According to Newton's law of universal gravitation, F(r) is given by[12]

${\displaystyle \mathbf {F} (\mathbf {r} )=-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }},}$

where ${\displaystyle {\hat {\mathbf {r} }}}$ is a unit vector lying along the line joining M and m and pointing from M to m. Therefore, the gravitational field of M is[12]

${\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}=-{\frac {GM}{r^{2}}}{\hat {\mathbf {r} }}.}$

The experimental observation that inertial mass and gravitational mass are equal to an unprecedented level of accuracy leads to the identity that gravitational field strength is identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle, which leads to general relativity.

Because the gravitational force F is conservative, the gravitational field g can be rewritten in terms of the gradient of a scalar function, the gravitational potential Φ(r):

${\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla \Phi (\mathbf {r} ).}$

Michael Faraday first realized the importance of a field as a physical quantity, during his investigations into magnetism. He realized that electric and magnetic fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy.

These ideas eventually led to the creation, by James Clerk Maxwell, of the first unified field theory in physics with the introduction of equations for the electromagnetic field. The modern version of these equations is called Maxwell's equations.

A charged test particle with charge q experiences a force F based solely on its charge. We can similarly describe the electric field E so that F = qE. Using this and Coulomb's law tells us that the electric field due to a single charged particle is

${\displaystyle \mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q}{r^{2}}}{\hat {\mathbf {r} }}.}$

The electric field is conservative, and hence can be described by a scalar potential, V(r):

${\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}$

A steady current I flowing along a path ℓ will create a field B, that exerts a force on nearby moving charged particles that is quantitatively different from the electric field force described above. The force exerted by I on a nearby charge q with velocity v is

${\displaystyle \mathbf {F} (\mathbf {r} )=q\mathbf {v} \times \mathbf {B} (\mathbf {r} ),}$

where B(r) is the magnetic field, which is determined from I by the Biot–Savart law:

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {\frac {Id{\boldsymbol {\ell }}\times {\hat {\mathbf {r} }}}{r^{2}}}.}$

The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a vector potential, A(r):

${\displaystyle \mathbf {B} (\mathbf {r} )={\boldsymbol {\nabla }}\times \mathbf {A} (\mathbf {r} )}$

The E fields and B fields due to electric charges (black/white) and magnetic poles (red/blue).[13][14] Top: E field due to an electric dipole moment d. Bottom left: B field due to a mathematical magnetic dipole m formed by two magnetic monopoles. Bottom right: B field due to a pure magnetic dipole moment m found in ordinary matter (not from monopoles).

In general, in the presence of both a charge density ρ(r, t) and current density J(r, t), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to ρ and J.[15]

Alternatively, one can describe the system in terms of its scalar and vector potentials V and A. A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J,[note 1] and from there the electric and magnetic fields are determined via the relations[16]

${\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}V-{\frac {\partial \mathbf {A} }{\partial t}}}$

${\displaystyle \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} .}$

At the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime.

The E fields and B fields due to electric charges (black/white) and magnetic poles (red/blue).[13][14] E fields due to stationary electric charges and B fields due to stationary magnetic charges (note in nature N and S monopoles do not exist). In motion (velocity v), an electric charge induces a B field while a magnetic charge (not found in nature) would induce an E field. Conventional current is used.

In general relativity, mass-energy warps space time (Einstein tensor G),[17] and rotating asymmetric mass-energy distributions with angular momentum J generate GEM fields H[18]

Einstein's theory of gravity, called general relativity, is another example of a field theory. Here the principal field is the metric tensor, a symmetric 2nd-rank tensor field in spacetime. This replaces Newton's law of universal gravitation.

Waves can be constructed as physical fields, due to their finite propagation speed and causal nature when a simplified physical model of an isolated closed system is set . They are also subject to the inverse-square law.

For electromagnetic waves, there are optical fields, and terms such as near- and far-field limits for diffraction. In practice though, the field theories of optics are superseded by the electromagnetic field theory of Maxwell.

A convenient way of classifying a field (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types:

Fields are often classified by their behaviour under transformations of spacetime. The terms used in this classification are:

• scalar fields (such as temperature) whose values are given by a single variable at each point of space. This value does not change under transformations of space.
• vector fields (such as the magnitude and direction of the force at each point in a magnetic field) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves contravariantly under rotations in space. Similarly, a dual (or co-) vector field attaches a dual vector to each point of space, and the components of each dual vector transform covariantly.
• tensor fields, (such as the stress tensor of a crystal) specified by a tensor at each point of space. Under rotations in space, the components of the tensor transform in a more general way which depends on the number of covariant indices and contravariant indices.
• spinor fields (such as the Dirac spinor) arise in quantum field theory to describe particles with spin which transform like vectors except for one of their components; in other words, when one rotates a vector field 360 degrees around a specific axis, the vector field turns to itself; however, spinors would turn to their negatives in the same case.

Fields may have internal symmetries in addition to spacetime symmetries. In many situations, one needs fields which are a list of spacetime scalars: (φ₁, φ₂, ... φ_N). For example, in weather prediction these may be temperature, pressure, humidity, etc. In particle physics, the color symmetry of the interaction of quarks is an example of an internal symmetry, that of the strong interaction. Other examples are isospin, weak isospin, strangeness and any other flavour symmetry.

If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an internal symmetry. One may also make a classification of the charges of the fields under internal symmetries.

Statistical field theory attempts to extend the field-theoretic paradigm toward many-body systems and statistical mechanics. As above, it can be approached by the usual infinite number of degrees of freedom argument.

Much like statistical mechanics has some overlap between quantum and classical mechanics, statistical field theory has links to both quantum and classical field theories, especially the former with which it shares many methods. One important example is mean field theory.

Classical fields as above, such as the electromagnetic field, are usually infinitely differentiable functions, but they are in any case almost always twice differentiable. In contrast, generalized functions are not continuous. When dealing carefully with classical fields at finite temperature, the mathematical methods of continuous random fields are used, because thermally fluctuating classical fields are nowhere differentiable. Random fields are indexed sets of random variables; a continuous random field is a random field that has a set of functions as its index set. In particular, it is often mathematically convenient to take a continuous random field to have a Schwartz space of functions as its index set, in which case the continuous random field is a tempered distribution.

We can think about a continuous random field, in a (very) rough way, as an ordinary function that is ${\displaystyle \pm \infty }$ almost everywhere, but such that when we take a weighted average of all the infinities over any finite region, we get a finite result. The infinities are not well-defined; but the finite values can be associated with the functions used as the weight functions to get the finite values, and that can be well-defined. We can define a continuous random field well enough as a linear map from a space of functions into the real numbers.