Classical field theory

The first field theory of gravity was Newton's theory of gravitation in which the mutual interaction between two masses obeys an inverse square law. This was very useful for predicting the motion of planets around the Sun.

Any massive body M has a gravitational field g which describes its influence on other massive bodies. The gravitational field of M at a point r in space is found by determining the force F that M exerts on a small test mass m located at r, and then dividing by m:[2]

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

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[2]

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

where ${\displaystyle {\hat {\mathbf {r} }}}$ is a unit vector pointing along the line from M to m, and G is Newton's gravitational constant. Therefore, the gravitational field of M is[2]

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

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

For a discrete collection of masses, M_i, located at points, r_i, the gravitational field at a point r due to the masses is

${\displaystyle \mathbf {g} (\mathbf {r} )=-G\sum _{i}{\frac {M_{i}(\mathbf {r} -\mathbf {r_{i}} )}{|\mathbf {r} -\mathbf {r} _{i}|^{3}}}\,,}$

If we have a continuous mass distribution ρ instead, the sum is replaced by an integral,

${\displaystyle \mathbf {g} (\mathbf {r} )=-G\iiint _{V}{\frac {\rho (\mathbf {x} )d^{3}\mathbf {x} (\mathbf {r} -\mathbf {x} )}{|\mathbf {r} -\mathbf {x} |^{3}}}\,,}$

Note that the direction of the field points from the position r to the position of the masses r_i; this is ensured by the minus sign. In a nutshell, this means all masses attract.

In the integral form Gauss's law for gravity is

${\displaystyle \iint \mathbf {g} \cdot d\mathbf {S} =-4\pi GM}$

while in differential form it is

${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho _{m}}$

Therefore, the gravitational field g can be written in terms of the gradient of a gravitational potential φ(r):

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

This is a consequence of the gravitational force F being conservative.

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 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 is given by the gradient of a scalar potential, V(r)

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

Gauss's law for electricity is in integral form

${\displaystyle \iint \mathbf {E} \cdot d\mathbf {S} ={\frac {q_{e}}{\epsilon _{0}}}}$

while in differential form

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{e}}{\epsilon _{0}}}\,.}$

A steady current I flowing along a path ℓ will exert a force on nearby 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:

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

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):

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

Gauss's law for magnetism in integral form is

${\displaystyle \iint \mathbf {B} \cdot d\mathbf {S} =0,}$

while in differential form it is

${\displaystyle \nabla \cdot \mathbf {B} =0.}$

The physical interpretation is that there are no magnetic 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 the electric charge density (charge per unit volume) ρ and current density (electric current per unit area) J.[3]

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[4]

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

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

Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation is a continuity equation, representing the conservation of mass

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0}$

and the Navier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid,

${\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} )=\nabla \cdot {\boldsymbol {\tau }}+\rho \mathbf {b} }$

if the density ρ, pressure p, deviatoric stress tensor τ of the fluid, as well as external body forces b, are all given. The velocity field u is the vector field to solve for.

Given a field tensor φ, a scalar called the Lagrangian density

${\displaystyle {\mathcal {L}}(\phi ,\partial \phi ,\partial \partial \phi ,\ldots ,x)}$

can be constructed from φ and its derivatives.

From this density, the action functional can be constructed by integrating over spacetime,

${\displaystyle {\mathcal {S}}=\int {{\mathcal {L}}{\sqrt {-g}}\,\mathrm {d} ^{4}x}.}$

Where ${\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x}$ is the volume form in curved spacetime. ${\displaystyle (g\equiv {\text{det}}(g_{\mu \nu }))}$

Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space.

Then by enforcing the action principle, the Euler–Lagrange equations are obtained

${\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \phi }}={\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\right)+\cdots +(-1)^{m}\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\phi )}}\right)=0.}$

Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the electromagnetic field. Maxwell's theory of electromagnetism describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the electric and magnetic fields. With the advent of special relativity, a more complete formulation using tensor fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together is used.

The electromagnetic four-potential is defined to be A_a = (-φ, A), and the electromagnetic four-current j_a = (-ρ, j). The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor

${\displaystyle F_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}.}$

To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have

${\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}F^{ab}F_{ab}\,.}$

We can use gauge field theory to get the interaction term, and this gives us

${\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}F^{ab}F_{ab}-j^{a}A_{a}\,.}$

To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potential A, and it's this potential which enters the Euler-Lagrange equations. The EM field F is not varied in the EL equations. Therefore,

${\displaystyle \partial _{b}\left({\frac {\partial {\mathcal {L}}}{\partial \left(\partial _{b}A_{a}\right)}}\right)={\frac {\partial {\mathcal {L}}}{\partial A_{a}}}\,.}$

Evaluating the derivative of the Lagrangian density with respect to the field components

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{a}}}=\mu _{0}j^{a}\,,}$

and the derivatives of the field components

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{b}A_{a})}}=F^{ab}\,,}$

obtains Maxwell's equations in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are

${\displaystyle \partial _{b}F^{ab}=\mu _{0}j^{a}\,.}$

while the other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that F is the 4-curl of A, or, in other words, from the fact that the Bianchi identity holds for the electromagnetic field tensor.[5]

${\displaystyle 6F_{[ab,c]}\,=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.}$

where the comma indicates a partial derivative.

After Newtonian gravitation was found to be inconsistent with special relativity, Albert Einstein formulated a new theory of gravitation called general relativity. This treats gravitation as a geometric phenomenon ('curved spacetime') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. Newtonian gravitation is now superseded by Einstein's theory of general relativity, in which gravitation is thought of as being due to a curved spacetime, caused by masses. The Einstein field equations,

${\displaystyle G_{ab}=\kappa T_{ab}}$

describe how this curvature is produced by matter and radiation, where G_ab is the Einstein tensor,

${\displaystyle G_{ab}\,=R_{ab}-{\frac {1}{2}}Rg_{ab}}$

written in terms of the Ricci tensor R_ab and Ricci scalar R = R_abg^ab, T_ab is the stress-energy tensor and κ = 8πG/c⁴ is a constant. In the absence of matter and radiation (including sources) the 'vacuum field equations,

${\displaystyle G_{ab}=0\,}$

can be derived by varying the Einstein–Hilbert action,

${\displaystyle S=\int R{\sqrt {-g}}\,d^{4}x}$

with respect to the metric, where g is the determinant of the metric tensor g^ab. Solutions of the vacuum field equations are called vacuum solutions. An alternative interpretation, due to Arthur Eddington, is that ${\displaystyle R}$ is fundamental, ${\displaystyle T}$ is merely one aspect of ${\displaystyle R}$, and ${\displaystyle \kappa }$ is forced by the choice of units.