Editing Classical field theory (section)

== Relativistic fields ==
Two of the most well-known Lorentz-covariant classical field theories are now described.

=== Electromagnetism ===
{{Main|Electromagnetic field|Electromagnetism}}

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]]. [[James Clerk Maxwell|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 {{math|1=''A<sub>a</sub>'' = (−''φ'', '''A''')}}, and the [[four-current|electromagnetic four-current]] {{math|1=''j<sub>a</sub>'' = (−''ρ'', '''j''')}}. The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank [[electromagnetic field tensor]]
<math display="block">F_{ab} = \partial_a A_b - \partial_b A_a.</math>

==== The Lagrangian ====
To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have
<math display="block">\mathcal{L} = -\frac{1}{4\mu_0}F^{ab}F_{ab}\,.</math>

We can use [[gauge field theory]] to get the interaction term, and this gives us
<math display="block">\mathcal{L} = -\frac{1}{4\mu_0}F^{ab}F_{ab} - j^aA_a\,.</math>

==== The equations ====
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,
<math display="block">\partial_b\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_b A_a\right)}\right)=\frac{\partial\mathcal{L}}{\partial A_a} \,.</math>

Evaluating the derivative of the Lagrangian density with respect to the field components
<math display="block">\frac{\partial\mathcal{L}}{\partial A_a} = \mu_0 j^a \,, </math>
and the derivatives of the field components
<math display="block">\frac{\partial\mathcal{L}}{\partial(\partial_b A_a)} = F^{ab} \,, </math>
obtains [[Maxwell's equations]] in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are
<math display="block">\partial_b F^{ab}=\mu_0 j^a \, . </math>
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.<ref>{{Cite web| url=http://mathworld.wolfram.com/BianchiIdentities.html|title=Bianchi Identities}}</ref>
<math display="block">6F_{[ab,c]} \, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0. </math>

where the comma indicates a [[partial derivative]].

=== Gravitation ===
{{Main|Gravitation}}
{{Further|General Relativity|Einstein field equation}}

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 (general relativity)|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,
<math display="block">G_{ab} = \kappa T_{ab} </math>
describe how this curvature is produced by matter and radiation, where ''G<sub>ab</sub>'' is the [[Einstein tensor]],
<math display="block">G_{ab} \, = R_{ab}-\frac{1}{2} R g_{ab}</math>
written in terms of the [[Ricci tensor]] ''R<sub>ab</sub>'' and [[Ricci scalar]] {{math|1=''R'' = ''R<sub>ab</sub>g<sup>ab</sup>''}}, {{math|''T<sub>ab</sub>''}} is the [[stress–energy tensor]] and {{math|1=''κ'' = 8''πG''/''c''<sup>4</sup>}} is a constant. In the absence of matter and radiation (including sources) the '[[vacuum field equations]]'',
<math display="block">G_{ab} = 0 </math>
can be derived by varying the [[Einstein–Hilbert action]],
<math display="block"> S = \int R \sqrt{-g} \, d^4x </math>
with respect to the metric, where ''g'' is the [[determinant]] of the [[metric tensor (general relativity)|metric tensor]] ''g<sup>ab</sup>''. Solutions of the vacuum field equations are called [[vacuum solution]]s. An alternative interpretation, due to [[Arthur Eddington]], is that <math>R</math> is fundamental, <math>T</math> is merely one aspect of <math>R</math>, and <math>\kappa</math> is forced by the choice of units.

=== Further examples ===
Further examples of Lorentz-covariant classical field theories are
* [[Klein-Gordon]] theory for real or complex scalar fields
* [[Dirac equation|Dirac]] theory for a Dirac spinor field
* [[Yang–Mills theory]] for a non-abelian gauge field