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{{general relativity}} | |||
In [[general relativity]], a '''manifestly covariant''' equation is one in which all expressions are [[tensor]]s. The operations of addition, [[tensor multiplication]], [[tensor contraction]], [[raising and lowering indices]], and [[covariant differentiation]] may appear in the equation. Forbidden terms include but are not restricted to [[partial derivatives]]. [[Tensor density|Tensor densities]], especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the [[determinant]] of the metric. | |||
Writing an equation in manifestly covariant form is useful because it guarantees [[general covariance]] upon quick inspection. If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation in [[special relativity]] when evaluated instantaneously in a [[local inertial frame]], then it is usually the correct generalization of the special relativistic equation in general relativity. | |||
== Example == | |||
An equation may be [[Lorentz covariance|Lorentz covariant]] even if it is not manifestly covariant. Consider the [[electromagnetic field tensor]] | |||
:<math>F_{ab} \, = \, \partial_a A_b \, - \, \partial_b A_a \,</math> | |||
where <math>A_a</math> is the [[electromagnetic four-potential]] in the [[Lorenz gauge]]. The equation above contains partial derivatives and is therefore not manifestly covariant. Note that the partial derivatives may be written in terms of covariant derivatives and [[Christoffel symbol]]s as | |||
:<math>\partial_a A_b = \nabla_a A_b + \Gamma^c_{ab} A_c</math> | |||
:<math>\partial_b A_a = \nabla_b A_a + \Gamma^c_{ba} A_c</math> | |||
For a [[Torsion tensor|torsion]]-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols | |||
:<math>\Gamma^c_{ab} - \Gamma^c_{ba} = 0,</math> | |||
which allows the field tensor to be written in manifestly covariant form | |||
:<math>F_{ab} \, = \, \nabla_a A_b \, - \, \nabla_b A_a .</math> | |||
==See also== | |||
*[[Lorentz covariance]] | |||
*[[Introduction to mathematics of general relativity]] | |||
*[[Introduction to special relativity]] | |||
==References== | |||
{{reflist}} | |||
* {{cite book|page=| author=C. B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill|edition=2nd| year=1994| isbn=0-07-051400-3}} | |||
* {{cite book|title=[[Gravitation (book)|Gravitation]]|author1=John Archibald Wheeler|author2=C. Misner|author3=K. S. Thorne|authorlink1=John Archibald Wheeler|authorlink2=Charles W. Misner|authorlink3=Kip Thorne|publisher=W.H. Freeman & Co|year=1973|isbn=0-7167-0344-0}} | |||
[[Category:General relativity]] | |||
[[Category:Tensors]] | |||
Latest revision as of 16:33, 16 September 2013
Diving Coach (Open water ) Dominic from Kindersley, loves to spend some time classic cars, property developers in singapore house for rent (Source Webpage) and greeting card collecting. Finds the world an interesting place having spent 8 days at Cidade Velha. In general relativity, a manifestly covariant equation is one in which all expressions are tensors. The operations of addition, tensor multiplication, tensor contraction, raising and lowering indices, and covariant differentiation may appear in the equation. Forbidden terms include but are not restricted to partial derivatives. Tensor densities, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the determinant of the metric.
Writing an equation in manifestly covariant form is useful because it guarantees general covariance upon quick inspection. If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation in special relativity when evaluated instantaneously in a local inertial frame, then it is usually the correct generalization of the special relativistic equation in general relativity.
Example
An equation may be Lorentz covariant even if it is not manifestly covariant. Consider the electromagnetic field tensor
where is the electromagnetic four-potential in the Lorenz gauge. The equation above contains partial derivatives and is therefore not manifestly covariant. Note that the partial derivatives may be written in terms of covariant derivatives and Christoffel symbols as
For a torsion-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols
which allows the field tensor to be written in manifestly covariant form
See also
- Lorentz covariance
- Introduction to mathematics of general relativity
- Introduction to special relativity
References
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