Vacuum solution

A vacuum solution is a solution of a field equation in which the sources of the field are taken to be identically zero. That is, such field equations are written without matter interaction (i.e.- set to zero).

## Contents

* 1 Examples
* 1.1 Maxwell's equations
* 1.2 Einstein field equations
* 1.2.1 Black hole vacuum solution
* 1.2.2 Kasner space
* 1.3 Kaluza-Klein theory
* 2 See also
* 3 Notes
* 4 References

## Examples

### Maxwell's equations

Main page: Physics:Maxwell's equations

In Maxwell's theory of electromagnetism, a vacuum solution would represent the electromagnetic field in a region of space where there are no electromagnetic sources (charges and electric currents), i.e. where the current[clarification needed] 4-vector vanishes:[1]

[math]\displaystyle{ J^a=0 }[/math]

### Einstein field equations

In Einstein's theory of general relativity, a vacuum solution[2] would represent the gravitational field in a region of spacetime where there are no gravitational sources (masses), i.e. where the energy–momentum tensor vanishes:[3]

[math]\displaystyle{ T_{ab}=0 }[/math]

#### Black hole vacuum solution

Main page: Astronomy:Kerr metric

#### Kasner space

Main page: Kasner metric

Kasner vacuum solution[4]

### Kaluza-Klein theory

In a Kaluza–Klein vacuum (static) field equations[5]

## See also

* Vacuum solution (general relativity)
* Topological defect

## Notes

1. ↑ Esposito, S. (1997), "Classical vgr? c solutions of Maxwell's equations and the photon tunneling effect", Physics Letters A 225 (4-6): 203–209, doi:10.1016/S0375-9601(96)00872-9, Bibcode: 1997PhLA..225..203E, http://linkinghub.elsevier.com/retrieve/pii/S0375960196008729, retrieved 2009-07-04
2. ↑ Stephani, H. (2003), Exact solutions of Einstein's field equations, https://www.loc.gov/catdir/samples/cam033/2002071495.pdf, retrieved 2009-07-04
3. ↑ Quevedo, H. (1990), "Multipole Moments in General Relativity-Static and Stationary Vacuum Solutions", Fortschritte der Physik 38 (10): 733, doi:10.1002/prop.2190381002, Bibcode: 1990ForPh..38..733Q, http://www3.interscience.wiley.com/journal/112587363/abstract, retrieved 2009-07-04
4. ↑ Chodos, A.; Detweiler, S. (1980), "Where has the fifth dimension gone?", Physical Review D 21 (8): 2167–2170, doi:10.1103/PhysRevD.21.2167, Bibcode: 1980PhRvD..21.2167C
5. ↑ Sorkin, R.D. (1983), "Kaluza-klein monopole", Physical Review Letters 51 (2): 87–90, doi:10.1103/PhysRevLett.51.87, Bibcode: 1983PhRvL..51...87S

## References

* Gott, J.R.; Richard, J. (1985), "Gravitational lensing effects of vacuum strings- Exact solutions", Astrophysical Journal 288 (Part 1), doi:10.1086/162808, Bibcode: 1985ApJ...288..422G
* Friedrich, H.; Nagy, G. (1999), "The initial boundary value problem for Einstein's vacuum field equation", Communications in Mathematical Physics 201 (3): 619–655, doi:10.1007/s002200050571, Bibcode: 1999CMaPh.201..619F, http://math.msu.edu/~gnagy/papers/hFgN99.pdf, retrieved 2009-07-04
* Appelquist, T.; Chodos, A. (1983), "Quantum dynamics of Kaluza-Klein theories", Physical Review D 28 (4): 772–784, doi:10.1103/PhysRevD.28.772, Bibcode: 1983PhRvD..28..772A

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