# 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).

## Examples

- 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 4-vector vanishes:
^{[1]}

- Einstein's theory of general relativity where 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]}

- Kaluza–Klein vacuum (static) field equations
^{[4]}

- Kasner vacuum solution
^{[5]}

## See also

- Einstein field equations
- Kasner metric
- Kerr metric
- Maxwell's equations
- Vacuum solution (general relativity)

## Notes

- ↑ Esposito, S. (1997), "Classical vgr? c solutions of Maxwell's equations and the photon tunneling effect",
*Physics Letters A*,**225**(4-6): 203–209, arXiv:physics/9611018, Bibcode:1997PhLA..225..203E, doi:10.1016/S0375-9601(96)00872-9, retrieved 2009-07-04 - ↑ Stephani, H. (2003),
*Exact solutions of Einstein's field equations*(PDF), retrieved 2009-07-04 - ↑ Quevedo, H. (1990), "Multipole Moments in General Relativity-Static and Stationary Vacuum Solutions",
*Fortschritte der Physik*,**38**(10): 733, Bibcode:1990ForPh..38..733Q, doi:10.1002/prop.2190381002, retrieved 2009-07-04 - ↑ Sorkin, R.D. (1983), "Kaluza-klein monopole",
*Physical Review Letters*,**51**(2): 87–90, Bibcode:1983PhRvL..51...87S, doi:10.1103/PhysRevLett.51.87 - ↑ Chodos, A.; Detweiler, S. (1980), "Where has the fifth dimension gone?",
*Physical Review D*,**21**(8): 2167–2170, Bibcode:1980PhRvD..21.2167C, doi:10.1103/PhysRevD.21.2167

## References

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

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