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]

\( J^a=0 \)

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]

\( T_{ab}=0 \)

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