Vacuum solution

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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).


Maxwell's equations[edit]

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]

Einstein field equations[edit]

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]

Black hole vacuum solution[edit]

Kasner space[edit]

Kasner vacuum solution[4]

Kaluza–Klein theory[edit]

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

See also[edit]


  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, arXiv:physics/9611018, Bibcode:1997PhLA..225..203E, doi:10.1016/S0375-9601(96)00872-9, retrieved 2009-07-04
  2. ^ Stephani, H. (2003), Exact solutions of Einstein's field equations (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, Bibcode:1990ForPh..38..733Q, doi:10.1002/prop.2190381002, retrieved 2009-07-04
  4. ^ 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
  5. ^ Sorkin, R.D. (1983), "Kaluza-klein monopole", Physical Review Letters, 51 (2): 87–90, Bibcode:1983PhRvL..51...87S, doi:10.1103/PhysRevLett.51.87