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Concept
Electrovacuum solution
Résumé
In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) source-free Maxwell equations appropriate to the given geometry. For this reason, electrovacuums are sometimes called (source-free) Einstein–Maxwell solutions.
In general relativity, the geometric setting for physical phenomena is a Lorentzian manifold, which is interpreted as a curved spacetime, and which is specified by defining a metric tensor (or by defining a frame field). The Riemann curvature tensor of this manifold and associated quantities such as the Einstein tensor , are well-defined. In general relativity, they can be interpreted as geometric manifestations (curvature and forces) of the gravitational field.
We also need to specify an electromagnetic field by defining an electromagnetic field tensor on our Lorentzian manifold. To be classified as an electrovacuum solution, these two tensors are required to satisfy two following conditions
The electromagnetic field tensor must satisfy the source-free curved spacetime Maxwell field equations and
The Einstein tensor must match the electromagnetic stress–energy tensor, .
The first Maxwell equation is satisfied automatically if we define the field tensor in terms of an electromagnetic potential vector . In terms of the dual covector (or potential one-form) and the electromagnetic two-form, we can do this by setting . Then we need only ensure that the divergences vanish (i.e. that the second Maxwell equation is satisfied for a source-free field) and that the electromagnetic stress–energy matches the Einstein tensor.
The electromagnetic field tensor is antisymmetric, with only two algebraically independent scalar invariants,
Here, the star is the Hodge star.
Using these, we can classify the possible electromagnetic fields as follows:
If but , we have an electrostatic field, which means that some observers will measure a static electric field, and no magnetic field.
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