Kerr–Newman metric
Most general asymptotically flat stationary electrovacuum solution of Einstein–Maxwell equations.
The Kerr–Newman metric describes the spacetime geometry around a mass that is electrically charged and rotating. It is a vacuum solution that generalizes the Kerr metric by additionally taking into account the energy of an electromagnetic field, making it the most general asymptotically flat and stationary solution of the Einstein–Maxwell equations in general relativity.
- type
- Exact solution in general relativity
- field
- General relativity, Einstein–Maxwell equations
- known_for
- Describing the spacetime of a rotating, electrically charged black hole
- related_solutions
- Kerr metric, Reissner–Nordström metric, Schwarzschild metric, Minkowski space
Lore & Background
About this time Ezra T. Newman found the solution for charged Kerr by guesswork. This formula for the metric tensor is called the Kerr–Newman metric. It is a generalisation of the Kerr metric for an uncharged spinning point-mass, which had been discovered by Roy Kerr two years earlier. The solution contains a singularity in the shape of a ring. The multipole structure of the solution suggests that the solution represents the field of a ring of charge rotating about its axis of symmetry. Similarly the Kerr solution represents the field of a ring of mass. However, for this simple view to be mathematically correct, charge (or mass in the Kerr case) needs to be distributed around the singular ring of the solution to break the multivalued behavior.
Reader's Guide
The Kerr–Newman metric is primarily of theoretical interest. Astronomical objects have axes for rotation and for magnetic fields, but the metric is only valid for co-aligned axes. Any Kerr–Newman source has its rotation axis aligned with its magnetic axis, which differs from commonly observed astronomical bodies such as the Sun or planets, where there is a substantial angle between the rotation axis and the magnetic moment. The model lacks description of infalling baryonic matter, light (null dusts) or dark matter, and thus provides an incomplete description of stellar mass black holes and active galactic nuclei. The solution however is of mathematical interest and provides a fairly simple cornerstone for further exploration. It reduces to the Kerr metric as charge goes to zero, to the Reissner–Nordström metric as angular momentum goes to zero, to the Schwarzschild metric as both charge and angular momentum go to zero, and to Minkowski space if mass, charge, and rotational parameter are all zero. Taking the gravitational constant G to be zero gives an electromagnetic field from a rotating charged disk with a boundary in Minkowski space. The solution is a special case of more general exact solutions that include a cosmological constant, a NUT parameter, and a magnetic charge.
Did You Know?
- The Kerr–Newman metric is an electrovacuum solution that only includes charges associated with the magnetic field; it does not include any free electric charges.
- The solution contains a singularity in the shape of a ring.
- The metric is only valid for co-aligned rotation and magnetic axes, unlike most astronomical bodies.
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