Reissner-Nordström Metric
Introduction
The Reissner-Nordström metric is a solution to Einstein's field equations which describes the spacetime geometry in the vicinity of a charged, non-rotating, spherically symmetric mass. This solution was first discovered independently by Hans Reissner and Gunnar Nordström, and thus bears their names.
Mathematical Formulation
The Reissner-Nordström metric is expressed in spherical coordinates (t, r, θ, φ), where t represents time, r is the radial distance, θ is the polar angle, and φ is the azimuthal angle. The metric is given by:
ds² = - (1 - 2M/r + Q²/r²) dt² + (1 - 2M/r + Q²/r²)⁻¹ dr² + r²(dθ² + sin²θ dφ²)
Here, M is the mass of the object, Q is its charge, and ds² is the spacetime interval. This equation describes the geometry of spacetime around the object.
Properties of the Reissner-Nordström Metric
The Reissner-Nordström metric has several interesting properties that distinguish it from the Schwarzschild metric, which describes an uncharged, non-rotating mass.
Event Horizons
The Reissner-Nordström metric possesses two event horizons, located at:
r± = M ± √(M² - Q²)
These are the radii at which the metric coefficient for the radial coordinate changes sign. For a black hole with charge Q, these represent the outer (r+) and inner (r-) event horizons.
Naked Singularity
If the charge Q of the black hole exceeds its mass M (i.e., Q² > M²), then the square root term in the event horizon equations becomes imaginary, and the event horizons disappear. This results in a naked singularity, a point in spacetime where the curvature becomes infinite and the laws of physics as currently understood cease to be applicable.
Comparison with Other Metrics
The Reissner-Nordström metric shares similarities with other solutions to Einstein's field equations, such as the Schwarzschild metric and the Kerr metric. However, it also exhibits unique features due to the presence of charge.
Schwarzschild Metric
The Schwarzschild metric describes the spacetime around an uncharged, non-rotating mass. When the charge Q in the Reissner-Nordström metric is set to zero, the metric reduces to the Schwarzschild metric.
Kerr Metric
The Kerr metric describes the spacetime around a rotating mass. The Reissner-Nordström metric can be seen as a special case of the Kerr-Newman metric (which includes both charge and rotation) where the rotation parameter is set to zero.
Applications and Implications
The Reissner-Nordström metric has various applications in the field of general relativity and theoretical physics.
Black Hole Physics
The Reissner-Nordström metric is used to model black holes with charge. While most astrophysical black holes are expected to be nearly neutral due to the abundance of both positive and negative charges in the universe, the Reissner-Nordström solution provides valuable insights into the properties of charged black holes.
Quantum Field Theory
In the context of quantum field theory in curved spacetime, the Reissner-Nordström metric is used to study the effects of charge and gravity on quantum fields.