Kerr–Newman metric

The Kerr–Newman metric describes the space and time around a mass that is both spinning and carrying an electric charge. It builds on the work of the Kerr metric, which looks at spinning masses without charge, by adding the effects of electric and magnetic fields.

This idea is mostly important for theory and math rather than for real objects we see in space. Real stars and black holes have their spinning and magnetic fields pointing in different directions, which the Kerr–Newman metric does not handle well. It also does not include details about normal matter, the light around black holes, or dark matter.

Even so, the Kerr–Newman metric is very useful for mathematicians and scientists. It gives a clear and simple starting point for studying more complex ideas about space, time, and gravity in the universe.

History

In 1963, scientists Roy Kerr and Alfred Schild found new ways to describe space and time around objects. In 1965, Ezra Newman found a way to describe space around a spinning object that also has an electric charge. This important discovery is called the Kerr–Newman metric. It builds on earlier work by Kerr, who had described uncharged, spinning objects.

Overview of the solution

Newman's result shows the simplest way that space and time can behave around a spinning object with an electric field, following Einstein's equations. This is sometimes called an "electrovacuum" solution.

This solution suggests the presence of a ring-shaped point where the math doesn't work out perfectly. It seems to describe the field around a spinning ring of charge. Unlike most objects we see in space, such as the Sun or planets in our Solar System, the spinning axis and the magnetic field in this solution line up perfectly.

Limiting cases

The Kerr–Newman metric can become simpler in special situations. When the charge Q becomes zero, it turns into the Kerr metric. If the spin J becomes zero, it changes into the Reissner–Nordström metric. When both the charge Q and spin J are zero, it becomes the Schwarzschild metric. Finally, if the mass M, charge Q, and spin a are all zero, it reduces to Minkowski space.

The four related solutions are shown in the following table:

Here, Q represents the body's electric charge and J is its spin angular momentum. If the gravitational constant G is set to zero, the Kerr–Newman solution describes an electromagnetic field from a rotating charged disk in Minkowski space. The Kerr–Newman solution is also a special case of more general solutions that include a cosmological constant, a Newman, Unti, Tamburino (NUT) parameter, and a magnetic charge.

Metric field

The Kerr–Newman metric describes the space around a rotating black hole that has mass, charge, and angular momentum. This helps us learn how such a black hole changes the space around it.

The formula for this metric changes depending on the coordinates used. Two common types of coordinates are Boyer–Lindquist coordinates and Kerr–Schild coordinates. To understand this better, we also look at the electromagnetic stress tensor, which explains the electric and magnetic fields around the black hole. Both the metric and the electromagnetic stress tensor are shown in the sections below.

Boyer–Lindquist coordinates

Main article: Boyer–Lindquist coordinates

This section talks about a special way to describe the space around a spinning, charged object using math. It uses special coordinates called Boyer–Lindquist coordinates to make the equations easier to understand. These coordinates help scientists study how gravity and electricity work together around such objects.

The math uses symbols like r, θ, and ϕ to show how time and space change near a spinning, charged mass. It also includes special numbers like a, ρ, and Δ that help simplify the equations. These numbers relate to the mass, spin, and charge of the object.

Kerr–Schild coordinates

The Kerr–Newman metric can be shown in a special way using coordinates called Kerr–Schild coordinates. These were created by Kerr and Schild in 1965.

This form uses special directions to describe the space around a spinning, charged object.

The description includes a few important pieces. There is a value for mass, a value for charge, and a number that shows how much the object spins. These values help us understand how space and time change near the object. The equations also include a special way to measure distance.

The description also includes how electric and magnetic fields behave around this object. These fields come from a special formula. Even though the math looks complex, it gives us clues about how charged, spinning objects shape the space around them.

Irreducible mass

The total mass of a spinning, charged object includes energy from its electric field and rotation. This total mass is always larger than the irreducible mass. When we take away energy, such as by spinning a black hole, the remaining mass can never go below this irreducible mass.

Important surfaces

Setting 1/g_rr to 0 and solving for r gives the inner and outer event horizon. This is found at the Boyer–Lindquist coordinate

r_H^± = r_s/2 ± √(r_s²/4 − a² − r_Q²).

Doing the same with g_tt gives the inner and outer ergosurface

r_E^± = r_s/2 ± √(r_s²/4 − a² cos²θ − r_Q²).

The area between the event horizon and the ergosurface is called the ergosphere. Inside the ergosphere, all light cones tilt toward the direction of rotation.^: 18

Equations of motion

The equations of motion tell us how a tiny charged particle moves around a spinning, charged black hole. These equations use special values like E for the particle's total energy and L_z for its spin around the black hole.

There are special rules that stay the same no matter where the particle is, called conserved quantities. These help scientists understand how particles move near black holes.

Extremal solutions and naked singularity

When some special rules are followed, such as when ( M^{2}-(J/M)^{2}-Q^{2}=0 ), the solution is called "extremal." If these values get too big, the object might not have an event horizon. This means it would not act like a normal black hole and might act in strange ways.

Dirac–Kerr–Newman electron model

Scientists found that the Kerr–Newman solution for a spinning, charged object gives a special number, called the gyromagnetic ratio. This number matches what we see with electrons. This idea helped create models of electrons using the rules of space and time. These models suggest electrons might have extra properties, but we have not found proof of these properties in experiments yet.