In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
and
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
, the
Reissner–Nordström metric is a
static solutionIn general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field which is irrotational, i.e., whose orthogonal distribution is involutive...
to the
Einstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Einstein's theory of general relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
in empty space, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass
M.
Discovered by
Hans ReissnerHans Jacob Reissner was a German aeronautical engineer whose avocation was mathematical physics. During World War I he was awarded the Iron Cross second class for his pioneering work on aircraft design....
and
Gunnar NordströmGunnar Nordström was a Finnish theoretical physicist who is best remembered for his theory of gravitation, which was an early competitor of general relativity....
, their metric can be written as
where
- τ is the proper time (time measured by a clock moving with the particle) in seconds,
- c is the speed of light
In physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
in meters per second,
- t is the time coordinate (measured by a stationary clock at infinity) in seconds,
- r is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,
- Ω is the solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point. A small object nearby may subtend the same solid angle as a larger object farther away...
,
- rs is the Schwarzschild radius
The Schwarzschild radius is a characteristic radius associated with every quantity of mass...
(in meters) of the massive body, which is related to its mass M by
-
- where G is the gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
, and
- rQ is a length-scale corresponding to the electric charge
Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
Q of the mass
-
- where 1/4πε0 is Coulomb's force constant
Coulomb's law, sometimes called the Coulomb law, is an equation describing the electrostatic force between electric charges. It was studied and first published in the 1780s by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
.
In the limit that the charge
Q (or equivalently, the length-scale
rQ) goes to zero, one recovers the
Schwarzschild metricIn Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or Sun...
.
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
and
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
, the
Reissner–Nordström metric is a
static solutionIn general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field which is irrotational, i.e., whose orthogonal distribution is involutive...
to the
Einstein field equationsThe Einstein field equations or Einstein's equations are a set of ten equations in Einstein's theory of general relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
in empty space, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass
M.
The Metric
Discovered by
Hans ReissnerHans Jacob Reissner was a German aeronautical engineer whose avocation was mathematical physics. During World War I he was awarded the Iron Cross second class for his pioneering work on aircraft design....
and
Gunnar NordströmGunnar Nordström was a Finnish theoretical physicist who is best remembered for his theory of gravitation, which was an early competitor of general relativity....
, their metric can be written as
where
- τ is the proper time (time measured by a clock moving with the particle) in seconds,
- c is the speed of light
In physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...
in meters per second,
- t is the time coordinate (measured by a stationary clock at infinity) in seconds,
- r is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,
- Ω is the solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point. A small object nearby may subtend the same solid angle as a larger object farther away...
,
- rs is the Schwarzschild radius
The Schwarzschild radius is a characteristic radius associated with every quantity of mass...
(in meters) of the massive body, which is related to its mass M by
-
- where G is the gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
, and
- rQ is a length-scale corresponding to the electric charge
Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields...
Q of the mass
-
- where 1/4πε0 is Coulomb's force constant
Coulomb's law, sometimes called the Coulomb law, is an equation describing the electrostatic force between electric charges. It was studied and first published in the 1780s by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
.
In the limit that the charge
Q (or equivalently, the length-scale
rQ) goes to zero, one recovers the
Schwarzschild metricIn Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or Sun...
. The classical Newtonian theory of gravity may then be recovered in the limit as the ratio
rs/
r goes to zero. In that limit, the metric returns to the Minkowski metric for
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
In practice, the ratio
rs/
r is almost always extremely small. For example, the Schwarzschild radius
rs of the
EarthEarth is the third planet from the Sun. It is the fifth largest of the eight planets in the solar system, and the largest of the terrestrial planets in the Solar System in terms of diameter, mass and density...
is roughly 9 mm (³⁄
8 inchAn inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...
), whereas a
satelliteIn the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavor. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....
in a
geosynchronous orbitA geosynchronous orbit is an orbit around the Earth with an orbital period matching the Earth's sidereal rotation period. This synchronization means that for an observer at a fixed location on Earth, a satellite in a geosynchronous orbit returns to exactly the same place in the sky at exactly the...
has a radius
r that is roughly four billion times larger, at 42,164 km (26,200
mileA mile is a unit of length in a number of different systems. In contemporary English, mile most commonly refers to the statute mile of 1,609.344 meters or the nautical mile of 1,852 meters...
s). Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close to
black holeIn general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
s and other ultra-dense objects such as
neutron starA neutron star is a type of remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and roughly the same mass as protons...
s.
Charged black holes
Although charged black holes with are similar to the Schwarzschild black hole, they have two horizons: the
event horizonIn general relativity, an event horizon is a boundary in spacetime, most often an area surrounding a black hole, beyond which events cannot affect an outside observer...
and an internal
Cauchy horizonIn physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem...
. As usual, the event horizons for the spacetime may be reliably located by analyzing the equation
This quadratic equation for
r has the solutions
These concentric
event horizonIn general relativity, an event horizon is a boundary in spacetime, most often an area surrounding a black hole, beyond which events cannot affect an outside observer...
s become
degenerateIn physics two or more different physical states are said to be degenerate if they are all at the same energy level. Physical states differ if and only if they are linearly independent. An energy level is said to be degenerate if it contains two or more different states...
for which corresponds to an
extremal black holeIn theoretical physics, an extremal black hole is a black hole with the minimal possible mass that can be compatible with the given charges and angular momentum....
. Black holes with are believed not to exist in nature because they would contain a
naked singularityIn general relativity, a naked singularity is a gravitational singularity without an event horizon. The singularities inside black holes are always surrounded by an area which does not allow light to escape, and therefore cannot be directly observed...
; their appearance would contradict
Roger PenroseSir Roger Penrose, OM, FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
's
cosmic censorship hypothesisThe weak and the strong Cosmic Censorship Hypotheses are two mathematical conjectures about the structure of singularities arising in general relativity....
which is generally believed to be true. Theories with
supersymmetryIn particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
usually guarantee that such "superextremal" black holes can't exist.
The electromagnetic potential is
If magnetic monopoles are included into the theory, then a generalization to include magnetic charge is obtained by replacing by in the metric and including the term in the electromagnetic potential.
External links