Planck force
Encyclopedia
Planck force is the derived unit of force
resulting from the definition of the base Planck units
for time, length, and mass. It is equal to the natural unit of momentum
divided by the natural unit of time.
:
,
,
where G is the gravitational constant
, c is the speed of light
, m is any mass and rG is half the Schwarzschild radius, rs, of the given mass.
Since the dimension of force is also a ratio of energy per length, the Planck force can be calculated as energy divided by half the Schwarzschild radius:
As mentioned above, Planck force has a unique association with the Planck mass. This unique association also manifests itself when force is calculated as any energy divided by the reduced Compton wavelength
(reduced by 2π) of that same energy:
Here the force is different for every mass (for the electron, for example, the force is responsible for the Schwinger effect (see page 3 here http://prst-ab.aps.org/pdf/PRSTAB/v5/i3/e031301). It is Planck force only for the Planck mass (approximately 2.18 × 10-8 kg). This follows from the fact that the Planck length is a reduced Compton wavelength equal to half the Schwarzschild radius of the Planck mass:
which in turn follows from another relation of fundamental significance:
, describing the properties of a gravitational field surrounding any given mass:
where
is the Einstein tensor
and
is the energy-momentum tensor.
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
resulting from the definition of the base Planck units
Planck units
In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...
for time, length, and mass. It is equal to the natural unit of momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
divided by the natural unit of time.
Other derivations
The Planck force is also associated with the equivalence of gravitational potential energy and electromagnetic energy http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/blahol.html#c2 and in this context it can be understood as the force that confines a self-gravitating mass to half its Schwarzschild radiusSchwarzschild radius
The Schwarzschild radius is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light...
:
,,where G is the gravitational constant
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...
, c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, m is any mass and rG is half the Schwarzschild radius, rs, of the given mass.
Since the dimension of force is also a ratio of energy per length, the Planck force can be calculated as energy divided by half the Schwarzschild radius:
As mentioned above, Planck force has a unique association with the Planck mass. This unique association also manifests itself when force is calculated as any energy divided by the reduced Compton wavelength
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...
(reduced by 2π) of that same energy:
Here the force is different for every mass (for the electron, for example, the force is responsible for the Schwinger effect (see page 3 here http://prst-ab.aps.org/pdf/PRSTAB/v5/i3/e031301). It is Planck force only for the Planck mass (approximately 2.18 × 10-8 kg). This follows from the fact that the Planck length is a reduced Compton wavelength equal to half the Schwarzschild radius of the Planck mass:
which in turn follows from another relation of fundamental significance:
General relativity
Planck force is often useful in scientific calculations as a ratio of electromagnetic energy per gravitational length. Thus for example it appears in the Einstein field equationsEinstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
, describing the properties of a gravitational field surrounding any given mass:
where
is the Einstein tensorEinstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
and
is the energy-momentum tensor.