Planck mass

# Planck mass

Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the Planck mass (mP) is the unit of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

in the system of natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

known as 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...

. It is defined so that≈ = , (or ),

where 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...

in a vacuum, 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...

, and ħ is the reduced Planck constant.

Particle physicists
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

and cosmologists
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

often use the reduced Planck mass, which is ≈ = 2.43 × 1018 GeV
GEV
GEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...

/c2.
The added factor of 1/ simplifies a number of equations in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

.

The name honors Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

because the unit measures the approximate scale at which quantum effects, here in the case of gravity, become important. Quantum effects are typified by the magnitude of Planck's constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

, .

## Significance

The Planck mass is approximately the mass of the Planck particle
Planck particle
A Planck particle, named after physicist Max Planck, is a hypothetical particle defined as a tiny black hole whose Compton wavelength is comparable to its Schwarzschild radius. Its mass is thus approximately the Planck mass, and its Compton wavelength and Schwarzschild radius are about the Planck...

, a hypothetical minuscule black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

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...

equals the Planck length.

The Planck mass is the smallest possible mass for a black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

.

Unlike all other Planck base 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...

and most Planck derived units, the Planck mass has a scale more or less conceivable to human
Human
Humans are the only living species in the Homo genus...

s. It is traditionally said to be about the mass of a flea
Flea
Flea is the common name for insects of the order Siphonaptera which are wingless insects with mouthparts adapted for piercing skin and sucking blood...

, but more accurately it is about the mass of a flea egg.

The Planck mass is an idealized mass thought to have special significance for quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

when general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

and the fundamentals of quantum physics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

become mutually important to describe mechanics.

### Dimensional analysis

The formula for the Planck mass can be derived by dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...

s ħ, c, and G, and attempt to combine them to get a quantity with units of mass. The expected formula is of the form
where are constants to be determined by matching the dimensions of both sides. Using the symbol L for length, T for time, M for mass, and writing "[x]" for the dimensions of some physical quantity x, we have the following:.
Therefore,
If we want dimensions of mass, the following equations must hold:.
The solution of this system is:
Thus, the Planck mass is:

### Elimination of a coupling constant

Equivalently, the Planck mass is defined such that the gravitational potential energy between two masses mp of separation r is equal to the energy of a photon (or graviton) of angular wavelength r (see the Planck relation), or that their ratio equals one.
Multiplying through,
This equation has units of energy times length and equals the value , a ubiquitous quantity when deriving the Planck units. Since the two quantities are equal their ratio equals one. From here, it is easy to isolate the mass that would satisfy this equation in our system of units:
Note in the second equation that if instead of planck masses the electron mass were used, the equation would no longer be unitary and instead equal a gravitational coupling constant
Gravitational coupling constant
In physics, the gravitational coupling constant, αG, is the coupling constant characterizing the gravitational attraction between two charged elementary particles having nonzero mass...

, analogous to how the equation of the fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

operates with respect to the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

and the Planck charge
Planck charge
In physics, the Planck charge , is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.The Planck charge is defined as:...

. Thus, the planck mass is an attempt to absorb the gravitational coupling constant into the unit of mass (and those of distance/time as well), as the planck charge does for the fine-structure constant; naturally it is impossible to truly set either of these dimensionless numbers to zero.

### Compton wavelength and Schwarzschild radius

The Planck mass can be derived approximately by setting it as the mass whose 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...

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...

are equal. The Compton wavelength is, loosely speaking, the length-scale where quantum effects
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

start to become important for a particle; the heavier the particle, the smaller the Compton wavelength. The Schwarzschild radius is the radius in which a mass, if confined, would become a black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

; the heavier the particle, the larger the Schwarzschild radius. If a particle were massive enough that its Compton wavelength and Schwarzschild radius were approximately equal, its dynamics would be strongly affected by quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...

. This mass is (approximately) the Planck mass.

The Compton wavelength is
Setting them equal:
This is not quite the Planck mass: It is a factor of larger. However, this is a heuristic derivation, only intended to get the right order of magnitude. On the other hand, the previous "derivation" of the Planck mass should have had a proportional sign in the initial expression rather than an equal sign. Therefore, the extra factor might be the correct one.

• Micro black hole
Micro black hole
Micro black holes are tiny black holes, also called quantum mechanical black holes or mini black holes, for which quantum mechanical effects play an important role....

• Orders of magnitude (mass)
Orders of magnitude (mass)
To help compare different orders of magnitude, the following list describes various mass levels between 10−36 kg and 1060 kg.-Units of mass:...

• Planck length
• Planck particle
Planck particle
A Planck particle, named after physicist Max Planck, is a hypothetical particle defined as a tiny black hole whose Compton wavelength is comparable to its Schwarzschild radius. Its mass is thus approximately the Planck mass, and its Compton wavelength and Schwarzschild radius are about the Planck...

## Sources

1. Sivaram C. WHAT IS SPECIAL ABOUT THE PLANCK MASS? PDF
2. Johnstone Stoney, Phil. Trans. Roy. Soc. 11, (1881)