Happy New Year 2013 to everyone and everywhere!
Let me apologize, first of all, by my absence… I have been busy, trying to find my path and way in my field, and I am busy yet, but finally I could not resist without a new blog boost… After all, you should know the fact I have enough materials to write many new things.
So, what’s next? I will dedicate some blog posts to discuss a nice topic I began before, talking about a classic paper on the subject here:
The topic is going to be pretty simple: natural units in Physics.
First of all, let me point out that the election of any system of units is, a priori, totally conventional. You are free to choose any kind of units for physical magnitudes. Of course, that is not very clever if you have to report data, so everyone can realize what you do and report. Scientists have some definitions and popular systems of units that make the process pretty simpler than in the daily life. Then, we need some general conventions about “units”. Indeed, the traditional wisdom is to use the international system of units, or S (Iabbreviated SI from French language: Le Système international d’unités). There, you can find seven fundamental magnitudes and seven fundamental (or “natural”) units:
5) Electric intensity:
6) Luminous intensity:
7) Amount of substance:
The dependence between these 7 great units and even their definitions can be found here http://en.wikipedia.org/wiki/International_System_of_Units and references therein. I can not resist to show you the beautiful graph of the 7 wonderful units that this wikipedia article shows you about their “interdependence”:
In Physics, when you build a radical new theory, generally it has the power to introduce a relevant scale or system of units. Specially, the Special Theory of Relativity, and the Quantum Mechanics are such theories. General Relativity and Statistical Physics (Statistical Mechanics) have also intrinsic “universal constants”, or, likely, to be more precise, they allow the introduction of some “more convenient” system of units than those you have ever heard ( metric system, SI, MKS, cgs, …). When I spoke about Barrow units (see previous comment above) in this blog, we realized that dimensionality (both mathematical and “physical”), and fundamental theories are bound to the election of some “simpler” units. Those “simpler” units are what we usually call “natural units”. I am not a big fan of such terminology. It is confusing a little bit. Maybe, it would be more interesting and appropiate to call them “addapted X units” or “scaled X units”, where X denotes “relativistic, quantum,…”. Anyway, the name “natural” is popular and it is likely impossible to change the habits.
In fact, we have to distinguish several “kinds” of natural units. First of all, let me list “fundamental and universal” constants in different theories accepted at current time:
1. Boltzmann constant: .
Essential in Statistical Mechanics, both classical and quantum. It measures “entropy”/”information”. The fundamental equation is:
It provides a link between the microphysics and the macrophysics ( it is the code behind the equation above). It can be understood somehow as a measure of the “energetic content” of an individual particle or state at a given temperature. Common values for this constant are:
Statistical Physics states that there is a minimum unit of entropy or a minimal value of energy at any given temperature. Physical dimensions of this constant are thus entropy, or since , , where t denotes here dimension of temperature.
2. Speed of light. .
From classical electromagnetism:
The speed of light, according to the postulates of special relativity, is a universal constant. It is frame INDEPENDENT. This fact is at the root of many of the surprising results of special relativity, and it took time to be understood. Moreover, it also connects space and time in a powerful unified formalism, so space and time merge into spacetime, as we do know and we have studied long ago in this blog. The spacetime interval in a D=3+1 dimensional space and two arbitrary events reads:
In fact, you can observe that “c” is the conversion factor between time-like and space-like coordinates. How big the speed of light is? Well, it is a relatively large number from our common and ordinary perception. It is exactly:
although you often take it as . However, it is the speed of electromagnetic waves in vacuum, no matter where you are in this Universe/Polyverse. At least, experiments are consistent with such an statement. Moreover, it shows that is also the conversion factor between energy and momentum, since
and is the conversion factor between rest mass and pure energy, because, as everybody knows, ! According to the special theory of relativity, normal matter can never exceed the speed of light. Therefore, the speed of light is the maximum velocity in Nature, at least if specially relativity holds. Physical dimensions of c are , where L denotes length dimension and T denotes time dimension (please, don’t confuse it with temperature despite the capital same letter for both symbols).
3. Planck’s constant. or generally rationalized .
Planck’s constant (or its rationalized version), is the fundamental universal constant in Quantum Physics (Quantum Mechanics, Quantum Field Theory). It gives
Indeed, quanta are the minimal units of energy. That is, you can not divide further a quantum of light, since it is indivisible by definition! Furthermore, the de Broglie relationship relates momentum and wavelength for any particle, and it emerges from the combination of special relativity and the quantum hypothesis:
In the case of massive particles, it yields
In the case of massless particles (photons, gluons, gravitons,…)
Planck’s constant also appears to be essential to the uncertainty principle of Heisenberg:
Some particularly important values of this constant are:
It is also useful to know that
Planck constant has dimension of . Physical dimensions of this constant coincide also with angular momentum (spin), i.e., with .
4. Gravitational constant. .
Apparently, it is not like the others but it can also define some particular scale when combined with Special Relativity. Without entering into further details (since I have not discussed General Relativity yet in this blog), we can calculate the escape velocity of a body moving at the speed of light
with implies a new length scale where gravitational relativistic effects do appear, the so-called Schwarzschild radius :
5. Electric fundamental charge. .
It is generally chosen as fundamental charge the electric charge of the positron (positive charged “electron”). Its value is:
where C denotes Coulomb. Of course, if you know about quarks with a fraction of this charge, you could ask why we prefer this one. Really, it is only a question of hystory of Science, since electrons were discovered first (and positrons). Quarks, with one third or two thirds of this amount of elementary charge, were discovered later, but you could define the fundamental unit of charge as multiple or entire fraction of this charge. Moreover, as far as we know, electrons are “elementary”/”fundamental” entities, so, we can use this charge as unit and we can define quark charges in terms of it too. Electric charge is not a fundamental unit in the SI system of units. Charge flow, or electric current, is.
An amazing property of the above 5 constants is that they are “universal”. And, for instance, energy is related with other magnitudes in theories where the above constants are present in a really wonderful and unified manner:
Caution: k is not the Boltzmann constant but the wave number.
There is a sixth “fundamental” constant related to electromagnetism, but it is also related to the speed of light, the electric charge and the Planck’s constant in a very sutble way. Let me introduce you it too…
6. Coulomb constant. .
This is a second constant related to classical electromagnetism, like the speed of light in vacuum. Coulomb’s constant, the electric force constant, or the electrostatic constant (denoted ) is a proportionality factor that takes part in equations relating electric force between point charges, and indirectly it also appears (depending on your system of units) in expressions for electric fields of charge distributions. Coulomb’s law reads
Its experimental value is
Generally, the Coulomb constant is dropped out and it is usually preferred to express everything using the electric permitivity of vacuum and/or numerical factors depending on the pi number if you choose the gaussian system of units (read this wikipedia article http://en.wikipedia.org/wiki/Gaussian_system_of_units ), the CGS system, or some hybrid units based on them.
High Energy Physicists use to employ units in which the velocity is measured in fractions of the speed of light in vacuum, and the action/angular momentum is some multiple of the Planck’s constant. These conditions are equivalent to set
Complementarily, or not, depending on your tastes and preferences, you can also set the Boltzmann’s constant to the unit as well
and thus the complete HEP system is defined if you set
This “natural” system of units is lacking yet a scale of energy. Then, it is generally added the electron-volt as auxiliary quantity defining the reference energy scale. Despite the fact that this is not a “natural unit” in the proper sense because it is defined by a natural property, the electric charge, and the anthropogenic unit of electric potential, the volt. The SI prefixes multiples of eV are used as well: keV, MeV, GeV, etc. Here, the eV is used as reference energy quantity, and with the above election of “elementary/natural units” (or any other auxiliary unit of energy), any quantity can be expressed. For example, a distance of 1 m can be expressed in terms of eV, in natural units, as
This system of units have remarkable conversion factors
A) of length is equal to
B) of mass is equal to
C) of time is equal to
D) of temperature is equal to
E) of electric charge in the Lorentz-Heaviside system of units is equal to
F) of electric charge in the Gaussian system of units is equal to
This system of units, therefore, leaves free only the energy scale (generally it is chosen the electron-volt) and the electric measure of fundamentl charge. Every other unit can be related to energy/charge. It is truly remarkable than doing this (turning invisible the above three constants) you can “unify” different magnitudes due to the fact these conventions make them equivalent. For instance, with natural units:
It is due to , and equations. Setting and or provides
, and .
Note that natural units turn invisible the units we set to the unit! That is the key of the procedure. It simplifies equations and expressions. Of course, you must be careful when you reintroduce constants!
It is due to , and again.
One extra bonus for theoretical physicists is that natural units allow to build and write proper lagrangians and hamiltonians (certain mathematical operators containing the dynamics of the system enconded in them), or equivalently the action functional, with only the energy or “mass” dimension as “free parameter”. Let me show how it works.
Natural units in HEP identify length and time dimensions. Thus . Planck’s constant allows us to identify those 2 dimensions with 1/Energy (reciprocals of energy) physical dimensions. Therefore, in HEP units, we have
The speed of light identifies energy and mass, and thus, we can often heard about “mass-dimension” of a lagrangian in the following sense. HEP units can be thought as defining “everything” in terms of energy, from the pure dimensional ground. That is, every physical dimension is (in HEP units) defined by a power of energy:
Thus, we can refer to any magnitude simply saying the power of such physical dimension (or you can think logarithmically to understand it easier if you wish). With this convention, and recalling that energy dimension is mass dimension, we have that
Using these arguments, the action functional is a pure dimensionless quantity, and thus, in D=4 spacetime dimensions, lagrangian densities must have dimension 4 ( or dimension D is a general spacetime).
In D=4 spacetime dimensions, it can be easily showed that
where is a scalar field, is a vector field (like the electromagnetic or non-abelian vector gauge fields), and are a Dirac spinor, a Majorana spinor, and are Weyl spinors (of different chiralities). Supersymmetry (or SUSY) allows for anticommuting c-numbers (or Grassmann numbers) and it forces to introduce auxiliary parameters with mass dimension . They are the so-called SUSY transformation parameters . There are some speculative spinors called ELKO fields that could be non-standandard spinor fields with mass dimension one! But it is an advanced topic I am not going to discuss here today. In general D spacetime dimensions a scalar (or vector) field would have mass dimension , and a spinor/fermionic field in D dimensions has generally mass dimension (excepting the auxiliary SUSY grassmanian fields and the exotic idea of ELKO fields). This dimensional analysis is very useful when theoretical physicists build up interacting lagrangians, since we can guess the structure of interaction looking at purely dimensional arguments every possible operator entering into the action/lagrangian density! In summary, therefore, for any D:
Remark (for QFT experts only): Don’t confuse mass dimension with the final transverse polarization degrees or “degrees of freedom” of a particular field, i.e., “components” minus “gauge constraints”. E.g.: a gauge vector field has degrees of freedom in D dimensions. They are different concepts (although both closely related to the spacetime dimension where the field “lives”).
i) HEP units are based on QM (Quantum Mechanics), SR (Special Relativity) and Statistical Mechanics (Entropy and Thermodynamics).
ii) HEP units need to introduce a free energy scale, and it generally drives us to use the eV or electron-volt as auxiliary energy scale.
iii) HEP units are useful to dimensional analysis of lagrangians (and hamiltonians) up to “mass dimension”.
In Physics, the Stoney units form a alternative set of natural units named after the Irish physicist George Johnstone Stoney, who first introduced them as we know it today in 1881. However, he presented the idea in a lecture entitled “On the Physical Units of Nature” delivered to the British Association before that date, in 1874. They are the first historical example of natural units and “unification scale” somehow. Stoney units are rarely used in modern physics for calculations, but they are of historical interest but some people like Wilczek has written about them (see, e.g., http://arxiv.org/abs/0708.4361). These units of measurement were designed so that certain fundamental physical constants are taken as reference basis without the Planck scale being explicit, quite a remarkable fact! The set of constants that Stoney used as base units is the following:
A) Electric charge, .
B) Speed of light in vacuum, .
C) Gravitational constant, .
D) The Reciprocal of Coulomb constant, .
Stony units are built when you set these four constants to the unit, i.e., equivalently, the Stoney System of Units (S) is determined by the assignments:
Interestingly, in this system of units, the Planck constant is not equal to the unit and it is not “fundamental” (Wilczek remarked this fact here ) but:
Today, Planck units are more popular Planck than Stoney units in modern physics, and even there are many physicists who don’t know about the Stoney Units! In fact, Stoney was one of the first scientists to understand that electric charge was quantized!; from this quantization he deduced the units that are now named after him.
The Stoney length and the Stoney energy are collectively called the Stoney scale, and they are not far from the Planck length and the Planck energy, the Planck scale. The Stoney scale and the Planck scale are the length and energy scales at which quantum processes and gravity occur together. At these scales, a unified theory of physics is thus likely required. The only notable attempt to construct such a theory from the Stoney scale was that of H. Weyl, who associated a gravitational unit of charge with the Stoney length and who appears to have inspired Dirac’s fascination with the large number hypothesis. Since then, the Stoney scale has been largely neglected in the development of modern physics, although it is occasionally discussed to this day. Wilczek likes to point out that, in Stoney Units, QM would be an emergent phenomenon/theory, since the Planck constant wouldn’t be present directly but as a combination of different constants. By the other hand, the Planck scale is valid for all known interactions, and does not give prominence to the electromagnetic interaction, as the Stoney scale does. That is, in Stoney Units, both gravitation and electromagnetism are on equal footing, unlike the Planck units, where only the speed of light is used and there is no more connections to electromagnetism, at least, in a clean way like the Stoney Units do. Be aware, sometimes, rarely though, Planck units are referred to as Planck-Stoney units.
What are the most interesting Stoney system values? Here you are the most remarkable results:
1) Stoney Length, .
2) Stoney Mass, .
3) Stoney Energy, .
4) Stoney Time, .
5) Stoney Charge, .
6) Stoney Temperature, .
The reference constants to this natural system of units (generally denoted by P) are the following 4 constants:
1) Gravitational constant.
2) Speed of light. .
3) Planck constant or rationalized Planck constant. .
4) Boltzmann constant. .
The Planck units are got when you set these 4 constants to the unit, i.e.,
It is often said that Planck units are a system of natural units that is not defined in terms of properties of any prototype, physical object, or even features of any fundamental particle. They only refer to the basic structure of the laws of physics: c and G are part of the structure of classical spacetime in the relativistic theory of gravitation, also known as general relativity, and ℏ captures the relationship between energy and frequency which is at the foundation of elementary quantum mechanics. This is the reason why Planck units particularly useful and common in theories of quantum gravity, including string theory or loop quantum gravity.
This system defines some limit magnitudes, as follows:
1) Planck Length, .
2) Planck Time, .
3) Planck Mass, .
4) Planck Energy, .
5) Planck charge, .
In Lorentz-Heaviside electromagnetic units
In Gaussian electromagnetic units
6) Planck temperature, .
From these “fundamental” magnitudes we can build many derived quantities in the Planck System:
1) Planck area.
2) Planck volume.
3) Planck momentum.
A relatively “small” momentum!
4) Planck force.
It is independent from Planck constant! Moreover, the Planck acceleration is
5) Planck Power.
6) Planck density.
Planck density energy would be equal to
7) Planck angular frequency.
8) Planck pressure.
Note that Planck pressure IS the Planck density energy!
9) Planck current.
10) Planck voltage.
11) Planck impedance.
A relatively small impedance!
12) Planck capacitor.
Interestingly, it depends on the gravitational constant!
Some Planck units are suitable for measuring quantities that are familiar from daily experience. In particular:
1 Planck mass is about 22 micrograms.
1 Planck momentum is about 6.5 kg m/s
1 Planck energy is about 500kWh.
1 Planck charge is about 11 elementary (electronic) charges.
1 Planck impendance is almost 30 ohms.
i) A speed of 1 Planck length per Planck time is the speed of light, the maximum possible speed in special relativity.
ii) To understand the Planck Era and “before” (if it has sense), supposing QM holds yet there, we need a quantum theory of gravity to be available there. There is no such a theory though, right now. Therefore, we have to wait if these ideas are right or not.
iii) It is believed that at Planck temperature, the whole symmetry of the Universe was “perfect” in the sense the four fundamental foces were “unified” somehow. We have only some vague notios about how that theory of everything (TOE) would be.
The physical dimensions of the known Universe in terms of Planck units are “dramatic”:
i) Age of the Universe is about .
ii) Diameter of the observable Universe is about
iii) Current temperature of the Universe is about
iv) The observed cosmological constant is about
v) The mass of the Universe is about .
vi) The Hubble constant is
The Schrödinger Units do not obviously contain the term c, the speed of light in a vacuum. However, within the term of the Permittivity of Free Space [i.e., electric constant or vacuum permittivity], and the speed of light plays a part in that particular computation. The vacuum permittivity results from the reciprocal of the speed of light squared times the magnetic constant. So, even though the speed of light is not apparent in the Schrödinger equations it does exist buried within its terms and therefore influences the decimal placement issue within square roots. The essence of Schrödinger units are the following constants:
A) Gravitational constant .
B) Planck constant .
C) Boltzmann constant .
D) Coulomb constant or equivalently the electric permitivity of free space/vacuum .
E) The electric charge of the positron .
In this sistem we have
1) Schrödinger Length .
2) Schrödinger time .
3) Schrödinger mass .
4) Schrödinger energy .
5) Schrödinger charge .
6) Schrödinger temperature .
There are two alternative systems of atomic units, closely related:
1) Hartree atomic units:
2) Rydberg atomic units:
There, is the electron mass and is the electromagnetic fine structure constant. These units are designed to simplify atomic and molecular physics and chemistry, especially the quantities related to the hydrogen atom, and they are widely used in these fields. The Hartree units were first proposed by Doublas Hartree, and they are more common than the Rydberg units.
The units are adapted to characterize the behavior of an electron in the ground state of a hydrogen atom. For example, using the Hartree convention, in the Böhr model of the hydrogen atom, an electron in the ground state has orbital velocity = 1, orbital radius = 1, angular momentum = 1, ionization energy equal to 1/2, and so on.
Some quantities in the Hartree system of units are:
1) Atomic Length (also called Böhr radius):
2) Atomic Time:
3) Atomic Mass:
4) Atomic Energy:
5) Atomic electric Charge:
6) Atomic temperature:
The fundamental unit of energy is called the Hartree energy in the Hartree system and the Rydberg energy in the Rydberg system. They differ by a factor of 2. The speed of light is relatively large in atomic units (137 in Hartree or 274 in Rydberg), which comes from the fact that an electron in hydrogen tends to move much slower than the speed of light. The gravitational constant is extremely small in atomic units (about 10−45), which comes from the fact that the gravitational force between two electrons is far weaker than the Coulomb force . The unit length, LA, is the so-called and well known Böhr radius, a0.
The values of c and e shown above imply that , as in Gaussian units, not Lorentz-Heaviside units. However, hybrids of the Gaussian and Lorentz–Heaviside units are sometimes used, leading to inconsistent conventions for magnetism-related units. Be aware of these issues!
In the framework of Quantum Chromodynamics, a quantum field theory (QFT) we know as QCD, we can define the QCD system of units based on:
1) QCD Length .
and where is the proton mass (please, don’t confuse it with the Planck mass ).
2) QCD Time .
3) QCD Mass .
4) QCD Energy .
Thus, QCD energy is about 1 GeV!
5) QCD Temperature .
6) QCD Charge .
In Heaviside-Lorent units:
In Gaussian units:
The geometrized unit system, used in general relativity, is not a completely defined system. In this system, the base physical units are chosen so that the speed of light and the gravitational constant are set equal to unity. Other units may be treated however desired. By normalizing appropriate other units, geometrized units become identical to Planck units. That is, we set:
and the remaining constants are set to the unit according to your needs and tastes.
This table from wikipedia is very useful:
i) is the fine-structure constant, approximately 0.007297.
ii) is the gravitational fine-structure constant.
Some conversion factors for geometrized units are also available:
Conversion from kg, s, C, K into m:
Conversion from m, s, C, K into kg:
Conversion from m, kg, C, K into s
Conversion from m, kg, s, K into C
Conversion from m, kg, s, C into K
Or you can read off factors from this table as well:
Advantages and Disadvantages of Natural Units
Natural units have some advantages (“Pro”):
1) Equations and mathematical expressions are simpler in Natural Units.
2) Natural units allow for the match between apparently different physical magnitudes.
3) Some natural units are independent from “prototypes” or “external patterns” beyond some clever and trivial conventions.
4) They can help to unify different physical concetps.
However, natural units have also some disadvantages (“Cons”):
1) They generally provide less precise measurements or quantities.
2) They can be ill-defined/redundant and own some ambiguity. It is also caused by the fact that some natural units differ by numerical factors of pi and/or pure numbers, so they can not help us to understand the origin of some pure numbers (adimensional prefactors) in general.
Moreover, you must not forget that natural units are “human” in the sense you can addapt them to your own needs, and indeed,you can create your own particular system of natural units! However, said this, you can understand the main key point: fundamental theories are who finally hint what “numbers”/”magnitudes” determine a system of “natural units”.
Remark: the smart designer of a system of natural unit systems must choose a few of these constants to normalize (set equal to 1). It is not possible to normalize just any set of constants. For example, the mass of a proton and the mass of an electron cannot both be normalized: if the mass of an electron is defined to be 1, then the mass of a proton has to be . In a less trivial example, the fine-structure constant, α≈1/137, cannot be set to 1, because it is a dimensionless number. The fine-structure constant is related to other fundamental constants through a very known equation:
where is the Coulomb constant, e is the positron electric charge (elementary charge), ℏ is the reduced Planck constant, and c is the again the speed of light in vaccuum. It is believed that in a normal theory is not possible to simultaneously normalize all four of the constants c, ℏ, e, and kC.
Fritzsch and Xing have developed a very beautiful plot of the fundamental constants in Nature (those coming from gravitation and the Standard Model). I can not avoid to include it here in the 2 versions I have seen it. The first one is “serious”, with 29 “fundamental constants”:
However, I prefer the “fun version” of this plot. This second version is very cool and it includes 28 “fundamental constants”:
The Okun Cube
Long ago, L.B. Okun provided a very interesting way to think about the Planck units and their meaning, at least from current knowledge of physics! He imagined a cube in 3d in which we have 3 different axis. Planck units are defined as we have seen above by 3 constants plus the Boltzmann constant. Imagine we arrange one axis for c-Units, one axis for -units and one more for -units. The result is a wonderful cube:
Or equivalently, sometimes it is seen as an equivalent sketch ( note the Planck constant is NOT rationalized in the next cube, but it does not matter for this graphical representation):
Classical physics (CP) corresponds to the vanishing of the 3 constants, i.e., to the origin .
Newtonian mechanics (NM) , or more precisely newtonian gravity plus classical mechanics, corresponds to the “point” .
Special relativity (SR) corresponds to the point , i.e., to “points” where relativistic effects are important due to velocities close to the speed of light.
Quantum mechanics (QM) corresponds to the point , i.e., to “points” where the action/angular momentum fundamental unit is important, like the photoelectric effect or the blackbody radiation.
Quantum Field Theory (QFT) corresponds to the point , i.e, to “points” where both, SR and QM are important, that is, to situations where you can create/annihilate pairs, the “particle” number is not conserved (but the particle-antiparticle number IS), and subatomic particles manifest theirselves simultaneously with quantum and relativistic features.
Quantum Gravity (QG) would correspond to the point where gravity is quantum itself. We have no theory of quantum gravity yet, but some speculative trials are effective versions of (super)-string theory/M-theory, loop quantum gravity (LQG) and some others.
Finally, the Theory Of Everything (TOE) would be the theory in the last free corner, that arising in the vertex . Superstring theories/M-theory are the only serious canditate to TOE so far. LQG does not generally introduce matter fields (some recent trials are pushing into that direction, though) so it is not a TOE candidate right now.
Some final remarks and questions
1) Are fundamental “constants” really constant? Do they vary with energy or time?
2) How many fundamental constants are there? This questions has provided lots of discussions. One of the most famous was this one:
The trialogue (or dialogue if you are precise with words) above discussed the opinions by 3 eminent physicists about the number of fundamental constants: Michael Duff suggested zero, Gabriel Veneziano argued that there are only 2 fundamental constants while L.B. Okun defended there are 3 fundamental constants
3) Should the cosmological constant be included as a new fundamental constant? The cosmological constant behaves as a constant from current cosmological measurements and cosmological data fits, but is it truly constant? It seems to be…But we are not sure. Quintessence models (some of them related to inflationary Universes) suggest that it could vary on cosmological scales very slowly. However, the data strongly suggest that
It is simple, but it is not understood the ultimate nature of such a “fluid” because we don’t know what kind of “stuff” (either particles or fields) can make the cosmological constant be so tiny and so abundant (about the 72% of the Universe is “dark energy”/cosmological constant) as it seems to be. We do know it can not be “known particles”. Dark energy behaves as a repulsive force, some kind of pressure/antigravitation on cosmological scales. We suspect it could be some kind of scalar field but there are many other alternatives that “mimic” a cosmological constant. If we identify the cosmological constant with the vacuum energy we obtain about 122 orders of magnitude of mismatch between theory and observations. A really bad “prediction”, one of the worst predictions in the history of physics!
Be natural and stay tuned!
Position or displacement and its various derivatives define an ordered hierarchy of meaningful concepts. There are special names for the derivatives of position (first derivative is called velocity, second derivative is called acceleration, and some other derivatives with proper name), up to the eighth derivative and down to the -9th derivative (ninth integral).
We are going to study the derivatives of position and their corresponding names and special meaning in Physmatics.
0th derivative is position
In Physics, displacement or position is the vector that specifies the change in position of a point, particle, or object. The position vector directs from the reference point to the present position.
A sensor is said to be displacement-sensitive when it responds to absolute position.
For example, whereas a dynamic microphone is a velocity receiver (responds to the derivative of sound pressure or position), a carbon microphone is a displacement receiver in the sense that it responds to sound pressure or diaphragm position itself. The physical dimension of position vector or the distance is length, i.e.,
1st derivative is velocity
Velocity is defined as the rate of change of position or the rate of displacement. It is a vector physical quantity, both speed and direction are required to define it. In the SI(metric) system, it is measured in meters per second (m/s).
The scalar absolute value (magnitude) of velocity is called speed. For example, “5 metres per second” is a speed and not a vector, whereas “5 metres per second east” is a vector. The average velocity (v) of an object moving through a displacement in a straight line during a time interval is described by the formula:
Therefore, velocity is change in position per unit of time. If the change is made “infinitesimally”, i.e., taking two very close points in time, we can define the instantanous velocity ( a.k.a, the derivative) as the limit of the average speed or two very close points when the time interval tends to zero:
Most piano-style music keyboards are approximately velocity-sensitive, within a certain specific, though limited range of key travel, i.e. to a first-order approximation, a note is made louder by hitting a key faster. Most electronic music keyboards are also velocity sensitive, and measure the time interval between switch contact closures at two different positions of key travel on each key.
The physical dimensions of velocity are
2nd derivative is acceleration
Acceleration is defined as the rate of change of velocity. It is thus a vector quantity with dimension . We can define average aceleration and instantaneous acceleration in the same way we did with the velocity:
In SI units acceleration is measured in . The term “acceleration” generally refers to the change in instantaneous velocity. Average acceleration can also be defined with the above formula.
The physical dimensions of acceleration are .
3rd derivative is jerk
Jerk (sometimes called jolt in British English, but less commonly so, due to possible confusion with use of the word to also mean electric shock), surge or lurch, is the rate of change of acceleration; more precisely, the derivative of acceleration with respect to time, the second derivative of velocity or the third derivative of displacement. Jerk is described by the following equations:
1) is the acceleration.
2) is the velocity.
3) is the position or displacement.
4) t is the time parameter.
Physical dimensions of jerk are .
4th derivative is jounce
Jounce (also known as snap) is the fourth derivative of the position vector with respect to time, with the first, second, and third derivatives being velocity, acceleration, and jerk, respectively; in other words, jounce is the rate of change of the jerk with respect to time.
Physical dimensions of snap are
5th and beyond: Higher-order derivatives
Following jounce (snap), the fifth and sixth derivatives of the displacement vector are sometimes referred to as crackle and pop, respectively. Dork has also been suggested for the sixth derivative. Although the reasons given were less than entirely sincere, dork does have an appealing ring to it, specially for geeks, freaks and dorks. The seventh and eighth derivatives of the displacement vector are sometimes referred to as lock and drop. Their respective formulae can be obtained in a simple way from the previous formalism.
In general, physical dimensions of higher order derivatives of position are defined to be quantities with , for any integer number greater or equal than zero.
-1st derivative (integral) of position is absement
Absement (or absition) refers to the -1th time-derivative of displacement (or position), i.e. the integral of position over time. Mathematically speaking:
The rate of change of absement is position. Absement is a quantity with dimension . In SI units, absement is measured in or metre seconds.
One meter-second corresponds to being absent from an origin or other reference point 1 meter away for a duration of one second. This amount of absement is equal to being two metres away from the origin for one half second, or being one half a metre from the origin for two seconds, or a 1mm absence for 1000 seconds, a 1km absence for 1 millisecond, and so on.
The word “absement” is a blend of the words absence and displacement.
The physical dimensions of absement are .
Useful applications of absement
Whereas most musical keyboard instruments, such as the piano, and many electronic keyboards, respond to velocity at which keys are struck, and some such as the tracker-organ, respond to displacement (how far down a key is pressed), flow-based musical instruments such as the hydraulophone, respond to the integral of displacement, i.e. to a time-distance product. Thus “pressing” a key (water jet) on a hydraulophone down for a longer period of time will result in a buildup of the sound level, as fluid (water) begins to fill the sounding mechanism (reservoir), up to a certain maximum filling point beyond which the sound levels off (along with a slow decay). Hydraulophone reservoirs have an approximate integrating effect on the distance or displacement applied by the musician’s fingers to the “keys” (water jets). Whereas the piano provides more articulation and enunciation of individual note-onsets than the organ, the hydraulophone provides a more continuously fluidly varying sound than either the organ or piano.
Of course all these models are approximate: hydraulophones are approximately presement-responsive, pianos are approximately velocity-responsive, etc..
The concepts of absement and presement originated in regards to flow-based musical instruments like hydraulophones, but may be applied to any area of physics, as they exist along the hierarchy of the derivatives of displacement.
A very slow-responding pipe-organ with tracker-action can often exhibit an effect similar to that of a hydraulophone, when it takes time for the wind and sound levels to build up, so that the sound level is approximately the product of how far down a key is pressed and how long it is held down for.
The concept of absement may also be applied to communications theory. For example, the difficulty in maintaining a communications channel (wired or wireless) increases with distance as well as with the time for which the channel must be kept active.
As a crude but simple example, absement may be used, very approximately, to model the cost of a long-distance phone call as the product of distance and time. A short-duration call over a long distance might, for example, represent the same quantity of absement as a long-duration call over a shorter distance.
Absement may also be used in sociological studies, i.e. we might express loneliness or homesickness as a product of distance from home and time away from home. Simply put, the old aphorism “absence makes the heart grow fonder” has been expressed as “absement makes the heart grow fonder”, to suggest that it matters both how absent one is (i.e. how far), as well as for how long one is absent.
Absement versus presement
Absement refers to the time-distance product (or more precisely the integral of displacement) away from a reference point, whereas the integral of reciprocal position, called presement, refers to the closeness, compounded over time.
The word “presement” is a portmanteau constructed from the words presence and displacement.
Placement (scalar quantity, nearness) is defined as the reciprocal of the position’s magnitude ( i.e., the reciprocal of the distance, an scalar quantity), and presement refers to the time-integral of placement. Most notably, with some high-pressure hydraulophones, it is physically impossible to fully obstruct a water jet, so position can never reach zero, and thus placement remains finite, as does its time integral, presement.
and where d is the distance , with the origin fixed to the zero vector. Simply put, absement is the time-integral of farness, and presement is the time-integral of nearness, to a given point (e.g. farness or nearness of a musicians finger to/from the exit port of a water jet in a hydraulophone).
Physical dimensions of placement are while the physical dimensions of presement are
Lower-order derivatives (higher-order integrals)
Some hydraulophones, such as the North Nessie (the hydraulophone on the North side of hydraulophone circle) at the Ontario Science Centre consist of cascaded hydraulophonic mechanisms, resulting in a double-integrating effect. In particular, the hydraulophone is linked indirectly to the North pipes, such that the water in direct physical contact with the fingers of the musician is not the same water in the organ pipes. As a result of this indirection, the instrument itself responds to presement/absement, the first integral of position whereas the pipes respond absemently to the action in the instrument, i.e. to the second integral of position of the player’s fingers. The time-integral of the time-integral of position is called absity/presity.
Absity is a portmanteau formed from the words absement (or absence) and velocity.
Following this pattern, higher time integrals of displacement may be named as follows:
1) Absement or absition is the integral of displacement.
2) Absity is the double integral of displacement.
3) Abseleration is the triple integral of displacement.
4) Abserk is the fourth integral of displacement.
5) Absounce is the fifth integral of displacement.
Likewise, presement, presity, preseleration, and similar words, are the integrals of reciprocal displacement (nearness).
Although there are no three-stage hydraulophones currently being manufactured as products, there are a number of three-stage (and some with higher numbers of stages) hydraulophone prototypes, in which some elements of the sound production respond to absity/presity, abseleration/preseleration, etc.
Derivatives of momentum
In Physics, momentum is defined as the product of mass and velocity, i.e.,
or mathematically speaking
Moreover, we define the concept of “force” as the rate of change of momentum with respect to time, i.e.,
It mass does not depend on the time, we get
Can we define names for the next derivatives of momentum with respect to time? Of course, we can. It is only a nominal issue. There is a famous “poem” about this:
“Momentum equals mass times velocity. Force equals mass times acceleration. Yank equals mass times jerk. Tug equals mass times snap. Snatch equals mass times crackle. Shake equals mass times pop.”
If mass is not constant, the common definitions of higher derivatives of momentum are as follows ( the last equality is obtained supposing the mass is constant with time):
0th time derivative of momentum is of course The Momentum itself ( I am sorry, Mom-entum is not related with your Mom).
1st time derivative of momentum is The Force ( I am sorry. It is a Star Wars joke).
2nd time derivative of momentum is The Yank ( I am sorry, it is not a tank or a yankie from USA).
3rd time derivative of momentum is The Tug ( I am sorry. It is not a bug in the deepest part of The Matrix).
4th time derivative of momentum is The Snatch ( I am sorry, it is not the golden Snitch).
5th time derivative of momentum is The Shake ( I am sorry, it is not the japanese sake or a sweet tropical milk-shake).
Notations for derivatives/integrals
Lebiniz operational notation: has a derivative with respect to x written as . Then, the derivative is denoted as the operator . Higher order derivatives and integrals can be defined recursively:
Newton dot notation: Derivatives are marked as dotted functions, e.g.,
and so on. Integrals are written in the usual form we do today.
Modern primed notation: Derivatives are marked as primed functions, e.g.,
and so on. Integrals are written in the usual form we do today.
Modern sublabel notation: Derivatives are marked with a subindex label denoting the variable with respect to we are making the derivative. Integrals are represented in the usual form. Thus,
and so on.
These notations have their own advantanges and disadvantanges, but if we use them carefully, any of them can be very powerful.
Physicists like to relate physical quantities in Mechanics/Dynamics to 4 main variables: force, power, action and energy. We can even dedude some interesting relationships between them and displacement, time, momentum, absement, placement, and presement.
1) Equations relating force and other magnitudes. Force dimensions are . Then, we have the identities:
2) Equations relating power and other magnitudes. Power dimensions are . We easily get:
3) Equations relating action and other magnitudes. Action dimensions are . We obtain in this case:
4) Equations relating energy and other magnitudes. Energy dimensions are . We deduce from this last case
In the same way, we can also deduce more fascinating identities:
since we easily get
and of course
Moreover, we also have
and the next interesting result as well:
Music, elements and Physics
The inspiring guide to the new names and variables was the theory of hydraulophones and music. In fact, there is a recent proposal to classify every musical instrument according to its physical origin instead of the classical element. It also makes sense to present the four states-of-matter in increasing order of energy: Earth/Solid first, Water/Liquid second, Air/Gas third, and Fire/Plasma fourth. At absolute zero, if it were possible, everything is a solid. then as things heat up they melt, then they evaporate, and finally, with enough energy, would become a ball of plasma, thus establishing a natural physical ordering as follows:
1) Earth/Solid played instruments. Geolophones. They produce sound pulsing the matter (“Earth”) of some object (string, membrane,…). Ordered in increasing dimension, from 1d to 3d, they can be: I) Chordophones (Played strings, streched objects with cross-section negligible respect to their length), II) Membranophones (Played membranes with thickness negligible respect to their area), III) Idiophones/Bulkphones (played 3d tensionless branes or higher).
2) Water/Liquid played instruments. Hydraulophones. These instruments produce vibrating sound pulsing jets of liquids (“Water”).
3) Air/Gas played instruments. Aerophones. These instuments produce vibrations and sound touching the flux of gases (“Air”).
4) Fire/Plasma played instruments. Ionophones. These instruments produce sonic waves playing the flux of plasma (“Fire”).
5) Quintessence/Idea/Information/Informatics played instruments. These instruments produce “sound” by computational means, whether optical, mechanical, electrical, or otherwise. We could name these instruments with some cool word. Akashaphones (from the sanskrit word/prefix “akasha”, meaning “aether, ether” or as Western tradition would say, “quintessence, fifth element”) will be the names of such instruments.
This classification matches the range of acoustic transducers that exist today (excepting the quintessencial transducer, of course) as well: 1) Geophone, 2) Hydrophone, 3) Microphone or speaker, and 4) Ionophone. In the same way I have never known a term for the akashaphones before, for the fifth transducer we should use a new term. Loakashaphone, from the same sanskrit origin than akashaphone, would be the analogue 5th transducer.
The following list is a summary of the derivatives of displacement/position:
A) Time integrals of position/displacement.
Order -9. Absrop. SI units . Time integral of absock. Dimensions: .
Order -8. Absock. SI units . Time integral of absop. Dimensions: .
Order -7. Absop. SI units . Time integral of absrackle. Dimensions: .
Order -6. Absrackle. SI units . Time integral of absounce. Dimensions: .
Order -5. Absounce. SI units . Time integral of abserk. Dimensions: .
Order -4. Abserk. SI units . Time integral of abseleration. Dimensions: .
Order -3. Abseleration. SI units . Time integral of absity. Dimensions: .
Order -2. Absity. SI units . Time integral of absement. Dimensions: .
Order -1. Absement. SI units . Time integral of position. Dimensions: .
Order 0. Position/Displacement. SI units . Dimensions: .
Remark: Integrals with respect to time of position measure “farness”.
B) Time derivatives of position/displacement.
Order 0. Position/Displacement. SI units . Dimensions: .
Order 1. Velocity. SI units . Rate of change of position. Dimensions: .
Order 2. Acceleration. SI units . Rate of change of velocity. Dimensions: .
Order 3. Jerk/jolt/surge/lurch. SI units . Rate of change of acceleration. Dimensions: .
Order 4. Jounce/snap. SI units . Rate of change of jerk. Dimensions: .
Order 5. Crackle. SI units . Rate of change of jounce. Dimensions: .
Order 6. Pop. SI units . Rate of change of crackle. Dork has also been suggested for the sixth derivative. Although the reasons given were less than entirely sincere, dork does have an appealing ring to it. Dimensions: .
Order 7. Lock. SI units . Rate of change of pop. Dimensions: .
Order 8. Drop. SI units . Rate of change of lock. Dimensions: .
Remark: Derivatives of position with respect to time measure “swiftness”.
C) Reciprocals of position/displacement and their time integrals.
Order 0. Placement. SI units . Placement (scalar quantity, nearness) is the reciprocal of position (scalar quantity distance), i.e., . Dimensions: .
Order -1. Presement. SI units . Time integral of placement. Dimensions: .
Order -2. Presity. SI units . Time integral of presement. Dimensions: .
Order -3. Preseleration. SI units . Time integral of presity. Dimensions: .
Order -4. Preserk. SI units . Time integral of preseleration. Dimensions: .
Order -5. Presounce. SI units . Time integral of preserk. Dimensions: .
Order -6. Presackle. SI units . Time integral of presounce. Dimensions: .
Order -7. Presop. SI units . Time integral of presackle. Dimensions: .
Order -8. Presock. SI units . Time integral of presop. Dimensions: .
Order -9. Presrop. SI units . Time integral of presock. Dimensions: .
Remark: Integrals of reciprocal displacement with respect to time measure “nearness”.
D) Time derivatives of momentum.
Order 0. Momentum. . SI units . Momentum equals mass times velocity. Dimensions: , where M denotes mass dimension.
Order 1. Force. . SI units are newtons. . Time derivative of momentum, or rate of change of momentum with respect to time. Dimensions: .
Order 2. Yank. . SI units . Time integral of presement. Rate of change of force with respect to time. Dimensions: .
Order 3. Tug. . SI units . Rate of change of yank with respect to time. Dimensions: .
Order 4. Snatch. . SI units . Rate of change of tug with respect to time. Dimensions: .
Order 5. Shake. . SI units . Rate of change of snatch with respect to time. Dimensions: .
Remark: Derivatives of momentum with respect to time measure “strengthness” or “forceness”.
So we have to remember 4 fascinating ideas,
i) Time integrals of position measure “farness”.
ii) Time derivatives of position measure “swiftness”.
iii) Time integrals of reciprocal position measure “nearness”.
iv) Time derivatives of momentum measure “forceness”.
And a fifth further great idea… Physics, Mathematics or more generally Physmatics own an inner “Harmony” or “Music” in their deepest principles and theories.
Some additional questions can be asked further:
0th. What about “infinite” order derivatives and integrals?
1st. What if time is not a continuous function?
2nd. What if time is not a scalar quantity?
3rd. What about fractional order/irrational order/complex order derivatives/X-order derivatives?
4th. What if (space) time/displacement does not exist?
5th. Can Mechanics/Dynamics of particles/fields/strings/branes/… be formulated in terms of integrals/reciprocals of “position” and “momentum” variables, i.e., as the power of negative and/or higher/lower derivatives? Would such a formulation of Mechanics/Dynamics be useful/meaningful for something deeper? That is, what are the right variables to study in Dynamics if some classical/quantum concepts are absent?
We could answer to some of these questions. For instance, the answer to the 0th question is interesting but it requires to know about jet spaces and/or path integrals. Moreover, the solution to the 3rd question would require the introduction of the fractional/fractal calculus. But that is another long story/log-entry to be told in a forthcoming future post!