LOG#053. Derivatives of position.

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., \left[\mathbf{x}\right]=\left[ d\right]=L

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 \Delta x in a straight line during a time interval \Delta t is described by the formula:

\mathbf{v}_m=\dfrac{\Delta \mathbf{x}}{\Delta t}

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:

\displaystyle{\mathbf{v}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \mathbf{x}}{\Delta t}\equiv \dfrac{d\mathbf{x}(t)}{dt}}

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  \left[\mathbf{v}\right]=LT^{-1}

2nd derivative is acceleration

Acceleration is defined as the rate of change of velocity. It is thus a vector quantity with dimension LT^{-2}. We can define average aceleration and instantaneous acceleration in the same way we did with the velocity:

\mathbf{a}_m=\dfrac{\Delta \mathbf{v}}{\Delta t}

\displaystyle{\mathbf{a}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta \mathbf{v}}{\Delta t}\equiv \dfrac{d\mathbf{v}(t)}{dt}}

In SI units acceleration is measured in m/s^2. 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 \left[\mathbf{a}\right]=LT^{-2}.

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) \mathbf{a} is the acceleration.

2) \mathbf{v} is the velocity.

3) \mathbf{x} is the position or displacement.

4) t is the time parameter.

Physical dimensions of jerk are \left[\mathbf{j}\right]=LT^{-3}.

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 \left[\mathbf{s}\right]=LT^{-4}

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 \left[\mathbf{Q}\right]=LT^{-r}, for any integer number r 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:

\displaystyle{\mathbf{A}=\int \mathbf{x}dt}

The rate of change of absement is position. Absement is a quantity with dimension LT. In SI units, absement is measured in ms 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 \left[\mathbf{A}\right]=LT.

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.

\mbox{Placement}\equiv \dfrac{1}{d}

\displaystyle{\mbox{Presement}=\int dt \dfrac{1}{d}}

and where d is the distance d=\sqrt{x^2+y^2+z^2}, 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 \left[\mbox{Placement}\right]=L^{-1} while the physical dimensions of presement are \left[\mbox{Presement}\right]=L^{-1}T

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 \mathbf{F}=m\mathbf{a}

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).

\mathbf{p}=m\mathbf{v}=\dfrac{d^0\mathbf {p}}{dt^0}.

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: f(x) has a derivative with respect to x written as \dfrac{df}{dx}. Then, the derivative is denoted as the operator D=\dfrac{d}{dx}. Higher order derivatives and integrals can be defined recursively:

D^2=\left(\dfrac{d}{dx}\right)^2\equiv \dfrac{d}{dx}\left(\dfrac{d}{dx}\right)=\dfrac{d^2}{dx^2}

D^r=\left(\dfrac{d}{dx}\right)^r\equiv \underbrace{\dfrac{d}{dx}\cdots\left(\dfrac{d}{dx}\right)}_\text{r-times}=\dfrac{d^r}{dx^r}, \;\; \forall r\geq 0

\displaystyle{D^{-1}=\int dx}

\displaystyle{D^{-2}=\int d^2x=\int (dx)^2=\int dx dx'}

\displaystyle{D^{-r}=\int d^rx=\int (dx)^r=\int dx\cdots dx^{(r)}=\int \underbrace{dx\cdots}_\text{r-times}}

Newton dot notation: Derivatives are marked as dotted functions, e.g.,

\dot{f}=\dfrac{df}{dx} \ddot{f}=\dfrac{d^2f}{dx^2} \dddot{f}=\dfrac{d^3f}{dx^3} and so on. Integrals are written in the usual form we do today.

Modern primed notation: Derivatives are marked as primed functions, e.g.,

f'=\dfrac{df}{dx} f''=\dfrac{d^2f}{dx^2} f'''=\dfrac{d^3f}{dx^3} 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,

f_x=\dfrac{df}{dx} f_{xx}=\dfrac{d^2f}{dx^2} f_{xxx}=\dfrac{d^3f}{dx^3} and so on.

These notations have their own advantanges and disadvantanges, but if we use them carefully, any of them can be very powerful.

Remarkable relationships

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 MLT^{-2}. Then, we have the identities:



2) Equations relating power and other magnitudes. Power dimensions are ML^2T^{-3}. We easily get:



3) Equations relating action and other magnitudes. Action dimensions are ML^2T^{-1}. We obtain in this case:

\mbox{Action}=\mbox{Energy}\times \mbox{Time}=\mbox{Displacement}\times\mbox{Momentum}=\mbox{Power}\times\mbox{(Time)}^2

\mbox{Action}=\mbox{Force}\times \mbox{Absement}=\dfrac{\mbox{Momentum}}{\mbox{Placement}}=\mbox{Mass}\times\begin{pmatrix}\mbox{Areolar}\\ \mbox{Velocity}\end{pmatrix}

4) Equations relating energy and other magnitudes. Energy dimensions are ML^2T^{-2}. We deduce from this last case


\mbox{Energy}=\mbox{Momentum}\times \mbox{Velocity}=\mbox{Power}\times\mbox{Time}


In the same way, we can also deduce more fascinating identities:



since we easily get


\mbox{Absement}=\mbox{Presement}\times \mbox{(Displacement)}^2=L^{-1}TL^2=LT


and of course


Moreover, we also have





and the next interesting result as well:

\boxed{(\mbox{Placement})(\mbox{Presement})=\begin{pmatrix}\mbox{Areolar}\\ \mbox{Velocity}\end{pmatrix}^{-1}=L^{-2}T}

or equivalently

\boxed{\begin{pmatrix}\mbox{Areolar}\\ \mbox{Velocity}\end{pmatrix}=v_A=\dfrac{1}{\mbox{Placement}\times \mbox{Presement}}=L^2T^{-1}}

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 ms^9.  Time integral of absock. Dimensions: LT^9.

Order -8. Absock. SI units ms^8. Time integral of absop. Dimensions: LT^8.

Order -7. Absop. SI units ms^7. Time integral of absrackle. Dimensions: LT^7.

Order -6.  Absrackle. SI units ms^6. Time integral of absounce. Dimensions: LT^6.

Order -5. Absounce. SI units ms^5. Time integral of abserk. Dimensions: LT^5.

Order -4. Abserk. SI units ms^4. Time integral of abseleration. Dimensions: LT^4.

Order -3. Abseleration. SI units ms^3. Time integral of absity. Dimensions: LT^3.

Order -2. Absity. SI units ms^2. Time integral of absement. Dimensions: LT^2.

Order -1. Absement. SI units ms. Time integral of position. Dimensions: LT.

Order 0. Position/Displacement. SI units m. Dimensions: L.

Remark: Integrals with respect to time of position measure “farness”.

B) Time derivatives of position/displacement.

Order 0. Position/Displacement. SI units m. Dimensions: L.

Order 1. Velocity. SI units m/s. Rate of change of position. Dimensions: LT^{-1}.

Order 2. Acceleration. SI units m/s^2. Rate of change of velocity. Dimensions: LT^{-2}.

Order 3. Jerk/jolt/surge/lurch. SI units m/s^3. Rate of change of acceleration. Dimensions: LT^{-3}.

Order 4. Jounce/snap. SI units m/s^4. Rate of change of jerk. Dimensions: LT^{-4}.

Order 5. Crackle. SI units m/s^5. Rate of change of jounce. Dimensions: LT^{-5}.

Order 6. Pop. SI units m/s^6. 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: LT^{-6}.

Order 7. Lock. SI units m/s^7. Rate of change of pop. Dimensions: LT^{-7}.

Order 8. Drop. SI units m/s^8. Rate of change of lock. Dimensions: LT^{-8}.

Remark: Derivatives of position with respect to time measure “swiftness”.

C) Reciprocals of position/displacement and their time integrals.

Order 0. Placement. SI units m^{-1}. Placement (scalar quantity, nearness) is the reciprocal of position (scalar quantity distance), i.e., 1/x. Dimensions: L^{-1}.

Order -1. Presement. SI units m^{-1}s. Time integral of placement. Dimensions: L^{-1}T.

Order -2. Presity. SI units m^{-1}s^2. Time integral of presement. Dimensions: L^{-1}T^2.

Order -3. Preseleration. SI units m^{-1}s^3. Time integral of presity. Dimensions: L^{-1}T^3.

Order -4. Preserk. SI units m^{-1}s^4. Time integral of preseleration. Dimensions: L^{-1}T^4.

Order -5. Presounce. SI units m^{-1}s^5. Time integral of preserk. Dimensions: L^{-1}T^5.

Order -6. Presackle. SI units m^{-1}s^6. Time integral of presounce. Dimensions: L^{-1}T^6.

Order -7. Presop. SI units m^{-1}s^7. Time integral of presackle. Dimensions: L^{-1}T^7.

Order -8. Presock. SI units m^{-1}s^8. Time integral of presop. Dimensions: L^{-1}T^8.

Order -9. Presrop. SI units m^{-1}s^9. Time integral of presock. Dimensions: L^{-1}T^9.

Remark: Integrals of reciprocal displacement with respect to time measure “nearness”.

D) Time derivatives of momentum.

Order 0. Momentum. \mathbf{p}. SI units kgms^{-1}. Momentum equals mass times velocity. Dimensions: MLT^{-1}, where M denotes mass dimension.

Order 1. Force. \mathbf{F}. SI units are newtons. N=kg\cdot ms^{-2}. Time derivative of momentum, or rate of change of momentum with respect to time. Dimensions: MLT^{-2}.

Order 2. Yank. \mathbf{Y}. SI units N\cdot s^{-1}=kgms^{-3}. Time integral of presement. Rate of change of force with respect to time. Dimensions: MLT^{-3}.

Order 3. Tug. \mathbf{T}. SI units N\cdot s^{-2}=kgms^{-4}. Rate of change of yank with respect to time. Dimensions: MLT^{-4}.

Order 4. Snatch. \mathbf{S}. SI units N\cdot s^{-3}=kgms^{-5}. Rate of change of tug with respect to time. Dimensions: MLT^{-5}.

Order 5. Shake. \mathbf{Sh}. SI units N\cdot s^{-4}=kgms^{-6}. Rate of change of snatch with respect to time. Dimensions: MLT^{-6}.

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!

Stay tuned!

LOG#049. Ludicrous speed.

We are going to learn about the different notions of velocity that the special theory of relativity provides.

The special theory of relativity is a simple wonderful theory, but it comes with many misconceptions due to bad teaching/science divulgation. It is not easy to master the full theory of relativity without the proper mathematical background and physical insight. In the internet era where knowledge is shared, a fundamental issue is to understand things properly. There are many people who thinks they understand the theory of relativity when they don’t. Even at the academia.

Moreover, you can find many people in the blogsphere/websphere trying to sell false theories and wrong theories. It is the same like the so-called alternative medicine: they are not medicine at all. Bad science is not science, it is simply a lie and not science at all. It is religion. Science can be critized, but nobody can critize that Earth revolves around the Sun, it is common knowledge and truth. So, we can make critics to scientist, but not the scientific method and well established theories. We can try to understand better or in a novel way, but we can not deny facts and experiments. Gerard ‘t Hooft, Nobe Prize, explain it in his web page www.phys.uu.nl/~thooft/.

It is important to remark that Science revolutions come when we extend the theories we know they are correct, like special relativity and not with a full destruction of the current and well-tested theories. Newtonian relativity is a limit of General Relativity. Galilean relativity is a limit of Special Relativity. Quantum Mechanics is a limit of QFT and so on. The issue is not that. Said these words, I am quite sure that scientists and particularly physicists wish to overcome current theories with new ones. However, the process to create a new theory is not easy. Specially, if you don’t understand the traps and theories that have passed every known test till now.

What is velocity? Classically, the answer is short and very clear/neat: velocity is the rate of change of position with respect to time. It is a vector magnitude. Mathematically speaking is the quotient between the displacement vector and the time interval, or in the infinitesimal limit, the derivative of the position vector with respect to time.

\boxed{\mathbf{v_m}=\dfrac{\Delta \mathbf{r}(t)}{\Delta t}\leftrightarrow \mbox{Average velocity}}

\boxed{\mathbf{v}=\dfrac{d\mathbf{r}(t)}{dt}\leftrightarrow \mbox{Instantaneous velocity}}

In the special theory of relativity, due to the fact that time is not universal but relative we can build different notions of velocity. And it matters. There are some clear concepts from relativity you should master till now:

a) You can attach a clock to any yardstick you could physically use for measurements of space and time.

b) You must distinguish the notions of coordinate velocity (map coordinate is another commonly used notion/concept) and proper velocity. The latter is sometimes called hyperbolic (or imaginary) velocity. These two notions are caused by the presence of two “natural” elections of time: the proper time and the coordinate time.

c) Due to the previous two facts, you must also distinguish between proper acceleration and geometric acceleration. Proper-accelerations caused by the tug of external forces and geometric accelerations caused by choice of a reference frame that’s not geodesic i.e. a local reference coordinate-system that is not ”in free-fall”. Proper-accelerations are felt through their points of action e.g. through forces on the bottom of your feet. On the other hand geometric accelerations give rise to inertial forces that act on every ounce of an object’s being. They either vanish when seen from the vantage point of a local free-float frame, or give rise to non-local force effects on your mass distribution that cannot be made to disappear. Coordinate acceleration goes to zero whenever proper-acceleration is exactly canceled by that connection term, and thus when physical and inertial forces add to zero.

People who are not aware of the previous comments, don’t understand relativity and the physics behind it. They even don’t undertand what experiments and their data say.

Let me review the main magnitudes, 3-vectors and 4-vectors which the special theory of relativity studies in the next tables:

The two notions of 3-velocity we do have from the special theory of relativity, i.e., from the 4-velocity \mathbb{U}=\dfrac{d\mathbb{X}}{d\tau},  are:

1) Coordinate velocity, \mathbf{v}:


It is the common notion of 3-velocity, measured from an inertial observer with respect to the coordinate time t. Note that the coordinate time is not a true invariant in SR!

2) Proper velocity (or the hyperbolic velocity/imaginary angle velocity related to it):

\mathbf{w}\equiv \dfrac{d\mathbf{r}}{d\tau}=\gamma \mathbf{v}

where \tau is the proper time. This velocity can intuitively defined as the distance per unit traveler-time, retains many of the properties that ordinary velocity loses at high speed. In addition to these two definitions, we also have:

1)Proper-acceleration \alpha, is the acceleration experienced relative to a locally co-moving free-float-frame, and it helps when we are accelerating, speeding, and in curvy space-time.

2) How some of the space-like effect of sideways ”felt” forces moves into the reference-frame’s time-domain at high speed, making the relatively unknown bound (from special relativity!)

\dfrac{dp}{dt}\leq m\alpha

With the above definitions, the relativistic momentum can be expressed in termns of coordinate velocity or proper velocity as follows:

\mathbf{P}=m\mathbf{w}=M\mathbf{v}=m\gamma \mathbf{v}



is the Lorentz factor. The last equal sign in the previous equation can be easily derived from the relativistic relationship:


and the definition of \gamma above.

Thanks to the metric-equation’s assignment of a frame-invariant traveler or proper-time \tau to the displacement between events in context of a single map-frame of comoving yardsticks and synchronized clocks, proper velocity becomes one of three related derivatives in special relativity (coordinate velocity \mathbf{v}, proper-velocity \mathbf{w}, and Lorentz factor \gamma) that describe an object’s rate of travel. For unidirectional motion, in units of lightspeed c (i.e. c=1 if we want to) each of these is also simply related to  a traveling object’s hyperbolic velocity angle or rapidity \eta by the next set of equations:

\eta=\sinh^{-1}\left( \dfrac{w}{c}\right)=\tanh^{-1}\left(\dfrac{v}{c}\right)=\pm \cosh^{-1}\left(\gamma\right)

The next table illustrates how the proper-velocity of w_0 \equiv c or “one map-lightyear per traveler-year” is a natural benchmark for the transition from a sub-relativistic coordinate frame to a (fake) auxiliary super-relativistic motion (in imaginary units of i=\sqrt{-1}). Note that the velocity angle or pseudorapidity \eta and the proper-velocity w run from 0 to infinity and track the physical coordinate-velocity when w<<c. On the other hand when w>>c, the (hyperbolic or imaginary) proper-velocity tracks Lorentz factor \gamma while velocity angle \eta is logarithmic and hence increases much more slowly:


 Hyperbolic velocities CAN exceed c! They can reach even the ludicrous speed of \infty when the coordinate velocity approaches c! However, you must never forget the fact that the velocity-angle/hyperbolic velocity IS imaginary in value. It is quite clear from the above table. Indeed, being somehow “trekkie” or a Sci-Fi “romantic” person, you could “define” warp-speeds as “imaginary/hyperbolic” velocities, i.e., in terms of proper velocity. In that case, you could get the correspondence

\mbox{WARP}0.25=\mbox{WARP}1/4=\dfrac{\sqrt{17}}{17}c\approx 0.24c

\mbox{WARP}0.5=\mbox{WARP}1/2=\dfrac{\sqrt{5}}{5}c\approx 0.45c

\mbox{WARP}1=\dfrac{\sqrt{2}}{2}c\approx 0.71c

\mbox{WARP}2=\dfrac{2\sqrt{5}}{5}c\approx 0.89c

\mbox{WARP}3=\dfrac{3\sqrt{10}}{10}c\approx 0.95c

\mbox{WARP}7=\dfrac{7\sqrt{2}}{10}c\approx 0.99c

\mbox{WARP}9=\dfrac{9\sqrt{82}}{82}c\approx 0.994c

\mbox{WARP}10=\dfrac{10\sqrt{101}}{101}c\approx 0.995c

\mbox{WARP}\infty\equiv c

In general, we can define the WARP speed as W=w/c and so, the proper velocity can be expressed in terms of the warp speed W in a very simple way w=Wc. Thus, the real or coordinate velocity would be connected with warp-speed through the relativistic equation:


Of course, the point is that, unlike the Sci-Fi franchise, the real velocity has never exceeded c, only the hyperbolic velocity and the proper velocity (note that in terms of SR, velocities approaching c imply very boosted frames, so despite we could travel to any point of the Universe in SR only approaching c very closely with respect to the traveler proper time-one human life-, but in terms of the “Earth” (or rest) reference frame millions of years would have passed away!).

When the coordinate-speeds approach c, the respective coordinate velocities deviate from this simple addition rule in that rapidities (hyperbolic velocity angle boosts) add instead of velocities, i.e. \eta_{12}=\eta_1+\eta_2. Coordinate velocities add non-linearly. And it is a well-tested consequence of the Special Theory of relativity.  For highly relativistic objects (i.e. those with momentum per unit mass much larger than lightspeed) the result of the coordinate-velocity expression  familiar from most textbooks is rather uninteresting since the coordinate-velocities all peak out at c, i.e., as everybody knows, in special relativity 1c\boxplus 1c=1c, because applying the relativistic addition of velocities rule, we get

c\boxplus c=\dfrac{ (c + c)}{(1 + 1)}=c

And it is a fact from both theory and experiment! It will remain as long as SR remains a valid theory. SR holds yet with an astonishing degree of precision and accuracy. So, you can not deny every data and experiment that confirms SR. That is completely nonsense but there are some people and pseudo-scientists out there building their own theories AGAINST the achievements and explanations that SR provides to every experiment we have done until the current time. I am sorry for all of them. They are totally wrong. Science is not what they say it is. Any theory going beyond SR HAS to explain every experiment and data that SR does explain, and it is not easy to build such a theory or to say, e.g., why we have not observed (apparently) superluminal objects. I will discuss more superluminal in a forthcoming post/log entry, some posts after the special 50th post/log that is coming after this one! Stay tuned!

Coming back to our discussion…Why is all this stuff important? High Energy Physics is the natural domain of SR! And there, SR has not provided ANY wrong result till, in spite that some researches going beyond the Standard Model include modified dispersion relationships that reduce to SR in the low energy regime, we have not seen yet ANY deviation from SR until now.

For unidirectional motion, at low speeds the coordinate velocity v_{13} of object 1 from the point of view of oncoming object 3 might be described as the sum of the velocity v_{12} of object 1 with respect to lab frame 2 plus the velocity v_{23} of the lab frame 2 with respect to object 3, that is:


Compare this expression to the previously obtained expression for rapidities! Rapidities always add, coordinate velocities add (linearly) only at low velocities. In conclusion, you must be careful by what you mean by velocity is a boosted system!

By the other hand, for relative proper-velocity, the result is:


This expression shows how the momentum per unit mass as well as the map-distance traveled per unit traveler time of object 1, as seen in the frame of oncoming particle 3, goes as the sum of the coordinate-velocities times the product of the gamma (energy) factors. The proper velocity equation is especially important in high energy physics, because colliders enable one to explore proper-speed and energy ranges much higher than accessible with fixed-target collisions. For instance each of two electrons (traveling with frames 1 and 3) in a head-on collision traveling in the lab frame (2) at


or equivalenty w_{12}=w_{23}=\gamma v\approx 88000 lightseconds per traveler second  would see the other coming toward them at coordinate velocity v_{13}\approx c and w_{13}=88000^2(1+1) \approx 1.55\cdot 10^{10} lightseconds per traveler second or \gamma_{13}mc^2\approx 7.9 \mbox{PeV}. From the target’s view, that is an incredible increase in both energy and momentum per unit of mass.

Other magnitudes and their frame dependence in SR can be read from the following table:

CAUTION: These results don’t mean that the “real” energy is that. Energy is relative and it depends on the frame! The fact that in colliders, seen from the target reference frame, the energy can be greater than the center of mass energy is not an accident. It is a consequence of the formalism of special relativity. A similar observation can be done for velocities. Coordinate velocities, IN THE FRAMEWORK OF SPECIAL RELATIVITY, can never exceed the speed of light. As long as SR holds, there is no particle whose COORDINATE velocity can overcome the speed of light. However, we have seen that PROPER velocities are other monsters. They serve as a tool to handle rotations along the temporal axis, i.e., to handle boosts mixing space and time coordinates. Proper (or hyperbolic) velocities CAN be greater than speed of light. But, it does not contradict the special theory of relativity at all since hyperbolic velocities ARE NOT REAL since they are imaginary quantities and they are not physical. We can only measure momentum and real quantities!  Moreover, remember that, in fact, group or phase velocities we have found before can ALSO be greater than c. So, you must be careful by what do you mean by velocity in SR or in any theory. Furthermore, you must distinguish the notion of particle velocity with those of the relative velocity between two inertial frames, since the particle velocities ( coordinate or proper) always refer to some concrete frame! In summary, be aware of people saying that there are superluminal particles in our colliders or astrophysical processes. It is simply not true. Superluminal objects have observable consequences, and they have failed to be observed ( the last example was the superluminal neutrino affair by the OPERA collaboration, now in agreement with SR).

Remark (I): From the last table we observe that in SR, the rotation angle is imaginary. Therefore, we are forced to use this gadget of hyperbolic velocity in order to avoid “imaginary velocities”.

Remark (II): Hyperbolic velocities would become imaginary velocities if we used the imaginary formalism of SR, the infamous ict=x_4.

Remark (III): Hyperbolic velocities are not coordinate velocities, so they are not physical at all. They are just a tool to provide the right answers in terms of rapidities, or the hyperbolic angle, whose units are imaginary radians! Hyperbolic velocities are measured in imaginary units of velocity!

Remark (IV): About the imaginary issues you can have now. The spacetime separation formula s^2=-c^2t^2+x^2+y^2+z^2 means that the time t can often be treated mathematically as if it were an imaginary spatial dimension. That is, you can define ct=iw so -c^2t^2=w^2, where i  is the square root of  -1, and w is a “fourth spatial coordinate”. Of course it is not at all. It is only a trick to treat the problem in a clever way.  By the other hand, a Lorentz boost by a velocity v can likewise be treated as a rotation by an imaginary angle. Consider a normal spatial rotation in which a primed frame is rotated in the wx-plane clockwise by an angle \varphi about the origin, relative to the unprimed frame. The relation between the coordinates (w',x') and (w,x) of a point in the two frames is:

\begin{pmatrix}w'\\ x'\end{pmatrix}=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}w\\ x\end{pmatrix}

Now set ct=iw and \theta=i\varphi, with t,\theta both real. In other words, take the spatial coordinate w to be imaginary, and the rotation angle \varphi likewise to be imaginary. Then the rotation formula above becomes

\begin{pmatrix}ct'\\ x'\end{pmatrix}=\begin{pmatrix}\cosh\theta & -\sinh\theta\\ -\sinh\theta & \cosh\theta\end{pmatrix}\begin{pmatrix}ct\\ x\end{pmatrix}

This agrees with the usual Lorentz transformation formulat if the boost velocity v and boost angle \theta are related by the known formula \tanh\theta=v/c=\beta. We realize that if we identify the imaginary angle with the rapidity, we are back to Special Relativity. Indeed, it is only the rotations involving the time axis which can cause confusion because they are so different from our everyday experience. That is, we experience rotations along some direction in our daily experience, so we are familiarized with rotations and their (real) rotation angles. However, rotations along a time axis mixing space and time is a weird creature. It uses imaginary numbers or, if we avoid them, we have to use hyperbolic (pseudo)-rotations.


A) Lorentz factor \gamma=\dfrac{E}{mc^2}

\boxed{\gamma \equiv \frac{dt}{d\tau}= \sqrt{1+\left(\frac{w}{c}\right)^2} = \frac{1}{\sqrt{1-(\frac{v}{c})^2}} = \cosh[\eta] \equiv \frac{e^{\eta} + e^{-\eta}}{2}}

B) Proper-velocity or momentum per unit mass.

\boxed{\frac{w}{c}\equiv \frac{1}{c} \frac{dx}{d\tau}=\frac{v}{c} \frac{1}{\sqrt{1-(\frac{v}{c})^2}}=\sinh[\eta]\equiv \frac{e^{\eta} - e^{-\eta}}{2} =\pm\sqrt{\gamma^2 - 1}}

C) Coordinate velocity v\leq c.

\boxed{\frac{v}{c} \equiv \frac{1}{c}\frac{dx}{dt}=\frac{w}{c}\frac{1}{\sqrt{1 + (\frac{w}{c})^2}} = \tanh[\eta] \equiv \frac{e^{2\eta} - 1} {e^{2\eta} + 1}= \pm \sqrt{1 - \left(\frac{1}{\gamma}\right)^2}}

D) Hyperbolic velocity angle or rapidity.

\boxed{\eta =\sinh^{-1}[\frac{w}{c}] = \tanh^{-1}[\frac{v}{c}] = \pm \cosh^{-1}[\gamma]}

or in terms of logarithms:

\boxed{\eta = \ln\left[\frac{w}{c} + \sqrt{\left(\frac{w}{c}\right)^2 + 1}\right] = \frac{1}{2} \ln\left[\frac{1+\frac{v}{c}}{1-\frac{v}{c}}\right] = \pm \ln\left[\gamma + \sqrt{\gamma^2 - 1}\right]}

E) Warp speed (just for fun):