In euclidean two dimensional space, rotations are easy to understand in terms of matrices and trigonometric functions. A plane rotation is given by:
where the rotation angle is , and it is parametrized by .
Interestingly, in minkovskian two dimensional spacetime, the analogue does exist and it is written in terms of matrices and hyperbolic trigonometric functions. A “plane” rotation in spacetime is given by:
Here, is the so-called hiperbolic rotation angle, pseudorotation, or more commonly, the rapidity of the Lorentz boost in 2d spacetime. It shows that rapidity are a very useful parameter for calculations in Special Relativity. Indeed, it is easy to check that
So, at least in the 2d spacetime case, rapidities are “additive” in the written sense.
Firstly, we are going to guess the relationship between rapidity and velocity in a single lorentzian spacetime boost. From the above equation we get:
Multiplying the first equation by and the second one by , we add the resulting equation to obtain:
From this equation (or the boxed equations), we see that corresponds to and . Setting , we deduce that
Since , and the pseudorotation seems to have a “pseudovelocity” equals to , the rapidity it is then defined through the equation:
If we remember what we have learned in our previous mathematical survey, that is,
We set in order to get the next alternative expression for the rapidity:
In experimental particle physics, in general 3+1 spacetime, the rapidity definition is extended as follows. Writing, from the previous equations above,
and using these two last equations, we can also write momenergy components using rapidity in the same fashion. Suppose that for some particle(objetc), its mass is , its energy is , and its (relativistic) momentum is . Then:
From these equations, it is trivial to guess:
This is the completely general definition of rapidity used in High Energy Physics (HEP), with a further detail. In HEP, physicists used to select the direction of momentum in the same direction that the collision beam particles! Suppose we select some orientation, e.g.the z-axis. Then, and rapidity is defined in that beam direction as:
In 2d spacetime, rapidities add nonlinearly according to the celebrated relativistic addition rule:
Indeed, Lorentz transformations do commute in 2d spacetime since we boost in a same direction x, we get:
This commutativity is lost when we go to higher dimensions. Indeed, in spacetime with more than one spatial direction that result is not true in general. If we build a Lorentz transformation with two boosts in different directions and , the Lorentz matrices are ( remark for experts: we leave one direction in space untouched, so we get 3×3 matrices):
and it is easily checked that
Finally, there is other related quantity to rapidity that even experimentalists do prefer to rapidity. It is called: PSEUDORAPIDITY!
Pseudorapidity, often denoted by describes the angle of a particle relative to the beam axis. Mathematically speaking is:
where is the angle between the particle momentum and the beam axis. The above relation can be inverted to provide:
The pseudorapidity in terms of the momentum is given by:
Note that, unlike rapidity, pseudorapidity depends only on the polar angle of its trajectory, and not on the energy of the particle.
In hadron collider physics, and other colliders as well, the rapidity (or pseudorapidity) is preferred over the polar angle because, loosely speaking, particle production is constant as a function of rapidity. One speaks of the “forward” direction in a collider experiment, which refers to regions of the detector that are close to the beam axis, at high pseudorapidity .
The rapidity as a function of pseudorapidity is provided by the following formula:
where is the momentum transverse to the direction of motion and m is the invariant mass of the particle.
Remark: The difference in the rapidity of two particles is independent of the Lorentz boosts along the beam axis.
Colliders measure physical momenta in terms of transverse momentum instead of the momentum in the direction of the beam axis (longitudinal momentum) , the polar angle in the transverse plane (genarally denoted by ) and pseudorapidity . To obtain cartesian momenta (with the z-axis defined as the beam axis), the following transformations are used:
Thus, we get the also useful relationship
This quantity is an observable in the collision of particles, and it can be measured as the main image of this post shows.
I have been fascinated (perhaps I am in love too with it) by Mathematics since I was a child. As a teenager in High School, I was a very curious student ( I am curious indeed yet) and I tried to understand some weird results I got in the classroom. This entry is devoted to some of those problems that made me wonder and think a lot out of class, at home. It took me some years and to learn complex variable function theory to understand one of the issues I could not understand before I learned complex variable. 3 years to understand a simple derivate! Yes, it is too much time. However, understanding stuff deeply carries time. Sometimes more, sometimes less.
This is the problem. A very simple problem indeed! The following “weird” (real) trigonometric-like function has a null derivative:
where the is the arctangent function ( the inverse of the tangent function ).
Since the derivative is zero, the two functions must differ by a constant. We can guess that constant with usual real variable calculus. It’s quite simple:
So, the difference is . However, this calculation does not explain why the two functions differ by a constant. The secret lies in the complex function origin of the arctangent. In complex variable function theory, it can be proved that
then and thus and therefore Q.E.D.
Now, we can calculate the functions in terms of complex variables:
We can make some algebra inside the logarithm function to get:
By the other hand, we also have
i.e.,the terms depending on x cancel to get a pure complex number! The number is
We have to calculate the logarith of the complex number
Then, . Of course, that is we were expecting to get since in this case
as before! That is, we recover the phase difference we also got with real calculus. The origin of the cancellation was in the complex origin of the arctangent function! Beautiful mathematics! The complex world is fascinating but very enlightening even for real functions!
What is a triangle? It is a question of definition in Mathematics. Of course you could disagree, but it is true. Look the above three “triangles”. Euclidean geometry is based in the first one. The second “triangle” is commonly found in special relativity. Specially, hyperbolic functions. The third one is related to spherical/elliptical geometry.
Today’s summary: some basic concepts in arithmetics, complex numbers and functions. We are going to study and review the properties of some elementary and well known functions. We are doing this in order to prepare a better background for the upcoming posts, in which some special functions will appear. Maybe, this post can be useful for understanding some previous posts too.
First of all, let me remember you that elementary arithmetics is based on seven basic “operations”: addition, substraction, multiplication, division, powers, roots, exponentials and logarithms. You are familiar with the 4 first operations, likely you will also know about powers and roots, but exponentials and logarithms are the last kind of elementary operations taught in the school ( high school, in the case they are ever explained!).
Let me begin with addition/substraction of real numbers (it would be also valid for complex numbers or even more general “numbers”, “algebras”, “rings” or “fields”, with suitable extensions).
Multiplication is a harder operation. We have to be careful with the axioms since there are many places in physics where multiplication is generaliz loosing some of the following properties:
Indeed the last rule can be undestood as the “division” rule, provided since in mathematics or physics there is no sense to “divide by zero”, as follows.
Now, we are going to review powers and roots.
Note that the identity is not true in general. Moreover, if then as well, as it can be easily deduced from the previous axioms. Now, the sixth operation is called exponentiation. It reads:
Sometimes you can read , where is the so-called “e” number. The definitions is even more general, since the previous property is the key feature for any exponential. I mean that,
We also get that for any , then . Finally, the 7th operation. Likely, the most mysterious for the layman. However, it is very useful in many different places. Recall the definition of the logarithm in certain base “a”:
Please, note that this definition has nothing to do with the “deformed” logarithm of my previous log-entry. Notations are subtle, but you must always be careful about what are you talking about!
Furthermore, there are more remarks:
1st. Sometimes you write . Be careful, some books use other notations for the Napier’s logarithm/natural logarithm. Then, you can find out there or even .
2nd. Whenever you are using a calculator, you can generally find and . Please, note that in this case is not the natural logarithm, it is the decimal logarithm.
Logarithms (caution: logarithms of real numbers, since the logarithms of complex numbers are a bit more subtle) have some other cool properties:
Common values of the logarithm are:
Indeed, logarithms are also famous due to a remarkable formula by Dirac to express any number in terms of 2’s as follows:
However, it is quite a joke, since it is even easier to write , or even
Are we finished? NO! There are more interesting functions to review. In particular, the trigonometric functions are the most important functions you can find in the practical applications.
Triangles are cool! Let me draw the basic triangle in euclidean trigonometry.
The trigonometric ratios/functions you can define from this figure are:
i)The function (sin), defined as the ratio of the side opposite the angle to the hypotenuse:
ii) The function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
iii) The function (tan), defined as the ratio of the opposite leg to the adjacent leg.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle ”A”. The ”’adjacent leg”’ is the other side that is adjacent to angle ”A”. The ”’opposite side”’ is the side that is opposite to angle ”A”. The terms ”’perpendicular”’ and ”’base”’ are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA ( a mnemonics rule whose derivation and meaning is left to the reader).
The multiplicative inverse or reciprocals of these functions are named the cosecant (csc or cosec), secant(sec), and cotangent (cot), respectively:
The inverse trigonometric functions/inverse functions are called the arcsine, arccosine, and arctangent, respectively. These functions are what in common calculators are given by . Don’t confuse them with the multiplicative inverse trigonometric functions.
There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to “co-“. From the goniometric circle (a circle of radius equal to 1) you can read the Fundamental Theorem of (euclidean) Trigonometry:
Indeed, from the triangle above, you can find out that the pythagorean theorem implies
So, the Fundamental Theorem of Trigonometry is just a dressed form of the pythagorean theorem!
The fundamental theorem of trigonometry can be rewritten too as follows:
These equations can be easily derived geometrically from the goniometric circle:
The trigonometric ratios are also related geometrically to this circle, and it can seen from the next picture:
Other trigonometric identities are:
The above trigonometric functions are also valid for complex numbers with care enough. Let us write a complex number as either a binomial expression or like a trigonometric expression . The famous Euler identity:
allows us to relate both two expressions for a complex number since
implies that and . The Euler formula is also useful to recover the identities for the sin and cos of a sum/difference, since
The complex conjugate of a complex number is , and the modulus is
Moreover, , , and if , then
We also have the so-called Moivre’s formula
and for the complex roots of complex numbers with the identity:
The complex logarithm (or the complex power) is a multivalued functions (be aware!):
The introduction of complex numbers and complex values of trigonometric functions are fun. You can check that
thanks to the Euler identity.
In special relativity, the geometry is “hyperbolic”, i.e., it is non-euclidean. Let me review the so-called hyperbolic trigonometry. More precisely, we are going to review the hyperbolic functions related to special relativity now.
We define the functions sinh, cosh and tanh ( sometimes written as sh, ch, th):
The fundamental theorem of hyperbolic trigonometry is
The hyperbolic triangles are objects like this:
The hyperbolic inverse functions are
Two specially useful formulae in Special Relativity (related to the gamma factor, the velocity and a parameter called rapidity) are:
In fact, we also have:
There are even more identities to be known. The most remarkable and important are likely to be:
You can also relate euclidean trigonometric functions with hyperbolic trigonometric functions with the aid of complex numbers. For instance, we get
and so on. The hyperbolic models of geometry/trigonometry are also very known in arts. Escher’s drawings are very beautiful and famous:
or the colorful variation of this theme
I love Escher’s drawings. And I also love Mathematics, Physics, Physmatics, and Science. Equations are cool. And hyperbolic functions, and other functions we have reviewed here today, will arise naturally in the next posts.