Say It With Science

Charge, Parity and Time Reversal (CPT) Symmetry

From our everyday experience, it is easy to conclude that nature obeys the laws of physics with absolute consistency. However, several experiments have revealed certain cases where these laws are not the same for all particles and their antiparticles. The concept of a symmetry, in physics, means that the laws will be the same for certain types of matter. Essentially, there are three different kinds of known symmetries that exist in the universe: charge (C), parity (P), and time reversal (T). The violations of these symmetries can cause nature to behave differently. If C symmetry is violated, then the laws of physics are not the same for particles and their antiparticles. P symmetry violation implies that the laws of physics are different for particles and their mirror images (meaning the ones that spin in the opposite direction). The violation of symmetry T indicates that if you go back in time, the laws governing the particles change.

There were two American physicists by the names of Tsunng-Dao Lee and Chen Ning Yang suggested that the weak interaction violates P symmetry. This was proven by an experiment which was conducted with radioactive atoms of colbalt-60 that were lined up and introduced a magnetic field to insure that they are spinning in the same direction. In addition, it was also found that the weak force also does not obey symmetry C. Oddly enough, the weak force did appear to obey the combined CP symmetry. Therefore the laws of physics would be the same for a particle and it’s antiparticle with opposite spin.

Surprise, surprise! There was a slight error in the previous experiment that was just mentioned. A few years later, it was discovered that the weak force actually violates CP symmetry. Another experiment was conducted by two physicists named Cronin and Fitch. They studied the decay of neutral kaons, which are mesons that are composed of either one down quark (or antiquark) and a strange antiquark (or quark). These particles have two decay modes where one will decay much faster than the other, even though they all have identical masses. The particles with the longer lifetimes will decay into three pions (denoted with the symbol π0), however the kaon ‘species’ with the shorter lifetimes will only decay into two pions. They had a 57 foot beamline, where they only expected to see the particles with slower decay rate at the end of the beam tube. In astonishment, one out of every 500 decays where from the kaons species that had a shorter lifetime. The main conflict with seeing the short-lived mesons at the end of the beam tube is because they are traveling relavistic speeds and therefore ignoring the time dilatationthat they are supposed to undergo. Thus, the experiment has shown that the weak force causes a small CP violation that can be seen in kaon decay.

(Source: aps.org)

Refraction

Light waves are part of the EM wave spectrum. When moving through an optical medium (i.e. air, glass, etc. …), the E field of the wave excites the electrons within the medium, causing them to oscillate, as a result, the light wave slows down slightly due to the loss of some of its kinetic energy. Its new speed is always less than that of the speed of light in a vacuum (v<c). Materials are characterized by their ability to bend as well as slow down light, which is known as optical refractive index (n).

         c
     n = -
         v
          speed of light in a vacuum
       = ----------------------------
         speed of light in the medium

n = 1 in a vacuum
n = more than 1 in all other media

Refraction itself occurs when light passes across an interface between two media with different indices of refraction. As a general rule (which can be derived by Snell’s law below), light refracts towards the normal when passing to a medium with a higher refractive index, and away from the normal when moving to a medium of lower refractive index.

Snell’s Law:

n₁sinα = n₂sinβ

where n₁ is the refractive index of the first medium

Reflection

One of the properties of a boundary between optical media is that some of the light that’s approaching the interface at the angle of incidence (α) is reflected back into the first medium, while the rest continues on into the second medium at the angle of refraction (β).

Angle of incidence = Angle of Reflection

The Hamilton-Jacobi Equation

This blog has posted more than a few times in the past about classical mechanics. Luckily, classical mechanics can be approached in several ways. This approach, which uses the Hamilton-Jacobi equation (HJE), is one of the most elegant and powerful methods.

Why is the HJE so powerful? Consider a dynamical system with a Hamiltonian H=H(q,p,t). Suppose we knew of a canonical transformation (CT) that generated a new Hamiltonian K=K(Q,P,t) which (for a local chart on phase space) vanishes identically. Then the canonical equations would give that the transformed coordinates (Q,P) are constant in this region. How easy it would be to solve a system where you know that most of the important quantities are constant!

The rub is in finding such a canonical transformation. Sometimes it can’t even be done analytically, but nevertheless this is the goal of the Hamilton-Jacobi method of solving mechanical systems. In the equation given above, S is the generating function of the CT. Coincidentally, it often comes out to just equal the classical action up to an additive constant! This is due to the connection between canonical transformations and mechanical gauge transformations; it turns out that the additive function used to define the latter is the generating function of the former. In general the HJE is a partial differential equation that might be solvable by additive separation of variables… but don’t get too hopeful! Oftentimes the value of the HJE comes not in finding the actual equations of motion but in revealing symmetry and conservation properties of the system.

Variable Star Astronomy

Variable stars are stars whose brightness changes because of physical changes within the star. There exist more than 30,000 variable stars in just the Milky Way. Variable star astronomy is a popular part of astronomy because amateur astronomers play a key role. They have submitted thousands of observed data and these data are logged onto a database. American readers can find information on it on the American Association of Variable Star Observers page. 

One of such variable stars are called Cepheids. Cepheids are pulsating variable stars because they undergo  a “repetitive expansion and contraction of their outer layers” [1]. In Cepheids, the star’s period of variation (about 1-70 days) is related to its luminosity; the longer the period, the higher the luminosity. In fact, when graphed, the relationship is shown by a straight line (as can be seen on the title image). Henrietta Swan Leavitt, an American astronomer, first discovered this and understood the significance of this knowledge.  Combined with understanding of the star’s apparent magnitude (a previously written post on this subject can be found here), astronomers can use this information to find a star’s distance from Earth. Cepheids are famously known for their usefulness in finding distances to far-away galaxies and other deep sky objects. Leavitt died early from cancer but was to be nominated for the Nobel Prize in Physics by Professor Mittag-Leffler (Swedish Academy of Sciences). 

Edwin Hubble used Leavitt’s discovery to prove that the Andromeda Galaxy (M31) is not part of the Milky Way, but was able to find the distance to the Andromeda Galaxy (between 2-9 million light years away). At first his calculation was incorrect (900,000 light years) because he observed Type I (classical) Cepheid Stars. Type I Cepheid stars are brighter, newer Population I stars. Hubble later used type II Cepheids (also called W Virginis stars), which are smaller, dimmer, Population II stars, and he was able to make more accurate calculations.

To determine the star’s distance, use the inverse square law of light brightness. 

A similar type of star are RR Lyrae Variable Stars. They are smaller than Cepheids and have a much shorter period (from a few hours to a day). On the other hand, they are far more common. Likewise, they can be used to solve for distances as well. Low mass stars live longer, and thus Cepheid stars are generally younger because they are more massive. 

Both Cepheids and RR Lyrae Variable stars are referred to as standard candles: objects with known luminosity. If you’ve ever wondered how astronomers came to those enormous figures when describing how far away galaxies and stars are from us, you can now better understand why and how. 

The Virial Theorem

In the transition from classical to statistical mechanics, are there familiar quantities that remain constant? The Virial theorem defines a law for how the total kinetic energy of a system behaves under the right conditions, and is equally valid for a one particle system or a mole of particles.

Rudolf Clausius, the man responsible for the first mathematical treatment of entropy and for one of the classic statements of the second law of thermodynamics, defined a quantity G (now called the Virial of Clausius):

G ≡ Σ_i(p_i · r_i)

Where the sum is taken over all the particles in a system. You may want to satisfy yourself (it’s a short derivation) that taking the time derivative gives:

dG/dt = 2T + Σ_i(F_i · r_i)

Where T is the total kinetic energy of the system (Σ  ½mv²) and dp/dt = F. Now for the theorem: the Virial Theorem states that if the time average of dG/dt is zero, then the following holds (we use angle brackets ⟨·⟩ to denote time averages):

2⟨T⟩ = - Σ_i(F_i · r_i)

Which may not be surprising. If, however, all the forces can be written as power laws so that the potential is V=arⁿ (with r the inter-particle separation), then

2⟨T⟩ = n⟨V⟩

Which is pretty good to know! (Here, V is the total kinetic energy of the particles in the system, not the potential function V=arⁿ.) For an inverse square law (like the gravitational or Coulomb forces), F∝1/r² ⇒ V∝1/r, so 2⟨T⟩ = -⟨V⟩.

Try it out on a simple harmonic oscillator (like a mass on a spring with no gravity) to see for yourself. The potential V ∝ kx², so it should be the case that the time average of the potential energy is equal to the time average of the kinetic energy (n=2 matches the coefficient in 2⟨T⟩). Indeed, if x = A sin( √[k/m] · t ), then v = A√[k/m] cos( √[k/m] · t ); then x² ∝ sin² and v² ∝ cos², and the time averages (over an integral number of periods) of sine squared and cosine squared are both ½. Thus the Virial theorem reduces to

2 · ½m·(A²k/2m) = 2 · ½k(A²/2)

Which is easily verified. This doesn’t tell us much about the simple harmonic oscillator; in fact, we had to find the equations of motion before we could even use the theorem! (Try plugging in the force term F=-kx in the first form of the Virial theorem, without assuming that the potential is polynomial, and verify that the result is the same). But the theorem scales to much larger systems where finding the equations of motion is impossible (unless you want to solve an Avogadro’s number of differential equations!), and just knowing the potential energy of particle interactions in such systems can tell us a lot about the total energy or temperature of the ensemble.

Hypercubes

What is a hypercube (also referred to as a tesseract) you say! Well, let’s start with what you know already. We know what a cube is, it’s a box! But how else could you describe a cube? A cube is 3 dimensional. Its 2 dimensional cousin is a square. 

A hypercube is just to a cube what a cube is to a square. A hypercube is 4 dimensional! (Actually— to clarify, hypercubes can refer to cubes of all dimensions. “Normal” cubes are 3 dimensional, squares are 2 dimensional “cubes, etc. This is because a hypercube is an n-dimensional figure whose edges are aligned in each of the space’s dimensions, perpendicular to each other and of the same length. A tesseract is specifically a 4-d cube). 

[source]

Another way to think about this can be found here:

Start with a point. Make a copy of the point, and move it some distance away. Connect these points. We now have a segment. Make a copy of the segment, and move it away from the first segment in a new (orthogonal) direction. Connect corresponding points. We now have an ordinary square. Make a copy of the square, and move it in a new (orthogonal) direction. Connect corresponding points. We now have a cube. Make a copy and move it in a new (orthogonal, fourth) direction. Connect corresponding points. This is the tesseract.

If a tesseract were to enter our world, we would only see it in our three dimensions, meaning we would see forms of a cube doing funny things and spinning on its axes. This would be referred to as a cross-section of the tesseract. Similarly, if we as 3-dimensional bodies were to enter a 2-dimensional world, its 2-dimension citizens would “observe” us as 2-dimensional cross objects as well! It would only be possible for them to see cross-sections of us.

Why would this be significant? Generally, in math, we work with multiple dimensions very often. While it may seem as though a mathematican must then work with 3 dimensions often, it is not necessarily true. The mathematician deals with these dimensions only mathematically. These dimensions do not have a value because they do not correspond to anything in reality; 3 dimensions are nothing ordinary nor special. 

Yet, through modern mathematics and physics, researchers consider the existence of other (spatial) dimensions.  What might be an example of such a theory? String theory is a model of the universe which supposes there may be many more than the usual 4 spacetime dimensions (3 for space, 1 for time). Perhaps understanding these dimensions, though seemingly impossible to visualize, will come in hand. 

Carl Sagan also explains what a tesseract is. 

Image: Peter Forakis, Hyper-Cube, 1967, Walker Art Center, Minneapolis

When describing the trajectory of a point particle in space, we can use simple kinematic physics to describe properties of the particle: force, energy, momentum, and so forth. But are there useful measures we can use to describe the qualities of the trajectory itself?

Enter the Frenet-Serret (or TNB) frame. In this post, we’ll show how to construct three (intuitively meaningful) orthonormal vectors that follow a particle in its trajectory. These vectors will be subject to the Frenet-Serret equations, and will also end up giving us a useful way to interpret curvature and torsion.

First, we define arc length: let s(t) = ∫₀^t ||x’(τ)|| dτ. (We give a quick overview of integration in this post.) If you haven’t encountered this definition before, don’t fret: we’re simply multiplying the change in position of the particle x’(τ) by the small time step dτ summed over every infinitesimal time step from τ=0 to τ=t=”current time”. The post linked to above also explains a short theorem that may illustrate this point more lucidly.

Now, consider a particle’s trajectory x(t). What’s the velocity of this particle? Its speed, surely, is ds/dt: the change in arc length (distance traveled) over time. But velocity is a vector, and needs a direction. Thus we define the velocity v=(dx/ds)⋅(ds/dt). This simplifies to the more obvious definition dx/dt, but allows us to separate out the latter term as speed and the former term as direction. This first term, dx/ds, describes the change in the position given a change in distance traveled. As long as the trajectory of the particle has certain nice mathematical properties (like smoothness), this vector will always be tangent to the trajectory of the particle. Think of this vector like the hood of your car: even though the car can turn, the hood will always point in whatever direction you’re going towards. This vector T ≡ dx/ds is called the unit tangent vector.

We now define two other useful vectors. The normal vector: N ≡ (dT/ds) / ( |dT/ds| ) is a vector of unit length that always points in whichever way T is turning toward. It can be shown — but not here — that T ⊥ N. The binormal vector B is normal to both T and N; it’s defined as B ≡ T x N. So T, N, and B all have unit length and are all orthogonal to each other. Since T depends directly on the movement of the particle, N and B do as well; therefore, as the particle moves around, the coordinate system defined by T, N, and B moves around as well, connected to the particle. The frame is always orthonormal and always maintains certain relationships to the particle’s motion, so it can be useful to make some statements in the context of the TNB frame.

The Frenet-Serret equations, as promised:

• dT/ds = κN
• dN/ds = -κT + τB
• dB/ds = -τN

Here, κ is the curvature and τ is the torsion. Further reading (lookup the Darboux vector) illustrates that κ represents the rotation of the entire TNB frame about the binormal vector B, and τ represents the rotation of the frame about T. The idea of the particle trajectory twisting and rolling nicely matches the idea of what it might be like to be in the cockpit of one of these point particles, but takes this depth of vector analysis to get to.

Bonus points: remember how v = Tv, with v the speed? Differentiate this with respect to time, play around with some algebra, and see if you can arrive at the following result: the acceleration a = κv²N + (d²s/dt²)T. Thoughtful consideration will reveal the latter term as the tangential acceleration, and knowing that 1/κ ≡ ρ = “the radius of curvature” reveals that the first term is centripetal acceleration.

—

Photo credit: Salix alba at en.wikipedia

Check out a previous post on gravity here, and another post on planetary orbit here.

Gravity and Astronauts in Orbit

Newton’s Law of Gravity states that the gravitational force between any two objects in the universe is proportional to the mass of the objects and inversely proportional to the square of the distance (measured from their centers) between them. That means that if you double the size of either of the masses, the gravitational force will double. Then also if you halve the distance between the two objects and their masses stay the same, the gravitational force between them will quadruple. The converse of both these statements is also true.

Despite gravity keeping everything around you steadily on the ground, it’s considered to be the weakest force in relation to the other four fundamental forces. After all, you can easily overcome the gravitational force of the whole entire Earth for a few seconds by simply jumping or throwing a ball up into the air.

A common misconception is that there’s no gravity in outer space. Anything with mass has an infinitely large gravitational field - but because it gets progressively weaker with distance, its effects quickly become negligible.

Then why do astronauts float around in the space shuttle? Why are they weightless? The only reason you can feel your weight on Earth is because the Earth is pushing back up at you. It’s the normal force from the Earth on your body. However within an orbiting spacecraft, you’re in free fall. There’s nothing to push back at you and make you feel your weight! 

Fun fact: Astronauts experience a lot of nausea and headaches during their first few weeks in orbit. This is probably because not only are they in free fall, but the liquids inside them are probably flowing around too (eg. fluids to the brain, nasal cavities, stomach contents, etc...)

Sources 1, 2 and 3.

Cosmic Rays 

As you are reading this sentence, thousands of cosmic rays are passing through your body. They have been your whole life… But what are they exactly?

The composition of cosmic rays is estimated as follows:

• 89% Simple protons
• 10% Alpha particles
• Most of the remaining 1% are electrons, positrons and antiprotons and other heavier nuclei (which are abundant end products of stars’ nuclear synthesis)

(However, the precise composition of cosmic rays outside the Earth’s atmosphere can vary depending on which part of the energy spectrum is observed.) 

There are two types of cosmic rays:

Primary cosmic rays

It is speculated that cosmic rays are formed when particles are accelerated by a blast waves of supernova remnants. They eventually gain so much energy from expanding clouds of gas and bouncing back and forth in magnetic fields that they continue to move through space at speeds very close to that of light. The supernova remnants are not able to contain them when they reach these speeds (and also have energies of over 10¹⁸ eV), so the cosmic rays escape into the galaxy. The maximum amounts of energy cosmic rays can gain depend on the size of the acceleration region and the strength of its magnetic field. 

Secondary cosmic rays

Primary cosmic rays interact with interstellar matter to produce a second type of cosmic ray. Heavy nuclei (mainly carbon and oxygen) that make up less than 1% of cosmic rays break up into lighter nuclei (of mainly lithium, beryllium and boron) upon penetration of the Earth’s atmosphere and surface by a process of cosmic ray spallation.  

Antiprotonic Helium

Antiprotonic helium consists of an electron and antiproton that orbit around a helium nucleus. The hyperfine structure of this exotic type of matter is studied very closely by a CERN experiment in Japan called ASACUSA (Atomic Spectroscopy And Collisions Using Slow Antiprotons) using laser spectroscopy.

To create antiprotonic helium, antiprotons are mixed with helium gas so that they spontaneously remove one of the electrons that orbit around each of the helium atoms and take their places. However, this reaction will only occur for 3% of the gas.

From the time that antiprotonic helium is created, the antiprotons orbiting the helium nucleus will only remain in orbit for a few micro seconds until they fall rapidly into the nucleus, causing a proton-antiproton annihilation. Surprisingly, antiprotonic helium has the longest lifetime of all the other antiprotonic atoms.

Laser Spectroscopy

ASACUSA physicists used a laser pulse (that if tuned correctly) will let the atom of antiprotonic helium absorb just enough energy so that the antiproton can jump from one energy level (aka orbit) to the other. Thus allowing physicists to determine the energy between orbits of an atom. Currently, laser and microwave precision spectroscopy of antiprotonic helium atoms is ASACUSA’s top priority. (Which is basically using two laser beams and pulsed microwave beams to further explore the ‘hyperfine energy levels’ of antiprotonic helium.)