Nabla in four dimensions

In special relativity, space-time is represented with one dimension of time and three dimensions of space : a four vector. In order to continue its production of grad, div, curl and laplacian, the 4 dimensional Nabla operator needs to adapt to its new environment.

How do we modify operations with our little upside-down triangle ?

To begin, the usual scalar product between vectors must be re-defined :

Our Nabla will have four components of partial derivatives, with the first component the derivative with respect to time, and the rest with respect to space (note the negative sign attached to the space components) :

We first examine the grad operation :

Next the divergence :

It’s easy to obtain the equivalent of the Laplacian in 4 dimension, called D’Alembertian :

Now to do the cross product so as to get the equivalent of the curl operator is a little more work. There are 16 combinations altogether. By analogy, we use the following idea for the 4-space cross product :

The 16 components that result from the cross product are :

The 16-components matrix is a antisymmetric tensor of 2^nd rank.

We use this tensor to house the electromagnetic field components of E and B.

Let’s see how this is done with our newly-discovered Fμν tensor.

We have chosen to work in the Cartesian coordinates system. If we were to use any other orthogonal or non-orthogonal systems, then it will be necessary to bring in a whole arsenal of tensor calculus.

 

 

 

Einstein’s less famous discoveries (2 of 5)

The failure of classical statistical mechanics to explain the anomalies in the variation of heat capacity with temperature was identified by Maxwell as a major crisis in Thermodynamics. It takes Quantum statistics to supply the theory for the sudden jumps in the values of the heat capacity as temperature rises.

Einstein made use of the the heat capacity formulation for diatomic gases to develop a model for calculating the heat capacity of solids. Here’s how it goes.

With this expression for the internal energy, the heat capacity at constant volume can be calculated :

The value of C_v approaches zero as T approaches zero, corresponding to experimental findings; and as T rises, C_v approaches 3Nk, in agreement with the Dulong-Petit model.

Einstein’s model works well for most solids but it is still not refined enough. An improved model is proposed by Debye which uses a spectrum of frequencies instead of just one single frequency for the harmonic oscillators.

Beautiful proof of Pythagoras’ theorem

The proof is by Euclid, who lived more than 2000 years ago in Greece.

We have first a right-angled triangle ABC. To each side we attach a square.

Euclid’s genius is to form up 2 lines : AD and CF, which immediately created 2 identical triangles ADB and FCB.

Now the area of ADB is ½ × BD × BK, which makes up half of the area of the rectangle BKLD, whereas the area of FCB is ½ × FB × BA, which makes up half of the square ABFG.

Therefore, area of square ABFG = area of rectangle BKLD.

Using similar constructions with lines AE and BI, and similar reasonings as above, it can be shown that area of square ACIH = area of rectangle CKLE.

So area of square ABFG + area of square ACIH = area of square BCED, which means :

AB² + AC² = BC²

And voilà!  Pythagoras’ theorem proven.

(Edited by: ADB)

 

 

Nabla in other coordinates systems

In the Cartesian system, the various operators like ∇, ∇·, ∇× or ∇² are easy to construct. How do we get the same set of operators when working in say, spherical coordinates system ?

Here we limit ourselves to orthogonal coordinates system, i.e., when the basis vectors are mutually perpendicular. To generalise the results, we use the symbols h_i for line element, e_i for unit vectors, and x_i for coordinates.

Let’s start with the grad operator :

Next, the divergence :

Combining the grad and the div, we have the laplacian :

Finally, the curl, which as expected, is computed with the use of cross products :

In non-orthogonal systems, these operators are constructed with tensors.

 

Einstein’s less famous discoveries (1 of 5)

Einstein coefficients are used today in the theory of laser.

Imagine an atom which can exist in 2 states : ground state (labeled 1) and excited state (labeled 2). There are 3 possible ways of transition between these states :

• Absorption of a photon to jump from state 1 to 2
• Spontaneous emission of a photon to jump from state 2 to 1
• Stimulated emission of a photon to jump from state 2 to 1

The process of Absorption and Stimulated Emission depends on the intensity of the radiation present whilst Spontaneous Emission is independent of it.

Einstein knows about Planck’s law of radiation, so he discovered that the two coefficients B must be equal : the probability of absorption and the probability of stimulated emission are the same!

He was also able to make out the ratio of A to B :

To know what A and B are exactly, one had to wait for the arrival of quantum electrodynamics.

Another way to understand this business of absorption and emission is by some funny   quantum mechanics symbols :

The first equation is the probability amplitudes of transition from the state with n photons to the state with n+1 photons (emission), and the second one from n+1 photons to n photons (absorption); a is the amplitude of transition when no photon is present.

The probability of emission and absorption when there are n photons present are then :

The emission probability consists of the contribution from n stimulated part and one spontaneous part. We can see that the stimulated emission probability and the absorption probability are the same, just as Einstein predicted.

Planck, Rayleigh-Jeans & Wien

The blackbody radiation law went through several revisions in the late 19th – early 20th century, culminating in the discovery of the quantisation of electromagnetic energy, and hence ushered in the era of quantum physics.

First, there was Rayleigh-Jeans law which can be derived as below :

For low frequency this law works well, but at high frequency it was a total failure, leading to the nickname “ultraviolet catastrophe”.

Wien’s law performs a little better but still didn’t fit the observed spectrum completely.

Then Planck had an epiphany one fine day and discovered that by tweaking a little on the energy of the electromagnetic waves and their frequency distribution, everything fit just right. His formula explained the Blackbody radiation spectrum, yet he was not aware that he had kicked off the quantum revolution in Physics !

We can back-derive Rayleigh-Jeans and Wien’s formula by taking limits on Planck’s law :

Stefan-Boltzmann constant

The energy radiated from a blackbody is proportional to the 4th power of its thermodynamic temperature. The constant of proportionality is called the Stefan-Boltzmann constant.

Let’s see how we can derive a formula for this constant.

Electromagnetic radiations are carried by photons. Photons are bosons, so they obey Bose-Einstein statistics.

The number of quantum states per energy level is deduced this way :

Planck’s law for blackbody radiation is found by combining the two terms above and the energy of a photon :

If we integrate this function over all frequencies, we obtain the energy density of the emitted radiation.

The formula for the Stephan-Boltzmann constant looks amusing : see how the power of the various terms run in sequence. The value of the constant is as amusing : 5,6,7,8 – making it an easy number to remember.

Oscillating dipole (3 of 3)

We will do 3 things here. First, find an expression for the Poynting vector generated by the dipole radiator, and the dipole’s average radiated power. Second, we will derive the amplitude of the oscillating dipole in relation to the incoming E-field strength which causes the dipole to shake in the first place. Third, we will look at the scattering cross-section under different circumstances, in particular Rayleigh scattering.

The Poynting vector, calculated using only the terms involving the acceleration of the dipole is :

The average power radiated by the dipole is :

Now, the amplitude of the oscillator is related to the incoming E field that energises it  :

Replacing the dipole amplitude in the formula for dipole power in the above  equation, we get an expression that can be broken down into several parts :

Thomson scattering occurs when the natural frequency  of the oscillator is negligible. This is applicable to scattering by free electrons.

Rayleigh scattering occurs when the oscillator’s natural frequency is much higher than the frequency of the incoming light. This is the scattering responsible for the colours of the sky.

Under Raleigh scattering, when ω doubles, the intensity of the scattered radiation goes up 16 times ! That’s why blue outshines red when light is scattered by air molecules, which act like oscillating dipoles.

The scattered intensity by N condensed water vapour molecules oscillating in phase is N² times more powerful than N independent molecules. So we see white clouds.

The wavelength of red light is about twice that of blue light. At sunset, the blues are scattered out more than the reds by the thicker atmosphere that sunlight needs to penetrate, and the red lightwaves outlast the blue to reach our tiny eyes.

Now we finally have a quantitative  explanation for our gorgeous blue sky, white clouds and red sunsets

Oscillating dipole (2 of 3)

After B, it’s time to get E.

We need the following 2 equations to get things rolling (the first one is called Lorenz Gauge, the second is simply the general solution for E):

After taking the divergence of A, integrate it with respect to t to get ϕ (not the azimuth angle but the electric potential !).

Then take the grad of ϕ and differentiate A with respect to t, we’ll get to E.

What a complicated-looking formula !

Again at large distance from the dipole, the really important term is the one with a double dot above the p.

We will make use of these terms only from E and B to produce our Poynting vector finally.

Oscillating dipole (1 of 3)

When an electric dipole shakes, electromagnetic waves are generated. We want to know the intensity of the emitted radiation, so we first look at the vector potential A that accompanies the shaking dipole :

With A obtained (as a function of the dipole velocity) , the next step is to get B, the magnetic field.

But before that, a little preparation work is in order. We would like to work in spherical coordinates, so let’s write down the components of A :

Now we shall get B by taking the curl of A :

To get the intensity of the scattered light at a large distance from the dipole, only the term in 1/r matters, i.e., the acceleration term.

By the way, we could also express B as a cross product between p and r :

Next time we will get E, then we’ll see what our Poynting vector E x B looks like.