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Largest Triangle inside a Curvilinear Triangle

June 20, 2015 1 comment

In the diagram shown we have part of a circle of radius $R$ whose center is at the point $(R,R)$ and which is tangent to the x and y axes — though the graph is drawn for R = 2, we want to work with general R.

Our focus is on the region under the circle above the x-axis. The question is: what is the maximum area that a triangle inside this region can have?

It may occur to you that there is a reasonable quick’ answer, but the point of the problem is to reason it out carefully so you more or less have a proof that you do indeed get a maximum area. Since the region is concave, the vertices of a triangle cannot be so that one is too close to the far right while another vertex close to the far top left (or else the triangle would not be fully inside the region).

Have fun!

Two Little Circles

In the diagram shown we see a big circle of radius R that is tangent to both the x and y axes. What is the radius of the little circle to its southwestern corner? Express its radius in terms of R. (The little circle is also required to be tangent to the axes. The diagram is draw for a circle of radius 2 but we’re working with a general radius R.)

After having solved this problem, the next question will be this:

You can easily see that we can repeat the process by looking at the southwestern circles for ever and ever because there will always be a gap, and you get smaller and smaller circles with small radii. What if you add up all their radii? Do they add up? If so to what number do they add up, in terms of the original radius R. If they don’t add up, why don’t they?

Cosmic Microwave Background

May 30, 2015 1 comment

The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around ${3000}^{\circ}$K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of $2.726^\circ$K.

$\displaystyle I(\nu) = \frac{8\pi h}{c^3} \cdot \frac{\nu^3}{e^{h\nu/kT} - 1}.$

In this formula, $\nu$ is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and $I(\nu)$ is the energy density as a function of frequency $\nu$.

The other variables are: $T$ is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), $c$ is the speed of light in vacuum, $h$ is Planck’s constant, and $k$ is Boltzmann’s constant from statistical mechanics.

If you plot the graph of this energy density function (against $\nu$) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:

Various Planck radiation density graphs depending on temperature T.

Here you see various plots of Planck’s function for different temperatures $T$. The horizontal axis labels the frequency $\nu$, and the vertical gives the energy density $I(\nu)$ per frequency. (Please ignore the rising black dotted curve.)

You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.

To those of us who know calculus, we can actually compute what frequency $\nu$ gives maximum density and give our answer in terms of the temperature $T$. All we do is compute the derivative of $I(\nu)$ with respect to $\nu$ and set it to zero and solve the resulting equation for $\nu$. You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:

$\displaystyle\nu_{\max} = 2.82 \frac{kT}{h}.$

(The equal sign here is an approximation!)

The $\nu_{\max}$ is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency $\nu_{\max}$ gets, and from Planck’s energy equation $E_{\max} = h\nu_{\max}$, so also does the energy of the radiation drop.

Now plug in the observed value for the temperature of the background radiation, which is $T = 2.726$ (degrees Kelvin), and working it out we get (approximately)

$\displaystyle\nu_{\max} = 160.2 \text{ GHz}$

This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)

Far back in time, when photons were released from their collision trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.

Homework Question: what was the max-frequency $\nu_{\max}$ at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! 😉 )

(It isn’t hard as it can be figured from the data above.)

Anyway, I thought working these out was fun.

The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being $3.5^\circ$ K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength $\lambda$ and frequency $\nu$ for electromagnetic waves we use the simple formula

$\displaystyle\lambda\nu = c$

where c is the speed of light (in vacuum).

And that’s the end of our little story for today!

Cheers, Sam Postscript.

The sacred physical constants:

Planck’s constant $h = 6.6254 \times 10^{-27} \text{ erg sec}$

Boltzmann’s constant $k = 1.38049 \times 10^{-16} \text{ erg/K}$

Speed of light $c = 2.9979 \times 10^{10} \text{ cm/sec}$.

Einstein summation convention

March 25, 2015 1 comment

Suppose you have a list of n numbers $A_1, A_2, A_3, \dots, A_n$.

Their sum $A_1 + A_2 + A_3 + \dots + A_n$ is often shorthanded using the Sigma notation like this

$\displaystyle\sum_{k=1}^n A_k$

which is read “sum of $A_k$ from k=1 to k=n.” This letter k that varies from 1 to n is called an ‘index’.

Vectors. You can think of a vector as an ordered list of numbers $A = (A_1, A_2, A_3, \dots, A_n) .$

If you have two vectors $A = (A_1, A_2, A_3, \dots, A_n)$ and $B = (B_1, B_2, B_3, \dots, B_n)$ their dot product is defined by multiplying their respective coordinates and adding the result:

$\displaystyle A\bullet B = A_1B_1 + A_2B_2 + A_3B_3 + \dots + A_nB_n.$

Using our summation notation, we can abbreviate this to

$\displaystyle A\bullet B = \sum_{k=1}^n A_k B_k.$

While working thru his general theory of relativity, Einstein noticed that whenever he was adding things like this, the same index k was repeated! (You can see the k appearing once in A and also in B.) So he thought, well in that case maybe we don’t need a Sigma notation! So remove it! The fact that we have a repeating index in a product expression would mean that a Sigma summation is implicitly understood. (Just don’t forget! And don’t eat fatty foods that can help you forget!)

With this idea, the Einstein summation convention would have us write the above dot product of vectors simply as

$\displaystyle A\bullet B = A_k B_k$

In his theory’s notation, it’s understood that the index k here would vary from 1 to 4, for the four dimensional space he was working with. That’s Einstein’s index notation where 1, 2, 3, are the indices for space coordinates (i.e., $A_1, A_2, A_3$), and k=4 for time (e.g., $A_4$). One could also write space-time coordinates using the vector $(x_1, x_2, x_3, t)$ where t is for time.

(Some authors have k go from 0 to 3 instead, with k=0 corresponding to time and the others to space coordinates.)

I used ‘k’ because it’s not gonna scare anyone, but Einstein actually uses Greek letters like $\mu, \nu, \dots$ instead of the k. The convention is that Greek index letters range over all 4 space-time coordinates, and Latin indices (like k, j, m,etc)  for the space coordinates only. So if we use $\mu$ instead of k the dot product of the two vectors would be

$\displaystyle A\bullet B = A_\mu B_\mu.$

So if we write $A_\mu B_\mu$ it means we understand that we’re summing these over the 4 indices of space-time. And if we write  $A_k B_k$ it means that we’re summing these over the 3 indices of space only. More specifically,

$\displaystyle A_\mu B_\mu = A_1B_1 + A_2B_2 + A_3B_3 + A_4B_4$

and

$\displaystyle A_k B_k = A_1B_1 + A_2B_2 + A_3B_3.$

There is one thing that I left out of this because I didn’t want to complicate the introduction and thereby scare readers! (I already may have! Shucks!) And that is, when you take the dot product of two vectors in Relativity, their indices are supposed to be such that one index is a subscript (‘at the bottom’) and the other repeating index is a superscript (‘at the top’).  So instead of writing our dot product as $A_\mu B_\mu$ it is written as

$\displaystyle A\bullet B = A_\mu B^\mu.$

(This gets us into covariant vectors, ones written with subscripts, and contravariant vectors, ones written with superscripts. But that is another topic!)

How about we promote ourselves to Tensors? Fear not, let’s just treat it as a game with symbols! Well, tensors are just like vectors except that they can involved more than one index. For example, a vector such as in the above was written $A_\mu$, so it involves one index $\mu$. What if you have two indices? Well in that case we have a matrix which we can write $M_{\mu \nu}$. (Here, the two indices are sitting side by side like good friends and aren’t being multiplied! There’s an imaginary comma that’s supposed to separate them but it’s not conventional to insert a comma.)

The most important tensor in Relativity Theory is what is called the metric tensor written $\large g_{\mu\nu}$. It describes the distance structure (metric = distance) on a curved space-time. So much of the rest of the geometry of space, like its curvature, how to differentiate vector fields, curved motion of light and particles, shortest path in curved space between two points, etc, comes from this metric tensor $\large g_{\mu\nu}$.

The Einstein ‘gravitational tensor’ is one such tensor and is written $G_{\mu \nu}$. Tensors like those are called rank 2 tensors because they involve two different indices. Another good example of a rank 2 tensor is the energy-momentum tensor often written as $T_{\mu \nu}$. This tensor encodes the energy and matter distribution in spaces that dictate its geometry — the geometry (and curvature) being encoded in the Einstein tensor $G_{\mu \nu}$. (If you’ve read this far, you’re really getting into Relativity! And I’m very proud of you!)

You could have a tensors with 3, 4 or more indices, and the indices could be mixed subscripts and superscripts, like for example $D_{\mu \nu}^{\alpha \beta}$ and $F_{\tau}^\gamma$.

If you have tensors like this, with more than 1 or 2 indices, you can still form their dot products. For example for the tensors D and F, you can take any lower index of D (say you take $\nu$ and set it equal to an upper index of F — and add! So we get a new tensor when we do this dot product! You get

$\displaystyle D_{\mu \nu}^{\alpha \beta} F_{\tau}^\nu = D_{\mu 1}^{\alpha \beta} F_{\tau}^1 + D_{\mu 2}^{\alpha \beta} F_{\tau}^2 +D_{\mu 3}^{\alpha \beta} F_{\tau}^3 +D_{\mu 4}^{\alpha \beta} F_{\tau}^4$

where it is understood that since the index $\nu$ is repeated, you are summing over that index (from 1 to 4) (as I’ve written out on the right hand side). Notice that the indices that remain are $\mu, \alpha, \beta, \tau$. So this dot product gives rise to yet another tensor with these indices – let’s give the letter C:

$\displaystyle C_{\mu \tau}^{\alpha \beta} = D_{\mu \nu}^{\alpha \beta} F_{\tau}^\nu$.

This process where you pick two indices from tensors and add their products along that index is called ‘contraction‘ – even though it came out of doing a simple idea of dot product. Notice that in general when you contract tensors the result is not a number but is in fact another tensor. This process of contraction is very important in relativity and geometry, yet it’s based on a simple idea, extended to complicated objects like tensors. (In fact, you can call the original dot product of two vectors a contraction too, except it would be number in this case.)

Thank you!
$\Sigma\alpha\mu$