My views on various issues/news of interest. I try not just to repeat what's already been published, but to add my take on them. Contrary opinions welcomed (but nicely).
Sincerely, Sam

Einstein summation convention

March 25, 2015 Leave a 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

Einstein’s Religious Philosophy

January 17, 2015 Leave a comment

Here is a short, sweet, and quick summary of some of Albert Einstein’s philosophy and religious views which I thought were interesting enough to jot down while I have that material fresh in mind. (I thought it’s good to read all these various views of Einstein’s in one fell swoop to get a good mental image of his views.) These can be found in most biographies on Einstein, but I included references [1] and [2] below for definiteness. (Throughout this note, ‘he’ refers, of course, to Einstein.) Let’s begin!

  1. Einstein began to appreciate and identify more with his Jewish heritage in later life (as he approached 50).
  2. He had profound faith in the order and discernible laws in the universe, which he said was the extent to which he calls himself ‘religious.’
  3. God had no choice but to create the universe in the way He did.
  4. He believed in something larger than himself, in a greater mind.
  5. He called nationalism an infantile disease.
  6. He received instruction in the Bible and Talmud. He is a Jew, but one who is also enthralled by “the luminous figure of the Nazarene.”
  7. He believed Jesus was a real historical figure and that Jesus’ personality pulsates in every word in the Gospels.
  8. He was not an atheist, but a kind of “deist.”
  9. He did not like atheists quoting him in support of atheism.
  10. He believed in an impersonal God, who is not concerned with human action.
  11. His belief in an impersonal God was not disingenuous in order to cover up an underlying ‘atheism’.
  12. He was neither theist nor atheist.
  13. He did not believe in free will. He was a causal determinist. (Not even God has free will! :-) )
  14. Though he did not believe in free will, nevertheless he said “I am compelled to act as if free will existed.”
  15. He liked Baruch Spinoza’s treatment of the soul and body as one.
  16. He did not believe in immortality.
  17. He believed that the imagination was more important than knowledge.
  18. He believed in a superior mind that reveals itself in world of experience, which he says represents his conception of God.
  19. He believed in a “cosmic religious feeling” which he says “is the strongest and noblest motive for scientific research.”
  20. “Science without religion is lame, religion without science is blind.”

There you have it, without commentary! ;-)

References.

[1] Albert Einstein, Ideas and Opinions.

[2] Walter Isaacson, Einstein: His Life and Universe. (See especially chapter 17.)

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Einstein and Pantheism

January 3, 2015 Leave a comment

Albert Einstein’s views on religion and on the nature and existence of God has always generated interest, and they continue to. In this short note I point out that he expressed different views on whether he subscribed to “pantheism.” It is well-known that he said, for instance, that he does not believe in a personal God that one prays to, and that he rather believes in “Spinoza’s God” (in some “pantheistic” form). Here are two passages from Albert Einstein where he expressed contrary views on whether he is a “pantheist”.

I’m not an atheist and I don’t think I can call myself a pantheist. We are in the position of a little child entering a huge library filled with books in many different languages.…The child dimly suspects a mysterious order in the arrangement of the books but doesn’t know what it is. That, it seems to me, is the attitude of even the most intelligent human being toward God.” (Quoted in Encyclopedia Britannica article on Einstein.)

Here, Einstein says that he does not think he can call himself a pantheist. However, in his book Ideas and Opinions he said that his conception of God may be described as “pantheistic” (in the sense of Spinoza’s):

This firm belief, a belief bound up with a deep feeling, in a superior mind that reveals itself in the world of experience, represents my conception of God. In common parlance this may be described as “pantheistic” (Spinoza)” (Ideas and Opinions, section titled ‘On Scientific Truth’ – also quoted in Wikipedia with references)

It is true that Einsteins expressed his views on religion and God in various ways, but I thought that the fact that he appeared to identify and not identify with “pantheism” at various stages of his life is interesting. Setting aside that label, however, I think that his general conceptions of God in both these quotes — the mysterious order in the books, a universe already written in given languages, a superior mind that reveals itself in such ways — are fairly consistent, and also consistent with other sentiments he expressed elsewhere.

A game with Pi

November 8, 2014 2 comments

Here’s an image of something I wrote down, took a photo of, and posted here for you. It’s a little game you can play with any irrational number. I took \pi as an example.

You just learned about an important math concept/process called continued fraction expansions.

With it, you can get very precise rational number approximations for any irrational number to whatever degree of error tolerance you wish.

As an example, if you truncate the above last expansion where the 292 appears (so you omit the “1 over 292″ part) you get the rational number 335/113 which approximates \pi to 6 decimal places. (Better than 22/7.)

You can do the same thing for other irrational numbers like the square root of 2 or 3. You get their own sequences of whole numbers.

Exercise: for the square root of 2, show that the sequence you get is
1, 2, 2, 2, 2, …
(all 2’s after the 1). For the square root of 3 the continued fraction sequence is
1, 1, 2, 1, 2, 1, 2, 1, 2, …
(so it starts with 1 and then the pair “1, 2″ repeat periodically forever).

Matter and Antimatter don’t always annihilate

October 11, 2014 2 comments

It is often said that when matter and antimatter come into contact they’ll annihilate each other, usually with the release of powerful energy (photons).

Though in essence true, the statement is not exactly correct (and so can be misleading).

For example, if a proton comes into contact with a positron they will not annihilate. (If you recall, the positron is the antiparticle of the electron.) But if a positron comes into contact with an electron then, yes, they will annihilate (yielding a photon). (Maybe they will not instantaneously annihilate, since they could for the minutest moment combine to form positronium, a particle they form as they dance together around their center of mass – and then they annihilate into a photon.)

The annihilation would occur between particles that are conjugate to each other — that is, they have to be of the same type but “opposite.” So you could have a whole bunch of protons come into contact with antimatter particles of other non-protons and there will not be mutual annihilation between the proton and these other antiparticles.

Another example. The meson particles are represented in the quark model by a quark-antiquark pair. Like this: p\bar q . Here p and q could be any of the 6 known quarks u, d, c, b, t, s and the \bar q  stands for the antiquark of q . If we go by the loose logic that “matter and antimatter annihilate” then no mesons can exist since p  and \bar q  will instantly destroy one another.

For example, the pion particle \pi^+ has quark content u\bar d  consisting of an up-quark u and the anti-particle of the down quark. They don’t annihilate even though they’re together (in some mysterious fashion!) for a short while before it decays into other particles. For example, it is possible to have the decay reaction

\pi^+ \to \mu^+ + \nu_\mu

(which is not the same as annihilation into photons) of the pion into a muon and a neutrino.

Now if we consider quarkonium, i.e. a quark and its antiquark together, such as for instance \pi^0 = u\bar u or \eta = d\bar d , so that you have a quark and its own antiquark, then they do annihilate. But, before they do so they’re together in a bound system giving life to the \pi^0, \eta particles for a very very short while (typically around 10^{-23} seconds). They have a small chance to form a particle before they annihilate. It is indeed amazing to think how such Lilliputian time reactions are part of how the world is structured. Simply awesome! ;-)

PS. The word “annihilate” usually has to do when photon energy particles are the result of the interaction, not simply as a result of when a particle decays into other particles.

Sources:

(1) Bruce Schumm, Deep Down Things. See Chapter 5, “Patterns in Nature,” of this wonderful book. :-)

(2) David Griffiths, Introduction to Elementary Particles. See Chapter 2. This is an excellent textbook but much more advanced with lots of Mathematics!

Comparing huge numbers

Comparing huge numbers is often times not easy since you practically cannot write them out to compare their digits. (By ‘compare’ here we mean telling which number is greater (or smaller).) So it can sometimes be a challenge to determine.

Notation: recall that N! stands for  “N factorial,” which is defined to be the product of all positive whole numbers (starting with 1) up to and including N. (E.g., 5! = 120.) And as usual, Mn stands for M raised to the power of n (sometimes also written as M^n).

Here are a couple examples of huge numbers (which we won’t bother writing out!) that aren’t so easy to compare but one of which is larger, just not clear which. I don’t have a technique except maybe in an ad hoc manner.

In each case, which of the following pairs of numbers is larger?

(1) (58!)2 and 100!

(2) (281!)2 and 500!

(3) (555!)2 and 1000!

(4) 500!  and 101134

(5) 399! + 400! and 401!

(6) 8200 and  9189

(The last two of these are probably easiest.)

Have fun!

 

Escher Math

https://i2.wp.com/upload.wikimedia.org/wikipedia/en/a/a3/Escher%27s_Relativity.jpg

Escher Relativity

You’ve all seen these Escher drawings that seem to make sense locally but from a global, larger scale, do not – or ones that are just downright strange. We’ll it’s still creative art and it’s fun looking at them. They make you think in ways you probably didn’t. That’s Art!

Now I’ve been thinking if you can have similar things in math (or even physics). How about Escher math or Escher algebra?

Here’s a simple one I came up with, and see if you can ‘figure’ it out! ;-)

(5 + {4 – 7)2 + 5}3.

LOL! :-)

How about Escher Logic!? Wonder what that would be like. Is it associative / commutative? Escher proof?

Okay, so now … what’s your Escher?

Have a great day!