Archive

Posts Tagged ‘Einstein’

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

Advertisements

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.)

======================================