In this post, we will derive the Einstein’s field equation from the least action principle. Let’s just construct an action with the Ricci scalar, Lagrangian and an invariant space-time volume element( ; g- metric) : All these quantities have the same value in all frames of reference.
Let’s call and also subsitute in the above equation.
To simplify further we employ Jacobi’s formula ** and also note that the boundary terms that arise out of vanish according to Stoke’s theorem.
This tells us that under a variation of the metric if we would like a stationary action then it has to be true that :
where – Riemann Tensor, – Ricci scalar, – Energy-momentum Tensor and – Metric Tensor
This is the Einstein’s field equation. Inserting the constants at the right places yields the more familiar form:
What would happen ?
As a fun little exercise you can start off with the invariant volume given by the inverse metric instead. i.e and you will obtain the same equation. BUT, you should be careful when employing the Jacobi’s formula else you might ending up with a extra minus sign with the Ricci scalar term.
Some more fun questions to ponder:
- Why is called the invariant space-time volume element ?
- Why is the action defined as ? Why not any other way?
- How do you prove the Jacobi’s formula ?
When one is learning about Laplace and Poisson equations it can be frustrating to remember its form in different coordinate systems. But when one is introduced to four vectors, special relativity and so on, here is a simple way to remember the Laplacian in any coordinate system.
where we is the inverse of the metric , is the determinant of the metric . And one specifies the coordinate system by mentioning the form of the metric. Let’s look at how this works out:
Noting that because they are independent variables, we get
Following the same approach as the Cylindrical and Cartesian coordinates, we get the following form for the Laplacian in Spherical coordinates,
When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions () are. Here’s one failsafe way using complex numbers that made things really easy :
From the above expression, we can obtain and
From this we can obtain and with absolute ease.
Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!
What ?!! There exists such an elegant decimal representation of the Fibonacci sequence? Well yes! and the only thing that you need to know to prove this is that if the Fibonacci numbers were the coefficients to a power series expansion, then the Fibonacci generating function is given as follows:
Subsituting the value of , we get :
To understand why this is true, we must start with the Fundamental Theorem of Vector calculus. If is a conservative field ( i.e ), then
What this means is that the value is dependent only on the initial and final positions. The path that you take to get from A to B is not important.
Now if the path of integration is a closed loop, then points A and B are the same, and therefore:
Now that we are clear about this, according to Stokes theorem the same integral for a closed region can be represented in another form:
From this we get that Curl = for a conservative field (i.e ). Therefore when a conservative field is operated on by a curl operator (), it yields 0.
Bravo Prof.Ghrist! Beautifully said 😀
So, I read this post on the the area of the sine curve some time ago and in the bottom was this equally amazing comment :
Diameter of the circle/ The distance covered along the x axis starting from and ending up at .
And therefore by the same logic, it is extremely intuitive to see why:
Because if a dude starts at and ends at , the effective distance that he covers is 0.
If you still have trouble understanding, follow the blue point in the above gif and hopefully things become clearer.
I really wish that in High School the math curriculum would dig a little deeper into Complex Numbers because frankly Algebra in the Real Domain is not that elegant as it is in the Complex Domain.
To illustrate this let’s consider this dreaded formula that is often asked to be proved/ used in some other problems:
Now in the complex domain:
And similarly for its variants like and as well.
Now if you are in High School, that’s probably all that you will see. But if you have college friends and you took a peak what they rambled about in their notebooks, then you might this expression (for ):
But you as a high schooler already know a formula for this expression:
where , are merely some numbers. Now you plot some of these values for lambda i.e () and notice that since integration is the area under the curve, the areas cancel out for any real number.
and so on….. Therefore:
This is an important result from the view point of Fourier Series!
Say what? This one blew my mind when I first encountered it. But it turns out Euler was the one who came up with it and it’s proof is just beautiful!
Say you have a quadratic equation whose roots are , then you can write as follows:
As for as this proof is concerned we are only worried about the coefficient of , which you can prove that for a n-degree polynomial is:
where are the n-roots of the polynomial.
Now begins the proof
It was known to Euler that
But this could also be written in terms of the roots of the equation as:
Now what are the roots of ?. Well, when i.e *
The roots of the equation are
Comparing the coefficient of y on both sides of the equation we get that:
* n=0 is not a root since
at y = 0
** Now if all that made sense but you are still thinking : Why on earth did Euler use this particular form of the polynomial for this problem, read the first three pages of this article. (It has to do with convergence)