After the previous book ‘The vital
question’, this book
took me to a ride at a whole different level from cells - several orders
heavier - the black holes and such (as the title may have given away).
A black hole is very heavy, it is so heavy that even electromagnetic radiation
(commonly referred to as ‘light’) cannot escape it. Basically, that kind of limits
the experimental validation of them. You just cannot directly see them in any
possible way. However, when heavy objects (read ‘very very heavy’) collide -
they change the spacetime around them very briefly, like a fly landing on the water,
to emit, what a brilliant man in his 30s called, gravitational waves. The disturbance in space is on the orders of
thickness of a human hair compared to the distance you cover by travelling 100
billion times around the earth. In other words, it is unnoticeable. Imagine
that these waves pass right through us, through the planets, through the
galaxies without registering any actual disturbance that we can perceive.
Some brilliant minds in late-1960s started thinking that it would be very cool
to actually see if that theory is true - in essence they wanted to feel a
gravitational wave. In the book, the author describes in detail the key people
involved in the LIGO project along with various challenges and solutions that
came by during the several decade journey to completion and the recent
detection of the gravitation wave. The book does not discuss a lot about the
technical details, hence it can be considered a good read even for layman.
Being a person in academia, Janna knows how things work and I like the fact
that she did emphasize the negative tones instead of suppressing them as most
science reporters do. One thing that she did manage very well is to create a
normal human environment even when talking about some of the world renowned
geniuses. It could have been very easy to over-dramatize in order to captivate
audience, but she did not and I appreciate that very much. It is a very easy
book to read, making it a seem like a short journal rather than an actual book.
After a recommendation from a friend, I picked this book almost immediately.
Let me be clear with the fact that this is not an easy read. It is packed with
information that gets more & more speculative as the chapters pass by. And set
aside atleast a couple of months to read and admire its entirety. Needless to
say, there is a lot of homework that can help in understanding the intricacies
of the book.
The book opens with a promise to explore one of the very possible ways that
life came to existence. That is exactly what the book delivers indeed. The
author takes on a wild journey all the way down to the cell level and starts
speculating how such mysterious building blocks of pretty much all life could
have born out of materials that existed a couple of billion years ago on earth.
I am very impressed by the way the author explains proton gradient as the fundamental
driving element of life. Then he ventures on a very impressive journey to explain the origins of
prokaryotes - leading to even the possible seperation of archea & bacteria. This part of the book is
well written and definitely is the highlight of the entire book.
Then he continues to speculate about the possible evolution of eukaryotes. It
gets very speculative and monotonous by this point. This part of the book is
nice, but it did not excite me at the same level as the prokaryotes section
did. Nevertheless, the discussion on energy being one of the fundamental driving forces is very intriguing.
The book sets out vast amounts of material & opinions for very interesting
thoughts. It is very engaging in the way it is written. However, the author’s
sense of humor is definitely not on par with his intellectual!
Recently, a discussion with a friend made me realize that a lot of students are
taught calculus as a bunch of formulae and not as a useful concept to understand
several phenomena in the universe. To show the beauty of calculus, I decided to
write a small tutorial explaining the basic concepts of limits, differentiation
and integration. Immediately connecting these concepts, I will also show how
basic motion laws in physics can be interpreted.
Slope, Limits, Differentiation & Integration
We will be using one and only one curve the whole tutorial, it will change its
purpose and meaning as we go by. But the curve will remain the same and between
the exact same points P1 = (x1,y1) and P2 = (x2,y2).
Figure 1: The one curve that will help us all.
When we graph something, we are always showing a relation between the variables
plotted on the different axes. In effect, we can say that y = f(x), where f
is some function. Some of the examples could be:
\[f(x) = 3x + 15x^2\]
Anyways, we all know that two points form a straight line and the slope of the
line (or inclination) is given by \(\frac{y2 - y1}{x2 - x1}\). This is simple!
Plotting this line along with the curve looks this way.
Figure 2: If we pay attention to the line, we can see that \(dx = (x2-x1)\), with \(dx = 10\)
Instead of one line, if I put a point (P3) in the middle of the curve and plot two
lines, they will have different slopes. The lines along with the plot look this
way.
Figure 3: Putting a point in the middle makes it two lines and \(dx = 5\)
We make them three, we make them four and on and on. As we increase the number
of points (essentially increasing the number of lines) between x1 and x2, we
decrease the distance between two adjacent intermediate points. This is what the concept
of limit really is. In our case, we reduce the dx to almost 0. If we write down the
slope with this concept along with putting (\(y = f(x)\) and \(x2 = x + dx\)), we get the following:
\[\lim_{dx\to0} \frac{f(x+dx) - f(x)}{dx}\]
Differentiation is pretty much finding the slope of a curve at a point!
At one time in the past, the biggest scientists in europe made a fuss about how to write this and in the modern
notation, it is commonly written as \(\frac{dy}{dx}\). Since, this may be a function as well - let us call this
\(g(x)\).
All this sounds cool! But then what is integration exactly? We will take a U-turn from here. Let us take a plot again,
this time a simple straight line with one constant value
which happens to be the slope that we measured of the line in figure 2. So, if we want the area under the line in figure 4, it is simply the area of the rectangle that will give us one value.
Figure 4: A single line that forms a rectangle with \(dx = 10\)
So, if we take the slopes of the 2 lines in figure 3 and plot them, they will look like the following image. The areas of these two rectangles will be two values.
Figure 5: The two lines that form two rectangles with \(dx = 5\)
Let us do the same thing again, let us take the dx all the way to 0, then we get a teeny-tiny wide rectangle that looks like the following:
Figure 6: A small rectangle with \(dx\to0\)
The area of this teeny-tiny rectangle has a specific value at that point x. And if we put all such values from 0 to x together,
we get the indefinite integral that is often written with capital letter of the function - G(x) at x. So, if we write that down with the fancy integral symbol :
That is it! If we plot all these values together, we will get a curve that kind of takes us back to figure 1 - saying :
\[f(x) = G(x) + c\]
*c is a random constant, which we will just ignore here!
A Dropping apple
Now, let us see how these concepts help us in a physical world. Let us take a random planet with g=2 and no atmosphere. If we drop an apple from some height and
see the change in height as a function of time elapsed. It should follow an interesting motion that puzzled a couple of people before, who eventually became really
really famous. Go ahead and press the button!
Of course, the first apple follows the laws of nature. But what is happening to the second and third apple? If we observe
this closely, the second apple travels at a constant speed for the whole duration - like the line in figure 2 and figure 4.
The third apple - it travels with one speed first and then picks up a second speed - like the lines in figure 3 and figure 5.
So, the only figures left now are figure 6 & figure 1 - and that is what our actual apple follows! It has a speed that slowly increases
as time goes by. And this is what we call an acceleration. So in essence, we have three different properties here - position, speed and acceleration.
They are all connected the exact same way a function, a derivative and its derivative are connected.
Even though all the apples start at the same place and finish at the same place at the exact same time, their behaviour during the movement is defined by these properties!
When it comes to real life, the derivatives and integrals are very nice ways to improve our understanding of trends. we have some places where micro-trends are easily modelled like cross-section area of some solids, immediate acceleration patterns and if we want to model the bigger trends, we then go for an integral. In the exact opposite way, some scenarios give us easy models for the macro-trends like measuring distances or any raw data and if we want to look at the finer trend, we go for a derivative. Often they are also used to find minima & maxima among the trends. There are some very fine tutorials out there for understanding more, this one is written only for entertaining people with the ideas.
Now given that the earth has g as acceleration, what will be the loss in height of an apple after t seconds? And yes!, the animation is a poor depiction of history :)
Easily one of the best books I read recently, The hunt for vulcan by Thomas
Levenson depicts an amazing story in the history of science. The book is about a
hypothetical planet named vulcan that is probably one of the best examples of
confirmation bias.
The story starts with Newton’s gravitational theory, that took the world by
surprise, explaining everything from an apple’s trajectory to the orbit of mars around
the sun. The theory kept mathematicians and astronomers busy for a couple of
centuries to predict and verify the trajectories of various bodies visible both in day &
night sky.
A french mathematician Le verrier came under spotlight when, using the
gravitational theory and a pen, figured out the existence & precise position of
Neptune - arguably the farthest planet in the solar system. He came up with that in
an attempt to explain the behavior of the-then farthest planet Uranus. In an
attempt to have a complete explanation of solar system, his focus turned to
Mercury - the-then closest planet to the sun in the solar system.
In a similar attempt to explain mercury’s observed trajectory, Le verrier
proposed the existence of a new planet vulcan between sun & mercury. For a
little under a century, astronomers ventured to utilize the short windows of
total solar eclipses with the single aim to see the phantom planet. Some very famous
astronomers even took the thought, that they saw vulcan, to their graves.
Then comes along a young clerk in a patent office using his exceptional
imagination abilities. His theories changed the world. But in the context of the
book, he destroyed vulcan. His theory of general relativity provided a
convincing explanation of why mercury acted that way.
Gravitational well around the sun
In summary, the book is a fascinating tale of how Le verrier created the
farthest planet in the solar system using the pen and Einstein destroyed the
supposedly-closest one to the sun using a pen too!
