The Perks of Infinity

The good thing about the universe is that there a lot of things. It has got something for everybody. Take a look at these artworks:

f_world_1

f_world_2

f_world_3

f_world_4

f_world_5

f_world_7

f_world_6

What I ultimately learned from this course was that, somewhere out there, these places have to exist.

They better do.

Image and Image and Image and Image and Image and Image

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Gravitational Slingshots

I always wondered why doesn’t the sun slow space probes down when they are leaving the Earth for outer planets. Isn’t there a risk that the probe might change its trajectory and fall into the sun? There is. You see, the more distant the space probe gets from the Sun, the more potential energy it gains. However, energy must be conserved at all costs. Therefore the probe loses its Kinetic energy (and therefore its speed) in order to get away from the sun. It is the same as when you throw a rock up into the air.

But there comes a point, as with the rock, when the probe loses all of its kinetic energy. At that time it has reached as far away from the sun as it can. Yes, you could add thrusters to make sure the probe continues its journey. But the extra weight and inefficiency of propellants known to us make it an unsuitable alternative.

Enter the Gravitational Slingshot! Nature’s way of compensating us (very marginally) for all the millions of years we’ve been dragged through the mud in the name of evolution. Through this method, space probes go into a partial orbit around a planet and emerge on the other side with a greater velocity. “No!”, some might say, because it is a violation of conservation of energy. Intuitively it seems that way, but it is all a matter of relativity.

slingshot

Imagine there is a probe approaching a planet with a velocity ‘u’. To an observer on the planet, the apparent velocity of the probe’s approach will be ‘V+u’, where ‘V’ is the planet’s and ‘u’ is the probe’s heliocentric velocity, i.e. velocity relative to the Sun. It will go into orbit at that speed. Now, when it comes out of orbit on the other side, it is still moving with a velocity ‘V+u’ relative to the planet’s surface. But the planet is also moving in the same direction at velocity ‘V’. So the final velocity as the probe leaves orbit will be ‘V+(V+u)’. Of course, some of that velocity will be reduced due to the planet’s potential, but in the end it will still be greater than the probe’s initial velocity.

If you look at what happened overall, ignoring how it happened, the probe approaches a moving planet at a certain velocity and “bounces off” at a higher velocity. It is just like when you throw a ball at the face of a moving train, the ball bounces off at a higher velocity. Now, the ball changes its momentum (first going in one direction, then another) and transfers that change to the train to ensure conservation. But the train is comparatively so massive that we do not notice the minuscule change in its velocity. That’s the same with planets and probes.

The effective increment in the probe’s velocity is due to the orbited body’s velocity relative to the Sun (analogously, the change in velocity of the rebounding ball depends on the train’s relative velocity to the ground). Of course, the Sun’s velocity relative to itself is zero. Therefore ‘V’ will be zero. So there will be no gravitational slingshot from the Sun (towards planets in its orbit) even though it is the most massive body in the solar system; just like there will be no increment in the velocity of the ball when you throw it at the ground.

Trous Noirs And trous noirs

For an explanation of the title, see the link at the end.

If you are seriously, irreconcilably frustrated by your significant other (or lack thereof) and you never want to see your significant other (or yourself) ever again, please accept a sincere piece of advice from me: Do not- I repeat: DO NOT throw them(or yourself) in a black hole. That would be a bad idea.

"I wasn't gonna push her!"

“I wasn’t gonna push her!”

Now the sensible will decimate my sagely wisdom because of the sheer improbability of a black hole ever crossing two recently uncrossed star crossed lovers. But the curious(and the willing) will ask: Why?

Because the face of the victim will adorn the cosmos for the rest of your life. That kind of kills the point of throwing someone into a hole which never spits anything out. Why?

In my previous article I wrote about how a black hole is formed. Now I will write about why black holes are great advertisement spots and potential reminders of every regretful thing you did in your life. Einstein pointed out in his general theory of relativity that gravity distorts space and time. So for an observer at an arbitrary distance, a clock near a massive object will appear to run slow. The nearer the clock is to the object’s center, the stronger will be the gravitational field, and the slower it will run (for less massive objects the clock will have just to be nearer to the center). However, there is a problem with heavy objects: they are usually very large. Therefore a clock won’t be able to get closer than the radius of the object. So the effects of time dilation won’t be apparent.

Black holes, however, have a zero radius. So objects can get close enough to experience significant relativistic effects. I will be using the case of you, and your significant other (real or imaginary) who recently lost his/her position of significance (and their balance on the space ship, apparently; sshhhh!):

time_dilation_gravity

In this picture, the green line represents the time measured by the observer(you) away from the influence of the black hole. The red line represents the time measured by the, uh, test subject. According to you, your time proceeds normally (the green line is not warped). The red line, even though it appears distorted to you, appears straight to the subject; just like you only have to walk straight without a care for the earth’s curvature to go to your destination, even though you appear to be moving in an arc to an observer in space. The numbers on both lines represent the hours elapsed since the break-up. Notice that the length between the hour intervals is the same for both green and red.

Now imagine that you both have clocks. Assume that the subject’s clock sends out a signal every hour. Also assume, for the sake of simplicity, that the signal reaches you instantaneously. As the warping of space time increases with decreasing distance to the black hole, you will get consecutive signals at ever increasing intervals, until at one point the next signal will take infinite time to reach you. However according to the subject, time will seem to pass normally because according to the subject, the red timeline is perfectly straight (just like with you and the earth). As you can see from the picture, no matter how far into time you progress, you will still get signals from the clock. That is to say, the simple act of disappearing forever will take your significant other an unimaginable long amount of time; and they wont even notice that you are getting impatient. As I said: bad idea.

To understand how the timelines work click here.

An explanation of the title here.

Relativistic Doodles

This post explains how my illustration of general relativistic time dilation works. This is the parent post.

Here is an image of a simple classical timeline:

timeline_1

There are 2 observers: green and red. They have their own watches. The red observer shoots an arrow towards the green observer. The position of the arrow vs. the time recorded by the two observers looks like the picture above. If we take a trace of the trajectory of the arrow, we get:

timeline_2

If we were to take the trace of a 2 signals 1 second apart that travel at constant speed, and 2 signals 1 second apart and traveling instantaneously, we would get:

timeline_3

You will notice that the components of all 4 traces (like the ones drawn in grey) are parallel and perpendicular to the space-time axes. This is always the case. So even when I distort the edge of the red timeline a bit, I get:

timeline_4

So if I am standing with the red observer, and I see him sending out signals, I will not notice a difference even when the red timeline distorts, because to me, the components of the signals are still parallel and perpendicular to my space and time. In the same way, when an observer is falling into a gravity well (like that of a black hole) and sends out signals, the observer does not notice the relativistic effects of gravity on the signal. However, the observer who is standing far away from such distortions notices an altered signal. So even though the signal at t=8 was instantaneous according to the red observer, it reached the green observer at t=9.

If you understood this, then you will know what is wrong with this set of traces of signals that a red observer sends as she falls into a black hole:

time_dilation_gravity_WRONG

To Stretch or Not To Stretch

Physics has the distinction of hosting the one of the weirdest concept hierarchies  Don’t get me wrong: physics is beautiful in its intricate connections. But sometimes, especially in the case of modern physics, one feels something like:

WTF_1

So, most of us know about special relativity. A quick summary for the unfortunate: Special relativity establishes the speed of light as constant in all inertial reference frames (that is, for all observers who are either at rest or moving at a constant velocity). One of its implications is that information (in the layman’s case:anything) cannot travel faster than light. This means that as one starts approaching the speed of light, stuff starts happening. Time slows down (according to an outside observer looking at you), your mass increases and weird lighting effects start taking place. I am concerned with length contraction: the shortening of length of objects which are moving at relativistic velocities.

After an unexpected abortion of its hiatus, my conscience prevented me from playing Ace Combat.  And so I was looking for something productive to do when I found this website. According to the article, even though relativistic speeds may cause measurable shortening of length, it most certainly is not observable. Instead the fast travelling object will actually “appear” elongated, while actually being “contracted” at the same time (Schrodinger’s cat, anyone?).  The more I progressed into the article, the more I was like:

WTF_2

But then, I went into scientist mode…

inception_meme__1_…and decided to do a little calculation of my own.

 Imagine there is an object moving towards you at a velocity ‘v’. The stationary length of the object is ‘l’. The distance between you and the farthest part of the object is ‘x’.

rel_exp_1

Just by looking at the image we can see that light from the back of the object takes longer to reach the observer. Mathematically:

t_back_eqnt_front_eqn

We also know that we see an image when photons belonging to the same “plane” reach our eyes. From the equations above, we can see that photons reflected from the front “l/c” seconds later will arrive at the same time (i.e. on the same “plane”) as photons reflected from the back. However in ‘l/c’ seconds, the object will have moved by the distance:

dist_eqn

So the image that will reach our eyes will be like:

rel_exp_2

The apparent length of the object will be:

obs_enlrg_fac_eqn

Let us now assume that the object’s velocity is actually relativistic. So the measured length of the object will shrink to:

rel_len_eqn

And it is this length that will undergo apparent distortion:

final_stretch_eqn

Where l(measured) is the stationary length of the object. The relativistic factor shrinks whereas the observational factor stretches the object. It all comes down to which one of those functions is more powerful. This is a graph of enlargement vs. speed. ‘1’ on the y axis represents no distortion.

plot_rel_factors

Back to Ace Combat. ibrahim2016 out.

Twinkle Twinkle Little Planet

Sounds wrong because the extra syllable blemishes the aesthetic quality of the symmetry of the rest of the poem.

trollface

Planets do twinkle, in fact. Its just that we do not notice. And it is the hallmark of a truly good (and bored) scientist to correct a misconception, even if it is only superhuman vision that can salvage the fallacy.

The atmosphere is a sea of air. Air is a fluid. Which means it is not vacuum. Therefore it has a refractive index. The density of air changes with altitude, so the refractive index also changes. Any light that enters the atmosphere (except from the normal vector, obviously) is bent.

Stars are very far away. They are so far away that they appear as point sources of light. So if we were to draw a ray diagram for a star, we would only need a single line to denote star light. It so happens that due to the constant ‘flow’ of the atmosphere, some times that star light is refracted so that it momentarily does not reach our eyes. That is called twinkling.

Simulation

Simulation

Planets are not that far away. Since they are nearer they can be resolved as light sources with a dimension. So a ray diagram for a planet will have multiple lines denoting light coming from different points on the planet. That light is also bent the same way as star light. And individual rays also sometimes bend enough so that they do not reach our eyes. But there are enough rays that do reach our eyes that we do not notice the slight change in the planet’s brightness. Hence we do not notice the planet twinkling.

Image Source