Mathematics and Sciences

Relativity (12)

2008-05-08T20:51:00.003+08:00

Now let us go back a little in 1917 and look at what Albert had discovered. Cosmology is the study of the whole Universe. Let us look at Isaac’s try of explaining the universe…

1 Isaac Newton has a theory: the Universe is one which is full of stars goes on endlessly in space and eternally in time.

2 The stars might go on forever in all directions. However, the problem with an infinite number of stars was that, there would be an infinite pull of gravity.

3 Even though space goes on forever in all directions, the stars gradually decrease in terms of number after a few zillion miles. As Albert put it, “The stellar Universe ought to be a finite island in the infinite ocean of space.

4 But Albert did not like 3 either. He decided that over long periods of time, the stars would drift away from one another.

5 Because the Universe was supposed to have been present forever, there should not be any stars left to see – they would have drifted out of sight by now.

6 However, Albert had an amazing idea. He already knew about General Relativity – matter curves space. So what would happen if there were lots of matter in space? Space would curve more and more and more and more and more until…

7 Until the Universe closed in by itself. It means that you could travel forever and not find an edge to the Universe, but you would run out of new places to see, and that a light ray sent out into space would eventually come back to its starting point (might take a long time though).

8 So, using General Relativity, Albert came up with an equation (another which is too complicated). In the equation, there is this “cosmological constant”. Albert had to put in the constant for it to make sense.

a. Imagine that you are wearing an elastic tie, stretched to its limit, with a pie at the end.
b. Now, pull the pie away from you. The tie will stretch and try to pull the pie back. Very luckily you arm pushes the pie away.
c. In the theory, the pie is like a star, your face is another and the elastic is gravity. The cosmological constant is your arm. What happens if there is no cosmological constant? Easy. Remove your arm away from the model of the Universe.
d. So Albert’s cosmological constant comes in handy to stop the Universe from collapsing.

9 However there is no other reason for the constant to be there. Albert was unsatisfied, but there seemed to be no other way round the constant. So it stayed in the equation for the next twelve years.

10 In 1929, an American astronomer by the name of Edwin Hubble came along. He spent his time looking at galaxies. It always seemed that the galaxies are going away from each other…

11 So it means that the Universe is expanding! Albert realizes his mistake. Since the Universe is expanding, it does not need a constant to stop it from collapsing.

12 The equations in General Relativity have always shown that the Universe is inflatable and deflatable. However Albert assumed that it was doing neither, as there was no evidence to suggest that it was inflating or deflating. Now that there is evidence, it proved that it was actually inflating.

a. Again, using the same illustration. There is a way to stop it from hitting you in the face. You can throw it away from you.
b. It will hit you later of course, but in the meantime you can have a rest from all that pushing.

13 Until recently, the Universe seemed to be expanding, but later on gravity would still win and snap it back together again. It seems that the Universe is being thrown apart so hard that it will never collapse again – which is like throwing the pie so hard that the tie is snapped.

14 So it seems to be like that. The cosmological constant was a big mistake. Perhaps.

15 Recently, scientists have found evidence that actually suggest the galaxies are moving as if there is a mysterious force pushing them apart. So now scientists are investigating this mysterious force.

Relativity (11)

2008-05-08T20:49:00.001+08:00

Now Albert has to find out exactly how matter curves space-time. Here he faces a problem. How could he measure the curvature of space-time? The instrument used to measure it would be inside the curvature too, so would the effort to measure it not be vain? He asked a friend, and he suggested Riemann’s equations for solving how metals bend when they are subjected to different temperatures. It is indeed very helpful.

Now, he faces another problem. Once you give up on ordinary geometry, there are many answers to one equation. Albert had to choose from a lot of ways to describe how exactly space-time is curved. Now the idea of Isaac Newton’s equations comes in. His equations are known to have only failed with the movement of the planet Mercury. So it would be nice if Albert’s equation can explain the movement of Mercury as well.

So, Albert in the end did come up with the equation though. It does explain the movement of Mercury as well as describe the curvature of space-time. However, the equation is very complicated and long, so I shan’t state it here.

From General Relativity Albert was also able to explain the fact that gravitational waves travel at the speed of light. Sadly, so far no scientist has detected gravitational waves using powerful detectors (they missed out a few opportunities like a massive star at a next-door galaxy had a powerful explosion).

In 1919, Einstein and Arthur Eddington along with other scientists planned two expeditions, one to Principe which is an island off the coast of West Africa, another to Sobral in Brazil. They have planned to view stars in an eclipse. No it is not an ordinary expedition, as you might have expected – the scientists wanted to prove that General Relativity is correct. They wanted to view the stars when there is Sun around as opposed to when the Sun is not around. When stars are near the Sun, the hypothesis would be that their arrangement in the sky would be a little different from the original calculations, as what the General Relativity predicts.

The result they managed to get was that indeed the stars appeared to be in a different position as opposed to original calculations. So Albert and his General Relativity were proved to be correct.

Relativity (10)

2008-05-08T20:46:00.003+08:00

When time is slowed down by gravity, the effect is too small to have an impact in our daily lives. The Earth slows down clocks by a second a century. Nevertheless, this effect can be measured. People have sent clocks revolving around the Earth on jet planes, and the clocks are found to have sped up by a teeny bit.

The Sun’s gravity is much greater than the Earth, but the impact would be still as small. It just slightly disturbs the path of light rays. It slows down clocks by a minute a year.

Now, heard of black holes? I’m sure you did. Black holes are scary things in the space which have immense gravity. Care to test out the hypothesis that gravity bends light with black holes? Yes, and the result would be that black holes actually pull light in completely so that it can never escape. Hence, they do not just slow time down – they stop it. It is just as if you are strong enough to grab that furry ball rolling past.

Imagine that you are watching someone approaching a black hole in a spaceship. You would see the spaceship getting slower and slower. Everything inside the spaceship slows down too. If the person goes into the black hole for a visit and comes out again (that is if it is even possible), you would find that less time has passed for them than the people on Earth.

Imagine it! If your parents have gone for such a trip, then they would think that their trip lasted for a week, while for you their trip lasted for twenty years! By that time you might be older than your parents!

So travelling in time might not seem impossible after all. All you need is lots of matter to put together and make a nice gravity-field and a vehicle which can take you in and out of the field. But there is one problem using this as a time machine, for even if you could travel into your future, you cannot travel back in time.

Next we shall explore the secret of spinning.

When you spin a bicycle wheel, it is not moving at a straight line. Hence General Relativity is supposed to deal with these non-smooth motions. How can General Relativity explain this motion?

Since acceleration and gravity affects both time and space, there is a way to shorten the wheel without shortening the spokes: to bend space so that there is a bit extra inside the wheel, so that the spokes would have more room.

To put it short and sweet, matter causes gravity and gravity curves space. So, we could actually say that matter curves space.

7: Matter curves space.

So we have now found out that matter slows down time and curves space. In other words, it curves space-time.

When there’s nothing else, moving objects move in straight lines. When there are other things, they cannot move in straight lines anymore. Planets and stars and everything else curve their motion with the curvature of space-time.

It is something like digging rings in the sand. If you dig rings around you, beach balls going by would tend to travel around you instead of travelling into you. It is because of what you have done to the sand. Similarly, matter curves space-time and it hence affects the motion of things going around them. In other words…

8: Light and matter follow the shape of space-time.

Relativity (09)

2008-05-08T20:35:00.002+08:00

Now we shall proceed on. Einstein has indeed accomplished a lot by inventing the Special Relativity. He actually proceeded to General Relativity, one which would deal with all sorts of motions, not just smooth motion. He came up with a great idea – which he called his happiest thought – to deal with the other motions. And that is: Gravity disappears when you fall off a house.

In other words, you cannot feel the pull of gravity when you are falling. Now this is odd. You would, usually, feel the pull when something pulls you, as the body has a natural resistance (i.e. inertia) to being pulled. However, experiments have shown that it is really true – the two forces (i.e. inertia and gravity) are of exactly the same pull.

So Albert decided to assume that gravity could do whatever acceleration could do, and vice versa.

Let us think about the moving train and the Principle of Relativity again. Now we need to prove that whenever the train accelerates (or decelerates), the passengers could not say for sure that they are moving or not. However we do usually sense these motions. In order to show that the Principle works for acceleration, there should be something that could cause the sensations. Now, all thanks to Albert’s happiest thought, that other “something” is gravity.

So, however light behaves under the influence of acceleration, gravity must also cause the same behaviour in light as well. How does light behave then? Let us do another thought experiment.

Imagine that there are two really fast people. They can run at nearly the speed of light, and hence are accustomed to such great speeds. Jerry has three big jellies, while Tom is prepared to fire a laser gun at the jellies.

When Tom fires the laser beam at the first jelly, Jerry does not move the jelly at all. Hence the jelly would look like this:

Figure 09.01

Next, when Tom fires the laser beam at the second jelly, Jerry moves the jelly at a constant speed ≠ 0. Hence the jelly would look like this:

Figure 09.02

Next, when Tom fires the laser beam at the third jelly, Jerry moves the jelly at a constant acceleration ≠ 0. Hence the jelly would look like this:

Figure 09.03

Note that the path of the laser beam is curved for the last jelly.

Hence, acceleration makes light curve. If gravity could do whatever acceleration could do, then it means that gravity would curve light rays.

Now let us do another thought experiment. Imagine that you are standing at a particular place. A big furry ball flies past you. You try to grab the ball, but because of its big size, you only managed to grab a few furs. Is there any other effect on the ball other than losing a few furs? Yes. You slow it down and change its direction.

So this is what happens to light: when gravity tries to “grab” it, it would slow down and curve as well. If light slows down, then light-clocks would too. Hence, gravity slows time down.

6: Gravity slows time down.

Relativity (08)

2008-03-22T13:30:00.003+08:00

Now, let us continue to wonder about the wonders of Relativity.

I shall now introduce the idea of a fourth dimension. Let us recap on the first three dimensions. They are up and down (length), left and right (height) and backwards and forwards (breadth). Together they represent space. Before we explore the idea of a fourth dimension, let us try to comprehend a world with fewer dimensions.

In this kind of world, it would look like the surface of a piece of paper. There are lines all over the place, but no lines go in or out of the paper. Imagine that a smart creature called Teenee live in this world.

Figure 08.01

Teenee can look around in his world, from the top to the bottom and from the left to the right of the piece of paper. However he cannot look out from the paper, so he cannot see you. You can see the objects around him as triangles, rectangles, circles, pointy shapes and curvy shapes. However he cannot see that; he can only see all the objects as lines as he sees them from the sides. To make this two-dimensional world you can even draw the above picture out on a cardboard, and then make a horizontal slit in the cardboard. Through the slit you slide in a small piece of paper at the corner.

What you could see is a small piece of paper passing through the cardboard. For our dear friend Teenee, he could see a line passing through his world, slowly getting longer and longer, before it gets shorter and shorter and eventually disappear.

Figure 08.02

Other creatures may see the line as it suddenly appear from nowhere, lengthened, shortened and then disappeared. Teenee would cleverly understand that the object comes from a three-dimensional world and hence it explains its behaviour.

Now, imagine that this happens in the three-dimensional world: something three-dimensional goes pass your world, as it suddenly appear, expand, shrink and disappear.

This sounds all very weird and spooky, but actually we are so used to it that we actually take it for granted. A better way to understand it is to imagine the way the traditional way of screening movies: a picture followed by another and showing them one by one very quickly.

A scientist, Hermann Minkowski, suggested that there is really a fourth dimension, it is just time. Together the four dimensions make up space-time. This idea is a big help in explaining what happens to objects moving at high speeds (they actually shrink).

It is also similar to holding a javelin under the Sun. When you hold it at different angle, its shadow changes its length. However, the javelin does not change its real length. Applying this concept in the understanding of the reason moving objects shrink; they just change their length in space, but they are actually just changing their angle in space-time.

So,

5: Space and time are linked.

Also, remember that all this shrinking actually depends on the observer. If you are moving with the javelin it would not appear to be shrinking. That is why it is called Relativity – the lengths of the objects depend on the speeds of the observers.

Relativity (07)

2008-03-04T21:48:00.004+08:00

After deciding that time is flexible after all, we proceed to experiment space. Let us do another thought experiment.

Imagine that there are 4 children, Amy, Ben, Charlotte and Danny, who are all armed with little laser guns. They also have their very accurate watches set to exactly the same time (precision is 1 second here). They mysteriously walk up to a train and start taking measurements.

First, they measure the train when it is stationary with rulers, and let us say that the reading is 12 metres long. Later, the train is moving at 1.2 × 10⁸ metres per second. Amy and Charlotte are standing at each end of the train and inside the train itself. Amy fires a ray of laser to Charlotte. Charlotte notes the time the ray hit the back of the train as 4 × 10^-8 seconds. So, from this, we know that:

Length of train = Time recorded × Speed of light = 4 × 10^-8 seconds × 3 × 10⁸ metres/second = 12 metres.

There is no surprise here, as the train does not suddenly expand or contract.

However, we proceed to Ben and Danny. Both of them are also standing at both ends of the train, but only outside the train. They proceed to measure the length of the train in the same way. Ben fires the ray of laser and Danny takes note of the time, which happens to be 3.7 × 10^-8 seconds. Why does this happen?

The simple reasoning is that by the time the laser ray has hit the end of the train, the back of the train has already moved some distance to the front, hence the actual distance of the path which the laser ray travelled is shorter. Here, we can conclude that for Ben and Danny, the moving train actually shrinks.

There is actually something familiar when you see this equation:
There is actually a real life example to illustrate this:

There is a very little particle called a muon. Muons are very small, even smaller than atoms. They have short but exciting life spans, lasting a few millionths of a second, as observed from laboratories. They are formed high above the Earth, more than 10 kilometres above a person’s height, where scary radiations from the space usually happen. The atmosphere prevents this radiation from getting too near to the Earth itself, which actually help in not letting you be burnt or anything by them.

There are many muons crashing onto Earth’s surface, but they could not be formed nearby for there are no fatal radiations that near to Earth. Let us work out the speed of the muons:

They have to travel at least 10 kilometres, and they have their whole life span to do it, which we would take to be 4 × 10^-6 seconds, as observed in 1 out of 6 muons in the labs. The mathematical calculation is: 10 km ÷ (4 × 10^-6) seconds, which would eventually turn out to be 2.5 × 10⁹ metres per second. **Note: This is definitely faster than the speed of light. However, is it not tested that there is nothing which could travel faster than the speed of light?

This would be quite easily understandable through the subject of relativity. If the muons could measure the speed themselves, the reading would show that they travelled that certain distance in 4 × 10^-6 seconds. However, they last for 4 × 10^-5 seconds according to the people on Earth!

The time differs because the muons are so fast that they go at nearly the speed of light. This goes to show that time do move slowly when things move fast.

4: Moving things shrink.

Mathematics: Probability

2008-03-04T21:45:00.000+08:00

This is about probability, everyone's favourite topic ^^. Probability is around everywhere in life, people use probability to calculate the chance of a person striking the lottery, probability of coin flips, etc. Probability is very closely related to permutations and combinations; it can be calculated by the number of good outcomes divided by the number of possible outcomes. For example, when flipping ten coins, permutations and combinations tells us that the chances (probability) of getting exactly 5 heads would be 10C5/2^10, or 252/1024, nearly 25%. Probability is also part of daily life. For example, imagine that you were late for work and wanted to get a pair of socks from a dark cupboard (let's imagine that the light bulb blew). Imagine that you have 10 pairs of socks, each one being different from the other in terms of length and colour. Say that you are not picky about which pair you wear, so long as both socks are identical. Probability tells you this:

The chance of taking out one sock is 1. Remember that you are not picky, so any sock you take will suit the requirement.
The chance of taking out the matching one on the next try would then be 1/19, as there are 19 socks left and only one sock matches the other. Therefore, the probability of picking out one pair of socks in the first try is 1/19.

Relativity (06)

2008-03-04T21:31:00.002+08:00

Now we just have to work out just how time slows down at certain speeds.

Einstein came up with this by using Pythagoras’ Theorem.

t = amount of time which passes for “you”;
T = amount of time that passes for moving object;
s = speed of moving object; (or rather, v = velocity if you prefer this);
c = speed of light.

We could work out how fast time travels for the situation stated above, as in Figure 05.04. Let’s say that the mirrors are 1 metre apart.

T = 3.3 nanoseconds;
s = 150,000,000 metres/second;
c = 300,000,000 metres/second.

So:

Until this Theory is invented, other people thought that time ticked away very smoothly and consistently at the same rate all over the whole Universe. However, Einstein discovered that time flowed at different rates all over the world, dependant on the speed which a certain object was moving.

So, you could not even safely say ‘meanwhile’.

3: There is no such thing as ‘MEANWHILE’.

*Note that the images are not really clear. I would try to upload better pictures if they are really illiterate.

Relativity (05)

2008-02-17T14:38:00.000+08:00

Figure 05.01

Figure 05.02

Figure 05.03

Figure 05.04

Here we begin to explore time. One way to explore time is... with a clock like this:

(Figure 05.01)

Then, imagine that you are with a friend, named Max. He is holding a light clock, similar to the one shown above. You are both standing in the dark.

(Figure 05.02)

Note that the light ray is moving up and down in a straight line inside the clock. So, each time the light ray bounces off a mirror, a tick could be heard. Of course, you could also see that it is moving at the speed of light.

Now, Max starts walking at a quarter of the speed of light, with the ticking light clock.

(Figure 05.03)

Note that the vertical distance travelled by the light ray is the same when Max was not moving.

Now, Max starts to run, at half the speed of light.

(Figure 05.04)

The vertical distance is still the same.

So... The point is that, the faster Max moves horizontally relative to you, the further the ray of light has to travel to get to the other mirror. Measure one of the zigs in Fig. 05.04, it is longer than the one in Fig. 05.03, and much longer than that in Fig. 05.02.

However, the light ray could not go faster to reach another mirror. Remember that light behaves normally to everyone?

So, if the light ray has to travel further and it could ot travel faster, it has to take a longer time to reach the other mirror. So there is a longer time between each tick. In other words, the clock slows down.

There are a few things to bear in mind:

•The light ray is not zigzagging relative to Max. It is still the same old distance, same old speed, same old time between ticks.

•This only happens to light. Even when you do the same thing to yo-yo, you would discover that it would travel faster. It has no problems, as there is no law which states that the speed of yo-yo is the same for everyone.

•Things always have to go faster before they become noticable.

So, if moving light clocks run slow, would other things be affected? The Principle states that there is no test that someone in a moving vehicle could do to find out if they are moving. However, the light clock slows down at high speeds. Would this slowing down not show the people in the moving vehicle that they are moving?

They woud not notice if they slow down too. Therefore, everything in the vehicle must slow down, including all clocks, machines, even the bodies and thoughts of travellers.

Time itself slow down in a moving vehicle.

Only then would everyone not know they are moving or not. This is the only way that the Principal of Relativity would still hold. No one would notice any changes to time at all, as everything they look inside the vehicle would be affected the same way as they are. Only to someone outside the vehicle would the sowing down be visible.

2: Time slows down in a moving vehicle.

Differentiation

2008-02-05T12:09:00.001+08:00

Now, on differentiation.

Thats the one about d/dx, dy/dx, d[]/d[], etc. Now for some facts:
d/dx(e^x)=e^x
d/dx(ax^b)=abx^(b-1)
d/dx(f(x)+g(x))=d/dx(f(x))+d/dx(g(x)), where f and g are functions of x.
d/dx(sin(x))=cosx
d/dx(cos(x))=-sinx
product rule: d/dx(ab)=a*d/dx(b)+b*d/dx(a)

Now:
d^2/dx^2(f(x))=d/dx(d/dx(f(x))) and is not equal to (d/dx(f(x))^2

Will update more next time, cya.

Physics: Relativity (04)

2008-02-01T23:27:00.000+08:00

Imagine that you are in a train, travelling at the speed of light, to Jupiter. You have some time to look at the back of the carriage.

Strangely, you cannot see anything there at all. Why?

As the train is travelling at the speed of light, the light at the back of the carriage cannot catch up. Does this show that this is a foolproof test to see that you are moving (at the speed of light) and at the same time, contradict the Principle of Relativity?

Surely the Principle cannot be wrong. So? Light must behave normally inside the light-speed train.

That means, light MUST behave normally to everyone, regardless of the speed of the moving vehicle.

1: The speed of light is the same for everyone.

Physics: Relativity (03)

2008-02-01T23:19:00.000+08:00

We shall touch on about the subject of Principle of Relativity. (Einstein invented the Theory of Relativity.)

This is what the Principle of Relativity is all about:
The laws of nature are the same whether you are moving in a straight line or not at all.

This is another way to phrase it:
There are no measurements that you can make to tell if you are moving in a straight line or not at all.

So? What has it got to do with Relativity?

Now, let us try to find out ways to contradict that.

Variation

2008-02-01T22:51:00.000+08:00

If y varies directly as x or is directly proportional to x,
y=kx, where k is a non-zero constant. If y varies inversely as x however, xy = k, k again being a non-zero constant.

Physics: Relativity (02)

2008-01-19T20:57:00.000+08:00

So, let us continue...

Einstein stunningly did away with the idea that there is no such thing as "absolute motion" or "absolute rest". He said that the only sort of motion is relative motion (which is why it is called relativity).

To help us understand the meaning of relative motion...

When you are saying that your car is travelling at the speed of 80 kilometres per hour, other people understand it as that your car is moving at the speed of 80 kilometres per hour relative to (compared to) the surface of the road. However, you can also say that your car is travelling at thousands of kilometres with the revolving Earth relative to the Sun!

So, the only sort of motion is RELATIVE motion.

Will continue next time... Stay tuned!

Physics: Relativity (01)

2008-01-17T21:34:00.000+08:00

I am going to talk about Relativity as one of my firsts write-ups. Most, if not all, of the information that is on this topic is obtained from Albert Einstein and his Inflatable Universe, by Dr Mike Goldsmith.

First of all, Albert Einstein is the one who is responsible for the Theory of Relativity. So we will start off by explaining why it is called Relativity.

Before Einstein, scientist thought that the things that were moving had "absolute motion", and those that were not were "absolutely at rest". It is obvious what this means. However, how do you know if you are "absolutely at rest"?

When you are sleeping, you are actually moving with the rotating Earth. Even if you are at the extreme poles of the Earth, the Earth is revolving around the Sun. The Sun is moving around in the Milky Way too. Then the galaxy is moving around other things. So, it seems that you cannot find absolute rest anywhere!

Will be continued later on... Stay tuned!

Mathema? Scientia?

2008-01-14T21:39:00.000+08:00

Why is this blog on the webpage called "mathema-scientia"?

First of all, "mathema" means "science" in Greek. "scientia" means "knowledge" in Latin.

And that is it.

Hello!

2008-01-14T21:26:00.000+08:00

This is the start of this blog called "Mathematics and Sciences".

The reason I wanted to start this blog is that I wanted everyone to share things about Mathematics and Sciences. So I shall open this blog to everyone's comments. If anyone wants to share about something they have read somewhere which is related to Mathenatics and Sciences, you may type it out and attach it in an email sent to my address: zhenjie_low@yahoo.com.sg.

So remember that "Mathematics and Sciences" had started on 14 January 2008!