The little red book of objective thought|
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Below are the 11 most recent journal entries recorded in
|Wednesday, June 11th, 2008|
|Derivatives and differential equations
In the last post, we have seen that for any object moving, we can define a quantity, the average speed, which is simply the distance between two positions of the object divided by the time elapsed for the object to go from one of the point to the other.
Moreover, we have seen that as the two points get closer, this average speed has a limit, which will call the speed.
Mathematically, the speed is the derivative of the position. For any measurement which can be described by real numbers, such as position, speed, length etc...and which depends on a variable, such as the time, we can define similarly the derivative.
For instance, consider an elastic whose length changes in time (for instance, there is a weight hanged to the elastic). At every time, we may measure its length. Between two times, we may measure how the length has evolved. For instance, we may compute the difference between the two lengths. If we now divide this quantity by the time elapsed, we will obtain the average change in length between the two times. If now we take the limit as the difference in time goes to zero, we obtain a quantity, the derivative of the length, which tells you how the length is changing at every instant.
The examples of the length or position where examples where the quantities we measure depends on one variable: the time.
Other quantities may depend on other variable. For instance, consider a tube, whose temperature at every point does not change in time, but changes when you move from one point of the tube to another. The temperature can then be viewed as a quantity which depends on the position. We may compute the average temperature between two points of the tube, and we may also compute the limit when the two points get very close. We will obtain the derivative of the temperature at every point, which tells you how the temperature changes when you move into the tube.
The conclusion is that many interesting physical measurable quantities admits what is called a derivative. This derivative tells you how the quantity in question changes when the variable it depends upon changes.
As seen in previous post, physical quantities are linked together by equations. Some of these equations contains not only quantities such as the position, but also, derivatives of other quantities, such as the speed. We call these equations differential equations.
The most famous differential equation of physics, is Newton's law. It may be easily stated as follows. Consider an object moving on the earth. Then we may measure its position, and thus we may compute its speed, i.e. the derivative of its position. We may also compute the derivative of the speed, we call this quantity the acceleration. Newton's law says that the mass of the object times the acceleration of the object is equal to the sum of the forces that act on the object.
Newton's law is a simple and wonderful differential equation.
Of course, to be of any interest, one need to be able to define and measure the forces acting on the object, as well as the mass of the object.
We'll be more precise in the following posts.
|Sunday, December 2nd, 2007|
|Position and speed
In the previous posts, we have seen that:
-Mathematics is the objective language needed to describe physics
-Different measurements in physics are linked together by equations.
Consider a car moving along a road. We can do a simple measurement: measuring the position of the car along the road at some time. Suppose that the car starts from a point A and arrived a point B after a time t. With measurements, we can calculate the distance between point A and point B. Then we can divide this distance by the time t. The distance between A and B divided by t is called the average speed of the car between A and B.
The average speed is an interesting quantity, however it does not tell you everything about the car. For instance, it could be that at the beginning the car was moving faster than at the end.
However, in average, the speed was the average speed.
However, suppose that I do the measurement for A and B very close to each other, that is to say, I note the position of the car at a point A and very quickly afterward, I note the position of the car and the time elapsed. The time will be small and the distance A and B will be small. We can compute the average speed again. Since the time elapsed between A and B is small, it is hard for the car to go very fast at A and slow at B, because it is hard to slow down a fast car very quickly.
Therefore, the average speed gives better information than for A and B far away and large time between A and B.
As the distance between A and B (and therefore the time elapsed) goes smaller and smaller, the average speed gives more and more information, as it gets more and more difficult for the car to go faster or slower.
Suppose now that we do three measurements of position A, B and C.
If A, B and C are all far away to each other, there is no reason a priori for the average speed between A and B to be close to the average speed between B and C. For instance, the car could go very fast between A and B, and then, it will have the time to slow down between B and C, resulting in a lower average speed between B and C.
However, if A, B and C are very close, it will be very hard for the car to go fast between A and B and then slow between C. Therefore, the average speed between A and B, will be close to the average speed between B and C. As A, B and C gets closer and closer, the average speeds of A and B and B and C will get closer and closer. Therefore, it would be a very good approximation to say the average speed between A and B and the average speed between B and C are equal.
Mathematically, we say that we take the limit of the average speed between A and B, as B goes to A.
By what has been said, this limit of the average speed is also the limit of the average speed between B and C, when B and C goes to A.
Therefore, this limit of the average speed depends only on the position A and we call it, the speed at A.
Let us review what we have said:
- the distance between two points divided by the time is called the average speed
- as the second point gets closer to the first point, the average speed goes to a quantity called the speed at the first point.
Mathematically, we say that the speed is the derivative of the position. We can compute other derivatives. The derivative of the speed is called the acceleration for instance.
The next post will be about the equation between derivatives, called differential equations.
|Tuesday, July 24th, 2007|
|Measures and physics
In the previous articles, we have defined a framework consisting of :
- an objective language: Mathematics
- a collection of facts: Experimental physics
- a set of predictions: Theoretical Physics
- a consistency check: verifying the predictions with experimental physics
We have said that Mathematics was the language used for our work. One important idea is the idea of quantifications or measures. In physics, we need to quantify or measure some objects. Recall that the aim of physics is to describe and understand the real world, the environment. However, once we start collecting facts, we realize that some facts can only be described by measuring them. For instance, how do we know that "A is taller than B"? By measuring A and B and comparing the results. The question of measure is fundamental in physics.
Associated with the measure, is the question of unit. The idea is simple. A unit is a standard for a measure.
There are a lots of quantities and lots of units useful in physics. Moreover, there are also lots of links between this quantities. For instance, consider a piece of metal, let's say, a cube of iron. Measure first the length of one side.
Measure then the mass of the cube. I can define a new quantity by the division between the mass and the length to the power 3. Take another cube and do the same, the new quantity will have the same value. I can do it for any cube of iron and therefore I can build a general rule, namely that the mass of a cube of iron divided by the length of one side to the power 3 is a constant.
This rule is what is called an equation in mathematics. It can be seen as a rule defining the links between different quantities.
Let us resume this post:
- to describe the world, we need to be able to measure some objects
- a unit system gives us a standard measure, therefore we can compare the results of our measurement with a previous measurement
- the results of the measurement are linked together by equations.
|Mathematics and physics
The language used in physics is the mathematical language. To explain why is it so, I believe it is sufficient to give basic definitions of mathematics and physics.
Let's define the mathematical language as the objective language. By this we mean a tool to describe exactly, using pure logic, a collection of things.
I would now define the content of physics by: a objective description of the collection of things we can interact with, sometimes called the world, the environment.
I guess it explains why we are using maths in physics.
|Predictability and physics
An interesting question which should be asked and answered at the beginning of any elementary course on physics is simply "What is the purpose of physics?". An easy answer is "to understand the real world".
How do we know that we do understand how the world is running ? The way physicists answer this question could be the following: we have at our disposition two different approaches. First we can make experiments. I can take an apple and let the apple fall down, once, twice, three times, hundreds of times. I always have the same conclusions, the apple falls. I can therefore build a rule: apple falls. Now this rule is very restrictive. We could call this a fact. Experimentalists collect facts. Suppose we have a lot of facts. The process is now to try to understand the link behind all these facts, to develop a general rule: apples falls, pear falls, anything not attached falls, objects falls. We have moved from the experiment to the theory. Once we have a general rule, we can verify it experimentally, taking new objects and trying to see if they fall. We are moving back from the theory to the experiment. It was possible to do so, because we were able to make predictions: all objects fall, not only apples and pears.
Let us resume this process:
- we collect facts through experiments
- we build general rules from these facts
- with these new general rules, we can make predictions
- we check our predictions with experiments again
In particular, one of the reason why the physicist believes he can understand better the world than other people is because he can make predictions. Without this prediction power, physics will be not be very useful, it will only be interesting from the point of view of education and academia but will have no place in industry for instance.
For these reasons, it is believed that any good physical theory must give predictions. If you build a theory, where you explain this and that but you cannot make any predictions, nobody will be able to tell if your theory is right or wrong, even if it fits well with the actual collection of facts. If you can make predictions, it does not mean your theory is right or wrong, but it will give more support to your theory.
|Tuesday, April 10th, 2007|
|Connecting your Sony Ericsson T68i phone to your GNU/linux operating system via IrDA
>>>Note: this post (contrary to the previous ones) has nothing to do with maths/physics and is not intended for true linux beginners. So do not bother read that if you can't understand most of it (but how do you know if you don't read it, a new paradox is born...). This does not mean I will not answer comments if they are appropriate.>>>
Since I had nothing else to do this evening, I tried and succeed to connect my phone to my laptop using the IrDA. IrDA is an infrared standard and the only mean to connect this phone to my computer as I don't have Bluetooth on it and I do not want to buy the cable to connect the phone to my laptop using a usb port.
Some basics about IrDA and the linux drivers. There exists two types of drivers for IrDA devices: FIR and SIR drivers. The first one gives the best result (best transfer speed) but is only working with some chipset. I don't have a chipset compatible with this driver and therefore had to use the other one: SIR.
You will need first to enable IrDA in your kernel (as modules for instance). This is well explained here: http://tuxmobil.org/Infrared-HOWTO/infrared-howto-s-kernel.html
Then you will need to install irda-utils. For instance, with a debian system:
# apt-get install irda-utils
There are some configurations questions during the installation. Just read carefully and answer.
Then you will need to set-up your modules properly. So for debian, it means editing files in /etc/modutils . Fortunately, this is already done by the debian package irda-utils, so you should have a file irda-utils in /etc/modutils containing everything that is needeed.
To finish the low level configuration you need just to tell linux which ttyS port has to attach to the IrDA stack of the kernel. One way to guess the correct ttyS is first to look at your kernel logs:
%dmesg | grep tty
serial8250: ttyS0 at I/O 0x3f8 (irq = 4) is a 16550A
serial8250: ttyS1 at I/O 0x2f8 (irq = 3) is a 16550A
My guess it that one these serial port is the modem, the other one is the infrared device. I just tried the second one randomly and it works.
To attach the serial port to the IrDa stack of the kernel:
irattach /dev/ttyS1 -s
If you're getting error, you should check that the modules are correctly charged in the kernel using lsmod and eventually use modprobe to put them by hand.
To check that the low-level part is working properly, switch on the infrared connexion on your phone, put it on front of the infrared port and try:#irdadump
3:01:41.394501 xid:cmd f4da8795 > ffffffff S=6 s=3 (14)
23:01:41.394556 xid:cmd 7b97d07f > ffffffff S=6 s=3 (14)
23:01:41.474494 xid:rsp 7b97d07f < 7f3d1bb7 S=6 s=3 T68 hint=9124 [ PnP Modem IrCOMM IrOBEX ] (20)
23:01:41.482504 xid:cmd f4da8795 > ffffffff S=6 s=4 (14)
(I removed some of the logs given by irdadump in the above). You can see that it actually does recognize the T68 phone!
If it doesn't work at that stage, it might be the serial port (ttyS1) that is wrong.
Ok, that's all for the low-level part.
We can get more information easily and play a bit:
IrLMP: Discovery log:
nickname: T68, hint: 0x9124, saddr: 0x7b97d07f, daddr: 0x7f3d1bb7
The last number is the IrDA address of the device. You can ping it using irdaping#irdaping
IrDA ping (0x7f3d1bb7 on irda0): 32 bytes
32 bytes from 0x7f3d1bb7: irda_seq=0 time=109.70 ms.
Now, we are going to use the obex protocol to connect to the phone:
First, install the appropriate package:#apt
-get install openobex-apps
Using IrDA transport
OBEX Interactive test client/server.
It will prompt you for some command. type c to connect to the phone:
Version: 0x10. Flags: 0x00
type g and then the name of the file to download it (in this example, the file is telecom/pb/0.vcf, it contains some information, just try cat telecom/pb/0.vcf to figure out what, after you have downloaded it):
GET File> telecom/pb/0.vcf
get_client_done() Found body
Filename = telecom/pb/0.vcf
Wrote 0.vcf (93 bytes)
type d to deconnect and q to quit openobex.
You know as much as me now. (or maybe more, or less)
|Thursday, February 22nd, 2007|
|Axioms and logic
Today, I would like to let General Relativity alone and explain some fundamental ideas in Mathematics. Modern Mathematics as we know them today contain theorems, definitions, propositions, lemma...These statements often take the form: if statement "A is true then statement B is true" or "A is equivalent to B" or "A is true". To develop a theory in Maths, we will therefore do the following: "A exists" will be assumed. Then will show lots of things like "A implies B", "B is equivalent to C" etc...and we will therefore have conclusion of the type "C exists".
In this intellectual game, we can distinguish between several ideas.
i) there are statements that will be fundamentals and will be the starting point of the whole theory: we will call them axioms.
ii) there are logical rules that defines a logical system and enable us to say that within the theory, some statements are true, others are false. These logical rules are of the form: implication (if A true, then B is true), equivalence (A is true if and only if B is true), negation (if A is false then the negation of A is true) etc... To define properly the logical system, we need to define a syntax (how we are going to write the rules) together with a semantic (how we are going to interpret the rules)
Let us apply this and re-discover mathematics:
- forget everything about maths
- define a system of rules and define how you interpret these rules
- defines some axioms
- work out everything you can from all of this,
The whole constitute a formal mathematical theory.
Note the following: this constitute a mathematical theory because we defined it like that. There are no reasons that a mathematical theory has to be what it is. No reasons in the formal, mathematical meaning. (it's a vicious circle, one has to start at some point)
Most of the mathematics that we use today use the "classical logic system", which is based on the intuitive idea of logic you all already know, together with some axioms which constitute a basis for the set theory.
There exists different types of logic systems. For example, in our intuitive logic, either A is true or A is false. So, for instance, consider the following proposition: (A is true) or (A is false)
In our intuitive logic, this proposition is always true.
However we might define a new logic where A is either true, false or in a indeterminate state: a state where A is neither true or false.
In that case, the previous proposition is not always true, it is true if A is true or if A is false, but it is false if A is in the indeterminate state.
There exists also different choice of axioms. The next time, I will try to describe the axioms of the set theory and in particular, to give some explanations about the axiom of choice, a controversial axiom of the set theory.
|Saturday, February 17th, 2007|
|Gravitational Collapse and the expected formation of black holes
Welcome back to the little red book of objective thoughts and our little description of black holes and General Relativity.
Today, we will look at an interesting phenomenum: gravitational collapse.
What happens into a star? There are lots of atoms together and the temperature is extremely high. Since the temperature is extremely high some fusion reactions between atoms will take place. At the beginning most of the atoms are hydrogen. Hydrogen atoms are light and easy (easier than other atoms) to fusion. After some time, there will be less hydrogen and more heavy atoms, the results of the fusion process.
When there are fusion reactions, a lot of energy is produced. This energy will, for instance, induce a very high temperature. Who says high temperature, says high pressure. So there will be a high pressure into the star. This pressure will induce a force that pushes the particles apart. However, on the other hand, the gravity will act on the particles and try to stick particles together, so that there is two forces-one coming from the pressure and one from the gravity-that act in opposite directions. Since we can see the stars this means that the two forces compensate and that an equilibrium is reached.
However, after some times, a lot of fusion reactions will have taken place, and that means less light atoms like hydrogen and more heavy atoms like iron. For the heavy atoms, one need a lot of energy to produce fusion, which means that the fusion reaction is less expected. So the overall effect will be that, after some times, the total energy released by the fusion reactions will get reduced and the temperature so the pressure will be reduced. So the equilibrium will be broken and the gravity will win. The star will start collapsing, its radius will get smaller and smaller and at some time, we will have a huge mass (the mass of the star) into a small enough area. It seems then natural to imagine that a black hole will be formed out of this process.
Ps: Note that this article is oversimplified, and that most of what is written there are expectations and not absolute truth that everybody agree ons (especially for the conclusion). In particular, I am not a specialist of stellar processes so I would appreciate if someone who is one could gives us his opinion.
|Friday, February 16th, 2007|
|Observers and Mass in Newtonian theory and General Relativity
First, let us agree on what an observer means:
An observer is somebody with a rule (so he can draw and make measurements on for instance, 3 axes x,y,z) and a clock (so he can write that event A occurs at time t, and event A' occurs at time t').
At first sight, there is no difference in the definition of observers between Newtonian theory and General Relativity.
Let us first look at observers in Newtonian Theory. In Newtonian Theory, we postulate (this is an axiom of the theory, something that we admit and do not try to prove) that there exists special observers for which the laws of physics will all be the same. You can think of these special observers as people being either fixed or traveling at constant speed with respect to each other. Now you should see the intrinsic problem with these definitions: fixed compared to who? Well, compared to an another observer. As you probably see, these definitions are not completely satisfactory because they sort of make reference to themselves. If you want to stop the self-referencing you have to add an absolute observer, that would be the reference for all other observers. But then this new absolute observer is still artificial, in the sense that there is no justification for it and no experimental evidence for such a preferred observer.
Forget about Newtonian Theory not being a satisfactory theory and focus on the following point: All observers that are fixed or moving at constant speed between each other will measure the time similarly. For instance, they will measure that event A occurs at the same time t and event A' at the time t'...
So the time is an absolute function. In particular, the special observers all measure the time similarly.
Now, because the special observers all measure the same thing at the same time, if two special observers try to measure the mass of something, they all agree on the value. So we can define the mass independently of the observer.
In General Relativity, there is no special observers. Moreover, two observers, being at different point of the space-time, will have different output if they try to measure let say, the time at which an event occurs. This is because, in General Relativity, there is no concept such as an absolute time.
However, for every particle, there exists a particular observer, the observer that is moving with the particle, at the same speed. For this observer, the particle seems fixed in the same way that when you are sitting in a train, you do not move compared to the train, even if you do move relatively to the outside. An observer moving with the particle can measure its mass, and we will call it the rest mass: the mass of the particle as measured in a frame where the particle is fixed. Now it is a postulate of General Relativity that light has rest mass 0. It is more than a postulate, it is the definition: a light ray is the trajectory of a particle which have zero rest mass at every point of the trajectory.
That's it for today!
|Friday, February 2nd, 2007|
|What is a black hole? session 2
The last time, we talked about General Relativity. Remember that it is a theory that aims to describe the dynamics (how things move) of the space-time, under a large scale, and that in the framework of this theory (that is to say, when this theory is applicable), heavy objects will bend light rays so that they will not be straight anymore.
Today we continue with this idea, and imagine a heavy object, like a star, which bends light rays. Now imagine another star, with roughly the same volume, but heavier than the first one. Then General Relativity predicts that the light rays will be more curved. If we continue to take heavier and heavier stars, the light rays will be more and more curved. At some point the light rays will be so curved that they will look like a circle or a spiral. In such a case, they will leave from the massive star, then they will turn, turn, turn and come back to the star.
Why do you we see stars in the sky ? Because they emit some light and send us this light. Now imagine a extremely heavy star, one that is so heavy that light rays leaving the star get bent into circles. This star will not be visible to us, because the light that it emits will turn into a circle and never hit us. So also the star do exist somewhere, we can't see it, it is just completely black. In a map of the sky, where there should be a shining star, there is just a black hole. Here we are!
|Thursday, January 25th, 2007|
|What is a black hole? session 1
To start, let's talk about something I know, or more precisely something I pretend to know: black holes.
To understand what a black hole is, it is important to understand some of the features of General Relativity. Waouh, sounds so easy!!
Ok, I'll just say the following: General Relativity is opposed to Newton theory. Both theories aimed to describe the physical world, which is assumed to be composed of matter that follows some rules. Nothing fancy so far. As all other theory of physics, they are described by some equations and therefore through the language of Mathematics. Of course, the maths involved in Newton theory (the one that you learned at high school) is far more simple than the maths behind General Relativity.
Both theories aimed to describe in particular a phenomenon often called gravitation. In Newtonian theory, gravity is looked as a force that acts on matter proportionally to its mass. Since light does not seem to be affected by gravitation (at least on earth), we were thinking that the mass of the light is null.
Look now at General Relativity. Imagine the following, take a sheet of paper, and raise one side, keeping it flat, and let some sand for example flows from the top to the bottom of the sheet. The sand is flowing in a straight line from the top to the bottom. Now do the same thing, raise one side of the sheet, but do not keep it flat, for example put something heavy in the middle, so that it looks bent. If you let the sand flows on the sheet, it will follow the curvature of the sheet and therefore it would not moved on a straight line.
Well, one idea (only one idea, because General Relativity can be approached under several angles) behind General Relativity is very similar to the sheet and the sand. The sheet is the space-time, i.e. the three dimension of the space plus the dimension of the time and the sand is the light. Without any matter, the space-time is flat and the light goes straight. If you add some matter to the space-time, then it get bent and the light move along a curve which is not straight anymore.
Now in reality, only extremely heavy object (like the sun) will be heavy enough to bend the space-time and therefore bend the light rays, so that it is very difficult to see this phenomenon. However, with the precision of today's telescopes, we can still measure it and the results agree with the prediction of General Relativity. Isn't it marvelous?
Ok, tomorrow, I'll talk about the black hole. I promise.