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Einstein's Theory Of Relativity
I have begun reading Einstein's book "The Special and General Theory Of Relativity," and I figured I'd make a thread with my notes and thoughts on what I read, as I read it. So far I have read the first three chapters, so let me share what I have written thus far, and I will continue to update. 
It is a natural to be proudly confident in the truth of geometrical principles. And yet these “truths” provided by geometry cannot be said to be ultimately, objectively true, but rather naturally, and logically, following the acceptence of the basic propositions, of things such as a “straight line” connecting two “points” on a “plane.” These things that you assume to be true by virtue of their very proposition are not inherently true, but rather useful assumptions to make. They are useful for the very fact that we CAN draw so many logical conclusions based on their initial acceptence.
“Rigid body” means that the distance between two points remains constant despite the bodies relative position or influence by outside forces. Such an object cannot physically exist, due to relativity, but we can safely assume an object to be perfectly rigid if it is not travelling near the speed of light.
NEXT TWO PARAGRAPHS ARE A QUOTE FROM THE BOOK: "The purpose of mechanics is to describe how bodies change their position in space with time. It is not clear what is to be understood here by ‘position’ and ‘space.’ I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the ground, without throwing it. Then, disregarding the influence of the air resistence, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices the stone falls to earth in a parabolic curve. I now ask: Do the ‘positions’ traversed by the stone lie ‘in reality’ on a straight line or on a parabola?’ Moreover, what is meant here by motion ‘in space’?
In the first place, we entirely shun the vague word “space,” of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by ‘motion relative to a practically rigid body of reference.’ If instead of “body of reference” we insert “system of coordinates,” which is a useful idea for mathematical description, we are in a position to say: The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an indepedently existing trajectory(path-curve), but only a trajectory relative to a particular body of reference."
ME AGAIN: Now, let us put this concept in terms that are useful in its comparion to quantum mechanics, as well as the relationship between consciousness and reality in general. The “shape” of the motion of the object as expressed in a sytem of coordinates, in this example, can only be determined relative to a particular coordinate system. The coordinate system could be likened to the essential viewpoint behind the observation of any event in space-time. Just as the coordinate system determines the perceived trajectory of the motion of the stone, with the perceived trajectory being the only ‘real’ trajectory in the first place, the essential viewpoints and personal bias/dispositions of the individual consciousness who bears witness to events ‘in reality’ determines the ‘reality’ of the ‘external events’ the individual goes through.
It is a natural to be proudly confident in the truth of geometrical principles. And yet these “truths” provided by geometry cannot be said to be ultimately, objectively true, but rather naturally, and logically, following the acceptence of the basic propositions, of things such as a “straight line” connecting two “points” on a “plane.” These things that you assume to be true by virtue of their very proposition are not inherently true, but rather useful assumptions to make. They are useful for the very fact that we CAN draw so many logical conclusions based on their initial acceptence.
“Rigid body” means that the distance between two points remains constant despite the bodies relative position or influence by outside forces. Such an object cannot physically exist, due to relativity, but we can safely assume an object to be perfectly rigid if it is not travelling near the speed of light.
NEXT TWO PARAGRAPHS ARE A QUOTE FROM THE BOOK: "The purpose of mechanics is to describe how bodies change their position in space with time. It is not clear what is to be understood here by ‘position’ and ‘space.’ I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the ground, without throwing it. Then, disregarding the influence of the air resistence, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices the stone falls to earth in a parabolic curve. I now ask: Do the ‘positions’ traversed by the stone lie ‘in reality’ on a straight line or on a parabola?’ Moreover, what is meant here by motion ‘in space’?
In the first place, we entirely shun the vague word “space,” of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by ‘motion relative to a practically rigid body of reference.’ If instead of “body of reference” we insert “system of coordinates,” which is a useful idea for mathematical description, we are in a position to say: The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an indepedently existing trajectory(path-curve), but only a trajectory relative to a particular body of reference."
ME AGAIN: Now, let us put this concept in terms that are useful in its comparion to quantum mechanics, as well as the relationship between consciousness and reality in general. The “shape” of the motion of the object as expressed in a sytem of coordinates, in this example, can only be determined relative to a particular coordinate system. The coordinate system could be likened to the essential viewpoint behind the observation of any event in space-time. Just as the coordinate system determines the perceived trajectory of the motion of the stone, with the perceived trajectory being the only ‘real’ trajectory in the first place, the essential viewpoints and personal bias/dispositions of the individual consciousness who bears witness to events ‘in reality’ determines the ‘reality’ of the ‘external events’ the individual goes through.
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Comments
The point is how the stone appears to behave in relation to the observer.
Law Of Inertia: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line.
The Law of Inertia holds true to a high degree of approximation with the visible fixed stars. And yet if we use a coordiante system that is relative to the earth, you obtain a result that is contrary to the statement of the law of inertia. So if we are to adhere to this law, we must refer these motions only to systems of coordinates relative tow hich the fixed stars do not move in a circle. The laws of the mechanics of Galilei-Newton can be regarded as valid only for a Galileian system of coordinates.
V
If you were standing on the ground and were to look up at a bird flying in a straight line at a constant speed, and then were to view that same event from another perspective, again from a train moving in a straight line at a constant speed, you would find that the bird was still flying in a straight line at a constant speed, but was moving in another direction.
This can be expressed thus: If, relative to K, K1 is a uniformly moving coordinate system devoid of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K. This statement is called the principle of relativity(in the restricted sense). So, relative to the earth, in any change in the direction of the movement of earth, measurements of the properties of events in terrestrial space would be the same RELATIVE TO the direction of the earth's motion. The direction of the earth's motion in and of itself would change throughout the course of the year, and therefore the direction of the motion of everything ON earth would change as well. So if relativity were not true then the properties of terrestrial space would have to change, since the direction of the earth's movement, and therefore everything on it, has changed. Yet it does not change, which is a proof of relativity, since although the direction of movement "in reality" has changed everywhere on earth along with the earth itself, everything on earth has stayed constant RELATIVE TO earth, hence their properties stay constant as well.
but if you want to learn more, Stanford university has many physic courses online for free from very distinguished professors (one of them is a good friend of Feynman).
Stanford youtube channel
http://www.youtube.com/user/StanfordUniversity/videos?view=pl
(those are courses, usually about 10, 2 hours lectures each)
the courses that may interest you:
Course | String Theory and M-Theory
Course | Astrobiology and Space Exploration
Course | Particle Physics: Standard Model
Course | Particle Physics: Basic Concepts
Course | Modern Physics: Statistical Mechanics
Course | Modern Physics: Cosmology
Course | Modern Physics: Einstein's Theory
Course | Modern Physics: Special Relativity
Course | Quantum Entanglements: 3 parts
Course | Modern Physics: Quantum Mechanics
Course | Modern Physics: Classical Mechanics
there are so many other physic lectures on youtube btw, if one is interested in such subject, one can indulge for many months all for free.
The internet had amazing potential, and sometime, like with this stuff, it came true!
@patbb: Thanks for the links.
muahahah
Imagine again a train travelling at a constant speed, towards a man standing still on an embankment. Imagine a man were to travel at a constant speed in the same direction as the train. How would you calculate the speed of the man walking, relative to the embankment? This would, it seems, be best represented by W = v + w, where W is the speed of the man on the train, v is the speed of the train, and w is the speed of the man walking in and of himself.
VII
The speed of light must be constant for all colors, and it also cannot vary based on the velocity of the motion of the body emitting the light. Now if a ray of light were to be sent along the embankment, it would move at c, the speed of light. Now we will say again that a train is moving at the velocity v, and that it is travelling in the same direction as the light. Let us determine the velocity of the propagation of the ray of light relative to the carriage. We can apply the consideration of the previous section(W=v+w), in this case replacing the man with light. Here we replace the velocity W of the man relative to the embankment with the velocity of light relative to the embankment. w is the required velocity of light with respect to the carriage, and we have w=c-v. This is because if we assume the laws of propogation of light to be true, and the speed of light to be constant, the only way you could maintain the constancy of the speed of light being set off on the train relative to the embankment is if you were to subtract the speed of the train from c, the speed of light.
The velocity of the propogation of a ray of light relative to the carriage, therefore, comes out smaller than c. This seems to be a direct contradiction of the principle of relativity in section V(If, relative to K, K1 is a uniformly moving coordinate system devoid of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K). If every ray of light propagated relative to the embankment has a velocity c, then it would appear another law of propogation of light must necessarily hold with respect to the train; this contradicts the principle of relativity.
He then says that, through analysis, it became evident that in reality there is no incompatibility between the principle of relativity and the law of propagation of light, and that by systematically holding fast to both these laws a logically consistent theory could be arrived at. This is the “special theory of relativity,” which he will get into later.
Forgot to take notes...don’t think anything particularly note-worthy was said, though. Mostly just discusses a hypothetical situation where we discuss the merits of two lightning strikes, taking place at A and B, happening simultaneously, with reference to a mid-point M. The question of what “simultanous” means is discussed, as well as how it follows that the constancy of the speed of light would have to be true with both of the lightning bolts in order to determine accurately that they did indeed occur simultaneously, and we would therefore have to verify that the light rays travelled at the same speed with both in order to make the claim of simultaneous occurance.
IX
A passenger on the train would use the train as their reference-body, seeing all events in reference to the train. The definition of simultaneity can be given relative to the train in the same way as relative to the embankment. The question, then, naturally arises: Are two events which are simultaneous with reference to the embankment also simultaneous relative to the train? This cannot be the case, he says.
If we say that lightning strikes A and B are simultaneous relative to the embankment, we mean that the rays of light emitted at the places A and B meet each other at the mid-point M of the embankment. But the events A and B also correspond to positions A and B on the train. M1 will represent the midpoint between A and B on the travelling train. An individual on the train, who sees reality in reference to the moving train, is hastening towards the beam of light coming from B, while riding away from the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than we will see that emitted from A. Therefore, an observer who takes the train as his reference body must come to the conclusion that lightning flash B took place earlier than lightning flash A. We thus arrive at an important result:
Events which are simultaneous with reference to the embankment are not simultaneous with reference to the train, and vice-versa. Every reference body has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event. This coincides with my thoughts that we are all on our own time-line, and how the assumption that everything and everyone functions on one time-line is the source of much of our stress and anxiety.
X
Basically, since time is relative to the point of reference, it follows that conceptions of distance over time would also be relative.