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| Thread: Understanding Relativity | |
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Svarog
Honorable
Supreme Hero
statue-loving necrophiliac
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posted November 26, 2004 01:02 AM |
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Understanding Relativity
The scientific development of physics in the XX century has been a remarkable dramatic revolution that changed the essence of how we percept the world and its physical laws. Einstein’s theory of relativity is one such revolutionizing concept that in the minds of common people borders with the realms of science fiction and common reasoning. I’ll write a short article revealing the basic concepts of relativity. If there are any interested in learning more, please don’t hesitate to ask.
There are two known theories of relativity: The General Theory of Relativity and the Special Theory of Relativity, as a special case of the GTR. General Theory of Relativity deals with time-space continuums, world lines, 4 and more dimensions, extremely complicated mathematics, and it’s an obscure, almost irrational theory, which is considered to be understood by only a handful of people on Earth. Naturally, not including me. 
Special Theory of Relativity is much more widely understood. In order to explain it, first there are some basic lessons you need to know before you even start reading on relativity. Whenever we observe something, we do it from a certain reference frame, from where we measure and observe the physical phenomena. There are two kinds of rf: inertial reference frames, and non-inertial reference frames. IRF* are conditionally taken to stand still, or move with a constant velocity, while NIRF are considered to have acceleration (change in velocity). STR** deals only with inertial reference frames.
Newtonian (classical) mechanics assumed an absolute space, independent of matter and time, so called Euklidian space (three-dimensional), unlimited, homogenous, isotropic etc. This means that an object of 1 meter is exactly 1 meter long no matter the reference frame from where we measure it; that 1 year lasts 1 year for all observers (no matter the reference frame). Velocities are arithmetically added or subtracted (e.g. If I move with velocity of 5 km/h with relation to a tree, and a bird moves with 4 km/h wrt*** to myself, then the bird moves with 5+4=9 km/h wrt the tree. Furthermore, mass is constant, and independent of object’s velocity etc.
All these postulates of classical physics have been rejected with STR. However, this doesn’t mean they arent applicable to everyday life, since SPR only gives observable results when dealing with velocities comparable to the speed of light (c=300,000km/s). In all other cases, the discrepancy between classical predictions and what actually happens (with relativity taken into account) is so minimal, almost non-existent. And this is the reason why people havent been aware of the inherent relativity of the world.
When scientists started dealing with light, they found anomalities which couldn’t be explained with the classical theory of absolute space and time. (the famous Michelson experiment)
In the early XX century, Einstein formulated his basic postulates for the theory of relativity:
1. Physical laws are the same in all IRFs.
2. Speed of light is constant in all IRFs.
Pay attention to the second one. That means that c doesn’t depend on the speed of its source or the observer of the light beam. Consequently, a guy that travels with half the speed of light (150,000km/s) with relation to a light source, will also see the light beam traveling with c. Light also travels with c, wrt the light source. Or expanding on the example with myself, the bird and the tree, we cannot arithmetically add/subtract velocities, because speed of light is always the same for all observers.
In addition, there’s no absolute space. All things are relative, and in order for any measurement to have sense, it has to be relative to some reference frame. In other words, measurements for all physical concepts are always different depending on the rf**** from where we observe.
These simple postulates imply a wide range of consequences, proven with mathematical equations (which I’ll leave out, unless someone wants the mathematical explanation).
1. Simultaneity of events
Events that happen at the same time in one reference frame, don’t happen at the same time in another rf, moving with near c velocity in relation to the first ref frame. They only happen at the same time for both ref frames, if they happen at the same place.
A demonstrative example: An observer from the earth reference frame observes two lightning bolts that happen at the same time, one on the east, the other on the west.. A train travels with near c velocity wrt the ground. A traveler from the train rf, observes one lightning bolt going earlier than the other. Note that this isnt because the speed of light changes to c+v and c-v, but because for him, one event actually happens earlier. Even if the observer was nearer the lightning bolt that strikes later, it’d still be the same order of events.
2. Time dilation
This means that time doesnt pass the same for all observers. The greater the velocity of a ref frame wrt our own rf, the slower the time rate is for events happening in that reference frame. For example, if we observed a watch on a space shuttle traveling with near c velocity wrt us, we’d see time passing slower on the shuttle, and all events (including aging) passing slower. But because there’s no absolute space (and all IRFs are equally valid), we’re in no position to determine who’s actually moving (the shuttle, or us, along with the entire planet). Therefore, from his ref frame, he observes time on earth passing slower (kind of like a replay).
Consequence of time dilation is that time travel is actually possible (though only in the “future” which on the other hand is a relative term, to begin with).
3. Length contraction
When measuring objects which move with near c velocity relative to us, we always measure them shorter (in the direction of movement) than someone would measure them from their own rf (i.e. if they stand still). Simple example: for a shuttle traveling from sun to earth with 0.866 the speed of light wrt the Earth, the Sun-Earth distance would be half from what we measure here, from Earth. This is called length contraction.
4. Velocity addition
Remember the example with the bird and the tree? Well, here’s how actually the velocities are added. If v is my own speed wrt the tree; u is the bird’s speed wrt the tree, and u1 is the bird’s speed wrt myself:
U=(u1+v)/(1+v*u1/c^2). So, not a simple addition.
A direct consequence of this is that no object can ever reach the speed of light. Here’s the proof: If u1 is c (the bird travels with c wrt to myself), and my speed is c wrt the tree (v=c), then under classical mechanics we’d see the bird traveling with 2c wrt the tree. But
U=(c+c)/(1+c*c/c^2)=2c/(1+1)=2c/2=c
5. Relativistic mass
Mass is not constant, but it increases for moving objects. The smallest mass an object can have, is when it’s not moving wrt us (Mo), but for every object moving with near c speed wrt an observer it appears much heavier for the observer. Thus, if an object reached c, it would have infinite mass, and no amount of energy can accelerate it above c (due to the infinite mass), so that’s why there isnt a velocity greater than c.
6. Relationship between mass and energy.
Ahh, the famous formula: E=mc^2. What it actually means is that mass and energy are two different forms of the same physical reality, and can easily convert from one to another. Therefore, when an object is moving with near c velocity wrt an observer, he increases in mass, which is actually an increase in total energy (since mass effectively is energy). Where did this energy come from? It’s the kinetic energy we’ve added to the object, converted to mass.
Also, in atomic bombs, when atoms disintegrate they release such big amounts of energy (in the form of gamma rays) precisely because they lose mass (which is converted to nuclear energy).
From all this said so far, one can conclude that there’s a time-space continuum, a mass-energy entity, and a world with no absolute, only relative perceptions. All is relative - a completely new page in the book of science thanks to the ingenious mind of Albert Einstein.
* IRF=Inertial reference frames
** STR=Special Theory of Relativity
*** wrt=with relation to
**** rf=reference frame
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The meek shall inherit the earth, but NOT its mineral rights.
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Wiseman
Known Hero
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posted November 26, 2004 12:50 PM |
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Ah, the good old highschool physics....
I would just like to add something:
Comparable to c means higher or equal to 0.2c.
That means we use relativistic aproach only if the speed
exceedes 60000 km/h.
Also we mustn`t give Einstein too much credit.
It was Hendrik Lorentz who first made new transormations, after it was obvios that old Galileo`s were obsolete. He derived most of new formulae, and Einstein just included them in his theory, while drawing
other important conclusions and implications.
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Truth may be out there, but lies are inside your head.
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Svarog
Honorable
Supreme Hero
statue-loving necrophiliac
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posted November 29, 2004 02:47 AM |
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Edited By: Svarog on 29 Nov 2004
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I tried to keep it as basic as possible. If anyone's interested in discussing more (including wise enough men), give a sign.
Meanwhile, here's a fairly simple example, which demonstrates most of the mentioned relativistic concepts at once. There are two pictures for the same event, as seen from both reference frames. I think they are self-explainatory, so I wont include text, but if its necessary, tell me.
Intro: Two twins, one is standing on the Earth and the other is traveling at constant speed from the Sun to the Earth with a space ship. The ship has a pole at the front, in order to trigger the start of the Earth clock.
In the end they both agree on the passed time they measured during the trip, but as you see the paths that lead them to the same result are rather different.
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The meek shall inherit the earth, but NOT its mineral rights.
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Conan
Responsible
Supreme Hero
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posted November 30, 2004 02:27 AM |
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if you agree that when you look at the stars, you are looking at the past, and that time is different if you move faster and faster, then do you agree with time travel?
Do you agree that if you move faster than light, you are going in the past?
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Your life as it has been is over. From this time forward, you will service.... us. - Star Trek TNG
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Svarog
Honorable
Supreme Hero
statue-loving necrophiliac
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posted November 30, 2004 02:38 AM |
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Quote:
if you agree that when you look at the stars, you are looking at the past, and that time is different if you move faster and faster, then do you agree with time travel?
Time travel is a viable concept, but it has nothing to do with looking at the stars and getting past info. It is possible if you travel with great speed, to leave earth and when you come back to find out that you have aged less (even several years less) than others.
Quote:
Do you agree that if you move faster than light, you are going in the past?
It's impossible to move faster than light. Under the present understanding od science. And I dont wish to speculate.
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The meek shall inherit the earth, but NOT its mineral rights.
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PinkFlamingo
Adventuring Hero
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posted August 09, 2011 06:34 AM |
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OK, I know this is an old thread, and I'm not even sure if the original poster will read this, but I am a bit confused.
Can you explain simultaneity more thoroughly? Also, I don't understand the math of adding velocities.
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Tsar-Ivor
Promising
Legendary Hero
Scourge of God
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posted August 09, 2011 03:09 PM |
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I have concerns when I post in a month old thread  , let alone a 7year old dead n' burried one, but now that you've resurrected it I'll take some time and read through it 
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"No laughs were had. There is only shame and sadness." Jenny
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Corribus
Hero of Order
The Abyss Staring Back at You
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posted August 17, 2011 09:48 PM |
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Edited by Corribus at 05:11, 19 Aug 2011.
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Pink Flamingo:
Quote:
Can you explain simultaneity more thoroughly? Also, I don't understand the math of adding velocities.
Well, I don't think Svarog posts here anymore, so maybe I can help you.
Simultaneity:
The idea is that the perception of flow of time changes depending on reference point. Einstein used a famous example of lightning bolts and trains to explain the concept of simultaneity.
Let's start with a single observer standing on one side of a long train (which is standing still for now). Another observer is standing outside the train some distance away, and he's situated equidistant from the two ends of the train. Imagine now that lightning strikes both ends of the train at the same time. The person outside the train perceives the two lightning strikes as occuring at the same time, because it takes the light from each one the same amount of time to reach his eyes. However, the guy standing at one end of the train would perceive one lightning bolt as striking before the other, because it takes the light from one strike much less time to reach him than the other. (Of course, the speed of light is so fast that you'd never be able to pick up this difference - we're speaking in principle here.)
Maybe a good analogy is if you're standing blindfolded in the middle of a field, 5 km away from one cannon and 10 km away from another. A friend of yours is standing at a different position, 7.5 km from both cannons. Neither of you know where you are in relation to the cannons. Now suppose the cannons are scheduled to fire at the same instant. Your friend will report both cannons as firing simultaneously. You will report that one fired before the other.
That's the idea.
Now back to the train. Suppose you're in the center of the train (on the train), and your friend is standing on a platform outside the train. The train is moving, and at the instant the train passes your friend (so that he's equidistant from the two ends of the train, just like you are), lightning strikes both ends of the train.
Your friend will perceive these bolts as striking simultaneously because the distance between him and each of the two points is the same - it takes the light from the bolts the same amount of time to reach him, and they're in the same frame stationary of reference that he is (stationary relative to the moving earth, of course. ). You, however, are in a frame of reference that is moving with respect to the frame in which the lightning strikes.
Let the train be on an imaginary x-axis. Call the two ends of the train A and B, and the train is moving in the +X direction -> directly toward B. When lightning strikes either end, light flashes in every direction. If the train wasn't moving, you'd see the light from the A strike that was moving toward B (+X direction) and the light from the B strike that was moving toward A (-X) direction. However, YOU are moving in the +X direction with some velocity, v.
Now lightning strikes both ends of the train at time 0. At time 0 + t (some tiny amount of time later), the light from strike A moves toward you in the +X direction at speed c and the light from strike B moves toward you in the -X direction at speed c. But you're moving in the +X direction at velocity v. So after a short time, t, you are now vt closer to where the B was when you started, and vt further away from where A was when you started.
You can do some basic math if you are given numbers for positions and velocity, but essentially because you're moving toward B, it takes the light from the A strike longer to reach you than the light from the B strike, and so you perceive the strikes to happen at different times, even though your friend perceives them to happen at the same time. The concept is related to the Doppler Effect, btw - you would also see the two strikes of lightning as slightly different colors in addition to slightly different times.
Of course, light is going to travel much, much, much faster than the speed of a train, so in this scenario both you and your friend would perceive the lightning to strike simultaneously (the time difference is smaller than your neural circuitry can process). However as the speed of the train increases to near light speed (what we call relativistic velocities), the effect of frame of reference on perceived simultaneity becomes more and more important.
Adding Velocities:
This math is rather complicated and involves some calculus. I'm not sure why Svarog included it here given that he was trying to keep things basic.
Essentially you have to start with Lorentz transformations of position from one frame of reference to another and then differentiate them with respect to time.
What does that mean?
Well, for two frames of reference - one moving and one not - you can express the points in space and time of one in terms of the points in space and time of the other, with the relative velocity being the mediating factor. Doing this in relativistic frames is called a Lorentz transformation, and lacking a suitable equation editor I'm not even going to try to do it here. Most college-level physics books can give you the requisite several pages of equations to treat velocity transformations properly.
I think basically what point Svarog was trying to do get across was the following. Let two people be standing on opposite ends of a speeding train, and you are standing on a platform watching the train go by. One of the people on the train throws a ball to the other one. In classical physics, the speed that you (stationary observer) would measure for the ball can be determined simply by adding the velocity of the ball and velocity of the train. However, as the speed of the train approaches that of light, this classical picture begins to fail. Velocities are no longer simply additive - or better put, the speed of the ball is different for different frames of reference. (If the velocities WERE simply additive, as in the classical picture, it'd be possible for the ball to travel faster than the speed of light, because the speed of the ball the observer would measure would be the speed of the train plus the speed at which the guy on the train throws the ball. The relativistic velocity transformation ensures that nothing can move faster than speed of light in any frame of reference.) Do note that in the equation Svarog provided, in the limit of small velocities, the denominator approaches one and the classical result is obtained.
If you're really interested in following the math more closely, you'd be better off looking at a physics book rather than having me butcher it on here. It's been too long since I tried to do that kind of stuff.
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I'm sick of following my dreams. I'm just going to ask them where they're goin', and hook up with them later. -Mitch Hedberg
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