Theory of Relativity

I wrote this writeup as a conversation piece.

I did not fail physics, like you all must think. I misunderstood a story, what actually happened is that they were able to pass a waveform of particles into one end of a superconductor, and reconstruct the entire waveform at the other end before the entire waveform got through the superconductor, effectively transmitting information faster than the speed of light, even though nothing actually travelled faster than the speed of light.

My bad, hats off to you, Pseudomancer, for dissecting my writeup.

Preserved for posterity, and so that Pseudomancer's writeup still makes sense, is my original writeup below.

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Albert Einstein's greatest thought was the theory of relativity -- that everything is relative and that the point of view you take makes a difference in any experiment, more or less. Extensions of the theory of relativity are how Einstein deduced that a person, travelling in a spaceship moving at really high speeds, would age less quickly than a person at rest, or even at a slower speed. Einstein also believed that the intergalactic speed limit was the speed of light. This, however, is erroneous, since recently we were able to accelerate a particle to several times the speed of light. Einstein believed this to be true since he theorized that the closer you got to the speed of light, the more relative mass the object had and greater acceleration was needed to move closer to the speed of light. At the speed of light, mass was infinite and negligible at the same time. If we consider the relative mass of an object on a graph, we could think of the speed of light as being the asymptote for that mass.. and that perhaps on the other side of that asymptote, relative mass decreases similarly. Thus, by this consideration, the faster you go after you jump over the speed of light, the easier it would be to accelerate. Oh, I do think that Einstein was right.. energy might not be able to travel faster than the speed of light.. but matter surely can. Interesting consideration, don't you think..?

Arrr, no! NO NO NO!
Let me dissect this, I hate to be an asshole but as informative nodes go... this is just all wrong.

Albert Einstein's greatest thought was the theory of relativity -- that everything is relative and that the point of view you take makes a difference in any experiment, more or less.

While indeed his greatest thought (or rather, rigorous system of thoughts), Einstein did not demonstrate that 'everything is relative', this is a misinterpretation of his theory. In German, the title of the relevant paper was Zur Invariantenztheorie or 'on the theory of invariance'. The title of 'relativity' is misleading, as what Einstein was really doing was establishing what was absolute. More on this later. The 'point of view' you refer to would be the frame of reference, or point of view with respect to relative state of motion of the observer. Again, more on this later.

Extensions of the theory of relativity are how Einstein deduced that a person, travelling in a spaceship moving at really high speeds, would age less quickly than a person at rest, or even at a slower speed.

The point is that of time dilation. An observer in one frame of reference, observing someone in a frame travelling at some velocity relative to them, would observe time to travel slower in the 'moving' frame than in their own 'stationary frame. Of course, an observer in the 'moving' frame would observe time to travel slower in the 'stationary' frame as form their point of view the states of motion of the two frames are reversed. To the 'moving' person, he himself would age at the normal rate of one second per second.

Einstein also believed that the intergalactic speed limit was the speed of light.

The what? Replace 'intergalactic' with 'absolute'. The idea here is that nothing can go faster than light (or more properly speaking, be accelerated beyond the speed of light - more on this soon), not just stuff travelling between galaxies.

This, however, is erroneous, since recently we were able to accelerate a particle to several times the speed of light.

BZZ, wrong. His theory's actually been confirmed by evidence rather than refuted, time dilation, length contraction, and mass increase have actually been observed and no particles have been made to travel even at the speed of light (I'll call it c from now on), let alone 'several times faster'. What has been done is that quantum entanglement has been used to propagate the state of quantum entities instantaneously, but this is a completely different branch of physics and has nothing to do with motion per se. Also, there are certain exotic particles that travel faster than c anyway, but they have no mass and travel backwards in time. Go figure, it's a controversial theory (but not so controversial that it's not mentioned in second year physics). They start out faster than light, and can't drop below c.

Einstein believed this to be true since he theorized that the closer you got to the speed of light, the more relative mass the object had and greater acceleration

You mean force.

was needed to move

You mean accelerate.

closer to the speed of light. At the speed of light, mass was infinite and negligible at the same time.

Nope, sorry. If an object with rest mass (it weighs something when it's standing still, unlike photons) is accelerated towards c, its mass increases until at the speed of light it is infinite, which is, well, not negligible. If an object (like a photon) has no rest mass, it will be accelerated to c by any force acting on it, and its energy will be determined entirely in terms of its wavelength. At c, it will still have no mass, other than that which we consider to be its energy.

If we consider the relative mass of an object on a graph, we could think of the speed of light as being the asymptote for that mass.. and that perhaps on the other side of that asymptote, relative mass decreases similarly. Thus, by this consideration, the faster you go after you jump over the speed of light, the easier it would be to accelerate.

Ummm, I just don't understand this bit. I'll have to take your word for it, cos I ain't read it anywhere else.

Oh, I do think that Einstein was right.. energy might not be able to travel faster than the speed of light.. but matter surely can.

Interesting consideration, don't you think..?

Two words:

E=mc²

Mass is energy, energy mass according to general relativity. How can you say that you think Einstein was right and then say that?

Anyway, onto the real stuff.

The Theory of Relativity

By Pseudomancer

Now, there are two theories of relativity. The first is special relativity, the second general relativity. Special merely systematizes certain observational phenomena and allows a mathematical basis for prediction, whereas general takes Minkowski's geometrical formulation of the special theory into account and sets forth a geometrical explanation of the phenomena systematzed under special relativity. Special relativity comes first, so...

Prior to the work of Albert Einstein, the prevailing cosmology was that of Isaac Newton. Now, the orthodoxy of the day was that matter existed in an absolute frame (ie. a frame of reference which was 'really' stationary) and though I think this was a misinterpretation of Newton's theory, that's how it was. Now, one consequence of thinking of light as a wave, as they did back then, was that it required some kind of medium through which to travel. You know, sound doesn't travel in a vacuum and so light, being a wave, can't either? Anyway, this medium was thought to be a luminiferous ether, which just means 'a thing we can't detect that light travels through'. Convenient, no? This ether was thought to be stationary in the absolute frame, and so numerous experimnets were done to try and detect Earth's motion through the ether by measuring the differences in the speed of light as it approached Earth from different directions. This was, of course, based on the same principle that makes a head-on between two cars travelling at 50 km/hr in opposite directions about as bad as a car+wall at 100 km/hr. Problem was, they returned null results. That is, no matter which direction the experimenters looked in, light came at them at the same speed. The most famous null result was from the Michelson-Morley team, but there is some doubt as to whether Einstein had any knowledge of this particular result when he started work on the theory. He undoubtedly knew of others though, and it was these null results that helped him formulate special relativity. Now, centuries before, Copernicus, Galileo et al had done away with the geocentric model, so nobody was happy to simply say 'ah, ok, that means Earth is stationary, just like the Church told us'. As far as anyone knew, Earth was moving around the Sun, and there was no reason for light to come at it at the same speed regardless of direction. Unless, of course, the speed of light were constant no matter how fast you were travelling...

The other principle on which special relativity was based was that the laws of physics should be formulable in such a way that leaves them unchanged when switching from one inertial frame to another. That is to say, if you come up with some laws of physics that describe, say, a game of pool, they have to work for the game whether it's played on Earth, Mars, a ship moving at constant velocity or wherever.

So, given

1. "The laws of physics should work across all inertial frames of reference" and
2. One law of physics is that the speed of light, c, is a constant

Einstein derived special relativity. Note that 2. is an assumption, though it has been borne out perfectly by observation of various frames of reference, not just that of Earth.

An important fact to note is that the earlier formulations of 1, such as Galileo's (that the mechanical laws of motion are the same for all observers travelling at constant velocity) implied that there was some stationary frame of reference against which such velocity could be measured. This idea was screwed up by 2, and so Einstein was looking for a way to reconcile the two.

Special relativity consisted of the equations governing time dilation, length contraction and mass increase. I can't remember the equations exactly, though they all feature the factor sqrt(1 - v²/c²). Put simply, the faster you go (according to somebody else travelling at a different speed), the shorter (in the direction of travel) you would appear (to that somebody else). The slower time would seem to travel for you (to the somebody else), and the more massive you would get (I don't know who this is relative to). The idea behing special relativity is that of no priviledged frame of reference, that is you can take any inertial frame and the rules will allow you to work out how things look from it, thus doing away with the need for an absolute frame.

Then along came Minkowski, and things got surreal. He generated a geometrical coordinate system that explained the phenomena of special relativity. We can't include pictures in our nodes yet, so I can't show you what it looks like, but imagine this:

A grid, representing the stationary coordinate system. On the x-axis, the units of distance, light seconds (distance light travels in a second. About 300 000 000 metres). On the y-axis, time, measured also in light seconds (think seconds. The axes are this way around to remind you not to think of time as an independent variable and they're both in light seconds so that you can use pythagoras' theorem to express any distance between any two points in the system, something that can't readily be done if the units are different.
Along the diagonal, from (0,0), through (1,1), (2,2) etc, is a line representing the path of a photon. Think of this line as representing the speed of light (it's really called a world line, which just means that a thing follows that path through the system). Because of the axes I've chosen, any line steeper than this line will represent objects travelling slower than c, any line shallower objects travelling faster than c. You won't see too many of these though so don't worry.
Ok, now imagine a line about twice as steep, ie the world line of something travelling at half the speed of light. What we;re going to do is try and draw this object's frame of reference onto the same graph the other one was on. The first thing we need is the axes. According to this object, anything travelling at the same speed as it will be stationary. In the first frame (call it F), stationary objects were along the y-axis, so the world line at half c will be the y-axis for the new frame (call it F'). So, having drawn in the y-axis already, it remains to find the x-axis. Now, you might think that the x-axis could be drawn by just drawing it at right angles to the y-axis, like in F, producing a rotated version, but if you look at the grid you'd have drawn, the world line for the photon is no longer indicating that it travels at the speed of light. You've violated principle 2 from above, so the x-axis must go somewhere else.
Perhaps you can see it. The photon's world line has to go along the diagonal in whatever coordinate system you use, to account for the null results. This means that the x-axis is just the y-axis, reflected along the photons world line. F' looks like someone got F and squeezed the corners, stretching it along the diagonal photon world line.
Ok, we've got the axes, now we need the units. Minkowski comes along and calculates mental stuff here, and basically the units are bigger in F' than F.

Einstein says "I don't understand what this man has done with my theory."

If you look at the two frames you've drawn, and you got the units right, you'll be able to see why things like length contraction happen. If you take something that's one light second across in F', and look at how it looks in F, the area represented by it on the coordinate system (a 'vertical' block) hits F at an angle, and therefore it looks to be less than 1 light second across. And vice-versa, because remember F' has its units stretched (by a hyperbolic function, incidentally. It possesses this particular value of 'stretch' so that principle 1 is not violated, as this distortion value means that translations in the other direction, from F' to F, give the same amount of alteration).

So, then Einstein took this geometry and played with it in his genius mind, eventually coming out with

E=mc²

Einstein's theory was not about how everthing is relative to the observer, but rather about what it is that is independent of them. Take a look at the coodrdinate system you've drawn. It features only one spatial dimension, and one temporal, but the universe has 3 spatial dimensions plus one temporal, so the real coordinate system we live in (so to speak), is a four-dimensional version of this sketch. Spacetime. Saying that everything is relative to the observer is true in the same sense that saying 'the cat is not visible' is true, if there is a wall between you and the cat. The relation between moving frames is geometrical, and observer-independent.

As you change your frame of reference (in the context of GR, this means changing your relative speed), spacetime does not change, but what you observe as being space-like and what you observe as being time-like about it does change. This leads to such phenomena as the relative nature of simultaneity, time dilation, length contraction and suchlike. The mass increase is owing to the mass-energy equivalence of E=mc² - as you supply more energy to increase the speed of the object, its mass also increases (you're essentially giving it more mass), so more energy is required to accelerate it further, giving it more mass and so on.

See general relativity for my exposition of the implication of a curved spacetime metric. That is, a 4-dimensional universe that bends when you put stuff in it.

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All material in this node fished out of Pseudomancer's grotesquely oversized head, where it was placed by a series of lectures by Philip Catton at the University of Canterbury, New Zealand.

The essence of time dilation with respect to the theory of relativity has been demonstrated experimentally. In one experiment, two atomic clocks were carried on high speed air plains. One traveled eastward (with the rotation of earth), and one westward (against the rotation of earth). After the flight, these clocks were compared and found to have either gained or lost timed (relative to the control clock which remained on the ground). The amount of time lost and gained was a function of their direction of travel (motion), and altitude (gravity). The results confirmed the predictions of time dilation from Einstein's theory of relativity.

"During October, 1971, four cesium atomic beam clocks were flown on regularly scheduled commercial jet flights around the world twice, once eastward and once westward, to test Einstein's theory of relativity with macroscopic clocks. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40+/-23 nanoseconds during the eastward trip and should have gained 275+/-21 nanoseconds during the westward trip ... Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59+/-10 nanoseconds during the eastward trip and gained 273+/-7 nanosecond during the westward trip, where the errors are the corresponding standard deviations. These results provide an unambiguous empirical resolution of the famous clock "paradox" with macroscopic clocks."

J.C. Hafele and R. E. Keating, Science 177, 166 (1972)

The Twin "Paradox" (not really a paradox) explains this idea also, and more completely in the three planets thought experiment. The key point being inertial reference frames.

Common sense likens photons to bullets that are shot out of a gun. A bullet shot out of a car in potion travels faster than one shot from a stationary refrence point. The difficultly with this analogy is that this does not apply to things moving at the speed of light (bullets move much slower than photons - which always travel at the speed of light for whatever medium they are in). The speed of light is constant. Period. Yes, it does get confusing with the concept of inertial reference frames.

Let us imagine a clock in which time is measured by a photon bouncing between two surfaces:

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  ^
  |
  o
  |
  v
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As your velocity approaches c, this clock will start to slow down. Why? Simple, because the distance between the two surfaces grows - they are no longer directly on top of each other:

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  \        /\
   \      /  \
    o    o    o
     \  /      \
      \/        \etc...
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However, you will not notice the clock slowing down because you are slowing down with it. To actualy see the slow down (or speed up) you will need to be in a diffrent frame of refrence.

Time dilation will happen in accordance to the Lorentz transformation. Yes, this is a big 'nasty' function with lots of primes and radicals and subscripts.

c is the speed-limit of light in that nothing can be accelerated to the speed of light with respect to you. Accelerating two objects to 75% of the speed of light in different directions giving a relative velocity greater than c just doesn't work (see the three planets thought experiment for more information on this). Space-time is interwoven in such a way that no observer ever sees an object moving toward them at greater than c. Once the Einstein velocity addition is applied, the result is always a relative velocity less than c. This may not agree with common sense, but this appears to be the way the universe works.