guthmund

Surprisingly, not for a paper, just some general....curiousity.

I've been reading a lot on physics lately. Brian Greene's two books, some stuff by Michio Kaku, Richard Feynman and the latest by Bodanis (I think that's who wrote it..) about E=mc^2, not to mention dozens of sites on the internet, including this one, mind you.

I understand bits and pieces, but I still have a few questions and I thought the gurus here might have the answers.

Okay...first. I understand that c is squared, what I don't understand is why. Bodanis remarks that s'Gravesande found that when he dropped his metal spheres into soft clay he noticed that if he propelled the second ball twice as fast it fell into the clay four times as far, three times as fast, it fell nine times as far into the clay. Kaku termed it "force multiplier" and further explained that in similar instances it worked the same. He wrote that a car increasing speed from 20 mph to 80 mph had increased in speed some four times and logically, the car moving at 80 mph should have four times as much 'energy' as the car moving at 20 mph, but in reality, the car moving at 80 mph has sixteen times as much 'energy' as the car moving at 20 mph.

Every book, I've read (unless I managed to miss it somewhere or misunderstand ) fails to explain "why." They just say it is or that's the nature of energy. So...why? Is just taken for granted that the force multiplier effect is there or is there a reason why?

One more...

Kaku and Greene explain that if I could watch a car travelling at about the speed of light (since you can't really travel at the speed of light, which, again, I sort of understand ) the car would look compressed toward the direction of motion. That the height of the car (or whatever) would stay the same, but lengthwise it would compress like an accordion. As I understand it, inside the car everything would be 'normal,' it's only from the outside looking in that it gets crunched. As the car slowed down and eventually stopped, everything would be back to 'normal' from everyone's point of view. Kaku asks who was really compressed, you? or the car? He further states that "According to relativity, you cannot tell, since the concept of length has no absolute meaning."

So, the question is...why compressed? I get that nobody really knows why it happens, but do they know why 'compressed?' I mean, if length has no absolute meaning, why can't the car look elongated? Why doesn't it stretch?

That's all for now, but I've got some questions on general relativity as well if anyone's interested.

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stingc

The point of E=mc^2 is that energy and mass are equivalent. Originally, the point of it was to say that a body's inertia (mass) depended on its energy content.

Now, saying energy and mass are equivalent in words doesn't quite work if you sit down to calculate anything. The two quantities have different units. Energy is expressed in joules, foot-pounds, ergs, etc. Mass is in kilograms or slugs. Equating these things directly is meaningless. If mass and energy are indeed equivalent, you need a conversion factor. This should be a constant of nature, and it needs to have units of velocity squared (you can easily check this by using the fact that 1 joule is by definition 1 kilogram-meter^2/second^2). c^2 is the natural choice.

This is the simplest way of looking at it, but there are others. Einstein certainly didn't arrive at the equation like this. He had to say why energy and mass should be equivalent in the first place. If you want to know why that is, I could refer you to some better (but harder) books than the ones you're reading.

The statement about spheres in clay is to help you understand why energy has the units it does. If you simply accept this, the example is irrelevant. In reality, energy was given its present definition (not that the basics are really new, or were even new 100 years ago) due to many complicated and interwoven discoveries. The concept of energy also encompasses so much more than just motion that I think the example is basically useless. I hope that your books contain a much more substantial explanation of what energy is than that.

Kaku's statement that there is no absolute length in relativity is misleading. He meant that one's most naive definitions of length are not absolute in relativity. Hence the length contraction. There does exist an absolute notion of length in relativity. It doesn't change size depending on how fast something goes (unless the object is physically stretching or compressing in the usual sense).

This is actually quite simple to define. If lengths are always different to people moving at different speeds, just define the absolute length to the be the one measured by someone moving at the same speed as the object itself. You might think that this is cheating, but you can set things up such that anyone will measure exactly the same length without actually changing speed to do so.

As for the why things don't expand, this can be shown intuitively through a couple of diagrams, but that would require too much background for me to describe here. If you're interested, I'd highly recommend that you get a relativity book which uses spacetime diagrams to explain things. The viewpoints given in most popular books - while correct - were basically abandoned by most physicists after 1908 or so. The geometric formulation of relativity is far easier to understand, and makes all the length contraction business really simple. It also removes most of the 'wackiness' and apparent paradoxes of 'traditional' relativity.

I hope that helps. I can answer your general relativity questions also, if you're still interested.

Martian

stingc covered the questions fairly well. All I have to add is a reminder that there's no reason to feel bad if you find general or special relativity or anything related in quantum mechanics confusing. It's a very counter-intuitive way of thinking and I know of no one personally who understood all of the concepts on the first try.

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KnifeMissile

Wow, it's been a while since a Relativity thread happened in this forum. We've probably been through all this before but this thread is so slow these days that I welcome another opportunity to show the simplicity and straightforwardness of an all too often convoluted subject. You do seem genuinely confused and I blame the sources you've chosen to learn off of. I personally find that the best source to learn Relativity (and other branches of Physics) is from a genuine text book. They can be purchased at relatively (ha ha) reasonable prices from used book stores, especially if they are not for any class of any academic program. My favourite is Young, although Giancoli is also quite good. The big secret that all your various sources probably neglected to tell you is that a firm understanding of Special Relativity requires nothing more than highschool mathematics, which is probably why a math flunky was able to formulate this theory.

Another thing you should know is that science doesn't explain why things happen, it merely describes how things happen. While many things can be explained in terms of other things, ultimately, things just behave the way they do and all we can do is make a note of it.

Now, I'm afraid to go into great length over details you already understand so I'm going to go really fast and you can come back with everything you know and what you don't understand, okay?

There are really only two things you need to know about Special Relativity. They are the two assumptions, or postulates of Special Relativity. Two simply facts that the entire theory is based on. They are:

1) The laws of Physics are the same regardless of your inertial reference frame.
2) Light travels at the same speed regardless of your inertial reference frame.

The first postulate is obvious. No one point of view is any more valid than another and, literally, the laws of Physics don't change simply because you're going somwhere else.

The second postulate is a little less obvious and tricks a lot of people up. We're used to the idea that Relativity dictates that if a car travelling at 20 kph is passed by another car travelling at 30 kph, the first car will see them pass at 10 kph. While this appears to be true for most objects, light doesn't behave that way. While it travels along the road at the speed of light, it will actually pass the 20 kph car at, also, the speed of light. Strange but true!

Now that we understand these fundamental observations of reality, lets try to answer your questions, starting with the second one. Length contracts (shrinks) because time dilates (shortens). The longer you travel, the farther you go. So, if time dilates, length contracts. Get it?
Of course, time dilates because light must travel a greater distance for the moving object so, as per our earlier understanding, it must have taken a longer time to do so. But, because of the first postulate, that longer time for us translates into a slower time for the moving object.

Your first question is a little more subtle. First of all, your concern for why c is squared is misguided. The simply fact is that the units would not be right if it weren't. Energy is force×distance, or mass×distance²/time². If you didn't square c, which is speed (distance/time), then the formula would come out mass×distance/time, which is momentum, a very different quality!
The derivation of the formula E=mc² comes from the integration of force over distance, the very defition of energy! If you accept that you can put an arbitrary amount of energy into moving an object yet that object's speed must remain finite, it follows that the speed of the object becomes asymptotic to some speed while its energy is unbounded. Integrating this asymptotic formula shows that as the speed approaches zero, the energy becomes a finite, non-zero value and that value is mc².

So, there you have it. The Relativity speed course with all the actual diagrams and formulas cut out. This is really an opportunity for you to specify exactly which part you didn't understand in your travels through the theory of Relativity. I can expand on any part to any amount of detail so that you will understand! I hope you have a desire to do so...

guthmund

Believe it or not, that made sense. A much better explanation than 'just because."

Yes, please do. Although I enjoy the simpler works because often they take the time to explain with analogies (easier to wrap my head around than straight mathematics), I sometimes feel as if I'm missing the bigger picture because the books have been 'dumbed' down.

Well, yeah. I think the example of s'Gravensande was really more an attempt to explain why the velocity term was squared rather than explain the concept of energy. I'm interested though, if you've got a more substantial explanation then I would like to hear it, if you don't mind.

[quote]This is actually quite simple to define. If lengths are always different to people moving at different speeds, just define the absolute length to the be the one measured by someone moving at the same speed as the object itself. You might think that this is cheating, but you can set things up such that anyone will measure exactly the same length without actually changing speed to do so. [quote]

Ha! I never thought of it like that, but, yeah, that makes all sorts of sense.

It has to do with Lorentz' doesn't it? At least, Lorentz Transformations are used to measure the amount of the contraction, right? Okay, spacetime diagrams...got it.

It helped immensely, stingc. I will certainly have some general relativity questions in the near future and I would certainly appreciate the help.

/off to read KnifeMissile's post..

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rsl12

stingc and knifemissile have already summed it up nicely. I will offer then just these two useful illustrations for time dilation and length contraction. The illustrations follow from the two facts described by knifemissile (i.e., physics works no matter from where you are looking, and light always travels at speed c).

TIME DILATION
Ok, you're sitting on your front porch, and your friend is driving past in his very fast car (at a constant velocity), and he is holding up to the window a see-through box containing a light beam, which is he has managed to trap between two perfectly polished mirrors (parallel to the floor). To your friend, the light beam is bouncing straight up and down, and the time it takes for the beam to go from one mirror to the next is d/c, where d is the distance between the mirrors.

To you, however, the light beam isn't going straight up and down--it's going ziggy zaggy because the car is moving! (Pretend you're in dark dark space, and all you can see is the light beam, like a flashlight waving around in the dark. You wouldn't see it go up and down if your friend is travelling past you--you'd see it go up-left, down-left, up-left, etc.) Nevertheless, light still travels at the speed of light, despite the fact that the distance it must travel is longer (the vertical distance between mirrors is the same, but now there's a horizontal component to the movement), so the time it takes for the light to go from one mirror to the next is longer for you than it is for your friend (i.e., for you, it appears that time is moving slower in the car!). Using some pythagorean theorem and rearranging terms using simple algebra, you can figure out that the time it takes is d/(c^2-v^2)^0.5, where d is the distance between mirrors and v is the speed of the car. (Note, however, that if you were holding the same box with the same light beam, your friend would see it moving ziggy zaggy. Therefore, to your friend, you are the one moving slower through time! But that's another matter altogether.)

LENGTH CONTRACTION
This is a bit trickier. Okay, now let's say that your friend is still in the car with the light beam in a box, but that the box now has the mirrors perpendicular to the floor (i.e., the light beam travels parallel to the floor, going back and forth in the same direction as the car's movement). The time it takes for the light beam to go from one mirror to the next is, for your friend, still d/c, or 2d/c to do one complete lap. For you, however, the light is travelling forward 2 steps, then going back one (I'm talking figuratively now) since the car and the mirrors are in motion. How long does it take to make one complete lap? Well, since TIME DILATION is occuring, we know that the time should be 2d/(c^2-v^2)^0.5. However, we also know that you can calculate the time by figuring out the time it takes to get to the 1st mirror, or d/(c-v), and the time it takes to get back, which is d/(c+v). So the time it takes is the sum of these. Using the fact that (c-v)(c+v)=(c^2-v^2), we can figure this time to be 2dc/(c^2-v^2). So if we set these two methods of calculating time to be equal, we have

2d/(c^2-v^2)^0.5=2dc/(c^2-v^2)

how is it possible? the left side is *almost* identical to the right side, except that the right is multiplied by a peksy term of c/(c^2-v^2)^0.5!!

The answer is that the 'd' in the left side is *not* the same as the 'd' on the right side. The 'd' on the left was calculated using the distance between the mirrors when the mirrors were parallel to the floor (i.e., the same distance that your friend measures between the mirrors), whereas the 'd' on the right is the distance you observe between the mirrors when the mirrors are *perpendicular* to the floor! Let's call this second distance d prime (d'). Therefore the relationship between d and d' is:

d = d'c/(c^2-v^2)^0.5

The lengths of things aligned with the direction of motion appear *shorter* to you!

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rsl12

Actually, now I have a puzzler for all of you:

Taking the above example, let's say you have two boxes of light in the car--once vertically aligned, one horizontal, but otherwise identical. Your friend shoots a beam of light out of both boxes at the same time. Photosensitive sensors are located on the far-end mirrors of both boxes, which immediately turn on flashlights located at the top of the boxes. These flashlights are pointed at another sensor right in the middle of these two boxes, which will display a big 'H' if the flashlight on top of the box with horizontally aligned mirrors reaches the sensor first, a big 'V' if the flashlight from the vertically aligned box does, or a big 'E' if the two reach simultaneously.

What letter does your friend see? What letter do you see? What would happen if, instead of flashlights, you had rubber balls that were kicked towards the sensor via some kind of kicking mechanism?

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stingc

Geroch's "Relativity from A to B" is probably a good start. I haven't read it myself, but he is an excellent writer, and the description looks good.

A somewhat different kind of book is Taylor and Wheeler's "Spacetime Physics." This is more of a (very) introductory textbook written rather whimsically. The math is no more complicated than algebra, though.

One of the rarely advertised facts of science (especially physics) is that most of its progress hinges on formulating questions (and their answers) in just the 'right' way. Using one quantity/coordinate system/mathematical technique over another often makes a tremendous difference in one's intuition. And this is ultimately what guides experimental design and analysis, as well as theoretical explorations. Anyway, energy is one of those things that was found to be very useful a long time ago. Its utility has only increased since.

Energy is closely related to capacity to do work (or the amount of work required to prepare a system from some reference state). Work is in turn defined to be the action of a force over a distance. It therefore has units of newton-meters or foot-pounds. But the units of force are really mass*length/time^2 (think of F=ma). So work has units of mass*(length/time)^2 = mass*velocity^2. By definition, energy also has these units.

The reason that the concept of work requires that a force act over a distance is that things can sometimes exert forces on each other forever if there is no displacement. The chair you're sitting on right now is pushing you upwards to counteract the force of gravity pulling you down (otherwise you'd fall through it). Despite this, it clearly isn't doing any 'work' in the intuitive sense of that word.

The real justification, though, is that the total energy of an isolated system never changes. It turns out that this is actually due to the universe's 'temporal translation symmetry.' It can be proven that energy conservation is a consequence of the fact that the fundamental laws of physics don't care what time it is. This result is a special case of something called Noether's theorem. This states that every symmetry automatically implies the existence of a conserved quantity. Linear and angular momentum happen to be due to the translational and rotational symmetries of space, respectively.

Interestingly, these symmetries often disappear in curved spacetimes. Finding useful conservation laws becomes extremely tricky in general relativity.

Yes, all of this can be done with Lorentz transformations. I find them annoying, though. Everything you need can be derived much more elegantly from the Minkowski metric/invariant interval (a generalization of the Pythagorean theorem that connects spatial and temporal 'directions').

fuzzybottom

ive been reading through all of this and its made me wonder about a few things

to clarify, is the speed of light, c, the speed of all electromagnetic waves, or just the visible spectrum?

do electromagnetic waves interfere with eachother like soundwaves do? like, if a wave of blue light and a wave of red light crossed paths would they combine to a purple light, or is that combination done by the eye, after registerting red and blue light separately but simultaneously?

EM waves have energy and momentum right? but momentum is p=mv. but EM waves dont have mass, and i guess E=mc² covers for that, so their momentum would be something like p=(E/c²)v where v = c, giving E/c, and since E = hc/L, it could be represented as p=h/L, or p=hf, or am i talking out of my ass. (L being lamda)

my main reason for the last question is, if they have momentum, in any reflection off a surface, conservation of momentum would have to be applied, and as such, the momentum of the wave would be affected, but since it doesnt have mass, its change would be exhibited by a change of energy, essentially a change of frequency, which would change its appearance if it happened to be a color, though only to some incredibly small degree.

and if anyone can explain why the c would be the same regardless of observers relative speeds, id love them forever =/

edit:
sorry, another question.
i know light only travels at c in a vacuum, but if it is released from a medium in which it moves slower into a vacuum, does it accelerate to c again, or maintain its lower velocity?

stingc

It's the same for all electromagnetic waves.

Yes, electromagnetic waves interfere (that's how people realized that light is a wave), but your conception of this seems a little off. Red and blue light have different wavelengths. When you combine the two, the waves are just summed. But it's impossible to add two sine waves with different frequencies to get something with a third frequency.

The perceived color of mixed light is a construction of your brain. The retina contains four main types of photodetectors (five in many women). One of these (the rods) works mainly in low light, and is pretty much irrelevant for seeing color. Then you have three (or four) basic types of cone cells. These work in relatively bright light, and are responsible for color vision. Each type of cell responds differently to different wavelengths of light. Roughly speaking, you have one type that responds well to red, another to blue, and another to green. The brain combines the brightness registered by each of these cells to form a single 'effective' color that you perceive.


You've just jumped over to quantum theory, but ok. E=hc/L applies to an individual photon, and the momentum of such a photon is h/L. This is not equal to hf, though. That's the energy again. p=h/L=hf/c. For realistic beams of light, the classical expressions for the energy and momentum of an electromagnetic wave are usually more useful, but can't be written down so easily.

I think you're talking about perfect reflection. If so, the magnitude of the momentum is unchanged. Its direction is reversed, of course. More generally, light can transfer momentum to a surface. But this happens in many different ways for realistic beams of light. The momentum and energy are not as simple as they'd appear from looking at single photons. So a color change isn't necessary.

Special relativity was discovered because the equations describing electromagnetic phenomena (obtained through extensive experiments) had this property. Nobody took it very seriously, though, and assumed that these equations only held in a special frame of reference (the aether) that we were very close to being in. Einstein decided to see what would happen if Maxwell's equations applied to all situations.

Rather than the Galilei transform satisfied by Newton's law of gravity, these equations satisfied the Lorentz transformation (which imply that c is universal). It was then possible to prove that all of the complicated electromagnetic phenomena discovered up to that time - light, induction (needed for electric motors and generators), and even the magnetic field itself - could be explained simply by knowing the force between two stationary charges. So the 18th century Coulomb's law combined with a Lorentz transformation (and some minor technicalities) reproduced the full richness of Maxwell's equations. This was too elegant to ignore, and the rest is history. Of course, the Michelson-Morley experiment and other results certainly helped these ideas gain acceptance.

It goes back to c.

fuzzybottom

thanks on many levels there.

and sorry, for some reason i was confusing wavelength with period, when i did the h/L = hf substitution, i just wasnt paying attention heh.

with the color, i still dont really understand, im really asking if theres any difference between seeing light with a wavelength halfway between red and blue, and seeing red and blue light in the same place.

guthmund

Wow.

There's quite a lot here to absorb, but I wanted to chime in to say just a couple of things real quick...

First, you guys are fantastic...really. Between the three of you, I think I've just about got this thing figured out. Although..that's just going to lead me somewhere else, no?

I've got my hands on some textbooks (not those mentioned, but just what I could conjure up before the weekend at school) and plan on rummaging through them sometime this weekend. I hope those will help.

HA! Yes! I was talking to one of the physics professors here on campus and he said much the same thing. He told me to read up on it and get with during lunch sometime next week.


/goes back to his stack of books...

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stingc

Yes, there's a difference. If you say that some beam of light is composed of a single wavelength, that means that its electric and magnetic fields are sinusoidal. Mixing two wavelengths together means that the electric field is the sum of two sine waves. Draw two such curves on a piece of paper with different wavelengths, and then try to draw their sum (or just compute it with some trig identities). You'll see that the result is not itself a sine wave. So mixing light with two distinct different wavelengths can never produce light having any single wavelength. The result is always a mixture.