To freeAtlast:
Good Lord! You're not only nasty, you're about the most ignorant self-proclaimed master of science of I've ever seen!
:: So when you say that "Einstein's equation does not preclude an object traveling FASTER than the speed of light", you're quite right, because it has nothing to do with velocity at all.
: quite right,
So your original claim was wrong.
Strike one.
: it deals with momentum actually.
Really. Let's see now, the equation is: E = m * c^2, or Energy = Mass * velocity^2. Last I heard, energy is not momentum and it has units of kg-m/s^2, whereas momentum is simply Mass * velocity and has units of kg-m/s.
Strike two.
: Usually, it is convient to square both sides of the equation.
Really. Then you get energy squared. That's much good for anything.
: I'm not cutting and pasting the whole damn thing into here and then trying to reformat it. Do a Goggle search.
I don't need to. I know the material.
:: Do you have any idea what an imaginary distance is? Or an imaginary time? Of course not, and neither does anyone else.
: Well, first off their NOT imaginary numbers, oh enlightened one. They would be complex numbers (with real and imaginary parts).
Of course they're pure imaginary numbers. Look at the equations I posted, stupid. Here's the one for the relativistic mass:
m = m0 / sqrt(1 - v^2/c^2) where m0 is the rest mass.
Here we have three real quantities on the right side: m0, v and c. If v > c, the square root yields a pure imaginary number which divides into the rest mass. Therefore, for v > c, the equation yields a pure imaginary number.
Strike three.
: You know just like the intelligence of this conversation has real (my part) and imaginary parts (what you say).
Your world is inverted from reality.
Strike four.
: A brief quote from Stephen Hawking taken from A Brief History of Time (don't have the phyiscal copy in front of me, found the txt OCR'd on the Internet, so no page number. look it up yourself if you think I'm making this up):
"one must use imaginary time... That is to say, for the purposes of calculation, one must measure
time using imaginary numbers, rather than real ones. This has an interesting effect on spacetime:
The distinction between time and space disappears completely... we may regard our use of Euclidean spacetime as merely a mathematical device (or trick) to calculate answers about real spacetime."
Your quote shows what kind of stupid conclusions the ignorant make when they try to understand things beyond their ken.
The quote is from page 134, and the parts you left out prove my point. In fact, the very last sentence you quoted proves my point: physics deals with real time in the end, and any use of "imaginary time" is "merely a mathematical device (or trick) to calculate answers about real spacetime."
Strike five.
Here is a more extended quotation along with my commentary that shows what Hawking was talking about:
We don't yet have a complete and consistent theory that combines quantum mechanics and gravity. However, we are fairly certain of some features that such a unified theory should have. One is that it should incorporate Feynman's proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle does not have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path in space-time, and with each of these histories there are associated a couple of numbers, one representing the size of a wave and the other representing its position in the cycle (its phase).
Hawking is talking in non-mathematician's terms about complex numbers, which as you seem to understand, have real and imaginary parts. As I stated, this is what falls out of solving Schroedinger's equation.
The probability that the particle, say, passes thrrough some particular point is found by adding up the waves associated with every possible history that passes through that point. When one actually tries to perform these sums, however, one runs into severe technical problems. The only way around these is the following peculiar prescription: One must add up the waves for particle histories that are not in the "real" time that you and I experience, but take place in what is called imaginary time.
This is the mathematical device or trick Hawking later referred to. Also note: the "sums" (actually the squares of the sums) represent a probability density function in real space-time. Such a function is not a physical quantity, but is yet another mathematical device that allows one to predict the probability that a particle will be found at some particular location.
Imaginary time may sound like science fiction but it is in fact a well-defined mathematical concept. [short lesson in multiplying imaginary numbers deleted] To avoid the technical difficulties with Feynman's sum over histories, one must use imaginary time. That is to say, for the purposes of the calculation one must measure time using imaginary numbers, rather than real ones.
This is precisely what we EE's do when we use complex numbers to represent sine waves of various frequencies and phases, and use Fourier analysis to break actual signals down into their Fourier components. These components have magnitude and phase, just like Feynman's "histories" are associated with numbers representing the magnitude and phase of quantum mechanical probabilities. Fourier analysis results in another "unreal" quantity called "negative frequency", which is just an artifact of the mathematical representation of the signal and has no physical significance.
This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean, after the ancient Greek Euclid, who founded the study of geometry of two-dimensional surfaces. What we now call Euclidean space-time is very similar except that it has four dimensions instead of two. In Euclidean space-time there is no difference between the time direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary, real values of the time coordinate, it is easy to tell the difference -- the time direction at all points lies within the light cone, and space directions lie outside.
In other words, in this four dimensional space-time called "Euclidean space-time", where for purposes of calculation time can have imaginary values, time can run backwards. In real space-time, however, time does not run backwards. That's why in real space-time "imaginary time", such as pops out of the various equations of special relativity when you plug in a velocity greater than the speed of light, is meaningless.
By now you should also see that the "imaginary time" that results from velocities greater than the speed of light has absolutely nothing to do with what Hawking is talking about, which proves that you don't understand what you read about science.
Strike six.
In any case, as far as everyday quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculate events about real space-time.
By this time interested readers will have seen your ridiculous error in trying to claim that this last quoted statement has anything at all to do with your point.
Strike seven.
: gee, AlanF, I guess the Lucasian (sp?) professor at Cambridge (a position once held by Issac Newton himself) is a regular nobody.
Not at all. But you've shown yourself to be a thoroughgoing dickhead who is too stupid to realize how stupid he is. More examples of that are forthcoming.
: You need imaginary time for singularities to exist. And you need singularities for black holes to exist, (which as far as we can tell they do since they have been observed in nature).
Here's a challenge, dickhead: present an equation that shows why imaginary time is necessary for black holes to exist. No hand waving permitted.
: Now, what do the imaginary parts of the complex roots exactly mean? As far as we can tell NOW, they have no accepted physical meaning. (just several discredited ones including stuff like negative mass) But, that doesn't mean they never will.
The imaginary numbers Hawking talks about do indeed have physical meaning. Not directly by themselves, any more than the negative frequencies that fall out of Fourier analysis have direct physical meaning, but when the mathematical devices and tricks are applied using these "unphysical" numbers, you get answers that represent reality very well. Thus, we again see you have no idea what you're talking about.
Strike eight.
: Inductors and capacitors have a complex impedance.
Wrong, they have pure imaginary impedances.
The impedance of an inductor is jwL and the impedance of a capacitor is 1/jwC, where "j" is the EE's equivalent of "i", and "w" (omega) is 2*PI*frequency.
Strike nine.
: Remember a complex number has real and imaginary part. Resistance is only the real part of impedance.
Inductors and capacitors have no resistive component, dickhead.
Strike ten.
: The imaginary part of the impedance is call the reactance. (I had a dash of EE in school too.)
You appear to have misunderstood all of it.
: Just curious AlanF, where oh where did you get your PhD in physics.
I never said I did. I said I took a lot of Ph.D. courses in math and physics.
You can't read. Strike eleven.
: You sound like you'd have trouble handling the intro physics class at your local community college, brightboy.
You statement shows just how stupid you are.
Strike twelve.
: And man that Stevie Hawking is just some crackpot fundie at Cambridge I guess with his imaginary time. btw, last I heard he was a devout atheist.
Pointless ad hominem.
Strike thirteen.
: well, it's safe to say you have no idea what you are talking about
HAHAHAHAHAHAHAHA!
Strike fourteen.
: and I await your Farkel-like response of picking out all my typos and grammatical hiccups
For you, you didn't do too badly this time. But your real errors are so ridiculous that it would be guilding the dung heap to add them to your score.
: b/c you know everything else I say is abso-freaking-lutely correct.
Right.
Strike fifteen.
: you better give up physics and go back to washing the windows. Oh, and please keep whining to Simon to get me banned after I showed you up earlier. Real manly of you I must say.
Bathory, you're about the biggest asshole I've ever come across. Several people have told me that you're not only braindead, but a true mental case. Give it up before those nice young men in their clean white coats come to take you away HA HA!
AlanF