#1




Hawking and imaginary time
This thread about Stephen Hawking reminded me of the portion of A Brief History Of Time that left me completely scratching my head: imaginary time.
Is Hawking suggesting that imaginary time is an actual property of the universe, similar to normal time? At some points it seems like he's treating the concept as more of a purely mathematical construct; i.e., to make the numbers work out by avoiding a singularity. I know I'm not really setting this question forth coherently, and any answer is liable to be over my head. Basically, I guess I'm asking: Is imaginary time "real"? And if not, then, well, what is it? 
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#2




I'll be interested to know what qualifies the second temporal dimension as a temporal dimension.
FrL 
#3




I think its all about the Complex Plane: a Real number line intersects an Imaginary number line.
IIRC, Imaginary and Complex values pop up all the time in Relativity and Quantum Mechanics. Also, since things like Trigonometry and rotation naturally arise from computing in the complex plane, perhaps this is a dimensional basis for quantum spin. 
#4




Quote:
Also, it's my understanding that while imaginary numbers are an inescapable fact of life for physicsyou can't do physics without themstill, no physical measurement ever has an imaginary value. This would lead me to think that even if there were a second temporal dimension as Hawking proposes, measurements along that dimension would not be imaginary. I don't knowand I eagerly wonderwhether it just happens that nothing ever measures imaginary, or rather whether it is known that nothing ever could measure imaginary for whatever reason. (Or maybe I'm wrong that nothing measures imaginary.) FrL 
#5




I've read several places that our universe couldn't exist in the form we see it if there was more than one time dimension; in particular it would make it possible for events to precede their causes. Or was Hawking talking about a different concept?

#6




From Hawking's lecture "The Beginning of Time" :
Quote:
Hawking also states in this lecture that this still an untested hypothesis. But yes, he using the term 'Imaginary Time' in the same sense as 'Imaginary Number' i.e existing on a plane perpendicular to 'Real Time' or a 'Real Number.' 
#7




Quote:
What?! I'm just not getting it, and I'm not necessarily saying that the problem lies with the book. Maybe I just don't have the background or spatial aptitude to see the "If A is X, then B is Y" that other readers seem to find so obvious. Unfortunately, I didn't have many Aha! moments with the book. Last edited by PoorYorick; 12222008 at 11:57 PM.. Reason: Tried to word OP as not to look like a doofus. Failed. 
#8




Quote:
If you're ever reading a science book for the general public, and you feel confused or less informed, you're reading a bad science book. Stop, and get a better book. Oh, and Herr Professor Doktor SoMuchSmarterThanYou Hawking has two whole equations in his books ostensibly for the general public. One is E=mc^{2}  and he does nothing else with it in the books. The other is the equation for the Schwarzchild radius of a black hole. He got it wrong! The units don't come out the same on both sides of his version of the equation. You get points off for that in Physics 101, Stevie. OK, I know he was using the form of the equation that professional theorists use, but neither he nor his editors remembered to put things right. And he did nothing else with *that* equation in his books. Last edited by Typo Knig; 12232008 at 01:03 AM.. 
#9




Quote:
Hawking has done several documentaries, and a couple of docu miniseries which make use of both simple and elaborate graphic effects to aid in illustrating the concepts he is attempting to explain, which may make a difference if you are more of a visual learner. Many of these are available in clips on youtube, and at least a few in full on google video. (For example, all six hours of "Stephen Hawking's Universe.") And I'm sure at least some of the related hits you'll find, if you make these searches, will be relevant as well. 
#10




Missed the edit window...
Also on google video, you can find an hour long doc about that which Hawking has been most off the mark. IIRC it's called "Hawking Paradox" ... though I may be mixremembering titles... there are a bunch there. 
#11




Quote:
There is nothing preventing us from speaking of complexvalued measurements just as well as we speak of realvalued measurements (or vectorvalued, or anglevalued, or...). Any quantity consisting of a scaling factor and a rotation is naturally described as a complexvalued measurement. Thus, for example, we can measure waves of a given frequency by their deviation from (or "ratio to") a unit reference wave; a wave of twice the amplitude and outofphase by 1/4 the period would correspond to the complex number 2i. Addition would correspond to superposition of waves; multiplication, per usual, would correspond to chaining such ratios. Or we could describe certain linear motions of a plane with complex numbers; the action of a 1/4 turn followed by a scaling by a factor of 2 would, then, be the correspondent of 2i here. Complex addition would correspond to addition of transformations (i.e., pointwise vector addition of outputs), while multiplication would correspond to composition of motions. You might object "But these aren't really complex values; they're just pairs of real values". Well, hell, complex numbers themselves can be seen as just pairs of real numbers; what more could we expect? One might just as well say there is no such thing as a negative measurement, just pairs of a "proper" nonnegative measurement and a further tag measurement indicating its sign. Or no such thing as a vector quantity, just tuples of reals! But that would be a pointless wordgame to play, and an obfuscating perspective to take. In short: it is silly to speak as though complex numbers are somehow less physically applicable than real numbers. The only thing wrong with "imaginary numbers" is the name. Some physics teachers may fail to emphasize this fact strongly, perhaps because their pedagogical goals don't impute a high enough payoff to the work required to do so (in the face of years of prejudice many students have developed), but it's a superstition worth fighting nonetheless, at least when the issue is explicitly raised. (I expounded on similar thoughts regarding complex numbers and "genuine" physical measurement in this thread, among other ones which coincidentally arose around the same time.) Last edited by Indistinguishable; 12232008 at 02:09 AM.. 
#12




I've been away from the boards for a few days. I just wanted to jump back in to say thanks to all.

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