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  #1  
Old 12-22-2008, 12:37 PM
Tom Tildrum Tom Tildrum is offline
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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  
Old 12-22-2008, 12:52 PM
Frylock Frylock is online now
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I'll be interested to know what qualifies the second temporal dimension as a temporal dimension.

-FrL-
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  #3  
Old 12-22-2008, 07:30 PM
Enola Straight Enola Straight is offline
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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.
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Old 12-22-2008, 09:05 PM
Frylock Frylock is online now
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Quote:
Originally Posted by Enola Straight View Post
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.
Imaginary time isn't really "imaginary" in the mathematica sense, as far as I can tell from poking around on the net.

Also, it's my understanding that while imaginary numbers are an inescapable fact of life for physics--you can't do physics without them--still, 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 know--and I eagerly wonder--whether 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-
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  #5  
Old 12-22-2008, 09:16 PM
Lumpy Lumpy is offline
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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?
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  #6  
Old 12-22-2008, 11:05 PM
Hap Shaughnessy Hap Shaughnessy is offline
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From Hawking's lecture "The Beginning of Time" :

Quote:
Quantum theory introduces a new idea, that of imaginary time. Imaginary time may sound like science fiction, and it has been brought into Doctor Who. But nevertheless, it is a genuine scientific concept. One can picture it in the following way. One can think of ordinary, real, time as a horizontal line. On the left, one has the past, and on the right, the future. But there's another kind of time in the vertical direction. This is called imaginary time, because it is not the kind of time we normally experience. But in a sense, it is just as real, as what we call real time.

The three directions in space, and the one direction of imaginary time, make up what is called a Euclidean space-time. I don't think anyone can picture a four dimensional curve space. But it is not too difficult to visualise a two dimensional surface, like a saddle, or the surface of a football.

In fact, James Hartle of the University of California Santa Barbara, and I have proposed that space and imaginary time together, are indeed finite in extent, but without boundary. They would be like the surface of the Earth, but with two more dimensions. The surface of the Earth is finite in extent, but it doesn't have any boundaries or edges. I have been round the world, and I didn't fall off.

If space and imaginary time are indeed like the surface of the Earth, there wouldn't be any singularities in the imaginary time direction, at which the laws of physics would break down. And there wouldn't be any boundaries, to the imaginary time space-time, just as there aren't any boundaries to the surface of the Earth. This absence of boundaries means that the laws of physics would determine the state of the universe uniquely, in imaginary time. But if one knows the state of the universe in imaginary time, one can calculate the state of the universe in real time. One would still expect some sort of Big Bang singularity in real time. So real time would still have a beginning. But one wouldn't have to appeal to something outside the universe, to determine how the universe began. Instead, the way the universe started out at the Big Bang would be determined by the state of the universe in imaginary time. Thus, the universe would be a completely self-contained system. It would not be determined by anything outside the physical universe, that we observe.

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.'
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Old 12-22-2008, 11:52 PM
PoorYorick PoorYorick is offline
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Quote:
Originally Posted by Stephen Hawking as quoted by Hap Shaughnessy
So real time would still have a beginning. But one wouldn't have to appeal to something outside the universe, to determine how the universe began. Instead, the way the universe started out at the Big Bang would be determined by the state of the universe in imaginary time.
See, this is the problem I had with A Brief History of Time. The words are in simple English, and I consider myself to be reasonably bright. I know how imaginary numbers work in analytic geometry, and I've even taken three semesters of undergrad physics (granted, it was physics for "Life Science Majors" over 20 years ago). I'm reading along, thinking I'm understanding every word and concept, then, BANG: "Thus, the universe would be a completely self-contained system. It would not be determined by anything outside the physical universe, that we observe."

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; 12-22-2008 at 11:57 PM.. Reason: Tried to word OP as not to look like a doofus. Failed.
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  #8  
Old 12-23-2008, 01:02 AM
Typo Knig Typo Knig is offline
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Originally Posted by PoorYorick View Post
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.
Alas, PoorYorick, the problem lies neither in our stars nor in ourselves, but in the fact that Hawking is a lousy science populizer. Hawking tossed around terms like "Imaginary Time" with no explanation at all about the square root of negative one. It makes the baby George Gamow cry.

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 So-Much-Smarter-Than-You Hawking has two whole equations in his books ostensibly for the general public. One is E=mc2 - 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; 12-23-2008 at 01:03 AM..
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  #9  
Old 12-23-2008, 01:20 AM
Hap Shaughnessy Hap Shaughnessy is offline
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Quote:
Originally Posted by PoorYorick
See, this is the problem I had with A Brief History of Time. The words are in simple English, and I consider myself to be reasonably bright. I know how imaginary numbers work in analytic geometry, and I've even taken three semesters of undergrad physics (granted, it was physics for "Life Science Majors" over 20 years ago). I'm reading along, thinking I'm understanding every word and concept, then, BANG: "Thus, the universe would be a completely self-contained system. It would not be determined by anything outside the physical universe, that we observe."

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.

Hawking has done several documentaries, and a couple of docu mini-series 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.
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  #10  
Old 12-23-2008, 01:29 AM
Hap Shaughnessy Hap Shaughnessy is offline
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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 mix-remembering titles... there are a bunch there.
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  #11  
Old 12-23-2008, 02:06 AM
Indistinguishable Indistinguishable is offline
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Originally Posted by Frylock View Post
I don't know--and I eagerly wonder--whether 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.)
Whether ratios of mass or intervals of time or so forth can be (usefully construed as) imaginary is one question. But undeniably, plenty of other things have imaginary measurements... as long as you don't gerrymander the definition of "measure" to exclude them.

There is nothing preventing us from speaking of complex-valued measurements just as well as we speak of real-valued measurements (or vector-valued, or angle-valued, or...). Any quantity consisting of a scaling factor and a rotation is naturally described as a complex-valued 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 out-of-phase 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 word-game 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; 12-23-2008 at 02:09 AM..
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  #12  
Old 12-26-2008, 01:36 PM
Tom Tildrum Tom Tildrum is offline
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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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