QUESTION THREE
3. What does Hawking mean by "imaginary" time versus "real" time? How is the concept of imaginary time useful in understanding the implications of Einstein's theory of relativity? In addition explain the "brane" theory and how it relates to a geometric understanding of space-time.
imaginary time is simply a mathematical construct to "fill in the gap" for the 4th dimension that cannot be physically seen. since space is three dimensional and is described by three lines perpendicular to one another, there's a big problem when taking time into account as the fourth dimension; having a fourth dimension means that one has to pull another line perpendicular to the three already-perpendicular-to-each-other lines. but this is geometrically impossible. thus, the concept of real and imaginary time is introduced.
space, or as hawking puts it: "history," moves along the real time line. now, just like in the cartesian coordinate, imaginary time line is perpendicular to real time line just like y axis is perpendicular to x axis. the center where they intercept each other would be the zero point. since the imaginary time is perpendicular to the real time axis, on which the three-dimensional space moves, then this concept of imaginary time "behaves like a fourth spatial direction," (hawking 60) and makes possible what was once geometrically impossible.
now, why would one need the fourth dimension? einstein's general theory of relativity requires all three spatial directions and one time direction. but the use of real time causes time to behave differently than space; namely because one can move up or down any of the spatial directions, but in real time, one can only move up, that is, to the future. imaginary time axis behaves differently; one can move up or down that axis, one can increase or decrease on that axis, so as hawking puts it, this imaginary time axis does behave "like a fourth spatial direction."
the imaginary time, however, has no physical interpretation. it's just simply a mathematical construct that is consistent with established theories. since hawking is what he calls "a positivist," he does not need physical interpretation of things; he claims that for a positivist, physical interpretation means little to none, since they "cannot determine what is real." (hawking 59) what is important for positivists like hawking is to "find which mathematical models describe the universe we live in." (hawking 59)
the brane theory, is a theory developed by cambridge's paul townsend. branes, or more formally known as "p-branes," are basically objects having lengths in p-direction. so a one dimensional objects would be a p=1 brane, a two dimensional object would be a p=2 brane, and so on. the introduction of the concept of branes has given deeper insights on how the universe works; hawking said that "all the p-branes could be found as solutions of the equations of supergravity theories in 10 or 11 dimensions." (hawking 54) this tells us that if the theoretical p-branes do exist, then it is possible that the universe we are living in is 10 or 11 dimensional. but doesn't that seem too much? the universe we are seeing with our eyes are three dimensional, even with time as the fourth dimension, it's already hard to comprehend, let alone ten or even eleven dimensions. hawking provides a brief explanation for this: "the idea was that the other 6 or 7 dimensions are curled up so small that we don't notice them." (hawking 54) the introduction of the branes completely alter our geometric understanding of the universe; there seems to be more of the universe than what the eyes can see.
imaginary time is simply a mathematical construct to "fill in the gap" for the 4th dimension that cannot be physically seen. since space is three dimensional and is described by three lines perpendicular to one another, there's a big problem when taking time into account as the fourth dimension; having a fourth dimension means that one has to pull another line perpendicular to the three already-perpendicular-to-each-other lines. but this is geometrically impossible. thus, the concept of real and imaginary time is introduced.
space, or as hawking puts it: "history," moves along the real time line. now, just like in the cartesian coordinate, imaginary time line is perpendicular to real time line just like y axis is perpendicular to x axis. the center where they intercept each other would be the zero point. since the imaginary time is perpendicular to the real time axis, on which the three-dimensional space moves, then this concept of imaginary time "behaves like a fourth spatial direction," (hawking 60) and makes possible what was once geometrically impossible.
now, why would one need the fourth dimension? einstein's general theory of relativity requires all three spatial directions and one time direction. but the use of real time causes time to behave differently than space; namely because one can move up or down any of the spatial directions, but in real time, one can only move up, that is, to the future. imaginary time axis behaves differently; one can move up or down that axis, one can increase or decrease on that axis, so as hawking puts it, this imaginary time axis does behave "like a fourth spatial direction."
the imaginary time, however, has no physical interpretation. it's just simply a mathematical construct that is consistent with established theories. since hawking is what he calls "a positivist," he does not need physical interpretation of things; he claims that for a positivist, physical interpretation means little to none, since they "cannot determine what is real." (hawking 59) what is important for positivists like hawking is to "find which mathematical models describe the universe we live in." (hawking 59)
the brane theory, is a theory developed by cambridge's paul townsend. branes, or more formally known as "p-branes," are basically objects having lengths in p-direction. so a one dimensional objects would be a p=1 brane, a two dimensional object would be a p=2 brane, and so on. the introduction of the concept of branes has given deeper insights on how the universe works; hawking said that "all the p-branes could be found as solutions of the equations of supergravity theories in 10 or 11 dimensions." (hawking 54) this tells us that if the theoretical p-branes do exist, then it is possible that the universe we are living in is 10 or 11 dimensional. but doesn't that seem too much? the universe we are seeing with our eyes are three dimensional, even with time as the fourth dimension, it's already hard to comprehend, let alone ten or even eleven dimensions. hawking provides a brief explanation for this: "the idea was that the other 6 or 7 dimensions are curled up so small that we don't notice them." (hawking 54) the introduction of the branes completely alter our geometric understanding of the universe; there seems to be more of the universe than what the eyes can see.
0 Comments:
Post a Comment
<< Home