The mathematical foundations of a theorem for 5-dimensional space-time, based on Stephen Hawking's no-boundary proposal and imaginary time.
Storified by Richard Jowsey ·
Thu, Apr 03 2014 17:30:52
The Beginning of Time - Stephen HawkingIn this lecture, I would like to discuss whether time itself has a beginning, and whether it will have an end. All the evidence seems to indicate, that the universe has not existed forever, but that it had a beginning, about 15 billion years ago. This is probably the most remarkable discovery of modern cosmology.
In 1976, during an undergrad physics tutorial at the University of Canterbury in NZ, I first noticed imaginary numbers appearing behind the event horizon of a black hole. Our tutor (a PhD candidate under Prof. Roy Kerr) was unpacking Schwarzschild's equation for time dilation, which goes to infinity at the event horizon, resulting in a "coordinate singularity". Being familiar with imaginary and complex numbers from years of studying electronics and radio engineering, on a hunch I tried multiplying the equation by i, the square root of minus one. Surprisingly, this simple trick "fixed" the equation inside the event horizon, but resulted in negative values for real time. Since backwards-time is impossible, the tutor snorted derisively, insisting that the spacetime inside a black hole's horizon is just "different", as if it were in some other dimension. I didn't understand his reasoning, but accepted the conventional wisdom. As one does.
Schwarzschild metric - Wikipedia, the free encyclopediaAccording to Birkhoff's theorem, the Schwarzschild solution is the most general spherically symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has no charge or angular momentum. A Schwarzschild black hole has a Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.
Decades later, I read Stephen Hawking's book, "A Brief History of Time" with fascination. Hawking describes a "no boundary" principle, like walking around the world. He also introduced the concept of "imaginary time", which he explained as another dimension of time, but itself timeless and eternal. One suspects he may have noticed hints of imaginary numbers "before" the big bang, similar to my suspicion, a decade earlier, that some kind of weird imaginary time began after stuff fell into a black hole. Or so it might seem, from the vantage point of a far-away observer.
Imaginary numberAn 5 imaginary number is a number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property . The square of an imaginary number bi is - b 2. For example, i is an imaginary number, and its square is −25.
Cite: Hawking, Stephen (1988)
“A Brief History of Time”. Bantam Books. ISBN 0-553-38016-8
Cite: J. B. Hartle, S. W. Hawking (1983) “Wave function of the Universe”. Physical Review D 28 (12): 2960
Conventionally, spacetime is conceptualised as having 4 dimensions, 1 time axis and 3 spatial axes (x, y, and z). In general relativity, the dimension of time is given "space-like" units, measured in light-years (of distance), by multiplying time by the speed of light, so ct. It is then mathematically rotated into the 4th dimension by multiplying the ct axis by the imaginary number, i, hence the "time axis" becomes ict. This 4-dimensional geometry has a "metric tensor" signature of (ict, x, y, z), also expressed like (-,+,+,+).
The only way to add an "imaginary time" dimension to this 4-dimensional spacetime is by repeating the process used historically to add the "real time" dimension. Multiplying ict by the imaginary number to construct a 5th spacetime dimension results in a -ct axis, since i-squared is -1. This 5-dimensional geometry has a metric signature of (-t, it, x, y, z). Hawking's no-boundary proposal implies that each dimension is vastly circular, thus appearing infinite, but having no end nor beginning. The geometry of this 5-dimensional spacetime can be conceived of as an enormous, symmetrically-spherical hypersphere.
Since a 5-dimensional geometry is almost impossible to imagine (good luck!), we simplify by considering all 3 dimensions of space to have collapsed into just 1. Then, laying the 2 dimensions of time across a flat "imaginary plane", we can easily conceptualise a single space axis intersecting the time plane at (0,0), and rotated into the 3rd dimension by 90 degrees.
Euler's Formula is one of the most beautifully simple and elegant ideas in mathematics. It has an imaginary time axis, and a real time axis. The circle of unit radius has a simple equation, based on Euler's number, e. This transcendental mathematical constant, approximately 2.7183, occurs everywhere in nature, and is used in equations describing exponential growth rates, compound interest, and the natural logarithms. The circle has 4 cardinal points, where theta (the angle of rotation) is 0, pi/2, pi, and 3pi/2 radians. When theta reaches 2pi radians, it returns to zero, and the circle is complete.
"Where the math leads, we follow..."
Euler's identityIn mathematics, Euler's identity (also known as Euler's equation) is the equality where Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered an example of mathematical beauty. Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x, where the values of the trigonometric functions sine and cosine are given in .