Greg Kuperberg on Sep 15th, 2005 at 12:11 pm
Conformal maps of the Earth are a great introduction to complex analysis. If you identify the Earth with the Riemann sphere, then the Mercator map is exp(i*z), while the quincuncial projection is a Weierstrass elliptic function. Or you could view it as a 2-to-1 conformal projection from a torus to a sphere with four ramified points. I imagine that it is relevant to one-loop calculations in string theory in that guise.
At what level has this map then progressed if we held such views to the "horizon and boundary conditions." That is now replaces what we talk about of earth, and now relay the mass consideration to events in the gravitational field? Has the mathematic hypothesized now, gone through a revision, and needed support of mathematical views?
Campbell's Soup Can A. Warhol
What mathematics would move our perception to the gravitational views seen there? Gary Horowitz relays the outside label of a can of a soup as the conformal surface, while the soup, the spacetime fabric?
On planet Earth, we tend to think of the gravitational effect as being the same no matter where we are on the planet. We certainly don't see variations anywhere near as dramatic as those between the Earth and the Moon. But the truth is, the Earth's topography is highly variable with mountains, valleys, plains, and deep ocean trenches. As a consequence of this variable topography, the density of Earth's surface varies. These fluctuations in density cause slight variations in the gravity field, which, remarkably, GRACE can detect from space.
So one would look at topography as something much different then what is laid out on this globe as "hills and valleys"?
So now this map, has this extra feature to it.
Holography encodes the information in a region of space onto a surface one dimension lower. It sees to be the property of gravity, as is shown by the fact that the area of th event horizon measures the number of internal states of a blackhole, holography would be a one-to-one correspondance between states in our four dimensional world and states in higher dimensions. From a positivist viewpoint, one cannot distinquish which discription is more fundamental.
Pg 198, The Universe in Nutshell, by Stephen Hawking
While on this topic it behooves me to think of the "horizon" and the mathematical construct that has taken us there. While we see to explain the nature of the effect in a fifth dimensional view, it had been reduced to "temperature" as a relation of this conformal view?
"D-branes provide the fundamental quantum microstates of a black hole that underlie black hole thermodynamics"
As much as one would try and ignore this position, you cannot get away from the mathematics or the approach and what this has culminated too.
I like Peter and his no nonsense views, but he has gone to far in rejecting the basis of "mathematical dialogue" in face of what D brane issue had been taken too?
Why would he reject mathematics on the one hand demonstrative of a particular point of view to which it has developed, then, ignore what position it had taken both string theory and Lee Smolins attempts at the disciption of the blackhole dynamics, from the views of that horizon?
With regards to the conformal field theory approach. While I am in my infancy, I recognize the views of Bekenstein Bound, and the hologrpahical approach. One must first learn to crawl, then walk I know, but how indeed does one get to the vision held, when he himself(who ever you like) cannot explain how such a mathematics like string theory, arose to help with our views of reality?
In 1919, Kaluza sent Albert Einstein a preprint --- later published in 1921 --- that considered the extension of general relativity to five dimensions. He assumed that the 5-dimensional field equations were simply the higher-dimensional version of the vacuum Einstein equation, and that all the metric components were independent of the fifth coordinate. The later assumption came to be known as the cylinder condition. This resulted in something remarkable: the fifteen higher-dimension field equations naturally broke into a set of ten formulae governing a tensor field representing gravity, four describing a vector field representing electromagnetism, and one wave equation for a scalar field. Furthermore, if the scalar field was constant, the vector field equations were just Maxwell's equations in vacuo, and the tensor field equations were the 4-dimensional Einstein field equations sourced by an EM field. In one fell swoop, Kaluza had written down a single covariant field theory in five dimensions that yielded the four dimensional theories of general relativity and electromagnetism. Naturally, Einstein was very interested in this preprint .
While one may use sites to give indicative values ot the information, can we ignore these assumptions mathematically driven. It paved the way for how we view things that we did not see before. Go ahead reject it then:)
Are we not looking for the Trigger?:)