Monday, April 21, 2014

An infinite circle is a straight line

This is how the Universe works.

A circle with infinite radius is a line

The edge of a circle with infinite radius is straight

"In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (loosely speaking, a circle with infinite radius). Many difficult problems in geometry become much more tractable when an inversion is applied. "


"A regular apeirogon has equal-length sides and equal corner angles, just like any regular polygon. Its Schläfli symbol is {∞}.

If the corner angles are 180°, the overall form of the apeirogon resembles a straight line, This line may be considered as a circle of infinite radius, by analogy with regular polygons with great number of edges, which resemble a circle."

Numberphile video "Epic Circles"

Hawking - It is wrong to take the Big Bang literally

Stephen Hawking
The Grand Design (2010)
pages 128-129

Stephen Hawking
The Grand Design (2010)
pg. 128-129

Measurements of helium abundance and the CMBR [Cosmic Microwave Background Radiation] provided convincing evidence in favor of the Big Bang picture of the very early Universe, but although one can think of the Big Bang picture as a valid description of early times, it is wrong to take the big bang literally, that is, to think of Einstein's theory as providing a true picture of the origin of the Universe. That is because General Relativity predicts there to be a point in time at which the temperature, density, and curvature of the Universe are all infinite, a situation mathematicians call a singularity. To a physicist this means that Einstein's theory breaks down at that point and therefore cannot be used to predict how the Universe began, only how it evolved afterwards. So although we can employ the equations of general relativity and our observations of the heavens to learn about the Universe at a very young age, it is not correct to carry the Big Bang picture all the way back to the beginning.

Singularities "cured" by String Theory

Briane Greene
The Hidden Reality
pg. 111

Absolute Spacetime and "Invariance Theory"

Brian Greene, 
The Fabric of the Cosmos (2004)
pg. 51

Singularities - When theories break down

 Stephen Hawking
A Brief History of Time
page. 49

page 63

The predictions of the theory 
"break down at the big bang singularity"
"Classical general relativity,  by predicting points of infinite density, predicts its own downfall, just as classical mechanics predicted its downfall by suggesting that atoms should collapse to infinite density"

Paul Steinhardt, Neil Turok
Endless Universe
pages 37, 38

page  126

pages 184, 185

Ask a Mathematician / Physicist

"In practice, when you’re calculating something in physics and you find a singularity in your calculation (this happens all the time), which usually looks like “1/x”, then that means that there’s a mistake somewhere, or you’re looking at something that never happens, or there are physical laws or effects that haven’t been taken into account."

BICEP2 links

BICEP2 (primordial gravitational waves) links:

Killing the Straw Man: Does BICEP Prove Inflation?

Lawrence Krauss, James Dent, Harsh Mathur

"While the Inflationary signal remains the best motivated source, the current measurement unfortunately still allows for the possibility that a comparable gravitational wave background might result from a self ordering scalar field transition that takes place later at somewhat lower energy."


Star dust casts doubt on recent big bang wave result

(new scientist)

"An imprint left on ancient cosmic light that was attributed to ripples in spacetime – and hailed by some as the discovery of the century – may have been caused by ashes from an exploding star.
In the most extreme scenario, the finding could suggest that what looked like a groundbreaking result was only a false alarm. Another possibility is that the stellar ashes could help bring the result in line with other cosmic observations. We should know which it is later this year, when researchers report new results from the European Space Agency's Planck satellite."


Have galactic 'radio loops' been mistaken for B-mode polarization?

 (physics world)

"Radio loop" emissions, rather than signatures of the early universe, could account for the observation of B-mode polarization announced by the BICEP2 collaboration earlier this year. That is the claim of a trio of cosmologists that has found evidence that local structures in our galaxy generate a polarized signal that was previously unknown to astronomers studying the cosmic microwave background (CMB). The new foreground, which can be detected in the radio and microwave frequencies, is present at high galactic latitudes and could potentially be misinterpreted as a B-mode polarization signal caused by primordial gravitational waves, thus casting doubt on the BICEP2 finding. 


Sixty Symbols

Professor Ed Copeland of SixtySymbols explains some serious problems with inflation and the Big Bang theory.