In this paper we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.
The main idea was to try to follow the classical definitions of a curve, jets of curves at a point and the geometric or kinematic definition of higher tangent bundles. One of the complications in the superworld is that supermanifolds are not just set theoretical objects. To overcome this the more categorical set-up of the “functor of points” and “internal Homs” is needed.
I posted a little about this before here.
Both special and general relativity have a reputation for being very complicated theories to understand. While it is true that one needs some mathematical machinery to really master these theories, Egdall does a great job in showing that it is possible with a bare minimum of high school mathematics to get an appreciation of the main ideas.
Edgall does not completely stay away from some mathematics, though as it is isolated somewhat from the main text, the mathematically shy should not be scared away from this book. The style is lighthearted and is full of thought experiments illustrated by short entertaining stories.
The author does an excellent job in highlighting the main features of special and general relativity in a way suitable for the lay reader to understand. Moreover, the development of the theory is presented in a chronological/historical context by trying to describe Einstein’s trail of thought and how he was influenced by the various problems with physics found in the late 1800’s. We get a good overview of Einstein the man from this book.
However, like all popular science books, there are the odd statements like “…the photon’s perspective” and “..spacetime curvature has energy” which any physicist will question. That said, I do consider the book well-written, entertaining and a useful introduction to the ideas of relativity for the lay person.
The book is divided into two chapters, an appendix with more mathematical details and a rather extensive list of notes with sources. Part I deals with special relativity describing the initial development of the theory starting from the conflict between classical mechanics and electromagnetic theory. The strange consequences of Einstein’s special relativity including the idea of space-time, time dilation, length contraction and his famous equation E=mc^2 are discussed.
Part II introduces us to general relativity which is the most accurate theory of gravity know to science. The fundamental idea is that gravity should be viewed as space-time curvature and some of the consequences of this are discussed in this book. Some aspects of modern cosmology and the big bang theory are also briefly discussed.
Note: This book is based on lay courses in modern physics Edgall teaches at Lifelong Learning Institutes at several universities in South Florida.
Paperback: 300 pages
Publisher: World Scientific Publishing Company (February 24, 2014)
Einstein Relatively Simple
|| Online nominations for the Breakthrough Prize in Fundamental Physics and the New Horizons Prize in Fundamental Physics are now open.
The Breakthrough Prize is a $3 million award and the New Horizons Prize is worth $100,000 and given to young researchers.
The Breakthrough Prize
||The UK government is launching a new scheme to encourage those with a PhD in mathematics or physics to teach in high schools in England.
Under the scheme researchers would be paid £40,000 a year for conducting master classes for pupils in networks of schools, setting up free online maths and physics resources for schools to use, and teaching lessons that stretch more advanced students.
However, there is some concern that some of the researchers will not have all the skills needed to teach children, especially those that are struggling with the subjects.
I share some of this concern, however getting researchers involved in schools is a good thing. I don’t know the details here, but I would hope the terms are based on some kind of extended leave from your university.
University researchers to teach pupils in government maths drive BBC News
A report commissioned by The Campaign for Science and Engineering (CaSE) provides quantitative economic evidence that Government funding in science and engineering can boost growth in the economy.
The report states that for every £1 spent by the UK government on R&D, the private sector R&D output rises by 20p.
The report was written by Prof. J.Haskel of Imperial College Business School, and Prof. A. Hughes and E. Bascavusoglu-Moreau, both of the University of Cambridge.
The Economic Significance of the UK Science Base
Astronaut Tim Peake. Image by the UK Space Agency.
The UK Space Agency is today (2 May 2014) launching a competition for school children between the ages of 7 and 14 to design a British-inspired, balanced and exciting meal for British ESA astronaut Tim Peake to eat during his 6 month mission to the International Space Station (ISS).
I suggest a rocket salad for Tim’s launch!
The Great British Space Dinner
It has been a while since I have produced any iterated function system fractals, so here is my latest effort. I have not coloured them, rather the opacity describes the number of times a point has been visited. This gives a nice effect for some, but not all of these fractals.
This is just a Julia set that I created. I was surprised at just how symmetric it is as well as showing clearly the fractal property of self-similarity.