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General Relativity as extrinsic curvature and other lies my professors told me

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This post is about how General Relativity (GR) is explained to the masses, specifically how one should picture curvature. Most popular science accounts use the 'rubber sheet' analogy.

Extrinsic curvature, curvature by embedding 2D in 3D. 

Consider the picture of the Earth rotating around the Sun. This is the classic picture most 'scientific american' type articles will throw your way to explain what curvature is.  It is the bending of a 2D surface in 3D. If you take a 2D rubber sheet and put some mass, it will deform and a particle will orbit around it.  This is a good picture in the sense that it is based on classic visual 3D intuition and reproduces the correct result for 2D sheets that deform.  But try generalizing it to 3D.

GR as extrinsic curvature by embedding 3D in 4D? 

A finer problem with this image is that it seems to imply that you should abstractly extend this construction from 3D to 4D.  The curvature is extrinsic coming from the bending of 3D in a higher (4…

Feynman on the use of imaginary numbers

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For anyone having done physics as a major, the use of imaginary number (i^2=-1) is as natural as breathing air.  In the 'shut up and calculate' sense, imaginary numbers are easy to work with, but only a fool would stop and ask 'why are we using imaginary numbers in the first place?". That part can be mysterious. Why would numbers that have no reality (no real number multiplied by itself is negative) find their way into physics? like most 'magic mysteries' of physics this one is hidden in plain sight. Most folks do not ever question the use of complex numbers in quantum theory.  Ask someone who knows a little and they will huff and puff with 'of courses', ask someone who knows a lot and some of them will pause and many will say "I don't know".

Enter Feynman: imaginary exponents
Whenever I want to get to deeper and more natural insights, I turn to Feynman. Feynman has a characteristic treatment of the imaginary exponents in his books (Lectu…

Wilczek Time Crystals

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When Frank Wilczek releases something, the physics community usually pays attention.  His latest paper on Time Crystals lead to a bit of controversy.  I recently attended a talk given at Georgia Tech by Al Shapere, the co-author.

What are time crystals? 
The paper is mathematical in nature.  Shapere considers langrangians that are quartic functions of a phase speed.  Crucially the term usually associated with kinetic energy (the square one) is negative. This 4th order potential leads to the swallowtail catastrophe of Thom.  As a catastrophe it can make claims of generality and structural stability.  Furthermore the swallowtail shape gives us several values that minimize the lagrangian, it is said to be 'multivalued'.  This multi-valuedness in the potential means there are several states in the ground state.  The system can oscillate between those states (since they have the same energy) and the ground state is thus dynamic and periodic. The periodicity is a bit of a 'leger…

Feynman "six not so easy pieces" notes on SR.

Like most grad student in physics, I wish I had had Richard Feynman as a teacher. For those who don't know the myth, he was a professor at Caltech, a member of the manhattan project, and a Nobel prize winner for the development of Quantum Electro Dynamics (QED).

Mostly, as an undergrad student, I admired him for his textbooks, the Feynman lectures, a collection of classes he gave at Caltech. It was love at first read. I remember being engrossed with the books sitting on the floor in a bookstore in Paris and just reading through the electromagnetic chapters. I was impressed at how clear and fluent the presentation was and how rigorously mathematical it was. Clearly Feynman was a man who liked to think through things, on his own terms.  Only when he had mastered  a thought would he teach it.  This is part of what makes him so interesting to physics students: the insights he developed. The magic has not changed and 25 years later, I still read Feynman when I want to get to the bottom…