Posts Tagged 'Relativity'

The 3-sphere: Extrinsic and Intrinsic Forms

Figure 1. An extract from Einstein’s 1917 paper on cosmology.

The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle {\mathbb{S}^1} in the plane {\mathbb{R}^2} and the sphere {\mathbb{S}^2} in three-space {\mathbb{R}^3}, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in {\mathbb{R}^4} but can also be envisaged as a non-Euclidean manifold in {\mathbb{R}^3}.

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Dynamic Equations for Weather and Climate

“I could have done it in a much more complicated way”,
said the Red Queen, immensely proud. — Lewis Carroll.

Books on dynamic meteorology and oceanography usually have a full chapter devoted to the basic dynamical equations. Since the Earth’s fluid envelop is approximately a thin spherical shell, spherical coordinates {(\lambda,\varphi, r)} are convenient. Here {\lambda} is the longitude and {\varphi} the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Fig 1. The momentum equations, as in Lorenz (1967). The metric terms are boxed.

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Poincare’s Square and Unbounded Gomoku

Poincare’s hyperbolic disk model.

Henri Poincar’e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved outward from the centre, everything got smaller in such a way that it would take an infinite time to reach the boundary.

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Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Lanczos240In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at].

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Gravitational Waves & Ringing Teacups

Newton’s law of gravitation describes how two celestial bodies orbit one another, each tracing out an elliptical path. But this is imprecise: the theory of general relativity shows that two such bodies radiate energy away in the form of gravitational waves (GWs), and spiral inwards until they eventually collide.


Warning sign, described by Thomas Moore as a “geeky insider GR joke” [image from Moore, 2013].

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A New Window on the World

The motto of the Pythagoreans was “All is Number” and Pythagoras may have been the first person to imagine that the workings of the world might be understood in mathematical terms. This idea has now brought us to the point where, at a fundamental level, mathematics is the primary means of describing the physical world. Galileo put it this way: the book of nature is written in the language of mathematics [TM102, or search for “thatsmaths” at].


Visualization of gravitational waves. Image credit MPI/Gravitational Physics/ITP Frankfurt/ZI Berlin.

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Where in the World?

Here’s a conundrum: You buy a watch in Anchorage, Alaska (61°N). It keeps excellent time. Then you move to Singapore, on the Equator. Does the watch go fast or slow? For the answer to this puzzle, read on. Continue reading ‘Where in the World?’

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