Dork Scratchings

  How do we change our minds?  We like to think we're reasonable people - that we listen to the argument and if it is good enough we change our minds!  According to Carl Sagan that barely ever happens outside of science: "In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion." -- Carl Sagan, 1987 CSICOP keynote address So what about the rest of the world?  People do change their minds, so how do they go about it?  It occurred to me that a good metaphor for the mechanism is a concept from general relativity.  The concept is Parallel Transport and the animation above illustrates ...

How fast would you have to fire a cannonball for it to never hit the ground? Newton's very first ideas about gravitational orbits are said to have come about from a thought experiment. A cannonball was known to lose 5 metres of altitude a second after being fired horizontally, but the Earth - being round - curves away from the cannonball as it flies forward. So it occurred to Newton to ask: How fast would the cannonball have to be fired for the curvature to completely compensate for the vertical loss? If a cannonball was fired at this speed it would never lose any altitude, and end up orbiting the Earth. The diagram above shows that the answer can be found using simply trigonometry and comes to $$ \begin{align} v &= \sqrt{gr} \\ &= \sqrt{9.81 ms^{-2} \times 6.371\times 10^6 m} \\ &= 7868\space ms^{-1} \\ &= 17603\space mph \end{align} $$ In general, (non-relativistic) orbits are elliptical The next stage was to look at more general orbit...

Is Our Universe Finite? A while ago I drew the pictures above to try to understand current ideas about the size of the universe. The diagrams are based on some pictures I saw in the book " Our Mathematical Universe ". The diagrams show two dimensional slices of four dimensional spacetime. The blue stuff is "inflationary material" which expands at an enormous rate. The current theory of inflation states that universes like ours form as bubbles in the inflationary material as some of the inflationary material changes phase and "evaporates" out as non-inflationary material. An important point is that the sides of this bubble are moving away from each other way too fast for anything - even light - to travel from one side to the other.  The 1st diagram illustrates the point that in this model there is room for more, far more, than one universe. The yellow region in the 2nd and 3rd diagrams is what is known as a light cone. The point in the middle...

A non-euclidean surface with coordinate system Metrics are important in both the calculus of surfaces and in General Relativity.  These objects are essentially NxN matrices whose values are defined by a) the coordinate system used to parametize the space, and b) the particular point $\boldsymbol{x} = (x^i)$.  In general, theorems can be categorized as either intrinsic or extrinsic , where extrinsic means that information about how the space is embedded in a higher dimensional space is used, and intrinsic means that nothing other than the metric and it's partial derivatives were used.  The word intrinsic comes from the idea that a being which "lives" in the space can calculate the metric. So how does such a being find the metric?   2 & 3 dimensional spaces In three dimensions it's pretty straightforward: Impose a coordinate system $x^i = x^1, x^2, x^3$ on the space Define the covariant basis $\boldsymbol{e_i} = \frac{\partial{\boldsymbol{R}}...