Insights Introducing Relativity on Rotated Graph Paper: A Graphical Motivation

robphy
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(based on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184 )

In my earlier Insight Spacetime Diagrams of Light Clocks,
I stated without proof that the areas of all light-clock diamonds are equal.
In my article, “Relativity on Rotated Graph Paper”, Am. J. Phys. 84, 344 (2016)
I provided algebraic proof.

In the penultimate draft, I had a non-algebraic motivating argument (which also motivates time dilation and length contraction)
that had to be left out because the article was already too long.
This argument now appears in Introducing relativity on rotated graph paper...

Continue reading...
 

Thread 'Euclidean geometry and gravity'

I asked a question here, probably over 15 years ago on entanglement and I appreciated the thoughtful answers I received back then. The intervening years haven't made me any more knowledgeable in physics, so forgive my naïveté ! If a have a piece of paper in an area of high gravity, lets say near a black hole, and I draw a triangle on this paper and 'measure' the angles of the triangle, will they add to 180 degrees? How about if I'm looking at this paper outside of the (reasonable)...
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Thread 'Is the variation of the metric ##\delta g_{\mu\nu}## a tensor?'

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
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