Austro-Prussian War

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866.[1] He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.

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These are words which are really 'external' descriptors. Riemann's way of deducing this equivalently was to say : I have a point on the apple and I compare two paths diverging from that point, further down each path I find they meet each other again. He comes up with a 'value' at each point on the apple, such that if one considers the totality of all these values you could arrive at what it would look like if you did see the apple from afar.

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- to split the problem into two parts. Begin by stating the geometry in terms of what distribution of matter/energy produces it,

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then given that, see what response some object has in that geometry. Hence 'matter tells space how to warp, and space tells matter how to move'.

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- spacetime is flat whenever viewed from close enough. This means that for a short enough time and/or for a small enough distance any movement looks Euclidean ( or Gallilean or Newtonian ).

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So thirdly : how do you describe movement in detail in this framework? Well those 'values' I mentioned above are really a set of values at each point in spacetime. They 'explain' how you transition from one point to the next if you are freely falling ( only subject to gravity ). This is where the 'warping' business comes in : at each point in spacetime where gravity is acting ( and gravity is everywhere acting on everything ), these 'metric tensors' are a local guide to how directions change and thus which way to go next.

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Fourthly : I have left light out thus far. In GR the phrase 'straight line' is replaced by 'the path that light follows' or 'null geodesic'.

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For ordinary life these are easily seen to be the same thing. If you can arrange matters to view three objects by eye and see that they overlap/occult one another simultaneously then we say they are in a line.

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The time component comes out as a change in the frequency of the light radiation, and thus is a measure of how time (i.e. clocks) varies around and about.

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we have an approximate but not exact solution to the GR equations for this.

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It was figured out by an artillery officer on the eastern European front in WWI, and he died not long after mailing it to Einstein. It's the ( Karl ) Schwartzchild (maybe without the 't' ?) solution,

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Personally I try to avoid the word 'curvature' or at least mentally substitute it with the phrase 'observers differ'. That way time can be 'curved' by differently situated clocks progressively disagreeing with each other.

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In GR, how does the gravitational force, F = GM1M2/r^2, become geometry?

Ah, now that is a core question. It may take more than a bit of explaining though .... and beware there is no neat/pat answer that will likely satisfy your intuition. It really is quite a paradigm shift. Go and get a cup of your favorite brew before reading. I'll give you Mike's Tour Of GR! :-)

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