I'm trying to find a hard sci-fi story where one of the major premises was that the world has a Riemannian geometry (x^2+y^2+z^2 + t^2) instead of the real world's Minkowski geometry (x^2+y^2+z^2 - t^2). The effects of this on relativity and how it affected the world were very significant. Here's some of the effects that I remember:

  • Stars, instead of looking like points of white light, looked like lines that were cross-sections of a rainbow
  • It was impossible for two objects to collide with certain velocities unless a third object was also present
  • It was possible to "turn around" your direction through time, just as it's possible to turn around the direction you're moving through space in the real world

I remember seeing this online several years ago, along with an out-of-character analysis of the setting that explained all of the above effects in detail. Does anyone know what this is and/or where I found it?

  • Welcome to Science Fiction & Fantasy! This question would be improved by going through the checklists here; How to ask a good story-ID question? – Stormblessed Nov 29 '18 at 4:49
  • @Loki it's indeed called Riemannian, not Euclidean. – Joseph Sible Nov 29 '18 at 23:28
  • @JosephSible Riemannian manifolds is the more general term, it contains both Minkowski and Euclidean metrics. Feel free to check Wikipedia. – Rebel-Scum Nov 30 '18 at 7:29
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    @Loki I meant that now that it's been answered, I've confirmed that Riemannian is the major term the author uses to describe it, so even if Euclidean would be more correct from a technical perspective, Riemannian is much more helpful for anyone else who would search for this question. – Joseph Sible Nov 30 '18 at 13:16

Sounds like Greg Egan’s Orthogonal trilogy. There is considerable technical detail (more than 80,000 words, so enough for another novel) on Egan’s website.

Orthogonal is a science fiction trilogy by Australian author Greg Egan taking place in a universe where, rather than three dimensions of space and one of time, there are four fundamentally identical dimensions. While the characters in the novels always perceive three of the dimensions as space and one as time, this classification depends entirely on their state of motion, and the dimension that one observer considers to be time can be seen as a purely spatial dimension by another observer.

Technically, the space-time of the universe portrayed in the novels has a positive-definite Riemannian metric, rather than a pseudo-Riemannian metric, which is the kind that describes our own universe.

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    Note that Egan has also written the novel Dichronauts, set in a universe with two space dimensions and two time dimensions, a space-time signature of (+ + - -). – Mike Scott Nov 29 '18 at 8:25
  • I find it hard to understand how the second law of thermodynamics would apply in such a universe. It's through burning away low-entropy states that time has an arrow in our universe. In a plain 4D block, there's no philosophical reason for anything living to exist. (I mean, sure, there's no (scientifically) identified-yet reason for the Big Bang to exist, either --nor, even if the Big Bang is just a local low-entropy fluctuation as I've seen mentioned as a possibility by Sean Carroll, for the wider fabric to exist-- but the low entropy initial state at least allows logical progressions.) – Jacob C. Nov 30 '18 at 20:33
  • Note: I've checked out gregegan.net/ORTHOGONAL/05/Thermodynamics.html a bit but I'm still confused. Perhaps reading the books themselves would make it clearer. – Jacob C. Nov 30 '18 at 20:35

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