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I've heard a few times that "the time was imaginary before the Big Bang". I am guessing it means that at this stage, the space-time was a Riemannian $4$-manifold, but I am not sure this guess is correct.

If this is right, then I would like to know more about this Riemannian manifold. But where a mathematician can learn about it? Is there by any chance a book adapted to geometers? (otherwise it will be very hard to read.)

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    $\begingroup$ Could you provide some references about the "imaginary time"? And what does the word "before" mean when it describes the relationship between two events, one within imaginary time and the other real? $\endgroup$
    – LeechLattice
    Commented Jul 18, 2022 at 9:05
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    $\begingroup$ @LeechLattice --- this is indeed the key problem with the "imaginary time" device: The paper I cite in my answer formulates the paradoxical situation via a question: "If some real time $t_r$ is later than some imaginary time $t_i$, then is $t_r$ later than $t_i$ in real time or in imaginary time?" There is no consistent answer to this question. $\endgroup$
    – Carlo Beenakker
    Commented Jul 18, 2022 at 9:45

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The introduction of imaginary time as a way to resolve the Big Bang singularity is a proposal by Hawking and others. I don't think it plays a role in modern cosmology, see Emerging from imaginary time. The main difficulty is how to join the 4-dimensional region with Euclidean metric to a 3+1 dimensional space-time region with a Lorentzian metric.

The modern understanding is that time is emergent, not "imaginary". Loop-quantum-gravity is one theory that attempts to formalize this. It is not formulated starting from a metric on space-time, the very existence of three space dimensions and one time dimension is derived from a more fundamental framework.

Thinking of time as an emergent property is a bit like how we think of temperature. The fundamental theory of gas at the molecular level does not have temperature as a variable. This variable arises ("emerges") when one tries to describe the statistical properties of many molecules. In this sense time did not exist at the Big Bang singularity. It emerged as the universe developed.

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  • $\begingroup$ There should be a metric with mixed signature that models the Big Bang. Do you know such examples? $\endgroup$
    – Anton Petrunin
    Commented Jul 18, 2022 at 14:47
  • $\begingroup$ something along these lines? arxiv.org/abs/1909.10669 $\endgroup$
    – Carlo Beenakker
    Commented Jul 18, 2022 at 15:04
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    $\begingroup$ I thought an explicit solution should be known, say for rotationally symmetric case. $\endgroup$
    – Anton Petrunin
    Commented Jul 18, 2022 at 17:06
  • $\begingroup$ I don't posit this answer suggests "imaginary" time to be at odds with reality (paragraph 2 sentence 1)? An imaginary-valued quantity in a valid model of time would interpret physically. $\endgroup$
    – OverLordGoldDragon
    Commented Jul 18, 2022 at 18:27
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    $\begingroup$ This is not an accurate answer. The Hawking no boundary proposal uses the Euclidean path integral to define a vector in a Hilbert space (for suitably physicsy meanings of the word ‘define’). I would also not say that the modern understanding of time is that it is emergent. Certainly people have made speculations in that direction, and certainly there are theories in which spacetime is emergent, but calling it thr “modern understanding” goes much too far. $\endgroup$
    – Aaron Bergman
    Commented Jul 19, 2022 at 0:41

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