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In general relativity spacetime is defined as manifold. Is this manifold the set of events? For example if our universe is constituted of only one non-interacting particle, should our manifold be a line?

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    $\begingroup$ You should think in terms of “possible events”. Something doesn’t necessarily happen at every point in spacetime. $\endgroup$
    – G. Smith
    Apr 3, 2021 at 6:20
  • $\begingroup$ @ G. Smith so in four dimensional manifold a free particle is constraint to four dimension. Isn't that the same as say that the manifold exist before any event? $\endgroup$ Apr 3, 2021 at 6:34
  • $\begingroup$ Such questions seem to me to be more about metaphysics than physics. Among other things, I don’t consider an event to “exist” but to “happen”, but I suppose that this is because I don’t think in terms of a “block” universe. My recommendation is to focus on the mathematics and not the metaphysics. $\endgroup$
    – G. Smith
    Apr 3, 2021 at 6:42
  • $\begingroup$ @G.Smith Knowing what formulae represent is genuine Physics, not metaphysics. $\endgroup$ Apr 3, 2021 at 6:49
  • $\begingroup$ @GiorgioP There is a difference between relating formula to physically observable things and giving an interpretation to formula. The former is physics, the latter is metaphysics. $\endgroup$
    – NDewolf
    Apr 3, 2021 at 12:46

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A manifold is not just a set, but it requires a set on which additional mathematical structures are built (topologies, coordinates, charts, atlas, metrics, and so on). Spacetime of GR can be considered as based on a set of points where each point is an event (something which happens). An event requires that there is something.

The implicit assumption of GR is that, whatever this something is, there are as many as required by a theory based on a continuum of events (more technically, enough to allow a separable topological space). It is not necessary that all the events are recorded. It is just required that at least in principle they could be identified. The initial chapters on space-time in the classic Misner Thorne Wheeler (MTW) Gravitation will provide a readable introduction to this concept.

As a consequence, your hypothesis of a set of events reduced to the world-line of just one particle has a drawback. If there would be only one particle how can we identify an event?

I would say that the best intuition for the events of the GR spacetime, consistent with MTW point of view, is as a concept analogous to test-particles in the definition of the electromagnetic field. They are required to assign an operative definition to the concept of field, and we may have good approximations for them at the macroscopic level, although, at the microscopic level, we may have a hard time going below the elementary charge. However, arrived at that scale, we need new theories.

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As long as you do all the math correctly you are free to use whatever English language words you want to describe it with. I would probably say that the spacetime manifold consists of points at which events could take place. I wouldn't say that the manifold itself consists of events, though.

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No, it's not.

A manifold is not just a set of points. It also has a topology and a smooth structure. In fact, there is such a thing as a pointless topology, where we only consider the topological structure and forgo the points. In a sense, here, points are extra structure.

Since QM suggests that points aren't physical, a pointless manifold is physically more realistic.

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  • $\begingroup$ what you are saying is that even in a empty universe the manifold exists? $\endgroup$ Apr 3, 2021 at 6:31
  • $\begingroup$ @amiltonmoreira I think physicists are perfectly happy to think about an empty universe “existing” as a manifold. That’s what Minkowski spacetime is. But in GR that manifold comes with a tensor field! $\endgroup$
    – G. Smith
    Apr 3, 2021 at 6:45
  • $\begingroup$ @amilton moreira: You mean a vacuum? Of course that exists. An outcome of QM, is that the vacuum structure of spacetime is actually quite complex with instantonic tunelling between vacua ... $\endgroup$ Apr 3, 2021 at 6:56
  • $\begingroup$ @ Mozibur Ullah This question is in the frame work of general relativity. I am assuming that quantum mechanics does not exist $\endgroup$ Apr 3, 2021 at 7:12
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    $\begingroup$ @GiorgioP: For me, physically speaking, the continuum structure breaks down and hence we need some other than the continuum structure that we find in the usual development of manifolds where having a set of points is essential. For example, there is Causal Set Theory. $\endgroup$ Apr 3, 2021 at 11:33
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The conclusion that I arrived, is that there is no manifold without objects in the universe. For example ,suppose our universe consist of a particle $\gamma$ and the a metric field $g$.The interaction between this metric field and this particle, is in such a way, that we can model this system as manifold, with a metric $g$ and curve $\gamma$ on it. So for example if this metric would act differently, between to different particles, $\gamma_1$ and $\gamma_2$ this model of universe as a four dimensional manifold with a metric $g$ on it, would be invalid.

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  • $\begingroup$ Why the down vote in this answer? $\endgroup$ Apr 3, 2021 at 9:55

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