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Is there a way, in Einstein's relativity (special and general) to view spacetime from a point of view that sits "outside" of it, to intuitively understand it?

We humans can only see 3 dimensions. But, if we were able to see 4 dimensions (or more) would the subject of relativity be simpler and more intuitive to us?

If for example, our Universe only had 1 dimension for space and 1 dimension for time, how would spacetime "look" in the special and general theory relativity of that Universe, if it had one?

Would we be able to picture such a 2D spacetime and intuitively understand it from some kind of "bird's eye view"?

I have heard that spacetime is called a "minkowski spacetime". But what if that spacetime only had 2 dimensions (1 for space and 1 for time as we saw earlier) how would that spacetime look like from the outside or from a-far?

Nuke
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    How does plain-old non-relativistic spacetime look "from the outside"? I don't think including relativity (special or general) is the issue here. Remember that spacetime is not like space. So how would you view non-relativistic 1+1 spacetime from the outside? What does it mean to "view" it from the outside? Do you mean illustrate worldlines and such? – Marius Ladegård Meyer Apr 01 '22 at 18:14
  • "If we were able to see 4 dimensions (or more) would the subject of relativity be simpler and more intuitive to us?" Unfortuantely we can't tell because we cannot even ask someone who can see 4 dimensions (or more). – Kurt G. Apr 01 '22 at 18:37
  • @KurtG. I was thinking that spacetime was hyperbolic, so maybe if spacetime only had 2 dimensions it would be easier for us to see its hyperbolic nature portrayed in the 3rd dimension. But now I'm not even sure that spacetime is hyperbolic and what that even means! – Nuke Apr 01 '22 at 19:06
  • The $1+1$ space time without matter would look like all those Minkovski diagrams we are very familiar with. – Kurt G. Apr 01 '22 at 19:16
  • @MariusLadegårdMeyer I was thinking that a 1+1 spacetime could just be a Cartesian grid with space on the x axis and time on the y axis at right angles with eachother... And then an object moving inside them having to follow some weird hyperbolic wordline that won't allow it to move faster (or slower) than C - and also that C vector when analyzed in its 2 components will yield the correct proper speeds for both the object's velocity in space and time... So an object moving at C at some angle, moves at 50% C in space and 86.6% C in time etc... So it has the correct Lorenz Transformations in it. – Nuke Apr 01 '22 at 19:19
  • @KurtG. Yes, I'm trying to understand these diagrams but they don't seem to intuitively explain why an object can't go faster than C inside them... Or maybe I just don't understand them as well as I should be! Can you give me a link to some explanation to how these Minkovski diagrams restrict an object from moving faster or slower than C in that 1+1 spacetime? Because for relativity to be intuitive all the things that we take for granted in relativity should be emergent properties from the underlying metric, and not just axioms that we take for granted because of our observational data. – Nuke Apr 01 '22 at 19:21
  • The Minkovski diagram does surely not explain intuitively or by itself why an object can't go faster than $c$. You need the extra information that there are timelike, spacelike, and lightlike world lines. The sign of the "length" tells you if the path connecting two events is timelike, spacelike or lightlike. Objects can have only timelike paths. Hence the speed limit. – Kurt G. Apr 01 '22 at 19:26
  • To understand the Minkowski spacetime diagram, one has to understand the geometry of the diagram. Its circles (curves of constant timelike interval) are the future-branch hyperbolas, not Euclidean-circles. (More generally, the spheres are the future-hyperboloids) Similarly, to understand a position-vs-time diagram (a Galilean spacetime diagram), one has to understand its geometry.... whose circles (curves of constant time-interval) are [vertical] lines [generally hyperplanes].) – robphy Apr 01 '22 at 19:48
  • @robphy Hello! So where can I start learning more about the Minkowski spacetime diagrams? I want to learn more about the things you explained! Any good place to start? – Nuke Apr 01 '22 at 23:12
  • @KurtG. Hello again! I saw this video https://youtu.be/sEDFHMLPaW8?t=768 about the Minkowski metric and I was wondering why does this metric have the -1? Is it an axiom (for example so that we can preserve casualty?) or an emergent property of the metric? Also in that metric we can have longer distances that are actually smaller in reality... Can we have a better visualization so that small real life distances are depicted by smaller distances in our spacetime visualization? – Nuke Apr 02 '22 at 00:56
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    @Nuke My favorite introductions are Bondi's Relativity and Common Sense and Geroch's General Relativity from A to B and Taylor and Wheeler's Spacetime Physics. I'd also suggest my own contribution https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/ and https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ . – robphy Apr 02 '22 at 01:00
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    It is hard to understand spacetime intuitively because it is truly different from classical space and time. My answer to Euclidean space to Minkowski spacetime is long, but it may help. – mmesser314 Apr 02 '22 at 02:35
  • @mmesser314 I have developed the idea that as long as space is flat, space follows the every day Euclidean geometry... But once you introduce time into the mix, then your metric becomes hyperbolic.

    Is that correct, or am I mistaken?

     – Nuke Apr 02 '22 at 18:04
  • @KurtG. I think I found something similar to what I was looking for https://youtu.be/1YFrISfN7jo?t=548 this is a great example of the spacetime interval being represented for a 1+1 spacetime – Nuke Apr 03 '22 at 00:44

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In some cases, it may help to view the spacetime "extrinsically" because the "geometry" might be more familiar to you.....

  • For example, if you were studying a flat (1+1)-dimensional spacetime with the "Asteroids or Pac-Man (video game) topology", you might find it useful to represent it with a torus (the 2D-surface of a donut with a hole) embedded in three dimensions.

However, (as we understand things today) such a higher-dimensional extrinsic viewpoint won't help you with "the physics" (as observers in that spacetime would deal with).... The physics (as we understand things today) is "intrinsic" to the spacetime.

So, in general, I don't think "thinking about spacetimes extrinsically" is useful for physics.

(P.S. Since I don't think that there is a unique extrinsic viewpoint, it seems to me that any "attempt to do physics extrinsically" depends on the choice of extrinsic viewpoint. It's akin to a choice of coordinates... )

robphy
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  • "You might find it useful to represent it with a torus (the 2D-surface of a donut with a hole) embedded in three dimensions." I would love to see an example of this, is there any visual examples of this?

    I wonder why that is? Why is spacetime is so mysterious? Such that we can only study it and represent it intrinsically? Is that a limit of our imagination? Or a limit of our knowledge about it?

     – Nuke Apr 01 '22 at 23:14
  • @Nuke There are some nice graphics at https://levsblog.quora.com/The-Topology-of-Pacman . In addition, these may be useful: https://users.monash.edu/~normd/documents/MATH-348-lecture-33.pdf and https://www.youtube.com/watch?v=z0J-ro9LvUM (Pac-Man and Donuts by Tyrone Ghaswala - TEDxUofTSalon) – robphy Apr 02 '22 at 01:06
  • wow those are going to make for some good read! Thanks for that, if I have any questions I might come back! – Nuke Apr 02 '22 at 03:02