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In Neil DeGrass Tyson's epic video, at 2:26:50

https://www.youtube.com/watch?v=AdHlVY8pEk0&list=PL1L9zQimONkUOxXJyM0xcJja9ItQu1P8l&src_vid=AdHlVY8pEk0&feature=iv&annotation_id=annotation_4179310099#t=12s

He mentions how if net energy is negative, the spacetime curvature is spherical, and if it's net positive, saddle-shaped.

He uses the terms "flat" and "saddle-shaped" which are 2d, but he's actually just using that as an analogy for 3d curvature, which we can't really visualize, correct? If so, what does it mean for 3d space to be "saddle-shaped" or "spherical"?

1) II'm guessing spherical implies that if we keep traveling in one direction in space, it would be possible to end up back where you started? (kind of like how you can end up in the same position traversing the surface of a planet if the 2d plane is curved?

2) If net energy is 0. But they also say there is far more dark energy and dark matter than energy and matter. So isn't this a contradiction? How does the accounting work out?

ahnbizcad
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In flat space the surface area of a sphere is $4\pi r^2$. In positively curved space the surface area of a sphere is less than $4\pi r^2$ and in negatively curved space the surface area of a sphere is greater than $4\pi r^2$. By $r$ I mean that if you start at the centre of the sphere with your trusty (infinitesimal) ruler and measure the distance to the sphere you'll be measuring the quantity $r$ (this is known as the proper distance). The area of the sphere is then measured by crawling over it and measuring it with the same ruler.

If you're looking for an intuitive way to visualise the different curvatures I can't help - please let me know if you find one!

Re your question 2: this is a somewhat vexed issue and different commentators have different views on whether it makes any sense to talk about the total energy of the universe. See for example the many and varied answers to Total energy of the Universe.

You specifically ask about dark energy: you need to appreciate that ordinary matter, dark matter and dark energy all have a positive energy density and you can add them all together to get the total energy density. Dark matter causes a repulsion because it has an unusual equation of state, not because it is exotic in any way. The Zero Energy Universe idea is that the negative gravitational potential energy exactly balances out the positive combined energy density of matter and dark energy so the net energy is zero.

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John Rennie
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  • The mathematical notion of proper distance really helps out a lot. So when people talk about the curvature, they're using a 2d mathematical construct that applies to the relativistic curvature of 3d spacetime (4d). Oh! When people say net zero energy, they're talking about only the gravitational potential energy, and NOT the sum total energy of all matter (including internal, kinetic, etc)? If everything has positive energy density, the sum can only be net positive, so we're talking about two separate concepts. – ahnbizcad Sep 06 '14 at 08:16
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    @gwho: The zero energy is potential energy + energy of matter/dark energy as give by $mc^2$. The energy of all the matter/energy is positive as you say, and the gravitational potential energy is negative, so the two cancel and sum to zero (although as I mentioned in the answer this is a contentious issue). – John Rennie Sep 06 '14 at 08:19
  • That certainly would account for there being more dark energy/matter than energy/matter, if both dark and "regular" matter/energy have positive energy density. I'll look more into the controversy. Thank you so much for explaining this. I am quite intrigued by finally being able to understand this, and seeing an elegant aspect to it. – ahnbizcad Sep 06 '14 at 08:25
  • Would tachyons be an example of what you mean by "unusual equation state"? – ahnbizcad Sep 06 '14 at 20:51
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    @gwho: no. See this Wikipedia article for what equation of state means. – John Rennie Sep 07 '14 at 04:49