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3 votes

Are there closed simply connected 2D manifolds that do not require a third dimension?

Aside from @mmesser314’s nice answer let me mention some other results. From the classification of surfaces, we know that the only (non-empty) 2-dimensional compact boundaryless simply connected ...
peek-a-boo's user avatar
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3 votes

Are there closed simply connected 2D manifolds that do not require a third dimension?

No. A $2$D manifold might fit perfectly well in a $3$D manifold if it exists. But that does not mean that it has to exist. There are a couple ways to approach this. First "exists". Manifold ...
mmesser314's user avatar
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3 votes
Accepted

Will expanding space rupture an empty box floating in outer space

The box would only be ripped apart if dark energy is "phantom energy", which increases in energy density as the Universe expands. In that case, the destruction of the box would occur right ...
Sten's user avatar
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2 votes

Viable values for the $K$ parameter in the FLRW metric

First, define a new time coordinate $\eta$ through integrating $$ d\eta = \frac{c dt}{a(t)}.$$ This time coordinate is often called conformal time. With this coordinate, the condition for a radial, ...
11zaq's user avatar
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2 votes

What is the gravitational field of a hole in an infinite perfect crystal?

The equations that govern gravitational fields and matter simply don't make sense when you apply them to an "infinite" volume of matter. Mathematically, we have $\nabla^2 \phi = 4 \pi G \...
Michael Seifert's user avatar
2 votes

What is the gravitational field of a hole in an infinite perfect crystal?

If you put overdensity next to underdensity, overdensity attracts underdensity and underdensity repels overdensity. They chase each other forever with runaway acceleration. Not true. A volume with ...
gandalf61's user avatar
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1 vote

What is the gravitational field of a hole in an infinite perfect crystal?

... overdensity attracts underdensity and underdensity repels overdensity ... Yes, in an empty universe a pair of matched positive and negative masses initially at rest would be self-accelerating ...
A.V.S.'s user avatar
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1 vote

Viable values for the $K$ parameter in the FLRW metric

$$c^2 d\tau^2 = c^2 dt^2 - \frac{a(t)^2}{(1-KX^2)} dX^2. $$ For this metric expression, if you set $K=1$, then the spatial surfaces are hyperspheres of radius $a$. Every position on these ...
Sten's user avatar
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1 vote

Can a light signal from Earth reach a galaxy outside the Hubble Horizon?

Here are some more reasons a light signal from Earth must be able to reach and pass the Hubble Horizon: The total velocity of a photon is given by its peculiar velocity and its recession velocity as ...
KDP's user avatar
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