Hyperbolic structures in the universe

[Loren Booda]

Most structures we see in the universe are of a spherical or circular nature. However, our universe is most likely hyperbolic (saddle-shaped) overall. Is it possible that there exist material structures on the large scale that manifest hyberbolic geometry?

 

[Garth]

The gravitational field around any spherical body, such as the Earth or Sun is spatially hyperbolic - this is called anti-de Sitter (ADS) space.

The visual characteristic of hyperbolic space is the curvature across the saddle-point has the opposite sense to the curvature along (normal to the first direction) the saddle-point.

Consider a point on the surface of the 'curved funnel' analogy to a spherically symmetric gravitational field. Move radially inwards and the curvature goes one way, move around the object in circular orbit and the curvature goes the other.

The intrinsic geometrical characteristics of such space is the cicumference of a circle > 2$\pi$r, the interior angles of a triangle sum to < 180⁰ and parallel lines diverge.

This geometry manifests itself in the deflection of light rays close to the Sun.

Consider two stars situated on the celestial sphere at different angles on a radius out from the Sun.

The light from the inner star would be deflected towards the Sun by a greater angle than the outer star. Therefore the angle subtended between the two stars has been increased by the presence of the Sun.

The 'straight lines', or geodesics, of the light rays from these stars have diverged, this is the property of hyperbolic ADS space.

Garth

 

[George Jones]

The gravitational field around any spherical body, such as the Earth or Sun is spatially hyperbolic - this is called anti-de Sitter (ADS) space.

I don't understand this. As topological spaces, ADS, the universal covering of ADS, and Schwarzschild are all distinct.

Or are you referring to a Schwarzschild-ADS solution?

 

[Garth]

I don't understand this. As topological spaces, ADS, the universal covering of ADS, and Schwarzschild are all distinct.

Or are you referring to a Schwarzschild-ADS solution?

I should have been more precise.

The Schwarzschild surface does not globally have ADS topology, I was just saying that locally, in any small enough region on a space-like surface (foliation) of the space-time around a gravitating mass, the curvature is hyperbolic.

If you take my light deflection example, two parallel light rays passing either side of the Sun would converge - the signature of spherical space.

However, when two parallel light rays pass close to, and both on the same side of, the Sun, so that one is closer than the other, then they will they diverge. Here the light rays are sampling the curvature of a local patch close to the Sun and they exhibit the signature of hyperbolic space.

Garth

 

[Chaos' lil bro Order]

I should have been more precise.

The Schwarzschild surface does not globally have ADS topology, I was just saying that locally, in any small enough region on a space-like surface (foliation) of the space-time around a gravitating mass, the curvature is hyperbolic.

If you take my light deflection example, two parallel light rays passing either side of the Sun would converge - the signature of spherical space.

However, when two parallel light rays pass close to, and both on the same side of, the Sun, so that one is closer than the other, then they will they diverge. Here the light rays are sampling the curvature of a local patch close to the Sun and they exhibit the signature of hyperbolic space.

Garth

Damn you are good at writing clearly, Garth. Very nice post.