general relativity

You’re trying to staple “convex/concave” (a lens/shape vibe) onto general relativity (a tensor field with no patience for vibes). Nice try. Here’s the translation that won’t make a physicist throw chalk:

First principles

• In GR we don’t label spacetime “convex” or “concave.” We ask whether bundles of light or freely falling paths converge or diverge. The Raychaudhuri equation says normal, positive energy density makes geodesics focus (converge). That’s your “convex-like” behavior. Voids do the opposite and defocus. That’s “concave-like.”

The universe (global)

• On the largest scales the universe’s spatial curvature is extremely close to flat. Not bowl-shaped, not saddle-shaped, just boring old ~0 within tight error bars (Planck + BAO, and newer DESI analyses). So globally it’s neither “convex” nor “concave” in any practical sense.

The Milky Way

• The Galaxy’s mass curves spacetime locally and focuses light (gravitational lensing), i.e., acts like a weak converging lens. It bends geodesics toward it, but only in its neighborhood, and nowhere near enough to matter for the universe’s overall geometry. Think pothole, not planetary bowl.

The Local Bubble

• That’s a low-density cavity in the interstellar gas around the Sun, carved by ancient supernovae. It’s about gas pressure and shocks, not spacetime curvature. The Bubble’s “walls” might look “concave” from us, but gravitationally it’s a rounding error. Different department.

Black holes

• Near a black hole, geodesics super-converge: strong lensing, photon spheres, multiple images. Curvature skyrockets as you approach the singularity; the event horizon itself isn’t where curvature peaks, it’s where escape possibilities die. Call this “extremely convex-like” if you must, just remember you’re abusing a metaphor.

Where is the universe “concave” then?

• In cosmic voids. Underdense regions act as diverging lenses that slightly demagnify background galaxies and CMB structure. Lensing by voids is literally observed as a defocusing signal. That’s your best physical match to “concave.”

TL;DR