# Measurable effects of localized negative curvature in spacetime

In the comments to my answer to this worldbuilding.se question, someone mentioned that a curvature sufficient to produce triangles whose angles add up to 179 degrees would produce a large perceived force.

My question, what is the minimum perceived force that someone would experience in a spherical region of spacetime that is curved enough that a geodesic triangle with side-length 100 m has an angle measure of 179 degrees?

I realize that several different metrics might produce such triangles, which is why I asked for the minimum perceived force. If there are degenerate solutions that provide no minimum perceived force, then I would be interested in the minimum non-trivial perceived force. Also, I am assuming that this negative curvature is being held constant by some unknown means, and so is not caused by any sort of energy or mass, and does not evolve over time.

The spacetime that meets your criteria is a static spatially open FRW metric. The Wikipedia page for the FRW metric lists general properties of the spacetime, but in your case you need two adjustable parameters: curvature sign $k=-1$ (meaning that space has a negative curvature), and scale factor $a(t)=\mathrm{const}=a_0$ (meaning that the spatial curvature of this metric does not change with time).

To determine the value of scale factor we can recall the following relation between angle defect ($\delta \phi$) and area for hyperbolic triangles: $\delta \phi \cdot a_0^2 = \text{Area}$. Since 179 degrees is very close to 180 we can expect that the area of such triangle is close to the area of equilateral triangle in flat space: $\approx 4330 \,\text{m}^2$. Taking the value of angle defect as $\pi/180$ we obtain $a_0\approx 500 \,\text{m}$.

Extended objects in a curved spacetime would experience tidal force, manifesting as a gravitational acceleration trying to separate parts of a single object. For a general spacetime the 4-acceleration from such force is given by a geodesic deviation equation. But since the metric we are considering is ultrastatic and thus Riemann tensor has only spatial components, for objects with velocity much smaller than the speed of light this is reduced to a 3-acceleration: $$\vec{w}=\frac{v^2 \, \vec{d}}{a_0^2} ,$$ where $\vec{d}$ is a separation vector (between parts of an object). For, say a human being moving in a car at 100 km/h, $|\vec{d}|\sim 1\,\text{m}$, the relative acceleration would be $0.3 \,\text{cm/s}^{2}$. While it is detectable, it is quite small and probably would not be felt in ordinary circumstances. For a high performance supersonic aircraft $v \approx 1\,\text{km/s}$, $|\vec{d}|=15\,\text{m}$, the acceleration would be $\approx 6\,g$, certainly quite large, however since such aircrafts are usually designed for high accelerations, a fighter jet flying in this space could probably survive such tidal forces. So, overall tidal effects on everyday objects would be relatively minor.

Another noticeable effect would occur if the region of such curvature is patched into an ordinary world. Then large rigid objects transported between regions of flat space and curved space would experience deformation, and would develop elastic stresses possibly destructive to their structure. Obviously, if we move rigid triangle with a side of 100 meters, its vertices would have to bend for about 1/3 rd of a degree, which depending on the material could destroy the object. Note, that to experience such a deformation an object must have a large size in two or all three dimensions. Effectively one dimensional objects (such as rails) would be perfectly fine. So a large rigid structure (such as big ship) could not safely enter from flat space into curved world.

Note, that such values of curvature are very large from the point of view of present day conditions that could be encountered in the universe, the effective energy needed to achieve such curvature is $-6\cdot 10 ^{20}\,\text{ kg/m}^3$ which is thousands of times larger (in absolute value) than a typical density in a neutron star. That it is a negative energy should be a hint that to the best of our understanding no such 'exotic matter' is possible. Such curvatures are causing such small tidal accelerations for slowly moving objects only because the curvature is directed only in spatial directions but not in the time direction, and a contribution from energy is precisely compensated by effective negative pressures. If there was a comparable curvature in the time direction then tidal acceleration experienced even by objects at rest would be of order $$|\vec{w}| \sim \frac{c^2 \, d}{a_0^2} ,$$ This would give $36\,000g$ of acceleration for $1\,\mu\text{m}$ of separation. So even the smallest objects would disintegrate.