In math, it’s a theorem based on certain assumptions and definitions about the distances between points, and what length means. You start with human-made assumptions and follow them wherever they lead. They might or might not describe physical reality though.
The assumptions are pretty well justified based on local observations of the everyday real world that we see. Are they true on a bigger scale, say at astronomical distances? People began to wonder this in the 1800’s, in the era of Gauss and Riemann. There’s another theorem that the interior angles of a triangle add up to 180 degrees, and Gauss (an astronomer as well as a math whiz) actually proposed testing that on astronomical observations. I don’t know if he tried any experiments though. A deviation from 180 degrees would mean that space was curved.
Lo and behold, it turns out that space actually is curved, in the presence of gravitational fields. That was figured out by none other than Einstein, who became world famous when Eddington did an observation during a solar eclipse in 1919 and saw the apparent motion of distant stars when they got lined up with the edge of the sun, confirming Einstein’s prediction. The eclipse was needed for the observation since otherwise the sun would have drowned out the distant stars. But, it was quite a sensitive experiment, maybe not possible in the era of Gauss.
Anyway, the “big” answer to your question is that the ratio being constant is in the end an empirically observed fact, but that on a cosmic scale is mostly only a close approximation, and (even Einstein didn’t foresee this) falls completely apart near very extreme regions like black holes.
Einstein’s theory (“general relativity”) was still an incredible work of genius. As the saying goes, they didn’t call him Einstein for nothing!
In math, it’s a theorem based on certain assumptions and definitions about the distances between points, and what length means. You start with human-made assumptions and follow them wherever they lead. They might or might not describe physical reality though.
The assumptions are pretty well justified based on local observations of the everyday real world that we see. Are they true on a bigger scale, say at astronomical distances? People began to wonder this in the 1800’s, in the era of Gauss and Riemann. There’s another theorem that the interior angles of a triangle add up to 180 degrees, and Gauss (an astronomer as well as a math whiz) actually proposed testing that on astronomical observations. I don’t know if he tried any experiments though. A deviation from 180 degrees would mean that space was curved.
Lo and behold, it turns out that space actually is curved, in the presence of gravitational fields. That was figured out by none other than Einstein, who became world famous when Eddington did an observation during a solar eclipse in 1919 and saw the apparent motion of distant stars when they got lined up with the edge of the sun, confirming Einstein’s prediction. The eclipse was needed for the observation since otherwise the sun would have drowned out the distant stars. But, it was quite a sensitive experiment, maybe not possible in the era of Gauss.
Anyway, the “big” answer to your question is that the ratio being constant is in the end an empirically observed fact, but that on a cosmic scale is mostly only a close approximation, and (even Einstein didn’t foresee this) falls completely apart near very extreme regions like black holes.
Einstein’s theory (“general relativity”) was still an incredible work of genius. As the saying goes, they didn’t call him Einstein for nothing!