Mathematical and physical differences between curved and flat surfaces in astrophysical observations

Abstract

The study of cosmology is based on Einstein's "general theory of relativity", and this leads to the consideration (to the imaginary) that we live in the middle of a curved space-time. This idea (the idea of ​​space-time curvature) is not easy at all, in other words, it is not easy to represent, and what is being done is a first stage to address and approach some points related to that idea.

             There are several difficulties to approach this idea, which is to first imagine four-dimensional space-time and then imagine that space-time is curved. But here we will be concerned with some simple situations: flat space (flat surface) and curved space (curved surface) in two and three dimensions. Any very simple situations and it is only space and not space and time (that is, not space-time) with only two dimensions and not a three-dimensional space where we currently live, the situation would be simpler. So, in other words, we will do what is called "the study of cosmology through the ant or the so-called ant technology", that is, an ant that lives in a two- or three-dimensional space. Here we don't care about time, we only care about space. And we will see what are the points that can be deduced from studying this, that is, what will the ant see when studying it, or in other words, what will we see if we are in the place of the ant?

             This is the first way to know that we live in a curved space. This is not easy to do in practical life, especially when studying cosmology, where the distances we are dealing with are very large, on the order of a light-year.

             Then the comparison will be made between flat surfaces and curved surfaces; And we will see what is the effect of curvature on the circumference of the circle, and this in turn leads to a physical effect or another physical effect, which is the concept of light flux.