/sci/ - Science & Math

>>8191166

Now, let’s determine the gravitational energy of a galaxy of mass m at the edge of a region with radius R:


Et = ½ mv^2 - GMm/r


We can multiply both sides by a positive number without changing anything and divide by m, as m is also a positive number, so we’ll multiply by 2:


2/m Et = v^2 - 2GM/r


As the velocity of a galaxy at a given distance is proportional to the distance, and the constant of proportionality is H, then v^2 is equal to H^2 R^2.


Now, what is the total mass of a sphere with radius R?


Well the volume of a sphere with radius R is 4/3 π r^3 and as the mass of a sphere with radius R is equal to volume times the average density, then we can write the equation as:


2/m Et = H^2 R^2 - 2G [(4π/3) ρR^3] /R


In order to further simplify, as R2 is a positive quantity we can divide by it, which leaves us with:


2/mR^2 Et = H^2 - 8 πG/3 ρ


Now, let’s refer to the constants 2, m and Et as minus kappa, so that 2/m Et is represented by –k:


-k/R^2 = H^2 - 8 πG/3 ρ


This is Einstein’s equation for an expanding universe, derived with all the factors of π and 3, which will ultimately determine the evolution of the universe.


Minus kappa is a constant related to the total energy of the galaxy, which in general relativity is the curvature of the universe.

>>8184501

Now, let’s determine the gravitational energy of a galaxy of mass m at the edge of a region with radius R:


Et = ½ mv^2 - GMm/r


We can multiply both sides by a positive number without changing anything and divide by m, as m is also a positive number, so we’ll multiply by 2:


2/m Et = v^2 - 2GM/r


As the velocity of a galaxy at a given distance is proportional to the distance, and the constant of proportionality is H, then v^2 is equal to H^2 R^2.


Now, what is the total mass of a sphere with radius R?


Well the volume of a sphere with radius R is 4/3 π r^3 and as the mass of a sphere with radius R is equal to volume times the average density, then we can write the equation as:


2/m Et = H^2 R^2 - 2G [(4π/3) ρR^3] /R


In order to further simplify, as R2 is a positive quantity we can divide by it, which leaves us with:


2/mR^2 Et = H^2 - 8 πG/3 ρ


Now, let’s refer to the constants 2, m and Et as minus kappa, so that 2/m Et is represented by –k:


-k/R^2 = H^2 - 8 πG/3 ρ


This is Einstein’s equation for an expanding universe, derived with all the factors of π and 3, which will ultimately determine the evolution of the universe.


Minus kappa is a constant related to the total energy of the galaxy, which in general relativity is the curvature of the universe..