Schwarzschild_Radius.html

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  • Background: Equivalence Of Work, Gravity-Work, And Energy:

    Work is measured in Ft-Lbf (English units) and is the product of force times distance. When a 10 Lbm object is lifted 3 feet height to place on a table, the work done is 3*10 or 30 Ft-Lbf. In this example the force is provided by the gravitational attraction of the Earth on the object. The 10 Lbm object gets infinitesimally lighter (about 2.85 micro pounds) as it is raised due to the slightly increased separation of the object to Earth's CG. The gravitational force of attraction is expressed as: F=GmMo/r2 (MKS unit of force is named after Sir Isaac Newton). If the object is dropped back to floor height, gravity force will accelerate the object to 13.89 ft per second which represents a kinetic energy of 30 Ft-Lb (actually ~29.9999957 Ft-Lb, average weight force times distance).

    Gravity Is A Conservative Field:

    This is an important concept in physics; Conservative field means that final work done on or by an object and final potential energy of that object is independent of the path taken and only dependent on the difference in final vertical height. No energy losses are incurred in repeatedly changing the height of an object; It takes work to raise the object, and the work is given up upon lowering it. This concept becomes important when considering an object falling into a 'gravity well' like a black hole. ('well' means a long drop into steep gravity gradient)

    Gravitational Work Done Over Long [Vertical] Distances

    Gravitational force is computed from F=GmMo/r2. The force is inversely proportional to the square of the separation distance (r) of mass (m) and mass (Mo). Here we need Issac Newton for the calculus. Gravity force varies significantly over distance for black hole problems, the work (W) done on a mass (m) falling into a gravity field can be found by integrating gravitational force (F) over distance (r).

    The Following Cartoon Depicts Discussion Of Gravity Work Done:

    The integrand (F dot dr) becomes (GmMo/r2dr). GmM constants pass through the integral leaving the integrand r(-2)dr. This integrates to W=GmMo(1/R1-1/R0) evaluated between R0 and R1. For black hole computations (Ro) is considered infinity, (1/infinity)=0. So Work=GmMo/Rfinal. If one allows (R1)=0 (At the singularity), the Force and Work become infinite; an untenable result where Relativity gets into trouble.

    Cartoon: Schwarzschild Radius Derivation

    BH Schwarzschild Event Horizon (Rs) Derivation:

    To derive (Rs), gravity work (GmMo/Rfinal) is equated to kinetic energy (½mv2/2): Where Rfinal=Rs (See above Cartoon). Notice (m) cancels leaving v2/2=GMo/R. The mass (m) of the object being dropped in is irrelevant [if it is very much less than mass (Mo) so that M0 does not move appreciably]. Solving for (R) yields: R=2GMo/v2. If we want gravity to accelerate us to velocity (v), plug in knowns (v), (G), and (M) and solve for (R). R will be the distance away from the mass (Mo) to attain the desired velocity (v). If (v) is set equal to (c), then (R) becomes (Rs), the Schwarzschild radius for a single mass black hole; Rs=2GM/c 2. All this is Newtonian physics.

    Sure hope you enjoy this as much as I! More physicsstuff coming soon. dac

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