Schwarzschild radius

— by 125 陳澔樂

ck1100535@gl.ck.tp.edu.tw 6/3/2022

https://slides.com/d/knW6GCc/live

Overview

History

Behind

Black Holes

Consequences &

Cool facts

Derivation

(Classical)

History

Black holes are where god divided by zero.

-Albert Einstein

The First Clue

F = \frac{GMm}{r^2}

Newton's Law of Universal Gravitation

(1687 Philosophiæ Naturalis Principia Mathematica)

The First Clue

If there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us; or if there should exist any other bodies of a somewhat smaller size, which are not naturally luminous; of the existence of bodies under either of these circumstances, we could have no information from sight; yet, if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability, as this might afford a clue to some of the apparent irregularities of the revolving bodies, which would not be easily explicable on any other hypothesis; but as the consequences of such a supposition are very obvious, and the consideration of them somewhat beside my present purpose, I shall not prosecute them any further.

— John Michell, 1784

To Henry Cavendish

A Whole New Era

instein

R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}

Einstein's Field Equation

General Relativity(1915)

Curvature

Tensor

Scaler Curvature

Metrics Tensor

Cosmological Constant

Stress

& Energy

Tensor

Interesting...

Said Schwarzschild

Karl Schwarzschild

1873-1916

symmetrical and simple

T_{\mu\nu}

symmetrical and simple

T_{\mu\nu}

R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}

r_s = \frac{2GM}{c^2}

Derivation

我不會GR

我超弱

- Me

First of All...

Escape\space velocity

Calculation

A planet of mass M
with an object of mass m
sitting on the surface
how fast should the object be to escape

if(distance>=    ){
 escape = true;
}

\infty

Energy!!

W = \Delta E = \vec{F} \cdot \vec{X}

\vec F = \frac{GMm}{r^2}

\vec{X} = r

\int_{R}^{\infty} \frac{GMm}{r^2}\cdot dr

=(-\frac{GMm}{\infty}) - (-\frac{GMm}{R}) = \frac{GMm}{R}

\frac{1}{2}mv^2 >= \frac{GMm}{R}

v^2 >= \frac{2GM}{R}

If c² < RHS...

c^2 < \frac{2GM}{R}

R<\frac{2GM}{c^2}

=r_s !!

QED.

Cool facts!

You know the rules, and so do I.

- Rick

r_s = \frac{2GM}{c^2}

r = \sqrt[3]{\frac{3M}{4\pi \rho}}

Not to scale

Schwarzschild(german) = Black Shield

hello?

小明在一個質量為M 密度為D 的均勻球狀星球htrae星上

此星球上除了重力沒有其他力作用，請問小明需要達到什麼速度才可以脫離這個星球

Q:

（A) : \frac{GM}{\zeta (\frac{-1}{12})D} \space (B):\sqrt[3]{\frac{GM}{\pi D e^{i\pi }}} \space (C):\sqrt{\frac{2GM}{\sqrt[3]{\frac{3M}{4\pi D}}}}

Ans:(C)

THANK YOU

ck1100535@gl.ck.tp.edu.tw