Schwarzschild radius
Overview
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
Q:
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
史瓦西半徑
By Willy Chan
