Looking at the metric ds 2 = 1 2m r dt 2 in Visual Studio .NET

Draw QR Code JIS X 0510 in Visual Studio .NET Looking at the metric ds 2 = 1 2m r dt 2

Looking at the metric ds 2 = 1 2m r dt 2
Recognizing QR-Code In .NET
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.
QR Code ISO/IEC18004 Generator In Visual Studio .NET
Using Barcode drawer for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.
The Schwarzschild Solution
QR Code Recognizer In .NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET applications.
Draw Bar Code In Visual Studio .NET
Using Barcode drawer for .NET Control to generate, create bar code image in VS .NET applications.
dr 2 r 2 d 2 + sin2 d 2 1 2m r
Read Bar Code In .NET Framework
Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications.
Generate QR-Code In C#.NET
Using Barcode generator for .NET framework Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications.
we see that there are no terms that explicitly depend on time t or the angular variable . Therefore we can de ne the following Killing vectors. The Killing vector that corresponds to the independence of the metric of t is = (1, 0, 0, 0) (10.45)
Draw QR Code ISO/IEC18004 In .NET
Using Barcode generator for ASP.NET Control to generate, create QR-Code image in ASP.NET applications.
QR Code ISO/IEC18004 Maker In VB.NET
Using Barcode generator for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications.
We associate the conservation of energy with the independence of the metric from time. We can construct another Killing vector that corresponds to the independence of the metric from . This is = (0, 0, 0, 1) (10.46)
UCC - 12 Maker In Visual Studio .NET
Using Barcode encoder for VS .NET Control to generate, create UPC-A Supplement 2 image in Visual Studio .NET applications.
Generating ECC200 In VS .NET
Using Barcode generator for VS .NET Control to generate, create DataMatrix image in .NET framework applications.
The independence of the metric with respect to is associated with conservation of angular momentum. The conserved quantities related to these Killing vectors are found by taking the dot product of each vector with the four velocity. The conserved energy per unit rest mass is given by e = u = gab a u b = 1 2m r dt d (10.47)
Bar Code Printer In .NET Framework
Using Barcode generation for .NET Control to generate, create bar code image in Visual Studio .NET applications.
ISBN Creation In .NET
Using Barcode creation for Visual Studio .NET Control to generate, create ISBN - 13 image in Visual Studio .NET applications.
The conserved angular momentum per unit rest mass is de ned to be l = u = gab a u b = r 2 sin2 d d = r2 d d (for = ) 2 (10.48)
EAN-13 Creation In None
Using Barcode creation for Online Control to generate, create UPC - 13 image in Online applications.
Print Matrix 2D Barcode In Visual Studio .NET
Using Barcode drawer for ASP.NET Control to generate, create Matrix 2D Barcode image in ASP.NET applications.
With these de nitions, and the choice of = /2, we can simplify (10.44) to read 1 2m r
Read ECC200 In .NET
Using Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Generating Code 39 Extended In None
Using Barcode drawer for Software Control to generate, create Code 39 image in Software applications.
e2 1
EAN128 Printer In None
Using Barcode creation for Excel Control to generate, create EAN / UCC - 14 image in Microsoft Excel applications.
ECC200 Creation In Java
Using Barcode generation for Java Control to generate, create ECC200 image in Java applications.
2m r
Code128 Creation In Java
Using Barcode printer for Eclipse BIRT Control to generate, create Code 128 Code Set B image in BIRT reports applications.
ANSI/AIM Code 128 Decoder In C#
Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications.
dr d
l2 =1 r2
(10.49)
The Schwarzschild Solution
2m r 2
Let s multiply through by 1 e2 1 2 2 dr d
and divide by 2. The result is 2m 1 l2 1 2 2r r = 1 m 2 r
Now we isolate the energy per unit mass term, which gives 1 e2 1 = 2 2 dr d
2m l2 1 2 2r r
If we de ne the effective potential to be Veff =
2m l2 1 2 2r r
1 and set E = e 2 , then we obtain an expression that corresponds to that used in classical mechanics
1 E = r 2 + Veff 2 which describes a particle with energy E and unit mass moving in the potential described by Veff . However, a closer look at the effective potential is warranted. Multiplying through by the leading term, we have Veff = l2 m l 2m + 2 3 r 2r r (10.50)
The rst two terms are nothing more than what you would expect in the newtonian case. In particular, the rst term correlates to the gravitational potential while the second term is the angular momentum term we are familiar with from classical orbital mechanics. The nal term in this expression is a modi cation of the potential that arises in general relativity. We nd the minimum and maximum values that r can assume in the usual way. The rst derivative is dVeff d = dr dr m l 2m l2 + 2 3 r 2r r = m l2 3l 2 m 3+ 4 r2 r r
Next we set this equal to zero:
The Schwarzschild Solution
l2 3l 2 m m 3 + 4 =0 r2 r r 2 2 mr l r + 3l 2 m = 0 Applying the quadratic formula, we nd the maximum and minimum values of r to be given by r1,2 = l2 l2 l 4 12 l 2 m 2 = 2m 2m 1 1 12 m2 l2 (10.51)
These values correspond to circular orbits. We can use a binomial expansion of the term under the square root to rewrite this as r1,2 = l2 2m 1 1 12 m2 l2
2 m2 l 1 1 6 2 = 2m l
The two values correspond to a stable and an unstable circular orbit, respectively. The stable circular orbit is one for which r1 l2 m
Meanwhile, for the unstable circular orbit, we take the minus sign and obtain m2 l2 1 1 6 2 r2 = 2m l Using m =
GM c2
l2 2m
m2 l2
= 3m
the orbit is given by r2 = 3 GM . For the Sun, this value is c2 6.67 10 11 m3 s2 /kg 1.989 1030 kg (3 108 m/s)2 = 4422 km
sun r2 = 3
The equatorial radius of the Sun is 695,000 km, and so we see that the unstable circular orbit lies well inside the Sun. Looking at (10.50), we can learn something about the behavior of the orbits for different values of r . First, consider the case where r = 2m, the Schwarzschild
Copyright © OnBarcode.com . All rights reserved.