The Schwarzschild Solution in Visual Studio .NET

Encoder QR Code in Visual Studio .NET The Schwarzschild Solution

The Schwarzschild Solution
Recognizing QR In Visual Studio .NET
Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications.
Making QR-Code In Visual Studio .NET
Using Barcode creation for VS .NET Control to generate, create QR-Code image in .NET applications.
Fig. 10-4. A light ray de ected by the Sun.
Recognize QR Code JIS X 0510 In Visual Studio .NET
Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications.
Barcode Drawer In .NET Framework
Using Barcode printer for .NET framework Control to generate, create barcode image in VS .NET applications.
In the case of the Sun, a de ection of 1.75 in. of arc is predicted. The interested reader can learn about the observational challenges and results in trying to measure this phenomenon, which has been done successfully.
Scan Bar Code In Visual Studio .NET
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Print QR-Code In C#
Using Barcode maker for .NET framework Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications.
Time Delay
QR Code JIS X 0510 Maker In Visual Studio .NET
Using Barcode drawer for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications.
QR Code Creation In Visual Basic .NET
Using Barcode generation for .NET Control to generate, create QR Code image in Visual Studio .NET applications.
The nal phenomenon that manifests itself in the Schwarzschild geometry is the time travel that is required for light to go between two points. The curvature induced in the spacetime surrounding a massive body like the Sun increases the travel time of light rays relative to what would be the case in at space. Again setting = /2 and using ds 2 = 0, we have 2m r 2m r
ECC200 Creation In Visual Studio .NET
Using Barcode generator for .NET framework Control to generate, create DataMatrix image in VS .NET applications.
Encode Code 3/9 In .NET Framework
Using Barcode printer for Visual Studio .NET Control to generate, create ANSI/AIM Code 39 image in Visual Studio .NET applications.
0= 1
Painting Code 128 Code Set C In .NET
Using Barcode drawer for Visual Studio .NET Control to generate, create Code128 image in VS .NET applications.
Identcode Encoder In VS .NET
Using Barcode creation for VS .NET Control to generate, create Identcode image in VS .NET applications.
dt 2 1
Code 39 Extended Drawer In VB.NET
Using Barcode drawer for .NET framework Control to generate, create Code 3 of 9 image in VS .NET applications.
ANSI/AIM Code 39 Reader In Visual Basic .NET
Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications.
dr 2 r 2 d 2
Barcode Generation In None
Using Barcode creator for Software Control to generate, create bar code image in Software applications.
Make Data Matrix In Java
Using Barcode generation for Java Control to generate, create Data Matrix image in Java applications.
Using the previous results for light rays, we write the last piece in terms of dr and obtain the following result:
Make Bar Code In Visual C#.NET
Using Barcode generation for .NET Control to generate, create bar code image in .NET applications.
Data Matrix 2d Barcode Printer In Java
Using Barcode creator for Java Control to generate, create Data Matrix 2d barcode image in Java applications.
2 dr 2 1 2mr0 /r 3 2 1 r0 /r 2 (1 2m/r )2
Matrix Barcode Creator In VS .NET
Using Barcode maker for ASP.NET Control to generate, create Matrix Barcode image in ASP.NET applications.
Create EAN128 In None
Using Barcode generation for Font Control to generate, create UCC-128 image in Font applications.
dt =
The Schwarzschild Solution
Taking the square root, expanding to rst order, and using conventional units (so we put ct in place of t), we obtain cdt = dr
2 1 r0 /r 2
2 2m mr0 3 r r
This result can be integrated. To consider the travel time of light between Earth and another planet in the solar system, we integrate between r0 to rp and r0 to re , where rp is the planet radius and re is the Earth radius. The result is
2 2 rp r0 + rp 2 r0 2 2 re r0 + re
ct =
2 2 rp r0 +
2 2 re r0 + 2m ln 2 2 re r0
m
2 2 rp r0
The ordinary at space distance between Earth and the planet is given by
2 2 2 2 the rst term, rp r0 + re r0 . The remaining terms indicate the increased distance caused by the curvature of spacetime (i.e., by the gravitational eld of the Sun). These terms cause a time delay that is measurable in the solar system. For example, radar re ections to Venus are delayed by about 200 s. Because of limited space our coverage of the Schwarzschild solution is incomplete. The reader is encouraged to consult the references listed at the end of the book for more extensive treatment.
Quiz
1. Using the variational method described in Example 4-10, the nonzero Christoffel symbols for the Schwarzschild metric are (a) t r t = d dr r 2( ) d , r rr = d , r = r e 2 , r = r e 2 sin2 tt = e dr dr 1 r = r , = sin cos 1 r = r , = cot
t rt r
The Schwarzschild Solution
= d dr
tt r
= e2( ) d , r rr = d , dr dr 1 r = r , = sin cos 1 r = r , = cot
= r e 2 ,
= r e 2 sin2
d , dr
= d , dr
r r
= r e 2 ,
= r e 2 cos2
1 = r, 1 = r,
= sin cos = cot
Suppose that we were to drop the requirement of time independence and wrote the line element as ds 2 = e2 (r,t) dt 2 e2 (r,t) dr 2 r 2 d 2 + sin2 d 2 The Rr t component of the Ricci tensor is given by 1 (a) Rr t = r d dt (b) Rr t = (c) Rr t =
1 d r dt r12 d dt
For the following set of problems, we consider a Schwarzschild metric with a nonzero cosmological constant. We make the following de nition: f (r ) = 1 We write the line element as ds 2 = f (r ) dt 2 + 1 dr 2 + r 2 d 2 + r 2 sin2 d 2 f (r ) 2m 1 r 3 r2
When you calculate the Ricci rotation coef cients, you will nd (a) (b) (c)
r r r tt tt tt
= = =
r3 9r 18m 3 r 3
3 3m r 9r 18m 3 r 3
3m r 3 9r 18m
The Schwarzschild Solution
When you calculate the components of the Ricci tensor, you will nd (a) Rt t = Rr r = R = R = (b) Rt t = Rr r = R = R = r 3 (c) Rt t = Rr r = R = R = 0 The Petrov type of the Schwarzschild spacetime is best described as (a) type O (b) type I (c) type III (d) type D
Copyright © OnBarcode.com . All rights reserved.