 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
Copyright 2006 by The McGrawHill Companies, Inc. Click here for terms of use. in .NET
Copyright 2006 by The McGrawHill Companies, Inc. Click here for terms of use. QR Code Recognizer In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Create QR In .NET Using Barcode drawer for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications. The EnergyMomentum Tensor
QRCode Reader In .NET Framework Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications. Bar Code Printer In .NET Using Barcode creator for .NET Control to generate, create bar code image in .NET framework applications. crossing the interface de ned by constant x b . In this case we re talking about the momentum four vector, so if a = t then we are talking about the ow of energy across a surface. Let s describe each type of component we can have in turn. These are T tt , T it , T ti , and T ij . Decode Barcode In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications. Create Quick Response Code In C#.NET Using Barcode creator for VS .NET Control to generate, create Quick Response Code image in .NET applications. Energy Density
QR Code Creator In VS .NET Using Barcode creation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Making QR Code In VB.NET Using Barcode creation for Visual Studio .NET Control to generate, create QR image in .NET applications. The T tt component represents energy density. To see why, consider the momentum four vector such that p = (E, p). Using the de nition we gave above, we see that in this case T tt is the p 0 component of the momentum four vector, or simply the energy, crossing a surface of constant time. This is energy density. In relativity, energy and mass are equivalent, and so we should really think of this as the massenergy density. In most applications energy density is denoted by u; however, we don t want to confuse that with the four velocity and so we denote the density of massenergy by . Therefore, for the stressenergy tensor, we can write = T tt UCC.EAN  128 Maker In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create EAN 128 image in .NET framework applications. Create GS1  13 In .NET Using Barcode printer for .NET framework Control to generate, create EAN13 Supplement 5 image in .NET framework applications. Momentum Density and Energy Flux
Make Code 39 Full ASCII In .NET Using Barcode drawer for .NET framework Control to generate, create Code 3 of 9 image in .NET framework applications. Generating ANSI/AIM Codabar In .NET Using Barcode printer for .NET framework Control to generate, create Ames code image in .NET applications. Momentum density is momentum per unit volume. If we call momentum density , then the momentum density in the idirection is i = T it This is the ow of momentum crossing a surface of constant time. Now consider T ti . This term (which is actually equal to T it since the energymomentum tensor is symmetric) represents the energy ow across the surface x i . ANSI/AIM Code 39 Scanner In Visual Basic .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET framework applications. Barcode Reader In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Stress
Code39 Creator In Java Using Barcode creator for Android Control to generate, create Code 39 Full ASCII image in Android applications. Drawing Code 128B In Java Using Barcode printer for Eclipse BIRT Control to generate, create Code 128 Code Set B image in BIRT applications. The nal piece of the stressenergy tensor is given by the purely spatial components. These represent the ux of force per unit area which is stress. We have T ij UCC.EAN  128 Drawer In None Using Barcode encoder for Software Control to generate, create EAN 128 image in Software applications. Printing Barcode In Java Using Barcode creator for BIRT Control to generate, create bar code image in Eclipse BIRT applications. The EnergyMomentum Tensor
Encoding GS1128 In None Using Barcode generator for Online Control to generate, create USS128 image in Online applications. Barcode Scanner In Visual Basic .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. time time component = energy density
time space components = energy flux
= T ab
space time components = momentum density spatial components represent stress
Fig. 71. A schematic representation of the stressenergy tensor. T 00 is the energy
density. Terms of the form T 0j (where j is a spatial index) are energy ux. Terms with T j0 are momentum density, while purely spatial components T ij are stress. This term is the ith component of force per unit area (which is stress) across a surface with normal direction given by the basis vector e j . Analogously, T ji is the jth component of force per unit area across a surface with normal given by the basis vector ei . Returning to the view of a tensor that maps vectors and one forms to the real numbers, we obtain these components of the stressenergy tensor by passing as argument the basis vectors; i.e., Tij = T ei , e j The organization of the components of the stressenergy tensor into a matrix is shown schematically in Fig. 71. We will consider two types of stressenergy tensor seen frequently in relativity: perfect uids and dust. Conservation Equations
Conservation equations can be derived from the stressenergy tensor using b T ab = 0 (7.1) The EnergyMomentum Tensor
This equation means that energy and momentum are conserved. In a local frame, this reduces to T ab =0 xb (7.2) In the local frame, when the conservation law (7.2) is applied to the time coordinate we obtain the familiar relation: T 00 T 0i T 00 T 0i + + + =0 = = i i t x t x t which is the conservation of energy. Dust
Later we will describe a perfect uid which is characterized by pressure and density. If we start with a perfect uid but let the pressure go to zero, we have dust. This is the simplest possible energymomentum tensor that we can have. It might seem that dust is too simple to be of interest. However, consider that the dust particles carry energy and momentum. The energy and momentum of the moving dust particles give rise to a gravitational eld. In this case, there are only two quantities that can be used to describe the matter eld in the problem the energy density and how fast (and in what direction) the dust is moving. The simplest way to obtain the rst quantity, the energy density, is to jump over to the comoving frame. If you re in the comoving frame, then you re moving along with the dust particles. In that case there is a number of dust particles per unit volume n, and each dust particle has energy m. So the energy density is given by = mn. The second item of interest is none other than the velocity four vector u. This of course will give us the momentum carried by the dust. Generally speaking, to get the stressenergy tensor for dust, we put this together with the energy density. So for dust, the stressenergy tensor is given by T ab = u a u b (7.3) For a comoving observer, the four velocity reduces to u = (1, 0, 0, 0). In this case, the stressenergy tensor takes on the remarkably simple form

