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The total number of particles is the same as viewed from both frames. in VS .NET
The total number of particles is the same as viewed from both frames. Decode Quick Response Code In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. QR Code JIS X 0510 Creator In .NET Framework Using Barcode drawer for .NET Control to generate, create QRCode image in VS .NET applications. More Complicated Fluids
Scanning Quick Response Code In VS .NET Using Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Generation In .NET Framework Using Barcode generator for .NET framework Control to generate, create barcode image in .NET framework applications. The most general form that the stressenergy tensor can assume in the case of a uid is that for a nonperfect uid that can have viscosity and shear. This is beyond the scope of this book, but we will describe it here so that you will have seen it before and can see how viscosity is handled. The stressenergy tensor in this case is T ab = (1 + ) u a u b + (P )h ab 2 ab + q a u b + q b u a Bar Code Reader In VS .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. Making Quick Response Code In C# Using Barcode encoder for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in Visual Studio .NET applications. The EnergyMomentum Tensor
QR Code Generation In VS .NET Using Barcode generation for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Denso QR Bar Code Maker In VB.NET Using Barcode generator for .NET framework Control to generate, create QR Code image in VS .NET applications. The quantities de ned here are as follows: P h ab ab qa speci c energy density of the uid in its rest frame pressure the spatial projection tensor, = u a u b + g ab shear viscosity bulk viscosity expansion shear tensor energy ux vector Encoding Data Matrix In VS .NET Using Barcode maker for .NET Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications. Making Matrix Barcode In .NET Framework Using Barcode creator for VS .NET Control to generate, create Matrix 2D Barcode image in VS .NET applications. The expansion describes the divergence of the uid worldliness. Therefore, it is given by = a u a The shear tensor is ab = 1 1 c u a h cb + c u b h ca h ab 2 3 Linear Barcode Maker In .NET Framework Using Barcode generation for VS .NET Control to generate, create 1D image in Visual Studio .NET applications. Create Bookland EAN In .NET Using Barcode creation for .NET Control to generate, create ISBN  10 image in .NET applications. Quiz
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a3 = 0 a 3 = 0 The EnergyMomentum Tensor
Using the correct result in Problem 3, if a 3 is taken to be volume V and E = a 3 , which of the following is found to be correct (a) dE + P dV = 0 (b) dE + T dS = 0 (c) dE P dV = 0 Given a perfect uid, we can write the spatial components of the stressenergy tensor as (a) T i j = ij P (b) T i j = ij (c) T i j = ij (P + ) CHAPTER
Killing Vectors
Introduction
We ve all heard the modern physics mantra and it s true: symmetries lead to conservation laws. So one thing you might be wondering is how can we nd symmetries in relativity when the theory is so geometric Geometrically speaking, a symmetry is hiding somewhere when we nd that the metric is the same from point to point. Move from over here to over there, and the metric remains the same. That s a symmetry. It turns out that there is a systematic way to tease out symmetries by nding a special type of vector called a Killing vector. A Killing vector X satis es Killing s equation, which is given in terms of covariant derivatives as b X a + a X b = 0 (8.1) Note that this equation also holds for contravariant components; i.e., b X a + a X b = 0. Killing vectors are related to symmetries in the following way: if X is a vector eld and a set of points is displaced by X a dxa and all distance Copyright 2006 by The McGrawHill Companies, Inc. Click here for terms of use.
Killing Vectors
relationships remain the same, then X is a Killing vector. This kind of distance preserving mapping is called an isometry. In a nutshell, if you move along the direction of a Killing vector, then the metric does not change. This is important because as we ll see in later chapters, this will lead us to conserved quantities. A free particle moving in a direction where the metric does not change will not feel any forces. This leads to momentum conservation. Speci cally, if X is a Killing vector, then X u = const X p = const along a geodesic, where u is the particle four velocity and p is the particle four momentum. Killing s equation can be expressed in terms of the Lie derivative of the metric tensor, as we show in this example. EXAMPLE 81 Show that if the Lie derivative of the metric tensor vanishes, then LX gab = 0 This implies Killing s equation for X , given in (8.1). SOLUTION 81 The Lie derivative of the metric is LX gab = X c c gab + gcb a X c + gac b X c Let s recall the form of the covariant derivative. It s given by c X a = c X a +

