 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
The EnergyMomentum Tensor in Visual Studio .NET
The EnergyMomentum Tensor QR Decoder In Visual Studio .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. QR Code 2d Barcode Creation In VS .NET Using Barcode drawer for VS .NET Control to generate, create QRCode image in VS .NET applications. 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 QR Code JIS X 0510 Decoder In .NET Framework Using Barcode decoder for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Encode Bar Code In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. T ab
Recognizing Bar Code In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. Print Denso QR Bar Code In C# Using Barcode encoder for .NET framework Control to generate, create QR image in .NET applications. (7.4) Drawing QR Code JIS X 0510 In VS .NET Using Barcode maker for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. QR Code JIS X 0510 Generation In Visual Basic .NET Using Barcode printer for .NET Control to generate, create QR Code 2d barcode image in VS .NET applications. Now consider the case of a stationary observer seeing the dust particles go by with four velocity u. In that case, we have u = ( , u x , u y , u z ) , where 1 the u i are the ordinary components of three velocity and = 1 v 2 . Looking at (7.3), we see that in this case the stressenergy tensor is 1 ux = 2 y u uz ux (u x )2 u yux uzux uy ux u y (u y )2 uzu y uz ux uz y z u u (u z )2 EAN 13 Drawer In Visual Studio .NET Using Barcode drawer for .NET framework Control to generate, create GS1  13 image in .NET applications. Drawing GTIN  128 In Visual Studio .NET Using Barcode creator for .NET framework Control to generate, create GTIN  128 image in Visual Studio .NET applications. T ab
Bar Code Printer In .NET Using Barcode creation for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. Intelligent Mail Encoder In VS .NET Using Barcode creation for .NET framework Control to generate, create OneCode image in .NET framework applications. (7.5) Bar Code Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Creating UCC.EAN  128 In Visual Basic .NET Using Barcode maker for VS .NET Control to generate, create GS1128 image in .NET framework applications. EXAMPLE 71 Show that the conservation equations for the energymomentum tensor in the case of dust lead to the equation of continuity of a uid. SOLUTION 71 The conservation equation is given by T ab =0 xb Setting a = t, we obtain T tx T ty T tz T tt T tb + + + =0 = xb t x y z Using (7.5), this becomes T tt T tx T ty T tz ( u x ) ( u y ) ( u z ) + + + = + + + t x y z t x y z = + ( u) t Barcode Generation In None Using Barcode creation for Office Word Control to generate, create bar code image in Word applications. Generate GS1128 In VS .NET Using Barcode drawer for ASP.NET Control to generate, create EAN / UCC  13 image in ASP.NET applications. and so we have
Painting Code 128 Code Set C In Java Using Barcode creation for Android Control to generate, create Code128 image in Android applications. ECC200 Encoder In None Using Barcode creation for Online Control to generate, create Data Matrix 2d barcode image in Online applications. The EnergyMomentum Tensor
EAN13 Maker In Java Using Barcode creation for Java Control to generate, create UPC  13 image in Java applications. Recognizing Barcode In Visual C#.NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. + ( u) = 0 t where u is the ordinary threedimensional velocity. This is the equation of continuity. Perfect Fluids
A perfect uid is a uid that has no heat conduction or viscosity. As such the uid is characterized by its mass density and the pressure P. The stressenergy tensor that describes a perfect uid in the local frame is 0 = 0 0 0 P 0 0 0 0 P 0 0 0 0 P T ab
(7.6) To nd the form of the stressenergy tensor in a general frame, we rst consider the at space of special relativity and boost to a frame of an observer with four velocity u. The stressenergy transforms to a general frame by Ta b = a c b dT cd
(7.7) However, we note that we can construct the most general form of the stressenergy tensor from the four velocity u, the metric tensor ab along with and p. Furthermore, the tensor is symmetric. This tells us that the general form of the stress energy tensor is T ab = Au a u b + B ab (7.8) where A and B are scalars. For this example, we assume that the metric is ab = diag (1, 1, 1, 1). Looking at (7.6), we notice that the only spatial components are T ii = P. Another way to write this is T ij = ij P (7.9) The EnergyMomentum Tensor
In the rest frame, we have u 0 = 1 and all other components vanish. Therefore, (7.8) takes the form T ij = B ij Comparison with (7.9) leads us to take B = P. Now we consider the time component. In the local frame it is given by T 00 = and so T 00 = = Au 0 u 0 + B 00 = Au 0 u 0 P = A P Therefore, we conclude that A = P + and write the general form of the stressenergy tensor for a perfect uid in Minkowski spacetime as T ab = ( + P) u a u b P ab For any metric gab , this immediately generalizes to T ab = ( + P) u a u b Pgab (7.11) (7.10) Note that the form of the stressenergy tensor in general will change if we take ab = diag ( 1, 1, 1, 1). In that case the equations become T ab = ( + P) u a u b + P ab T ab = ( + P) u a u b + Pgab EXAMPLE 72 Consider the RobertsonWalker metric used in Example 53: ds 2 = dt 2 + a 2 (t) dr 2 + a 2 (t)r 2 d 2 + a 2 (t)r 2 sin2 d 2 1 kr 2 (7.12) Suppose we take the Einstein equation with nonzero cosmological constant. Find the Friedmann equations in this case. SOLUTION 72 In the last chapter we found that in the local frame, the components of the Einstein tensor for this metric were given by 3 k + a2 a2 1 a G r r = G = G = 2 2 k + a 2 a a G t t = (7.13)

