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Four Vectors in .NET framework
Four Vectors Decode QRCode In .NET Framework Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications. Painting Denso QR Bar Code In .NET Using Barcode generation for VS .NET Control to generate, create QR Code JIS X 0510 image in VS .NET applications. In special relativity we work with a uni ed entity called spacetime rather than viewing space as an arena with time owing in the background. As a result, a vector is going to have a time component in addition to the spatial components we are used to. This is called a four vector. There are a few four vectors that are important in relativity. The rst is the four velocity, which is denoted by u and has components u= dt dx dy dz , , , d d d d QR Code 2d Barcode Reader In VS .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Create Barcode In .NET Framework Using Barcode encoder for VS .NET Control to generate, create barcode image in .NET applications. We can differentiate this expression again with respect to proper time to obtain the four acceleration a. The norm or magnitude squared of v v tells us if a vector is timelike, spacelike, or null. The de nition will depend on the sign convention used for the line element. If we take ds 2 = c2 dt 2 dx 2 dy 2 dz 2 , then if v v > 0 we say that v is timelike. If v v < 0, we say v is spacelike. When v v = 0, we say that v is null. The four velocity is always a timelike vector. Following this convention, we compute the dot product as v v = (v t )2 (v x )2 v y Bar Code Scanner In .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET framework applications. Drawing QR Code In Visual C# Using Barcode printer for .NET framework Control to generate, create QRCode image in VS .NET applications. (v z )2 Drawing QR Code In VS .NET Using Barcode generation for ASP.NET Control to generate, create QR image in ASP.NET applications. Painting QR Code ISO/IEC18004 In Visual Basic .NET Using Barcode encoder for .NET framework Control to generate, create QR image in Visual Studio .NET applications. The dot product is an invariant, and so has the same value in all Lorentz frames. If a particle is moving in some frame F with velocity u, we nd that energy and momentum conservation can be achieved if we take the energy to be E = m 0 c2 and de ne the four momentum p using p = m 0u (1.27) EAN13 Creator In .NET Framework Using Barcode encoder for VS .NET Control to generate, create GS1  13 image in Visual Studio .NET applications. Bar Code Generator In .NET Using Barcode drawer for .NET Control to generate, create barcode image in Visual Studio .NET applications. where m 0 is the particle s rest mass and u is the velocity four vector. In a more familiar form, the momentum four vector is given by p = E/c, px , p y , pz . Printing Matrix Barcode In .NET Framework Using Barcode generation for VS .NET Control to generate, create Matrix 2D Barcode image in .NET applications. Generating ISSN  10 In .NET Using Barcode creation for .NET Control to generate, create ISSN  10 image in .NET framework applications. Special Relativity
Bar Code Maker In VS .NET Using Barcode generation for Reporting Service Control to generate, create barcode image in Reporting Service applications. ANSI/AIM Code 39 Creation In ObjectiveC Using Barcode printer for iPhone Control to generate, create USS Code 39 image in iPhone applications. For two frames in the standard con guration, components of the momentum four vector transform as px = px E/c py = py pz = p z E = E cpx Using our sign convention for the dot product, we nd DataMatrix Maker In ObjectiveC Using Barcode printer for iPad Control to generate, create Data Matrix ECC200 image in iPad applications. Making Code 3 Of 9 In None Using Barcode generator for Font Control to generate, create Code 3 of 9 image in Font applications. 2 2 p p = E 2 /c2 px p 2 pz = E 2 /c2 p 2 y
Draw Code39 In Java Using Barcode printer for Java Control to generate, create Code39 image in Java applications. ECC200 Encoder In ObjectiveC Using Barcode generation for iPhone Control to generate, create Data Matrix 2d barcode image in iPhone applications. (1.28) Printing Barcode In .NET Framework Using Barcode generation for Reporting Service Control to generate, create bar code image in Reporting Service applications. DataMatrix Recognizer In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Remember, the dot product is a Lorentz invariant. So we can nd its value by calculating it in any frame we choose. In the rest frame of the particle, the momentum is zero (and so p 2 = 0) and the energy is given by Einstein s famous formula E = m 0 c2 . Therefore p p = m 2 c2 0 Putting these results together, we obtain E 2 p 2 c2 = m 2 c4 0 (1.29) Relativistic Mass and Energy
The rest mass of a particle is the mass of the particle as measured in the instantaneous rest frame of that particle. We designate the rest mass by m 0 . If a particle is moving with respect to an observer O with velocity v then O measures the mass of the particle as m= m0 1 v 2 /c2 = m0 (1.30) Now consider the binomial expansion, which is valid for x < 1, (1 + x)n 1 + nx
For n = 1/2, Special Relativity
1 (1 x) 1/2 1 + x 2 Setting x = v 2 /c2 in (1.30), we obtain m= m0 1 v 2 /c2 m0 1 + 1 v2 2 c2 1 v2 = m0 + m0 2 2 c Multiplying through by c2 we obtain an expression that relates the relativistic energy to the rest mass energy plus the Newtonian kinetic energy of the particle: 1 mc2 = m 0 c2 + m 0 v 2 2 Quiz
1. An inertial frame is best described by (a) one that moves with constant acceleration (b) a frame that is subject to constant forces (c) a frame that moves with constant velocity (d) a frame that is subject to galilean transformations The proper time d 2 is related to the interval via (a) d 2 = ds 2 (b) d 2 = ds 2 (c) d 2 = c2 ds 2 2 2 (d) d = ds 2 c 3. The principle of relativity can be best stated as (a) The laws of physics differ only by a constant in all reference frames differing by a constant acceleration. (b) The laws of physics change from one inertial reference frames to another. (c) The laws of physics are the same in all inertial reference frames. 2. 4. Rapidity is de ned using which of the following relationships (a) tanh = v c (b) tan = v c (c) tanh = v c (d) v tanh = c

