Contents. Contents. Matrices. Contents. Objectives. Matrices

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1 9/8/7 Physics for Majors Class 8 Matrices and Lorentz s Space-time Four- Last Class Test Review Scalars and vectors Three-vectors and four-vectors The energy-momentum four-vector Rotations about the z axis Relativistic kinematic problems Today s Class Matrices Classical boosts Lorentz boosts Space-time four-vectors Space and time problems in relativity Section Matrices A Math Problem You have probably seen matrices in this context. Given the equations what are x and y? One method is to write this in terms of matrices :

2 9/8/7 A Math Problem We can write this symbolically as If we can find a matrix such that then we can solve for : A Math Problem You don t need to know this! Once you learn matrix multiplication, this will make sense. The inverse of a matrix is: The last matrix is called the identity matrix since So for and 3 8 6, Matrix Operators But we won t use matrices to solve equations! We will uise matrices as operators. Operators do things to vectors or otjer operators. Examples include rotations and boosts. Matrix Operators In 3-dimensional space (configuration space), matrices are collections of nine numbers each of which depends on two different sets of directions,,, and,,. We occasionally write something like the following for a matrix More often, we write this as Scalar or Dot Product Dot Product Problem Evaluate 3 3 Note that the product is a scalar!

3 9/8/7 Matrix Vector This is really a collection of three dot products! Matrix Matrix This is really a collection of nine dot products!! Section Matrix Operations Z-Rotation What does the vector represent? cos sin The matrix sin cos rotates a vector counterclockwise by an angle around the z axis. Z-Rotation cos sin sin cos What matrix rotates a vector by 9 o? Apply this rotation to the unit vectors,,. Section 3 Lorentz Transform 3

4 9/8/7 Energy-Momentum Four-Vector We construct a four-dimensional vector as follows: The Lorentz There are two non-accelerating observers, and. And there is an object both see moving. If we know the velocity measured by, what is the velocity measured by? The Lorentz There are three velocities!, velocity of as measured by, velocity measured by, velocity measured by The Lorentz There are three velocities!, velocity of as measured by, velocity measured by, velocity measured by The Lorentz The ation takes a fourvector measured by one observer and gives the four-vector measured by a second observer moving with respect to the first. But first, let s do. 4

5 9/8/7 The Classical There are three velocities!.6, velocity of as measured by.7, velocity measured by? velocity measured by The Classical There are three velocities!.6, velocity of as measured by.7, velocity measured by. velocity measured by The Classical or more generally The Classical We want to rewrite this in terms of fourvectors where according to both observers. The Classical If all we need is, why write The Classical If -- It s much more like what we must do in relativity. Now we apply relativity by magic. 5

6 9/8/7 The Lorentz The Lorentz If The masses are all multiplied by! But there are three different s. What does each equal? v o in the x direction If the Observer is moving in the x direction with velocity v o, the transformation matrix is given by: Synchronization The coordinate systems used by Observer and Observer must be synchronized so that: At t = the origins are at the same point in space. The x, y, and z directions must be the same for both observers. Example Observer sees a spaceship of rest energy E moving at ½ the speed of light in the x direction. Observer moves in the same direction at ¼ the speed of light. What speed does he measure? Example /4 /

7 9/8/7 Example Observer sees a spaceship of rest energy E moving at ½ the speed of light in the y direction. Observer moves in the x direction at ¼ the speed of light. What velocity does he measure? Example General Form If you re interested, the general form of the ation in three dimensions is: The Inverse What is the transformation that takes a fourvector measured by Observer and give us the four-vector measured by Observer? Think about relative motion! Section 4 Space-time Four-vectors Space three-vectors Normal vectors in three spatial dimensions are: So why not just add one dimension of time? Think about units! 7

8 9/8/7 Space-time four-vectors We multiply time by c, the speed of light, to make a vector with units of length. Abraham Facsimile Fig.. Kolob, signifying the first creation, nearest to the celestial, or the residence of God. First in government, the last pertaining to the measurement of time. The measurement according to celestial time, which celestial time signifies one day to a cubit. 5.9/ or 5.7 Frames of Reference There are two non-accelerating observers, and. We wish to define events in and see how they compare to. Length A ruler lying along the x-direction has length L. (L is the rest length.) How long does an observer on the spaceship measure the ruler to be? Define two events in the rest (platform) frame: Event A, one end of ruler at time : Event B would be what? Length: Try This Event A: On the spaceship, we get Event B : so is the ruler longer? What s wrong? Length Event A: On the spaceship, we get Now what do we do? Event B : 8

9 9/8/7 Length If we measure both ends at time on the spaceship, Length So the length is: The length of moving objects contracts by a factor of! Now try doing the width of the ruler. Clocks A clock at the origin of the platform frame ticks once at time and next at time T. How long is it between ticks of the moving clock, measured on the spaceship? Clocks Define two events in the rest (platform) frame: Event A, first tick: Event B, second tick, would be what? The time between ticks is, so moving clocks run slowly. What do we need to know? The definitions of and and their general characteristics. How to make energy-momentum and spacetime four-vectors. Use the ation to see what moving observers measure, but just in simple cases. Know that moving rods contract in the direction of motion, moving clocks tick slowly, moving masses increase all by a factor of. Section 5 Matrices in 9

10 9/8/7 Making a Vector V={,,3} V//MatrixForm {,,3} 3 doesn t distinguish between row and column vectors. Dot Products V={,,3}; V={3,,}; V.V Making a Matrix M={{,,3},{4,5,6},{7,8,9}}; M//MatrixForm Multiplication with Matrices M={{,,3},{4,5,6},{7,8,9}}; P={{3,4,5},{6,7,8},{9,,}}; M.P//MatrixForm V={3,6,9}; M.V//MatrixForm Lorentz x-boost To get β, type ESC beta ESC β =.8; γ = /Sqrt[- β^]; L={{γ,- β γ,,},{- β γ, γ,,},{,,,},{,,,}}; LI={{γ,β γ,,},{β γ, γ,,},{,,,},{,,,}}; L.LI//MatrixForm Section 6 Recap

11 9/8/7 Big Ideas The ation relates how different observers moving at constant speed view the energy and momentum of the same object. Schedule Do Post-Class Quiz #8 Do Pre-Class Quiz #9 HW #7 is due Friday Quiz # is due Saturday Lab #3 is set up, but not due yet! Midterm # is on Oct. 3 in class!