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MCDB 110

by: Erika Kuvalis

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MCDB 110 MCDB 110

Erika Kuvalis
UCSB
GPA 3.82

Staff

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This 3 page Class Notes was uploaded by Erika Kuvalis on Thursday October 22, 2015. The Class Notes belongs to MCDB 110 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/226907/mcdb-110-university-of-california-santa-barbara in Molecular, Cellular And Developmental Biology at University of California Santa Barbara.

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Date Created: 10/22/15
Physics 110C Relativity Notes Michael Johnson April 11 2008 1 Motivation The goal of relativity is to reformulate physical principles so that they are independent of our choice of coordinates However it is observed that the speed of light is constant in all inertial frames so we have to be a little careful about what we mean by the sought coordinate independence Fully understanding the answer requires some differential geometry but the result is that we seek physical laws which are invariant under a particular class of coordinate transformations Lorentz transformations To understand these we must rst introduce the notion of a 4Vectori 2 4Vectors Vectors are usually de ned as quantities that transform in the same way as coordinates 11 12 13 In relativity we make time a coordinate as well and multiply by c in order to obtain consistency between units Thus an event is spacetime is described by the coordinates I 10 1112 13 ct z y We also need to introduce a metric which tells us how lengths and distances behave in our space The metric is just a generalization of the ordinary dot product only it is written as a tensor In the class we will be primarily interested in flat spacetime the domain of special relativity In this case the metric tensor is universally denoted 77 and has the following matrix representation 1 77 0 7 0 77 7 1 0 HOOD 0 1 0 0 GOD The last thing we need to de ne is our transformation group which is the set of all Lorentz transformations These are de ned as transformations which obey ATnA 77 Solutions can be classi ed as ordinary spatial rotations which arenlt very interesting and boosts we also need to include re ections for full generality Letls put these in a form that youlve seeni Suppose you want to transform a stationary frame into one where all coordinates are identical except that the modi ed frame is moving at velocity v BC in the positive xdirection with respect to the rst The corresponding Lorentz transformation tensor is 7 776 0 0 H 776 7 0 0 AV 0 0 10 0 0 01 Where 7 This allows us to write the ordinary Lorentz transformations between coordinate systems in the following way I Ag 1 Here llve used the Einstein summation convention of implicitly summing over all repeated indices This concisely tells us how position vectors change under Lorentz transformations We extend this concept by saying that a general vector v 1quot is any quantity that transforms as vVIM AZUVW Such a 1 one with an upper index is called contravariant There is another type of vector that transforms in a slightly different way wVz l Ali10 1quot In this case 10 is said to be a covariant vector Example The 4velocity of a particle is a vector but for it to transform properly it picks up factors of the Lorentz factor 7 lf17 is the ordinary velocity then we have v 757 717 3 Tensors A tensor is just a generalization of the familiar concepts of scalars vectors and matrices A scalar is a rank0 tensor a vector is a rank1 tensor a matrix is a rank2 tensor and so forth A scalar is just a quantity which is invariant under a particular type of transformation in special relativity it is a quantity which is invariant under Lorentz transformations As we saw above vectors can either be covariant or contravariant and are de ned by how they transform A rankn tensor has n indices each of which can either be up contravariant or down covariant In general a tensor of rank or type pq is an object with p contravariantly transforming indices superscripts and q covariantly transforming indices subscripts Example The Electromagnetic Stress Energy Tensor T043 is a tensor of type 20 Suppose we want to transform T043 using a particular Lorentz transformation AZ We must transform each index so we get the following equation T 3 A A2 T043 Note The above equation involves summation over a and B which is why they don7t appear on the left Also all these letters are just dummy indices the letters themselves have no importance Finally it is crucial that we keep track of whether indices are up or down since changing them gives different equations as we shall see later 4 Index Gymnastics Although it initially appears confusing unneccessary and cumbersome the index notation is ac tually used to make things simpler By following a set of simple rules it is possible to do tedious calculations in a way that requires no deep understanding of the mathematics Here are the basic ones to remember Index Rules 1 Repeated indices must come in updown pairs and are summed 2 The same index cannot appear more than twice in an equation the summation is ambiguous 3 lndices appear in the same position up or down on both sides of an equation 4 lndices may be raised or lowered using the metric tensor 77 Rather than dwell on the meaning of these let s just do a few quick examples Example Letls lower the indices of the electromagnetic stressenergy tensor T043 To do this we just use the equation T043 nmmgyTW Note how this obeys all the above rules One nice thing about index notation is that we donlt have to worry about noncommutativity since everything is implicitly written as a sum of terms which do commute So we can start by summing over the 11 index which just looks like matrix multiplication between new and TM Recall that 6E2i32 S S S 71 0 0 0 n 0 1 0 0 TM 1 7T 7T 7T W 0 0 1 0 5y iTy iTyy 7T 0 0 0 1 S 7sz 7sz 7T2 75E2 71032 is is is 7T 7T 7T V W x w my 1 gt Ta GMT Sy iTyT iTyy iTyl 5 sz sz 7T2 Now to lower the other index we must multiply by 77 but our summed index is the second one on T One way to get around this is just move our equation to make it look like matrix multiplication again T043 770M133 NWTWW NWTWM Tofwa Here llve used the fact that 77 is symmetric to swap the indices Now we just multiply matrices 7 6E271032 75 75y 75 71 0 0 0 S 7T 7T 7T 0 1 0 0 Ta Taym x m 1y 1 3 3 5y iTy iTyy 7T 0 0 1 0 S 7T 7sz 7T 0 0 0 1 6E2 32 is is is is 7T 7T1 7T 043 x m x m T Sy Tyr Tyy Tyz is 7sz 7sz 7T

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