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# INTRO TO ACTUARIAL SCI MATH 489

Texas A&M

GPA 3.6

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This 8 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 489 at Texas A&M University taught by Stephen Fulling in Fall. Since its upload, it has received 21 views. For similar materials see /class/226008/math-489-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 489 Secs 201 and 501 Spring 2008 Christoffel Symbols and Curvature Tensors for Two Classic Geometries March 28 Calculate the Christoffel symbols for the cosmological metric dr2 1 W r2 d62 r2 sin2 a d 2 d52 7dt2 Rt2 where ROS is an arbitrary twice differentiable positive function and k is an arbitrary constant Cf pp 3247325 of Schutz April 4 Calculate the Christoffel symbols for the static spherically symmetric metric d2 7621 dt2 62ATd7 2 7 2 d62 1 7 2 sin 62 d 2 where 10 and r are arbitrary functions Cf Exercise 635 of Schutz Exercise 1120 is a special case April 11 Calculate the Riemann tensor for the cosmological metric April 18 Calculate the Riemann tensor for the static spherically symmetric metric Other announcements March 6 Colloquium by Andrew Strominger String Theory Black Holes and the Fun damental Laws of Nature 400 pm in ENPH 202 Attendance not mandatory of course April 9 Test through Chapter 8 and possibly part of Chapter 12 to be decided later April 14 No class read the article on Topology and the Cosmic Microwave Background77 by Janna Levin Physics Reports 365 2002 2517333 Math 489 Secs 201 and 501 Spring 2008 Special Relativity and Electromagnetism The following problems composed by Prof P B Yasskin will lead you through the construction of the theory of electromagnetism in special rela tivity Please write your response as a connected essay similar to a chapter of a textbook If possible use TEX or a word processor so that you can make revisions easily See the course handout for due dates You may consult books and have discussions with other students but outright copying except from the problems themselves when appropriate is not allowed We regard spacetime as the vector space R4 with a Lorentz signature metric pseudo inner product Thus if we choose the orthonormal basis to be 6017050507 61 07150707 62 05071707 53 0707051 so that all indices run from 0 to 3 and the dual basis to be 60 then the metric is 71 0 0 0 a 0 1 0 0 n 710436 8 63 where 71043 0 0 1 0 0 0 0 1 and the inverse metric is 71 0 0 0 1 a a 0 1 0 0 n 1n ea e3 where 713 0 0 1 0 0 0 0 1 Derivatives Will be denoted by 8a 7 and indices Will be raised and lowered using 71 83W and n l ln problems 1711 we will study the electromagnetic field which is the 2 form 0 7E1 7E2 7E3 E1 0 Egg 732 E2 733 0 31 E3 32 731 0 the electromagnetic potential which is the 1 form A Ao o where Aa A1A2A3 and the electromagnetic current which is the vector J Jaea where JO p J1J2J3 F F043 90 93 where Fa H N to 5 QTY Page 2 Show that the rank 3 tensor Son37 BVFa l 804F137 l 813F704 is totally antisymmetric and hence is a 3 form Hint There are three pairs of indices to transpose Show that this implies that the components of S are all zero except for those for which 04 and y are distinct Write out the components of the equations aVFa BQFM a Fw 0 and a Fa 4an to see that these are the Maxwell equations 0 E4Wp xE8t1 0 x i t 4wj Write out the components of the equations Fa 80133 7 83140 to find expressions for E and g in terms of b and Note b is the negative of the usual scalar potential Show in 4 dimensional notation that if Fa 80133 7 83140 is satis ed then BVFO BQFM 813F704 0 is automatically satisfied Note The 3 dimensional version of these equations is a pair of identities 6 6 gtlt 14f 0 and 6 gtlt W 0 Consequently this subset of the Maxwell equations is actually an identity it is sometimes referred to as the electromagnetic Bianchi identity Substitute Fa 804113 7 8310 into the remaining Maxwell equation a Fai 4an to obtain the Maxwell equation for Al 9 gt1 00 D O Page 3 Let X be a function Also let 141 Al Bax and F 801A 7 81314 This is called a gauge transformation Relate F to F043 to see that the electromag netic field is gauge invariant Given Aa show that there always exists a function X such that A satisfies BO A 0 Note You may assume that it is always possible to solve a wave equation with an arbitrary but speci ed source Perhaps you can find a reference for this fact Write out the Maxwell equations for A to see that the Maxwell equations may always be solved for any arbitrary but speci ed current JO Write out the components of the equation BQJO 0 to see that this is conservation of electric charge Show that if a Fai 4an is satis ed then BQJO 0 is automatically satisfied This is sometimes referred to as an automatic conservation law Write out the function a 7F75FV5 in terms of E and This function is called the Lagrangian density for the vacuum electromagnetic field It is sometimes interpreted as the difference between the kinetic energy and the potential energy lBl2 Also write out the Lagrangian density in terms of b and Write out the components of the tensor 1 5 T043 E FO VFi i na FV F75 in terms of E and g to see that this consists of the electromagnetic energy density momentum density energy current and momentum current or stress This tensor is called the Maxwell energy momentum stress tensor 11 Page 4 ln 4 dimensional notation compute the divergence of the energy momentum tensor and use the Bianchi identities and the Maxwell equations to show that a Ta iJ Fa current is DODZSIO Problem 11 shows that the electromagnetic energy momentum is not conserved if the The reason for this is that we have ignored the energy momentum of the charged particles producing the current In problem 12 we study the motion of a charged particle Then in problem 13 we study the energy momentum tensor of a uid of charged particles 12 H to A particle of mass m with electric charge 1 is moving on the parametrized path 017 where 739 is the proper time Consequently it has unit timelike tangent vector 8x0 U Uo ea where U0 E mvvlmvaws and where 1 l 7 17m2 Further its 4 momentum is pa mUO Write out the components of the equations 353090 q U Faa to obtain the Lorentz force and power laws Hints Don7t expand pa Be careful with the factors of y Consider a uid of charged particles of rest mass m and charge 1 with uid velocity U01 and energy density p in the instantaneous local rest frame Then the charge density in the instantaneous local rest frame is 7p and the electromagnetic current is m JO i p UO m Assuming that the particles are non interacting except for their electromagnetic forces then i the energy momentum tensor for the uid is that for dust Til3 pU j U uid 7 Page 5 ll eaC par 1C 8 mOVeS aCCOr lng 0 e Oren Z OrCe equa lOn quot h t39 l d39 t th L t f t39 U BBWUQ qU FQB iii the energy momentum tensor for the electromagnetic field is Til FmF v 1 Ct 471 f7 F SF39M iv and the electromagnetic field satis es the Bianchi identities and the Maxwell equations with current JO Then as seen in problem 8 the electromagnetic current is conserved BQJO 0 and as seen in problem 11 the electromagnetic energy momentum tensor satisfies a Tgf iJ Fa Now use the Lorentz force equation and the conservation of electromagnetic current to show that the uid energy momentum tensor satisfies a Tf d J Fa Hint Factor Tf i d as l3 Tflluid Ua PU3 and use the product rule Thus the total energy momentum is conserved 813ng1 T31 039 ln problems 14 and 15 we study the behavior of the electromagnetic field under rotations and Lorentz boosts Under a general Lorentz transformation A0 7 the electro magnetic field transforms according to We then write FM FV5A 1VaA 153 0 7E1 7E2 7E3 0 7E1 7E2 7E E1 0 133 7132 E1 0 133 7132 F75 E2 7B3 0 B1 and Fa E2 iBS 0 Bl E3 B2 7131 0 ES 132 7131 0 14 First assume that the Lorentz transformation is a rotation about the z axis 5 0 0 0 1 cos l sin l 0 A0 7 0 R where RZ j 7 sin l cos l 0 0 0 0 Show that E and g transform as vectors Ri jEJ39 and Bi Ri jBJ39 Show that this generalizes to arbitrary rotation matrices Page 6 15 Now assume that the Lorentz transformation is a boost in the z direction With velocity 17 v z cosh A 0 0 sinh A 0 1 0 0 V 0 0 1 0 sinh A 0 0 cosh A 1 7 7 V where coshA y v2 and sinhA i M i Find expressions for E 1 L and B in terms of E and g and either A or v v V1712 ln problems 16 and 17 we study the Lagrangian and Hamiltonian formulations of electromagnetism Each problem begins With a discussion of the analogous formulation of classical mechanics and the situation for a general field theory With elds 1AA for A l N Then the special case of electromagnetism is treated With 1AA replaced by Al 16 In classical particle mechanics the Lagrangian is L T i V ml17l2 i Vf dz In discussing this Lagrangian it is useful to regard a and 17 a as independent variables One then computes 8L BL 8112 i iiiV and pi i 801 mvi The quantity pi is called the momentum conjugate to Then the EuleriLagrange equations for this Lagrangian are 2ampM dt 811i 815i 0 Or d Which is Newton7s equation With the force identified as E BiV the gradient of the potential Similarly in field theory in discussing a Lagrangian density it is useful to regard the fields 1AA and their derivatives BOAZJA as independent variables One then computes B B 7 and 7r 0 7 W A 8804114 gt1 Page 7 The quantity 77A E 7TA0 is the conjugate momentum to wA while the 4 vector 71340 is sometimes called the conjugate multimomentum to 1AA Then the EuleriLagrange equations for the Lagrangian density are B 8a 7 B aiJA Both sets of EuleriLagrange equations given above can be derived from appro priate variational principles In this exercise we apply the field theory version to the vacuum Maxwell Lagrangian density B Wio z i FV SFw Compute B B 7 d 043 7 9A0 an 7T 90310 Identify the conjugate momenta to A0 b and to AZ 710 7 a0 BBQla Compute the EuleriLagrange equations a B B i 0 3 90310 9A0 and verify that these are the vacuum Maxwell equations Hints Explicitly write out all metrics in but do not use 04 or Q as dummy indices At the first step do not expand F in terms of derivatives of A Use the chain rule Then compute the derivatives of F using formulas such as 885A 883140 626 In classical mechanics the Hamiltonian is H pivi 7 Lf17 but expressed as a function of f and 17 Hij wyy 2 P a 7 7 V T V m 2m 2m Similarly for a field theory the Hamiltonian density is H Mam1A 7 5W aw but expressed as a function of wA 814 4 and WA For electromagnetism express the Hamiltonian density H 7704801404 7 Aa B Aa as a function of Al 81140 and the non zero components of 7r0 Then express H as a function of E and B to see that the Hamiltonian density H is equal to the energy density T00

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