Announcement Course webpage http://highenergy.phys.ttu.edu/~slee/2402/ Textbook PHYS-2402 Lecture 6 Quiz 1: Thursday 30 min; Chap.2 Feb. 3, 2015 HW1 (due 2/10) 20, 26, 36, 41, 46, 51, 55, 70, 76, 87, 92, 111 Doppler Effect Chapter 2 Special Relativity 1. Basic Ideas 6. Velocity Transformation 2. Consequences of Einstein’s Postulates 7. Momentum & Energy 3. The Lorentz Transformation Equations 8. General Relativity & a 1 st Look at Cosmology 4. The Twin Paradox 9. The Light Barrier 5. The Doppler Effects 10. The 4 th Dimension
Relativistic Dynamics E = INTERNAL 2 ENERGY E INTERNAL = mc Outline: • Relativistic Momentum • Relativistic Kinetic Energy E = TOTAL • Total Energy ENERGY • Momentum and Energy in Relativistic Mechanics 2 • General Theory of Relativity 2 2 2 E p c ( mc ) = + Expressions for (total) Energy Kinetic Energy = KE and Momentum of a particle of mass m , moving at velocity u
# " ! # # " " ! ! u u u # # " " ! ! u u Is there Absolute Causality? Might cause precede effect in one reference frame but effect precede cause in different reference frame(s)? e.g. can someone see you first die, and then see you get born?
Let’s assume that the order of events is Is there Absolute Causality? changed in some reference frame S’ ! t > 0 Might cause precede effect in one reference frame but effect precede cause in different t ' 0 ! < reference frame(s)? Is that Possible? e.g. can someone see you first die, and then ! ! ! ! t and ! ! t’ are the time intervals between ! ! see you get born? the same two events observed in S and S’, respectively Let’s assume that the order of events is Using Lorentz transformations…. changed in some reference frame S’ & # v $ ! t ' x t ! = # " ! + ! v % 2 " c ! t > 0 t ' 0 ! < & # v x ! $ ! t ' t 1 ! = # ! " + v % 2 " c ! t Is that Possible? t 0 ! > if then ! ! ! t and ! ! ! ! ! t’ are the time intervals between t ' 0 ! < the same two events observed in S and S’, respectively
Using Lorentz transformations…. Using Lorentz transformations…. & # & # v v $ ! $ ! t ' x t t ' x t ! = # " ! + ! ! = # " ! + ! v v % 2 " % 2 " c c & # & # v x v x ! ! $ ! $ ! t ' t 1 t ' t 1 ! = # ! " + ! = # ! " + v 2 > c 2 ?? v v % 2 " % 2 " c ! t c ! t & # & # v ! x v ! x $ ! $ ! " + 1 < 0 " + 1 < 0 % 2 " % 2 " t 0 t 0 ! > c t ! > c t ! ! if then if then 2 2 c c t ' 0 t ' 0 ! < ! < ! x > ! t ! x > ! t Impossible v v YES Is there Absolute Causality? Relativistic Dynamics Might cause precede effect in one reference Some Examples NO frame but effect precede cause in different reference frame(s)? e.g. can someone see you NO first die, and then see you get born?
Problems Problems 2 2 2 2 4 E p c m c = + 1. What is the momentum of an electron with K = mc 2 ? 1. What is the momentum of an electron with K = mc 2 ? 2 2 E mc 2 K ! + " ! " 2 2 2 2 2 2 2 2 p m c m c 4 m c m c 3 mc = − = − = − = $ % $ % c c & ' & ' 2. How fast is a proton traveling if its kinetic energy is 2/3 of its total energy? 2. How fast is a proton traveling if its kinetic energy is 2/3 of its total energy? 2 2 ( 2 ) 2 K E mc K E 3 mc = = + = 2 3 3 1 V 1 8 ! " 3 1 V c = − = = 2 mc $ % c 9 3 2 E 1 V c / & ' = ( ) − 2 1 ( V c / ) − Problem Problem An electron initially moving with momentum p=mc is passed through a retarding potential difference An electron initially moving with momentum p=mc is passed through a retarding potential difference of V volts which slows it down; it ends up with its final momentum being mc/2 . (a) Calculate V in of V volts which slows it down; it ends up with its final momentum being mc/2 . (a) Calculate V in volts. (b) What would V have to be in order to bring the electron to rest? volts. (b) What would V have to be in order to bring the electron to rest? 2 2 2 2 2 ( 2 ) ( 2 ) ( 2 ) 2 (a) p=mc : E p c mc mc mc 2 mc = + = + = 1 2 1 5 ! " 2 E mc 2 ( mc 2 ) mc 2 p=mc/2 : = + = # $ 2 2 2 % & 5 ! " 2 2 ( 6 ) 5 E E E 2 mc 0.3 mc 0.3 0.5 10 eV 1.5 10 eV Δ = − = − ≈ ≈ × = × ' ( 1 2 ' ( 2 ) * 5 V 1.5 10 V Thus, the retarding potential difference = × ( ) (b) 2 2 2 5 5 E 2 mc E mc E 2 1 mc 2.1 10 eV V 2.1 10 V = = Δ = − ≈ × = × 1 2
Problem Problem An unstable particle of mass m moving with velocity v relative to an inertial lab RF disintegrates into An unstable particle of mass m moving with velocity v relative to an inertial lab RF disintegrates into two gamma-ray photons. The first photon has energy 8 MeV in the lab RF and travels in the same two gamma-ray photons. The first photon has energy 8 MeV in the lab RF and travels in the same direction as the initial particle; the second photon has energy 4 MeV and travels in the direction direction as the initial particle; the second photon has energy 4 MeV and travels in the direction opposite to that of the first. Write the relativistic equations for conservation of momentum and opposite to that of the first. Write the relativistic equations for conservation of momentum and energy and use the data given to find the velocity v and rest energy, in MeV, of the unstable energy and use the data given to find the velocity v and rest energy, in MeV, of the unstable particle. particle. photon 2 photon 1 photon 2 photon 1 E E mv momentum ph 1 ph 2 = − conservation c c 2 1 v c / after after ( ) before before − 2 mc energy E E = + ph 1 ph 2 conservation 2 1 ( v c / ) − mvc (a) E E 8 MeV 4 MeV 4 MeV = − = − = ph 1 ph 2 v ( ) a 4 1 2 1 ( v c / ) − = = = c ( ) b 12 3 mc 2 E E 8 MeV 4 MeV 12 MeV (b) = + = + = ph 1 ph 2 2 1 v c / ( ) − 2 ( ) mc 2 E E 1 v c / 12 MeV 1 1/ 9 11.3 MeV ( ) ( ) = + − = − ≈ ph 1 ph 2 Problem A moving electron collides with a stationary electron and an electron-positron pair comes into being General Relativity as a result. When all four particles have the same velocity after the collision, the kinetic energy required for this process is a minimum. Use a relativistic calculation to show that K min =6mc 2 , where m is the electron mass. p 4 p 1 2 • General relativity is the geometric theory of gravitation published by 2 E mc 4 E energy conservation + = 1 2 Albert Einstein in 1916. before after p 4 p = momentum conservation 1 2 1 2 ! 2 " • It is the current description of gravitation in modern physics. 2 2 2 2 ( 2 ) ( 2 ) 2 ( E ) 2 E mc mc 16 ( E ) 16 mc ( p c ) 2 + + = = + E 2 ( mc 2 ) p c ( ) = + 1 1 2 # 1 $ 16 1 1 % & 2 2 2 2 2 2 2 ( 2 ) 2 ( 2 ) ( 2 ) E mc ( p c ) ( E ) ( p c ) 2 E mc mc 16 mc = + − + + = • It unifies special relativity and Newton's law of universal gravitation, 2 2 1 1 1 2 ( 2 ) and describes gravity as a geometric property of space and time. mc 2 E mc 4 E + = 1 2 2 ( 2 ) 2 2 2 2 E 14 mc / 2 mc 7 mc K E mc 6 mc = − = = = p 4 p = 1 1 1 1 2 • In particular, the curvature of space-time is directly related to the four-momentum (mass-energy and momentum). In the center-of-mass RF: 2 ( 2 ) 2 E ' 4 mc 2 E 1 ' ' mc 2 mc ' 2 = = γ = → γ = 1 p 1 ' p 1 ' v ' 2 v ' ( ) 1 3 relative speed V = 1 v ' c − = → = • The relation is specified by the Einstein’s field equations, 2 2 c 4 2 before after a system of partial differential equations. (graduate level course) 2 mc v ' V 3 4 3 48 + K mc 2 6 mc 2 v = − = = = = = 1 ( 2 ) 1 3/ 4 7 49 1 vv c '/ 1 ( 48/ 49 ) + + −
General Relativity Special Theory of Relativity The two postulates: • Many predictions of general relativity differ significantly from those of classical physics. • Examples of such differences include gravitational time dilation, the gravitational red-shift of light, and the gravitational time delay. • General relativity's predictions have been confirmed in all observations and experiments to date. Accellerating BUT : earth frames • However, unanswered questions remain, solution is the quantum gravity!! ----- sounds quite complicate.. ? Special Theory of Relativity: Special Theory of Relativity: Deals exclusively with globally INERTIAL FRAMES - v = constant General Theory of Relativity: General Theory of Relativity: locally locally Deals also with Accelerating - LOCALLY INERTIAL FRAMES
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