PHYS-3301 Lecture 4. Chapter 2. Announcement. Sep. 7, Special Relativity. Course webpage Textbook

Transcription

1 Announement Course webage htt:// Textbook PHYS-330 Leture 4 HW (due 9/4 Chater 0, 6, 36, 4, 45, 50, 5, 55, 58 Se. 7, 07 Chater Seial Relativity. Basi Ideas. Consequenes of Einstein s Postulates 3. The Lorentz Transformation Equations 4. The Twin Paradox 5. The Doler Effets 6. Veloity Transformation 7. Momentum & Energy 8. General Relativity & a st Look at Cosmology 9. The Light Barrier 0. The 4 th Dimension

2 PARADOX: Seemingly absurd or self-ontraditory thought, often true statement (Oxford Ditionary The Key: Round Tri

3 The Key: Round Tri ACCELERATION Change of inertial frames Minkowski diagram of the twin aradox. There is a differene between the trajetories of the two twins: the trajetory of the shi is equally divided between two different inertial frames, while the Earth-based twin stays in the same inertial frame Veloity Transformation We may now relate veloity in different frames. We know that the lassial transformation u = u-v is wrong. The orret one is a straightforward aliation of the Lorentz transformation eq. S (x,u,t S (x,u,t u u u,u = veloity of an objet moving relative to a frame; quantities (osition, veloity, time have different value in different frame * We reserve the symbol v exlusively for the relative seed between the two frames u v + u Classial transformation is wrong

4 Lorentz Transformations NOTE that oordinates orthogonal to the diretion of motion stay the same What about y and z oordinates? (x - diretion of motion NO (Lorentz Length Contration in diretions other than along the diretion of relative motion Lorentz Transformations Lorentz Transformations Relativisti veloity Transformations Relativisti veloity Transformations u x, u y u z BUT ALL Comonents of the Veloity Vetor Transform BUT ALL Comonents of the Veloity Vetor Transform WHY?

5 Beause: The time transforms indeendently of the diretion of motion, oordinates do not, and veloity ombines both u (veloity of an objet in frame S is the differential dislaement in that frame divided by the differential time interval in that frame dt / dt u

6 Parallel to the Diretion of Relative motion Orthogonal to the Diretion of Relative motion ux = Classial Limit? (both u and v << O.K. ux = uy = 0 uz = 0

7 The Doler Effet The Doler Effet So far we learned about the transformation of: sae oordinates and time veloity Now, let s study the relativisti transformation of frequeny (how does the light aear in a moving referene frame? So far we learned about the transformation of: sae oordinates and time veloity Now, let s study the relativisti transformation of frequeny (how does the light aear in a moving referene frame? Doler Effet Seial Case: = 0 The soure of light moves away from the observer (shift to lower frequenies Red Shift

8 f = = β = f β T T 0 Seial Case: = 90 deg. The soure of light moves erendiular with reset to the observer 0 Transvere Red Shift The origin of the transverse Doler effet is time dilation, this is a ure relativisti effet, no ounterart in lassial mehanis. = m v Newton s nd Law: Relativisti Momentum No lassial analog the momentum of a artile, m is invariant (does not deend on the veloity F = d dt = m d v dt = m a exressed in terms of 3-vetors, invariant under G.Tr. (but not L.Tr. Outline: Relativisti Dynamis Relativisti Momentum Relativisti Kineti Energy Total Energy Momentum and Energy in Relativisti Mehanis General Theory of Relativity Next Week Quantum Physis Relativisti Kineti Energy Relativisti form of the nd Law (introdued by Einstein: F = d = γ mv dt = d m dt v ( where = m v v = γm v definition of the momentum in relativisti mehanis β = V / Examle: Calulate the momentum of an eletron moving with a seed of ( 0.98 me = 4.9m e 0.98 By ignoring relativisti effets, one would get = 0.98m e

9 Relativisti Kineti Energy In relativisti mehanis, the onet of energy is more useful than the fore : d K = F dl = dt dl = d dl = vd mv dt ( = # vd % $ (integration by arts x dy = xy y dx, x = v y = + * mv v f vdv # m = % d v / v / 0 v / $ mv, + m v /. v / - # m % % $ v f = v f / & + v v f / f / ( ( m = ' m K m m v / = = 0 ( γ mv & ( = v / ' ( = vdv mv f v f / + m m v f / m mv = v / & ( = v / ' v f / m = v=v f v=0 kineti energy of a artile of the mass m moving with seed v Relativisti Kineti Energy Show that E = + m 4 follows from =γ u mu and E = γ u m for momentum and energy in terms of m and u E E x x y y E z z = INVARIANT = m = E' = ( m x ' y ' 4 z ' What about E = m E = + ( m? E = + ( m Revolutionary Conet E = TOTAL ENERGY E = + (m

10 E = INTERNAL ENERGY (when = 0 E = m E = INTERNAL ENERGY E INTERNAL = m? E = TOTAL ENERGY E = + (m E = TOTAL ENERGY E = + (m Exressions for (total Energy and Momentum of a artile of mass m, moving at veloity u Classial Limit mu << mu E m +

11 Classial Limit Kineti Energy = KE mu << NEW mu E m + FAMILIAR kineti energ Energy Matter Energy Matter Atomi Bomb (Chater 0: Energy is CREATED From the Mass of Nulei (Internal energy is transformed into kineti energy

12 # u " # " # u # u # "