University:
Boston University
Course:
CAS PY 521 | Electromagnetic Theory I
Academic year:

2023
Views:

381

Pages:

21
Author:

granitt3c2r

Special Theory of Relativity Outline Introduction Relativity The speed of light Einstein’s Solution Time Space Conclusion Introduction The Special Theory of Relativity is the theory put forth by Albert Einstein in his 1905 paper, “On the Electrodynamics of Moving Bodies” “Special” because it applies the theory of relativity to inertial reference frames in flat space-time (no gravity) Relative Motion •Who is moving? •How do we know? •Up until Einstein we used addition and subtraction to calculate the observed speed to other objects Relative Motion Motion is relative as long as you are not accelerating! There is no experiment that you can do in an inertial reference frame that will distinguish it from another inertial reference frame vobject − ground = vobject −train + vtrain − ground The Speed of Light 670 million miles per hour (in a vacuum) 186,000 miles per second Light always moves at the same speed You can never catch a beam of light! Postulates of Special Relativity 1. 2. “The laws of electrodynamics and optics will be valid for all frames of reference in which the laws of mechanics hold good (non-accelerating frames).” “Light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body; here the velocity of light c is defined as the two-way velocity, determined with a single clock.” Simultaneous Events Events that are simultaneous for observers in one reference frame are not necessarily simultaneous to observers in another reference frame Time Dilation– Light Clock What is time? What is a clock? c = 3.0 x 108 m/s mirror Separation h = 15 cm Period = Ts = 1 billionth of a second h mirror Moving Light Clock •What happens if the light clock moves with a velocity v? vTm 2 vTm 2 h ⎛ vT ⎞ L1 = h + ⎜ m ⎟ ⎝ 2 ⎠ 2 The ball travels a distance: 2 ⎛ vT ⎞ L2 = h + ⎜ m ⎟ ⎝ 2 ⎠ 2 2 ⎛ vT ⎞ LT = 2 h 2 + ⎜ m ⎟ ⎝ 2 ⎠ 2 Time Dilation ⎛ vTm ⎞ 2 2 h +⎜ ⎟ L ⎝ 2 ⎠ Tm = T = c c Tm = 2h c2 − v2 2 2h Ts = c Time Dilation Tm = Tm = γTs Ts v2 1− 2 c γ= From one inertial reference frame it seems as though a clock moving in another frame ticks slower! 1 v2 1− 2 c What about Length? v2 Lm = L0 1 − 2 c Space-Time We can think of 4 dimensions, three space and time Objects move through space-time at the speed of light! If an object is “stationary” its motion is used to travel through time If the object “moves” in space, it travels more slowly through the time dimension Relative Motion vobject − ground vobject −train + vtrain − ground = vobject −train vtrain − ground 1+ c2 What happens if the velocity of the object with respect to the train is the speed of light? vobject − ground = c What is a Muon? Semi-stable fundamental particle Muon mass is 207 times the mass of the electron Decays into a positron and two neutrinos Evidence of Time Dilation - Muons Non relativistic Evidence of Time Dilation - Muons Relativistic Earth Frame Evidence of Time Dilation - Muon Relativistic Muon Frame Twin Paradox Twins – Earth Man and Rocket Man Rocket Man takes off at a velocity 60 % of the speed of light Rocket Man travel out 12 hours (by his watch) and then turns around and returns to Earth at the same speed that he left with. When Rocket Man returns to Earth he has aged 24 hours. How much has Earth Man Aged? Answer: Earth Man has aged 30 hours! 2 E=mc E = mc = 2 m= m0 c 2 v2 1− 2 c Erest = m0 c m0 2 v2 1− 2 c p = γm0 v 2 E= p 2c 2 + m 2c 4