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General Relativity General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Einstein’s Theories of Relativity Special Relativity Reminder: College Physics B - PHY2054C Time Dilation Length Contraction Special & General Relativity General Relativity Relativistic Momentum Mass Equivalence Principle 11/12/2014 My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building College Physics B Outline Einstein’s Theories of Relativity Special Relativity 1 Einstein’s Theories of Relativity Reminder: Time Dilation Special Relativity Length Contraction General 2 Reminder: Time Dilation Relativity Relativistic Momentum Mass Equivalence 3 Length Contraction Principle 4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Galilean Relativity and Light Einstein’s Theories of Relativity Galilean Relativity and electromagnetism do predict different Special Relativity results for observers in different inertial frames: Reminder: Time Dilation Experiments showed that Maxwell’s theory was correct. • Length The speed of light in the vacuum is always c. Contraction • General Galilean relativity for how the speed of light depends on Relativity • Relativistic the motion of the source is wrong. Momentum Mass ➜ Equivalence Einstein developed theory of relativity: Special Relativity. Principle Two Postulates 1 All laws of physics are the same in all inertial reference frames. 2 The speed of light in the vacuum is a constant. College Physics B Inertial Reference Frames Einstein’s Theories of Relativity Special Relativity Reminder: Time Dilation Length Contraction General Relativity Relativistic Momentum Mass The modern definition of an inertial reference is one in Equivalence Principle which Newton’s First Law holds: If a particle moves with a constant velocity, then the reference frame is inertial. ➜ Earth’s acceleration is small enough that it can be ignored (can be considered an inertial system). College Physics B Outline Einstein’s Theories of Relativity Special Relativity 1 Einstein’s Theories of Relativity Reminder: Time Dilation Special Relativity Length Contraction General 2 Reminder: Time Dilation Relativity Relativistic Momentum Mass Equivalence 3 Length Contraction Principle 4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Light Clock Einstein’s Theories of Relativity The two postulates lead to a surprising result concerning the Special Relativity nature of time. Reminder: Time Dilation Length A light clock keeps time by using a pulse Contraction General of light that travels back and forth between Relativity two mirrors: Relativistic Momentum Mass The time for the clock to “tick” once is Equivalence • Principle the time needed for one round trip: College Physics B Moving Light Clock Einstein’s Theories of The clock moves with a constant velocity v relative to the Relativity Special Relativity ground: Reminder: From Ted’s reference frame, the light pulse travels up and Time Dilation • Length down between the two mirrors: ∆t 0 = 2l/c. Contraction General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Moving Light Clock Einstein’s Theories of The clock moves with a constant velocity v relative to the Relativity Special Relativity ground: Reminder: From Ted’s reference frame, the light pulse travels up and Time Dilation • Length down between the two mirrors: ∆t 0 = 2l/c. Contraction Alice sees the light pulse travel a longer distance, but the General • Relativity speed of light is the same for Alice as for Ted. Relativistic Momentum Mass ➜ Because of the longer distance, according to Alice the Equivalence Principle light will take longer to travel between the mirrors. College Physics B Moving Light Clock Einstein’s Theories of Relativity For Alice, the time for one tick of the clock is: Special Relativity Reminder: ∆t Time Dilation ∆t = 0 Length 1 v 2 Contraction q − c2 General Relativity ➜ Relativistic The time for Ted is different from the time for Alice. Momentum Mass Equivalence The operation of the clock depends on the relative motion. Principle College Physics B Time Dilation Einstein’s Theories of Relativity Special Relativity Reminder: Time Dilation Length Contraction General Relativity Relativistic Special relativity predicts that moving clocks run slow. Momentum Mass Equivalence Principle This effect is called Time Dilation. For typical terrestrial speeds, the difference between ∆t and ∆t 0 is negligible. ∆t 0 is called the proper time: ∆t ∆t ∆t ∆t = 0 = 0 0 2 (100 mph)2 1 1 v 1 ≈ q − c2 q − c2 College Physics B Time Dilation Einstein’s Theories of Relativity Special Relativity Reminder: Time Dilation Length Contraction General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Outline Einstein’s Theories of Relativity Special Relativity 1 Einstein’s Theories of Relativity Reminder: Time Dilation Special Relativity Length Contraction General 2 Reminder: Time Dilation Relativity Relativistic Momentum Mass Equivalence 3 Length Contraction Principle 4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Lorentz Contraction Einstein’s Theories of Relativity 1 Special Relativity γ = : 2 2 Reminder: 1 v /c Time Dilation p − Length L0 = v∆t = L = v∆t 0 Contraction 6 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Lorentz Contraction Einstein’s Theories of Relativity Special Relativity Reminder: When measuring the length of the moving meterstick, Time Dilation you do so by noting the positions of the two ends at Length Contraction the same time, according to your clock. General Relativity Relativistic Momentum However, those two events – the two measurements Mass Equivalence you make – do not occur at the same time as seen by Principle the moving observer. In relativity, time is relative, and simultaneity (the idea that two events happen “at the same time”) is no longer a well-defined concept. College Physics B Question Einstein’s Theories of Relativity The short lifetime of muons created in the upper atmosphere Special Relativity of the Earth would not allow them to reach the surface of the Reminder: Time Dilation Earth unless their lifetime increased by time dilation. From Length the reference system of the muons, the muons can reach the Contraction surface of the Earth because: General Relativity Relativistic Momentum A Time dilation increases their velocity. Mass Equivalence Principle B Time dilation increases their energy. C Length contraction decreases the distance to the Earth. D The relativistic speed of the Earth toward them is added to their velocity. College Physics B Question Einstein’s Theories of Relativity The short lifetime of muons created in the upper atmosphere Special Relativity of the Earth would not allow them to reach the surface of the Reminder: Time Dilation Earth unless their lifetime increased by time dilation. From Length the reference system of the muons, the muons can reach the Contraction surface of the Earth because: General Relativity Relativistic Momentum A Time dilation increases their velocity. Mass Equivalence Principle B Time dilation increases their energy. C Length contraction decreases the distance to the Earth. D The relativistic speed of the Earth toward them is added to their velocity. College Physics B Special Relativity Einstein’s Theories of Relativity Special Relativity Reminder: Time Dilation Length Contraction General Relativity Relativistic 1 The speed of light is the maximum possible Momentum Mass speed, and it is always measured to have the Equivalence Principle same value by all observers. 2 There is no absolute frame of reference, and no absolute state of rest. 3 Space and time are not independent, but are unified as spacetime. College Physics B Special Relativity Einstein’s Theories of Relativity Special Relativity Reminder: Time Dilation Length Contraction Relativistic Addition of Velocities: General Relativity Relativistic ′ v1 + v2 Momentum v = v1 v2 Mass 1 + 2 Equivalence c Principle 1 When two velocities are much less than the speed of light, the relativistic addition of velocities gives nearly the same result as the Newtonian equation. ➜ Okay for speeds less than 10 % of the speed of light! ∼ 2 Experiments with particles moving at very high speeds show that the relativistic result is correct. College Physics B Outline Einstein’s Theories of Relativity Special Relativity 1 Einstein’s Theories of Relativity Reminder: Time Dilation Special Relativity Length Contraction General 2 Reminder: Time Dilation Relativity Relativistic Momentum Mass Equivalence 3 Length Contraction Principle 4 General Relativity Relativistic Momentum Mass Equivalence Principle College Physics B Momentum Einstein’s Theories of Relativity Conservation of momentum is one of the most fundamental Special Relativity conservation rules in physics and is believed to be satisfied by Reminder: Time Dilation all the laws of physics, including the theory of special relativity. Length Contraction According to Newton’s mechanics, a particle with a mass m0 General moving with speed v has a momentum given by: Relativity Relativistic Momentum Mass p = m0 v Equivalence Principle College Physics B Momentum Einstein’s Theories of Relativity Conservation of momentum is one of the most fundamental Special Relativity conservation rules in physics and is believed to be satisfied by Reminder: Time Dilation all the laws of physics, including the theory of special relativity. Length Contraction According to Newton’s mechanics, a particle with a mass m0 General moving with speed v has a momentum given
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