Simplifying Einstein's Thought Experiments
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The Special Theory of Relativity Lecture 16
The Special Theory of Relativity Lecture 16 E = mc2 Albert Einstein Einstein’s Relativity • Galilean-Newtonian Relativity • The Ultimate Speed - The Speed of Light • Postulates of the Special Theory of Relativity • Simultaneity • Time Dilation and the Twin Paradox • Length Contraction • Train in the Tunnel paradox (or plane in the barn) • Relativistic Doppler Effect • Four-Dimensional Space-Time • Relativistic Momentum and Mass • E = mc2; Mass and Energy • Relativistic Addition of Velocities Recommended Reading: Conceptual Physics by Paul Hewitt A Brief History of Time by Steven Hawking Galilean-Newtonian Relativity The Relativity principle: The basic laws of physics are the same in all inertial reference frames. What’s a reference frame? What does “inertial” mean? Etc…….. Think of ways to tell if you are in Motion. -And hence understand what Einstein meant By inertial and non inertial reference frames How does it differ if you’re in a car or plane at different points in the journey • Accelerating ? • Slowing down ? • Going around a curve ? • Moving at a constant velocity ? Why? ConcepTest 26.1 Playing Ball on the Train You and your friend are playing catch 1) 3 mph eastward in a train moving at 60 mph in an eastward direction. Your friend is at 2) 3 mph westward the front of the car and throws you 3) 57 mph eastward the ball at 3 mph (according to him). 4) 57 mph westward What velocity does the ball have 5) 60 mph eastward when you catch it, according to you? ConcepTest 26.1 Playing Ball on the Train You and your friend are playing catch 1) 3 mph eastward in a train moving at 60 mph in an eastward direction. -
Physics 200 Problem Set 7 Solution Quick Overview: Although Relativity Can Be a Little Bewildering, This Problem Set Uses Just A
Physics 200 Problem Set 7 Solution Quick overview: Although relativity can be a little bewildering, this problem set uses just a few ideas over and over again, namely 1. Coordinates (x; t) in one frame are related to coordinates (x0; t0) in another frame by the Lorentz transformation formulas. 2. Similarly, space and time intervals (¢x; ¢t) in one frame are related to inter- vals (¢x0; ¢t0) in another frame by the same Lorentz transformation formu- las. Note that time dilation and length contraction are just special cases: it is time-dilation if ¢x = 0 and length contraction if ¢t = 0. 3. The spacetime interval (¢s)2 = (c¢t)2 ¡ (¢x)2 between two events is the same in every frame. 4. Energy and momentum are always conserved, and we can make e±cient use of this fact by writing them together in an energy-momentum vector P = (E=c; p) with the property P 2 = m2c2. In particular, if the mass is zero then P 2 = 0. 1. The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time di®erence between the events according to an observer moving at u = 0:8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0:8c. Answer: According to the formula for a Lorentz transformation, ³ u ´ 1 ¢tobserver = γ ¢tEarth-Sun ¡ ¢xEarth-Sun ; γ = p : c2 1 ¡ (u=c)2 Plugging in the numbers gives (notice that the c implicit in \light-minute" cancels the extra factor of c, which is why it's nice to measure distances in terms of the speed of light) 2 min ¡ 0:8(8:3 min) ¢tobserver = p = ¡7:7 min; 1 ¡ 0:82 which means that according to the observer, event B happened before event A! If we reverse the sign of u then 2 min + 0:8(8:3 min) ¢tobserver 2 = p = 14 min: 1 ¡ 0:82 2. -
A Tale of Two Twins
A tale of two twins L.Benguigui Physics Department and Solid State Institute Technion-Israel Institute of Technology 32000 Haifa Israel Abstract The thought experiment (called the clock paradox or the twin paradox) proposed by Langevin in 1911 of two observers, one staying on earth and the other making a trip toward a Star with a velocity near the light velocity is very well known for its very surprising result. When the traveler comes back, he is younger than the stay on Earth. This astonishing situation deduced form the theory of Special Relativity sparked a huge amount of articles, discussions and controversies such that it remains a particular phenomenon probably unique in Physics. In this article we propose to study it. First, we lookedl for the simplest solutions when one can observe that the published solutions correspond in fact to two different versions of the experiment. It appears that the complete and simple solution of Møller is neglected for complicated methods with dubious validity. We propose to interpret this avalanche of works by the difficulty to accept the conclusions of the Special Relativity, in particular the difference in the times indicated by two clocks, one immobile and the second moving and finally stopping. We suggest also that the name "twin paradox" is maybe related to some subconscious idea concerning conflict between twins as it can be found in the Bible and in several mythologies. 1 Introduction The thought experiment in the theory of Relativity called the "twin paradox" or the "clock paradox" is very well known in physics and even by non-physicists. -
A Short and Transparent Derivation of the Lorentz-Einstein Transformations
A brief and transparent derivation of the Lorentz-Einstein transformations via thought experiments Bernhard Rothenstein1), Stefan Popescu 2) and George J. Spix 3) 1) Politehnica University of Timisoara, Physics Department, Timisoara, Romania 2) Siemens AG, Erlangen, Germany 3) BSEE Illinois Institute of Technology, USA Abstract. Starting with a thought experiment proposed by Kard10, which derives the formula that accounts for the relativistic effect of length contraction, we present a “two line” transparent derivation of the Lorentz-Einstein transformations for the space-time coordinates of the same event. Our derivation make uses of Einstein’s clock synchronization procedure. 1. Introduction Authors make a merit of the fact that they derive the basic formulas of special relativity without using the Lorentz-Einstein transformations (LET). Thought experiments play an important part in their derivations. Some of these approaches are devoted to the derivation of a given formula whereas others present a chain of derivations for the formulas of relativistic kinematics in a given succession. Two anthological papers1,2 present a “two line” derivation of the formula that accounts for the time dilation using as relativistic ingredients the invariant and finite light speed in free space c and the invariance of distances measured perpendicular to the direction of relative motion of the inertial reference frame from where the same “light clock” is observed. Many textbooks present a more elaborated derivation of the time dilation formula using a light clock that consists of two parallel mirrors located at a given distance apart from each other and a light signal that bounces between when observing it from two inertial reference frames in relative motion.3,4 The derivation of the time dilation formula is intimately related to the formula that accounts for the length contraction derived also without using the LET. -
Quantum Mechanical Laws - Bogdan Mielnik and Oscar Rosas-Ruiz
FUNDAMENTALS OF PHYSICS – Vol. I - Quantum Mechanical Laws - Bogdan Mielnik and Oscar Rosas-Ruiz QUANTUM MECHANICAL LAWS Bogdan Mielnik and Oscar Rosas-Ortiz Departamento de Física, Centro de Investigación y de Estudios Avanzados, México Keywords: Indeterminism, Quantum Observables, Probabilistic Interpretation, Schrödinger’s cat, Quantum Control, Entangled States, Einstein-Podolski-Rosen Paradox, Canonical Quantization, Bell Inequalities, Path integral, Quantum Cryptography, Quantum Teleportation, Quantum Computing. Contents: 1. Introduction 2. Black body radiation: the lateral problem becomes fundamental. 3. The discovery of photons 4. Compton’s effect: collisions confirm the existence of photons 5. Atoms: the contradictions of the planetary model 6. The mystery of the allowed energy levels 7. Luis de Broglie: particles or waves? 8. Schrödinger’s wave mechanics: wave vibrations explain the energy levels 9. The statistical interpretation 10. The Schrödinger’s picture of quantum theory 11. The uncertainty principle: instrumental and mathematical aspects. 12. Typical states and spectra 13. Unitary evolution 14. Canonical quantization: scientific or magic algorithm? 15. The mixed states 16. Quantum control: how to manipulate the particle? 17. Measurement theory and its conceptual consequences 18. Interpretational polemics and paradoxes 19. Entangled states 20. Dirac’s theory of the electron as the square root of the Klein-Gordon law 21. Feynman: the interference of virtual histories 22. Locality problems 23. The idea UNESCOof quantum computing and future – perspectives EOLSS 24. Open questions Glossary Bibliography Biographical SketchesSAMPLE CHAPTERS Summary The present day quantum laws unify simple empirical facts and fundamental principles describing the behavior of micro-systems. A parallel but different component is the symbolic language adequate to express the ‘logic’ of quantum phenomena. -
Time Dilation and Length Contraction Shown in Three-Dimensional Space-Time Frames
TIME DILATION AND LENGTH CONTRACTION SHOWN IN THREE-DIMENSIONAL SPACE-TIME FRAMES Tower Chen Division of Mathematical Sciences University of Guam e-mail:[email protected] Zeon Chen e-mail: zeon [email protected] (Received 9 December 2008; accepted 18 February 2009) Abstract In classical Newtonian physics, space and time are abso- lute quantities. Hence, space and time can be treated inde- pendently and discussed separately. With his theory of rel- ativity, Einstein proved that space and time are dependent and must not be treated separately. To maintain this in- terdependency and inseparability of space and time, a three- Concepts of Physics, Vol. VI, No. 2 (2009) 221 dimensional space-time frame, where time is embedded into a spatial coordinate system, has been conceptualized. Utiliz- ing this new type of frame, the concepts of time dilation and length contraction can easily be visualized with intuitive, ge- ometric graphs. The proposed three-dimensional space-time frame is an alternate mathematical frame that can be used to describe the motion of objects. It may also prove to be a use- ful tool in facilitating the understanding of Special Relativity and providing additional insights into space and time. 222 Concepts of Physics, Vol. VI, No. 2 (2009) TIME DILATION AND LENGTH CONTRACTION... 1 Introduction In classical Newtonian physics, the concepts of space and time are absolute. Space is composed of three orthogonal dimensions, and time is represented as a fourth dimension, perpendicular to each of the spatial axes. Both space and time are free to be discussed in- dependent of the other. -
The Special Theory of Relativity
The Special Theory of Relativity Chapter II 1. Relativistic Kinematics 2. Time dilation and space travel 3. Length contraction 4. Lorentz transformations 5. Paradoxes ? Simultaneity/Relativity If one observer sees the events as simultaneous, the other cannot, given that the speed of light is the same for each. Conclusions: Simultaneity is not an absolute concept Time is not an absolute concept It is relative How much time does it take for light to Time Dilation Travel up and down in the space ship? a) Observer in space ship: 2D ∆t = proper time 0 c b) Observer on Earth: speed c is the same apparent distance longer ν 2 = v∆t Light along diagonal: 2 D2 + 2 2 D2 + v2∆t 2 / 4 c = = ∆t ∆t 2D ∆t = c 1− v2 / c2 ∆t This shows that moving observers ∆t = 0 = γ∆t 2 2 0 must disagree on the passage of 1− v / c time. Clocks moving relative to an observer run more slowly as compared to clocks at rest relative to that observer Time Dilation Calculating the difference between clock “ticks,” we find that the interval in the moving frame is related to the interval in the clock’s rest frame: ∆t ∆t = 0 1− v2 / c2 ∆t0 is the proper time (in the co-moving frame) It is the shortest time an observer can measure 1 with γ = ∆t = γ∆t 1− v2 / c2 then 0 Applications: Lifetimes of muons in the Earth atmosphere Time dilation on atomic clocks in GPS (v=4 km/s; timing “error” 10-10 s) On Space Travel 100 light years ~ 1016 m If space ship travels at v=0.999 c then it takes ~100 years to travel. -
Lecture 17 Relativity A2020 Prof. Tom Megeath What Are the Major Ideas
4/1/10 Lecture 17 Relativity What are the major ideas of A2020 Prof. Tom Megeath special relativity? Einstein in 1921 (born 1879 - died 1955) Einstein’s Theories of Relativity Key Ideas of Special Relativity • Special Theory of Relativity (1905) • No material object can travel faster than light – Usual notions of space and time must be • If you observe something moving near light speed: revised for speeds approaching light speed (c) – Its time slows down – Its length contracts in direction of motion – E = mc2 – Its mass increases • Whether or not two events are simultaneous depends on • General Theory of Relativity (1915) your perspective – Expands the ideas of special theory to include a surprising new view of gravity 1 4/1/10 Inertial Reference Frames Galilean Relativity Imagine two spaceships passing. The astronaut on each spaceship thinks that he is stationary and that the other spaceship is moving. http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ Which one is right? Both. ClassMechanics/Relativity/Relativity.html Each one is an inertial reference frame. Any non-rotating reference frame is an inertial reference frame (space shuttle, space station). Each reference Speed limit sign posted on spacestation. frame is equally valid. How fast is that man moving? In contrast, you can tell if a The Solar System is orbiting our Galaxy at reference frame is rotating. 220 km/s. Do you feel this? Absolute Time Absolutes of Relativity 1. The laws of nature are the same for everyone In the Newtonian universe, time is absolute. Thus, for any two people, reference frames, planets, etc, 2. -
Length Contraction the Proper Length of an Object Is Longest in the Reference Frame in Which It Is at Rest
Newton’s Principia in 1687. Galilean Relativity Velocities add: V= U +V' V’= 25m/s V = 45m/s U = 20 m/s What if instead of a ball, it is a light wave? Do velocities add according to Galilean Relativity? Clocks slow down and rulers shrink in order to keep the speed of light the same for all observers! Time is Relative! Space is Relative! Only the SPEED OF LIGHT is Absolute! A Problem with Electrodynamics The force on a moving charge depends on the Frame. Charge Rest Frame Wire Rest Frame (moving with charge) (moving with wire) F = 0 FqvB= sinθ Einstein realized this inconsistency and could have chosen either: •Keep Maxwell's Laws of Electromagnetism, and abandon Galileo's Spacetime •or, keep Galileo's Space-time, and abandon the Maxwell Laws. On the Electrodynamics of Moving Bodies 1905 EinsteinEinstein’’ss PrinciplePrinciple ofof RelativityRelativity • Maxwell’s equations are true in all inertial reference frames. • Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s =300,000km/s = 300m/µs . • Therefore, light travels at speed c in all inertial reference frames. Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other. Special Theory: Inertial Frames: Frames do not accelerate relative to eachother; Flat Spacetime – No Gravity They are moving on inertia alone. General Theory: Noninertial Frames: Frames accelerate: Curved Spacetime, Gravity & acceleration. Albert Einstein 1916 The General Theory of Relativity Postulates of Special Relativity 1905 cxmsms==3.0 108 / 300 / μ 1. -
The Pole Vaulter and the Barn Picture a Pole Vaulter Running Towards a Long Barn (With Doors at Each End) at Maximum Speed
Special Relativity Dr. Kevin Kelly Saturday Morning Physics October 5th, 2019 Rewind back to the 1890s ▸ Let’s take a minute and think about what we (they) knew about the world in the late 1800s. Rewind back to the 1890s ▸ Let’s take a minute and think about what we (they) knew about the world in the late 1800s. ▸ Newton (1687): Gravity governs the motion of bodies on earth as well as in space. Newton’s Principia also gave us his famous “three laws” for motion of objects. Rewind back to the 1890s ▸ Let’s take a minute and think about what we (they) knew about the world in the late 1800s. ▸ Newton (1687): Gravity governs the motion of bodies on earth as well as in space. Newton’s Principia also gave us his famous “three laws” for motion of objects. ▸ Thermodynamics (1700s/1800s): contributions of many physicists. Determined how heat transfer works, led to the development of engines, etc. Rewind back to the 1890s ▸ Let’s take a minute and think about what we (they) knew about the world in the late 1800s. ▸ Newton (1687): Gravity governs the motion of bodies on earth as well as in space. Newton’s Principia also gave us his famous “three laws” for motion of objects. ▸ Thermodynamics (1700s/1800s): contributions of many physicists. Determined how heat transfer works, led to the development of engines, etc. ▸ Electricity & Magnetism (1700s/1800s): a collection of newly-discovered laws were determined to be related, brought together as “Maxwell’s Equations” In terms of describing what we see around us, and how we interact with the world, we have a pretty good handle on everything! This is a VERY famous quote, often attributed to Lord Kelvin (one of the titans of Thermodynamics). -
Albert Einstein - Wikipedia, the Free Encyclopedia Page 1 of 27
Albert Einstein - Wikipedia, the free encyclopedia Page 1 of 27 Albert Einstein From Wikipedia, the free encyclopedia Albert Einstein ( /ælbərt a nsta n/; Albert Einstein German: [albt a nʃta n] ( listen); 14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics.[2] He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect". [3] The latter was pivotal in establishing quantum theory within physics. Near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led to the development of his special theory of relativity. He Albert Einstein in 1921 realized, however, that the principle of relativity could also be extended to gravitational fields, and with his Born 14 March 1879 subsequent theory of gravitation in 1916, he published Ulm, Kingdom of Württemberg, a paper on the general theory of relativity. He German Empire continued to deal with problems of statistical Died mechanics and quantum theory, which led to his 18 April 1955 (aged 76) explanations of particle theory and the motion of Princeton, New Jersey, United States molecules. He also investigated the thermal properties Residence Germany, Italy, Switzerland, United of light which laid the foundation of the photon theory States of light. In 1917, Einstein applied the general theory of relativity to model the structure of the universe as a Ethnicity Jewish [4] whole. -
Introducing Quantum and Statistical Physics in the Footsteps of Einstein: a Proposal †
universe Article Introducing Quantum and Statistical Physics in the Footsteps of Einstein: A Proposal † Marco Di Mauro 1,*,‡ , Salvatore Esposito 2,‡ and Adele Naddeo 2,‡ 1 Dipartimento di Matematica, Universitá di Salerno, Via Giovanni Paolo II, 84084 Fisciano, Italy 2 INFN Sezione di Napoli, Via Cinthia, 80126 Naples, Italy; [email protected] (S.E.); [email protected] (A.N.) * Correspondence: [email protected] † This paper is an extended version from the proceeding paper: Di Mauro, M.; Esposito, S.; Naddeo, A. Introducing Quantum and Statistical Physics in the Footsteps of Einstein: A Proposal. In Proceedings of the 1st Electronic Conference on Universe, online, 22–28 February 2021. ‡ These authors contributed equally to this work. Abstract: Introducing some fundamental concepts of quantum physics to high school students, and to their teachers, is a timely challenge. In this paper we describe ongoing research, in which a teaching–learning sequence for teaching quantum physics, whose inspiration comes from some of the fundamental papers about the quantum theory of radiation by Albert Einstein, is being developed. The reason for this choice goes back essentially to the fact that the roots of many subtle physical concepts, namely quanta, wave–particle duality and probability, were introduced for the first time in one of these papers, hence their study may represent a useful intermediate step towards tackling the final incarnation of these concepts in the full theory of quantum mechanics. An extended discussion of some elementary tools of statistical physics, mainly Boltzmann’s formula for entropy and statistical distributions, which are necessary but may be unfamiliar to the students, is included.