Special relativity

Remember the Michelson and Morley experiment in 1887? experiment

Michleson and Morley thought it was a failure because it could not detect the aether wind, but the most important conclusion of this experiment is: The speed of the light is not influenced by the motion of earth (that moves at a considerable speed of ~100,500 kmh= 27,916 m/sec around the sun)

Lots of scientists from all over the world tried to explain this phenomenon…

Henry Poincare’ JulesJules Henri Poincaré ( 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as The Last Universalist since he excelled in all fields of the discipline during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, “Every simply connected, closed 3 manifold is homeomorphic to the 3-sphere.”simply connectedclosed manifoldhomeomorphic3-sphere Which was solved in 2002–2003. In his research on the three-body problem, Poincaré became the first person to lay the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Hendrik Lorentz (1853–1928) in He obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity. The Poincaré group used in physics and mathematics was named after him.

George Fitzgerald George Francis FitzGerald (3 August 1851 – 22 February 1901) was an Irish professor of "natural and experimental philosophy" (i.e., physics) at Trinity College in Dublin, Ireland, during the last quarter of the 19th century. He is known for his work in electromagnetic theory and for the Lorentz–FitzGerald contraction, which became an integral part of Einstein's Special Theory of Relativity. The FitzGerald crater on the far side of the Moon is named for him..Trinity CollegeLorentz–FitzGerald contractionSpecial Theory of RelativityFitzGerald craterfar side of the Moon

Sir Joseph Larmor Sir Joseph Larmor (11 July 1857 – 19 May 1942) was a British physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influential work was Aether and Matter, a theoretical physics book published in Larmor proposed that the aether could be represented as a homogeneous fluid medium which was perfectlyincompressible and elastic. Larmor believed the aether was separate from matter.physicistdynamicsthermodynamicsaetherhomogeneousfluidincompressibleelastic Parallel to the development of Lorentz ether theory, Larmor published the Lorentz transformations in thePhilosophical Transactions of the Royal Society in 1897 some two years before Hendrik Lorentz (1899, 1904). In 1903 he was appointed Lucasian Professor of Mathematics at Cambridge, a post he retained until his retirement in Lorentz ether theoryLorentz transformationsPhilosophical Transactions of the Royal SocietyHendrik LorentzLucasian Professor of Mathematics

Woldemar Voigt Woldemar Voigt (2 September 1850 – 13 December 1919) was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 1914 by Peter Debye, who took charge of the theoretical department of the Physical Institute. In 1921, Debye was succeeded by Max Born. Voigt worked on crystal physics, thermodynamics and electro-optics. His main work was the Lehrbuch der Kristallphysik (textbook on crystal physics), first published in He discovered the Voigt effect in The word tensor in its current meaning was introduced by him in Voigt profile and Voigt notation are named after him. He was also an amateur musician and became known as a Bach Expert. In 1887 Voigt formulated a form of the Lorentz transformation between a rest frame of reference and a frame moving with speed v in the x direction. However, as Voigt himself said, the transformation was aimed at a specific problem and did not carry with it the idea of a general coordinate transformation, as is the case in relativity theory.

Hendrik Lorentz Hendrik Antoon Lorentz (18 July 1853 – 4 February 1928) was a Dutch physicist who won the 1902 Nobel Prize in Physics. He was born in Arnhem, (The Netherlands). In 1875 Lorentz earned a doctoral degree. In 1878, at only 24 years of age, Lorentz was appointed to the newly established chair in theoretical physics at the University of Leiden, in the Netherlands. Arnhem

In 1895, with the attempt to explain the Michelson-Morley experiment, Lorentz proposed that moving bodies contract in the direction of motion (as other scientists had proposed before him) While working on describing electromagnetic phenomena (the propagation of light) in reference frames that moved relative to each other he discovered that the transition from one to another reference frame could be simplified by using a new time variable which he called local time. Lorentz's publications made use of the term “local time” without giving a detailed interpretation of its physical relevance.

… bute that was the beginning of the theory of relativity …

At the time of Newton and Galileo, the laws of motion were simple...

First law (or law of inertia). Every object continues in its state of rest, or of uniform motion in a straight line, unless compelled to change that state by external forces acted upon it. This law does not change if motion occurs in aframe of reference that moves with constant velocity (that is, no acceleration). Such frame is called inertial

V’p=V T +Vp If the train moves with constant velocity v T, and a passenger inside the train walks with velocity vp (say, in the direction of the motion of the train) the conductor on the platform will see the passenger speeding by him at velocity v’ p = v T +v p.

Δ’s= v T Δt + Δs The displacement of the passenger inside the train in the interval of time Δt is, (according to an observer on the ground) Δ’s= v’ p Δt = (v T +v p ) Δt So if v p Δt = Δs is the displacement of the passenger inside the train, the relation between Δ’s and Δs is Δ’s= v T Δt + Δs

Example: a train travels with a constant velocity of 100 km/h. A passenger walks inside the train with velocity of 1 km/h in the direction of the motion of the train. What is the displacement of the passenger according to an observer on the platform of the train in 1 min? How would your answer change if the passenger moves in the opposite direction of the train?

Answer. According to the conductor, the velocity of the passenger relative to the sidewalk is 101 km/h. So, if Δt= 1 minute= 1/60 hr, the displacement is 1.68 km. If the passenger moves in the opposite direction of the train, its velocity relative to the sidewalk is 99 km/h. So his displacement is 99x 1/60= 1.65 km

Example: a train stops in the station, and after loading the passengers, it starts moving with initial velocity of 6 km/h. A passenger is still talking to a friend standing on the sidewalk, and when the train moves, she walks in the train with velocity 2 km/h so to keep talking to her friend. How fast and in which direction should the friend on the sidewalk move so to keep talking to the passenger on the train?

Answer: The problem does not say in which direction the passenger on the train moves, but since she wants to talk to a friend in the station, she has to go toward the station! So her friend will see her moving with velocity v’ p = 6-2= 4km/h away from him, and if he wants to keep talking to her, he should run at 4 km/ h in the direction of the train.

To summarize: The velocity of a passenger in a train with respect to the sidewalk is v’ p = v T + v p Where v T is the velocity of the train and vp is the velocity of the passenger in the train.. The displacement of a passenger with respect to the sidewalk is Δ’s = v T Δt + Δs. The same equation are valid also in 2 or 3 dimension. We only need to use vectors instead of scalars.

In these problems we assume that the time elapsed in the train is the same as the time elapsed in the station. That is, Δt= time elapsed in the train= Δt’= time elapsed in the station.

You’ve never thought that there is any difference between the time measured on the surface of earth and the time measured inside a moving vehicle, right?

But Michelson and Morrey discovered that the speed of light c is the same if measured in the direction of the motion of earth and in the direction opposite to the motion of earth …

So, what’s going on here? … The earth is like a train carrying a passenger (a beam of light). The speed of light measured in the direction of the motion of the earth should be c+ ve, (Ve= Velocity of Earth) and in the direction perpendicular to the motion of the earth should be c. But instead it is c both times.

A better example are the headlight and the tail light of a train in motion. You will see the light travelling at the same speed, regardless if the train comes toward you or goes away from you.

Einstein’s Postulates (1905) First Postulate –The laws of physics are the same in any inertial frame of reference (principle of relativity) Second Postulate –The speed of light in vacuum is the same in all inertial frames That is, the speed of light is c (~3x10 8 m/s) and is independent of the motion of the source For example, If the light comes from the headlight of a train moving at with velocity v Newton: speed =c+vEinstein: speed still =c !!! c v

From now on, Δ’t and Δ’s are the time and the displacement of a moving object P on the “train” (i.e., an inertial frame that moves with constant speed v T ) measured by an observer in the station, (i.e., from a stationary inertial frame) and Δt and Δs are the time and the displacement of a moving object measured on the train (i.e., within the moving inertial frame)

For small velocities, v’ p = v T + v p that is Δ’s/ Δ’t= v T + Δs/ Δt But if P moves at the speed of light c, it should be Δ’s/ Δ’t = v’ p = v T + c =c Δs/ Δt= Δ’s/ Δ’t = c So, either Δ’t has to change, or Δ’s has to change, or both have to change.

Time dilation: Δ’t > Δt That is, time in the station appears to pass slower than on the train, or: an observer at the station that looks at a moving clock on the train measures a longer time on her watch than an observer on the train. Let’s prove this: v observer in the train Observer at the station light mirror both observers agree that light is travelling at speed=c but they disagree on the distance (path) the light has travelled S’S Observer S: Time taken d

Observer S’ L L Vt’ d The train travels with velocity v. The distance travelled by light is 2L So: the time taken by the light to come back to the mirror (measured in S’) is By Pythagoras, From these equations In S, Solving both equations for d^ 2 and equating

Vt’ d=c/(2t) Hence, time measured by an observer in S’ (the station) So, t’ > t because the “ correction factor” Hence, we can rewrite time as Is always >1 L=c/(2t’)

Example: b-quark decay The b-quark is an unstable sub-atomic particle. The b quark travels ~ 4mm at 0.99c so, (  =9) before decaying When the particle travels, its life span (i.e., as observed in S’) 4mm Image reconstructed by DELPHI particle physics Experiment at CERN (1 pico-s is 1x10-12 s) However, the average lifetime at rest (i.e., in S) of a b-quark is t=1.5 ps The discrepancy is explained by the time dilation