 # Special Relativity Einsteins Postulates Special Relativity Postulates Special

• Slides: 25 Special Relativity: Einstein’s Postulates Special Relativity: Postulates • Special Relativity: One of 2 major (revolutionary!) advances in happened in the early 20 th Century! • The other is Quantum Mechanics, of course! • Of the 2, Quantum Mechanics is more relevant, by far, to everyday life & it also has spawned many more physics subfields. • However (in my opinion), Special Relativity is by far the most elegant & “beautiful” of the 2. In a (relatively) simple mathematical formalism, It unifies classical mechanics with E&M! • The historical reasons that Einstein & others developed Special Relativity, the history of its development & eventual acceptance by physicists are interesting. • But (due to time) we will discuss this only briefly. • The philosophical implications of Special Relativity, the various “paradoxes” it seems to have, etc. are interesting. • But (due to time) we’ll discuss this only briefly. • In our classical mechanics review, we’ve stressed that Newton’s Laws of Motion are valid only in an Inertial Reference Frame: • Defined by Newton’s 1 st Law. • A frame that is not accelerating with respect to the “fixed stars”. Any frame moving with constant velocity with respect to an inertial frame is also an inertial frame! The Galilean Transformation Galilean Relativity! • Consider 2 coordinate systems: 1. Frame S, with time & space coordinates (t, x, y, z) & • 2. Frame S´, with time & space coordinates (t´, x´, y´, z´). • Frame S´ moves relative to S with constant velocity v in the +x direction. See figure • It must be true that t´= t, x´ = x - vt, y´= y, z´ = z • Transforming from one coordinate system to another in this way is called a Galilean Transformation. • Newton’s Laws of Motion are unchanged by a Galilean Transformation (t´= t, x´ = x - vt, y´= y, z´ = z) • Newton’s 2 nd Law: F = (dp/dt) F´ = (dp´/dt´) An implicit Newtonian assumption is that t´= t • In the equations of motion, the time t is an independent parameter, playing a different role in classical mechanics than the coordinates x, y, z. • Newtonian Mechanics: Frame S´ is moving relative to Frame S with constant velocity v in the +x direction. Let u = velocity of a particle in Frame S, u´ = velocity of the particle in Frame S´. A Galilean Transformation Gives u´ = u - v • By Contrast: in Special Relativity, the position coordinates x, y, z & time t are on an equal footing. • Physicists of the late 1800’s knew that Newton’s Laws of motion are unchanged by a Galilean transformation. • But, they also required that not just Newton’s Laws but ALL physical laws should be unchanged by a Galilean transformation from one inertial coordinate system to another. • In addition to Newton’s Laws of Motion, in classical physics, Maxwell’s Equations (ME) are used to describe Electromagnetic phenomena. • Also, ME contain a universal constant c The speed of Light in Vacuum • It can be shown that ME are NOT unchanged by a Galilean Transformation! • That is, ME & Newton’s Laws of motion ARE NOT COMPATIBLE! • In addition, it was assumed that, like mechanical waves, light waves must travel through a medium that was called the “Ether” which permeates all space. • They also reasoned that if this Ether exists, then it ought to be possible to detect the Earth’s motion through it. For example, the Earth should “drag” through the Ether & that should slow Earth down in it’s orbit. (The Ether “wind” theory). The Ether “Wind” Theory. • The Earth travels at a speed of more than v = 105 km/hour around the sun. If ether exists, the Earth moving through it would cause a "wind" in the same way that there seems to be a wind outside a moving car. • To a person in the car, the air outside the car would seem like a moving substance. In the same way, ether should seem like a moving substance to things on Earth. Michelson-Morley Experiment 1887 • Michelson & Morley did an experiment to try to prove theory that the ether existed. They did this by optical means with a device called an interferometer • The interferometer was designed to measure the speed and direction of the "ether wind" by measuring the difference between the speed of light traveling in different directions. Michelson-Morley Experiment 1887 • The interferometer was supposed to measure the speed & direction of the "ether wind" by measuring the difference between the speed of light traveling in different directions. It measured this difference by shining a beam of light into a mirror that was only partially coated in silver. Part of the beam would be reflected one way, and the rest would go the other. • Those two parts would then be reflected back to where they were split apart, and recombined. By looking at interference patterns in the recombined beam of light, any changes in speed because of the ether wind could be seen. They found that there was, in fact, no substantial difference in the measurements. That is, they failed to detect the ether! • This failure to detect the ether was very puzzling to the physics community at the time, & led to the creation of various new theories to explain the result. • The most important was the Lorentz Factor, which is used in Albert Einstein's special theory of relativity. Einstein Either Newtonian Mechanics or ME need to be modified. He modified Newtonian Mechanics. Einstein’s 2 Postulates Þ 1. THE POSTULATE OF RELATIVITY: The laws of physics are the same to all inertial observers. • This is the same as in Galilean-Newtonian Classical mechanics! Einstein’s 2 Postulates 2. THE POSTULATE OF THE CONSTANCY OF THE SPEED OF LIGHT: The speed of light, c, is independent of the relative motion of its source. • A revolutionary idea! Requires modifications of Newton’s mechanics at high speeds. Einstein’s 2 Postulates 1. RELATIVITY 2. CONSTANT LIGHT SPEED Definition: A Covariant Theory Any formulation of physics which satisfies both 1 & 2. The speed of light c is the same in all inertial coordinate systems. Einstein’s 2 Postulates 1. RELATIVITY 2. CONSTANT LIGHT SPEED • A Covariant Theory Any formulation of physics which satisfies both 1 & 2. • 1 & 2 tell us that we should consider Space & Time as 2 aspects of a single Spacetime. A 4 d geometric framework (“Minkowski Space”) • So, the division of space & time is different for different observers. Einstein’s 2 Postulates 1. RELATIVITY 2. CONSTANT LIGHT SPEED • These two hypotheses result in a mechanics in which Space & Time are considered 2 aspects of a single Spacetime. A 4 d geometric framework. The division of space & time is different for different observers. Einstein’s 2 Postulates 1. RELATIVITY 2. CONSTANT LIGHT SPEED • These two hypotheses result in a mechanics in which The meaning of “simultaneity” is different for different observers. • Space & time get “mixed up” in transforming from one inertial frame to another.