Chapter R2 Synchronizing Clocks

Newtonian time Time was held to be universal. Clocks ran the same in all inertial systems. In these systems if one system is moving at velocity β with respect to the home system, then the velocity of an object shot with a velocity v x in the home system would be v x – β in the other inertial system

All our experience tells us that time is Newtonian, but is it? In 1873, James Clerk Maxwell combined the work of many people in the area of electricity and magnetism in four equations, known as “Maxwell’s equations” These completely and accurately describe all electrical and magnetic phenomena.

Maxwell’s equations We will study these equations in detail in Unit “E”. One result is to give the velocity of all electromagnetic waves as a constant that depends on the electric and magnetic properties of free space. This is true for all inertial reference systems so this means that the speed of light must be identical in all inertial systems.

The problem of the “Ether”. The ether was thought to be the substance that carried electromagnetic waves in the same way that air carries sound. Experiments (Mickelson-Morley) showed that there was no ether, that pure space “vacuum” was capable of carrying the electromagnetic waves (light is an electromagnetic wave).

Which is true, the principle of relativity or Newtonian time? If Maxwell’s equations are valid in all inertial coordinate systems, then the speed of light must be the same in all systems. Our Galialean principle of relativiy tells us that If a spaceship traveling a velocity β away from us receives a pulse of light sent from the home (at rest) system, the speed of light measured by the ship should be c - β! c is the speed of light in the home system. Home β c

What do we need to change to keep the principle of relativity? There are three possibilities Maxwell’s equations are wrong. Principle of relativity is wrong The Galilean (Newtonian) velocity transformation equations are wrong Einstein simply asked, “What would be necessary to keep Maxwell’s equations and the Principle of Relativity?” That is all there is to the theory of relativity!

What is wrong with the Galilean velocity transformations? The way time is measured! Time cannot be absolute! To keep the speed of light the same in all inertial systems, time must depend on β, the relative speed with which the inertial systems are moving. Clocks in different inertial systems must run at different rates to keep the speed of light constant in all systems. We need a way to synchronize clocks in each system that is constituent with Maxwell’s equations, and with the principle of relativity.

How do we synchronize our clocks so that the speed of light is always “c” (299,792,458 m/s) in our system? A light at the origin flashes in all directions. A clock at any place in the reference system is considered synchronized if it gives the time as d/c where d is the distance the clock is from the origin.

SR units We define a unit of distance called the light- second This is the distance light travels in one second 299,792,458 m Example: A clock at the origin of an inertial reference system reads 7.3 s. After traveling 4.3 s, it reaches another clock. What time will this clock read? 11.6 seconds (if the clocks are synchronized.) Problem: 25 km = ? Seconds (SR units?) 83 μs

Spacetime diagrams To keep the spacetime diagram two- dimensional we will restrict ourselves to motion in the x direction. The y axis will be the time axis. These diagrams will be used to mark “events” (a certain place, a certain time). The path of the system is called its “world line”.

Spacetime diagrams x t World line of an object at rest World line of an object traveling at constant velocity World-line of an accelerating object. Slope of the lines is 1/v x If the units of the x-axis are light seconds, the slope of light is always 1.

Spacetime diagram of Earth rotating around the sun. This is a 3 dimensional curve. A helix.

Spacetime diagram of two light flashes moving toward one another. The slope of the light world line is always 1. s s The flashes meet at a distance of 1 s and at a time of 2½ s. (distance) (time)

Problems R2B.1(a,b), R2B.4 and R2B.7 Due Monday