Lecture 22 Special theory of relativity Galileo’s principle of relativity Michelson-Morley experiment Speed of light is special Lorentz transformation

Galileo’s principle of relativity In Galileo’s principle, the space and time are separate The time is absolute and the same in all inertial frames. 𝑡= 𝑡 ′ The space transforms according to 𝑥 ′ =𝑥−𝑢𝑡 𝑦 ′ =𝑦 𝑧 ′ =𝑧 If the primed system is moving toward the positive x direction

Consequence The velocity of light is different in different frames Suppose that it has velocity c in one frame, it will have 𝑐 ′ =𝑐−𝑢 in another frame The velocity has no top limit, it can be very very large 1 googolplex( 10 10 100 ) km/sec

Speed of light The speed of light is a very large number It was first measure by Galileo, before 1638. The first quantitative estimate of the speed of light was made in 1676 by Romer. From the observation that the periods of Jupiter’s innermost moon lo appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth’s orbit to obtain an estimate of speed of light of 2200000km/s, 26% lower than the actual value.

electromagnetism In Electromagnetism, Maxwell unified all theories in 1860’s, wrote done a set of fundamental equations: Maxwell equation. The Maxwell Eq. has a remarkable prediction: electromagnetic wave and its speed just equal to the speed of light So it was concluded that the light is an electromagnetic wave!

Speed of light wave According to the wave theory, a wave needs a medium to propagate. Furthermore, it is in the frame that the media is at rest, the wave speed can be calculated from the fundamental theory. What is the medium that supports the propagation of electromagnetic wave? Assume there is an “ether” In the “ether”, the light travels with THE speed of light.

Earth travelling in ether The Earth’s cannot be static in ether, otherwise, it will be to remarkable. Thus the speed of light travelling vertically will be different from that travelling horizontally. This difference can be measured, must be at least 30km/s. Michelson and Morley experiment was designed to make that measurement.

Michelson-Morley experiment (1887)

Einstein’s hypothesis Michelson-Morley experiment did not find any noticeable motion of the Earth relative to ether! This negative result prompted Einstein to make the following assumption: The speed of light is the largest speed of all motion, and is the same in all inertial frame. Furthermore, he made another additional assumption replacing Galileo’s relativity All laws of physics is the same in all inertial frames. This the foundation of special relativity.

Lorentz transformation Using Einstein’s assumption, one can derive the law of transformation 𝑥 ′ = 𝑥−𝑢𝑡 1− 𝑢 2 / 𝑐 2 𝑦 ′ =𝑦 𝑧 ′ =𝑧 𝑡 ′ = 𝑡−𝑢𝑥/ 𝑐 2 1− 𝑢 2 / 𝑐 2 This was first derived by H. A. Lorentz who showed that the Maxwell equation is invariant under this transformation!