# PHY6200 W07

## Chapter 15: Special Relativity

In Taylor, read sections 15.1 - 15.13 for this week and 15.13 to 15.18 next week.

Special relativity can be treated at a number of levels. I will try to present special relativity in a way that is consistent with what is done in general relativity. General relativity is beyond the scope of most undergraduate curricula, but many people are interested to learn the basics, and I hope this approach to special relativity makes that easier.

The difference between special and general relativity is that general relativity deals with gravity while special relativity doesn't. This seems somewhat trivial, but incorporating gravity brings a whole mathematical framework and numerous interesting phenomena.

### The Postulates of Special Relativity

The consequences of special relativity can be derived from a couple of laws, and ingenious thinking. First, a definition of an inertial frame that is more appropriate for this work.

#### Definition of an Inertial Frame

An inertial frame is any reference frame (that is, a system of coordinates x, y, z and time t) in which all the laws of physics hold in their usual form.

This definition is more precise than Newton's first law, as it includes the laws of electromagnetism and quantum mechanics as well as mechanics. Of course, these other laws of physics arrived well after Newton's time.

#### First Law of Relativity

If S is an inertial frame and if a second frame S' moves with constant velocity relative to S, then S' is also an inertial frame.

#### Second Law of Relativity

The speed of light (in vacuum) has the same value c in every direction in all inertial frames.

This rule has been tested to considerable precision.

### The Lorentz Transformation

In E&M we find the wave equation for light to be the pair of partial differential equations

∇²E = (1/c²)(∂²E/∂t²)    and    ∇²B = (1/c²)(∂²B/∂t²)

or, limiting ourselves to light propagating along the z direction

∂²Ex/∂z² = (1/c²)(∂²Ex/∂t²)    and    ∂²Bx/∂z² = (1/c²)(∂²Bx/∂t²),

and likewise for the y components, where c is the speed of light (≈ 3×108m/s). Now imagine the light viewed from a different reference frame moving with respect to the first along the z direction with speed v. We relate the coordinates in the new frame (prime) to those in the original (unprimed) by the Galilean transformation, z' = z - vt and t' = t (so z = z' + vt'). Then in this primed coordinate system, the terms in the wave equation transform as

∂Ex/∂z = (∂Ex/∂z')(∂z'/∂z + ∂z'/∂t) = (∂Ex/∂z')(1 - v)
∂²Ex/∂z² = (∂Ex/∂z')(∂z'/∂z + ∂z'/∂t) = (∂²Ex/∂z'²)(1 - v)²
∂Ex/∂t = (∂Ex/∂t')(∂t'/∂t) = ∂Ex/∂t'
∂²Ex/∂t² = ∂²Ex/∂t'²

The wave equation transforms as

(∂²Ex/∂z²)(1-v)² = (1/c²)(∂²Ex/∂t²),

and likewise for the others. The extra piece that is carried along doesn't cancel out, and presents a problem. While Newton's laws appear the same in inertial reference frames differing by a Galilean transformation, Maxwell's equations do not. Either Maxwell's equations are not the most general form for the equations of electromagnetism, or the Galilean transformation is not the correct way to connect inertial frames. There is strong evidence that Maxwell's equations are correct, so let's look at modifying the Galilean transformation.