In Taylor, read sections 15.1 - 15.13 for this week and 15.13 to 15.18 next week.
In E&M we find the wave equation for light to be the pair of partial differential equations
or, limiting ourselves to light propagating along the z direction
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
The wave equation transforms as
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.
The transformation that leaves Maxwell's equations invariant is called the Lorentz transformation. For the reference frames used above, it has the form z' = (z - vt)/√(1 - v²/c²) = γ(z - βct) and t' = (t - vz/c²)/√(1 - v²/c²) = γ(t - βz/c) where β = v/c and γ = (1 - β²)-½. Now the terms of the wave equation transform as
and
Under the Lorentz transformation, the wave equation transforms as
This simplifies to
Although I won't show it, the Lorentz transformation leaves Maxwell's equations (in vacuum) unchanged in form, not just the wave equation derived from them.
The laws of mechanics are well tested for motion on earth, and orbits of planets around the sun, and the predictions derived from them are in good agreement with measurements. Maxwell's equations are also well tested, and, if anything, the predictions are in even better agreement with measurements. That is, we are able to test the laws of electromagnetism to greater levels of precision than the laws of mechanics. At the velocities normally encountered in mechanics, the Lorentz transformation is little different from the Galilean transformation -- the Galilean transformation and non-relativistic mechanics can be thought of as the lowest order term in an expansion of the full relativistic form.
Notice that the Lorentz transformation mixes space and time coordinates. This will lead us to define four-vectors to use in relativistic calculations the way that we use three-vectors for non-relativistic mechanics. The following discussion is intended to illuminate the origin and definition of four-vectors.
The laws of classical mechanics are independent of the particular coordinate system they are applied in. A coordinate system serves as a framework for performing a calculation, but the rule that force equals the rate of change of momentum is applicable in any (inertial) coordinate system. The force exists independent of the coordinate system. It has a magnitude and direction. If we have two coordinate systems, S and S', both with the same origin, but where S' is rotated with respect to S, then the components of the force are different in the two coordinate systems, Fx &neq; Fx', and likewise for y znd z. By Newton's second law, the components of the momentum will differ as well, since dpx/dt = Fx, and likewise for y and z. The equations of motin carry an apparent dependence on the choice of coordinates, yet, the motion itself exists independent of any coordinate system -- when thrown, a stone will follow a parabolic path without being told the directions of x, y, and z!
We express this fact by writing the rule in three-vector form, F = dp/dt, where this form holds in any inertial frame. The square of the magnitude of F is the same, irrespective of our choice of coordinates, F'⋅F' = F⋅F = F² = constant. In fact, the dot product of any two vectors is a scalar, and scalars are independent of the choice of coordinates. It is very useful to find a similar mathematical object that has such a property under relativity.
Consider two nearby points in frame S. The square of the distance between them is dr² = dx² + dy² + dz² and is equal to the square of the speed of light times the time it takes for light to travel between the locations, dr² = (cdt)²
Consider another frame S' moving with respect to S with velocity V along the z axis. The square of the distance in that frame is different, given by dr'² = dx'² + dy'² + dz'² = dx² + dy² + γ²(dz - βcdt)². The time for light to travel between the locations is also different, given by (cdt')² = γ²(cdt - βdz)², but this still must equal dr'². Expanding both of these we find dr'² = dx² + dy² + γ²dz² - 2γ²βcdtdz + γ²β²(cdt)² and (cdt')² = γ²(cdt)² -2γ²βcdtdz + γ²β²dz². If we equate these and move terms around we find dx² + dy² + γ²(1 - β²)dz² = γ²(1 - β²)(cdt)². Note that γ² = 1/(1 - β²), and we recover the original equality for frame S. Therefore, although the length of a three-vector is not the same in all inertial frames, the difference ds² = dx² + dy² + dz² - (cdt)² = dr² - (cdt)² is a constant.
To have a mathematical object that behaves like vectors in three-dimensions, we are led to take a four-dimensional object in special relativity. These are known as four-vectors. The expression for the distance between two nearby points, known as the metric for the space, contains the seed for our four-vector. (The expression used here is the metric for "flat space" and isn't terribly exciting. In general relativity, space can be curved, and the curvature is expressed in the metric.) We will define the position four-vector as x = (x, y, z, ct) = (x, ct), where I use the boldface x to mean the usual three-dimensional vector for position, and add the fourth component of ct. We'll often refer to the four components with the notation x = (x1, x2, x3, x4) where the first three components are the usual spatial coordinates and the fourth component is the time component.
The square of the length of a four vector is chosen to match the expression for the separation for two nearby points,
The negative sign for the fourth component seems odd, and has led some authors to write the fourth component as the imaginary number ict. While this may look correct in the present context, it is not consistent with the notation used in E&M, relatvistic quantum mechanics, and general relativity, and we will avoid its use.
With this four-vector we can write the Lorentz transformation as
This can be represented compactly with matrix notation, introducing the "boost" Λ given by the 4×4 matrix:
| &Lambda = | γ | 0 | 0 | -γβ |
| 0 | 1 | 0 | 0 | |
| 0 | 0 | 1 | 0 | |
| -γβ | 0 | 0 | γ |
and then writing the Lorentz transformation as the matrix expression