Taylor 15.5 -- 15.7 for today and 15.8 -- 15.10 for Friday.
The Lorentz transformation between two frames moving with relative velocity in the z direction v is
and
where
I assume you are familiar with the relativistic velocity addition formula. Here I want to present a quick derivation of the formula that will allow you to derive the result quickly for an exam.
Start from the infinitesimal Lorentz boosts for position and time
and
Take the ratio, and
cancel the common γ factor on the right,Lastly, divide the numerator and denominator on the right by dt, take the limit to yield velocities and express βc as V
We can do the same for the x and y components, directions perpendicular to the relative motion of the frames, finding
This is expressed as a velocity difference, but just switch the sign of V to get velocity addition. This result has the property that as V → 0, v' → v - V, and V → c, v' → c.
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 ≠ Fx', and likewise for y and 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 motion 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
If we equate these and move terms around we find
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
is a constant. The expression for the square of the distance between two nearby points in space--time is known as the metric for the space. The metric given above is for flat space--time.
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". In general relativity, space can be curved, and the curvature is expressed in the metric.) We will define the position four-vector as
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.
The Lorentz transformation of a four-vector between two frames in relative motion along the 1 direction is written as
This can be represented compactly with matrix notation, introducing the "boost" Λ given by the 4×4 matrix:
| Λ = | 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