CP1 Revision Guide: Classical Mechanics and Special Relativity

Classical Mechanics

Special Relativity

Remember that it is easiest to use units in which \(c = 1\). In these units, the algebra is much simpler and you can always put the \(c\) back in at the end by dimensional analysis.

Basics of Special Relativity

The two postulates of special relativity are:

1. The laws of physics are the same in all inertial frames of reference.
2. The speed of light in a vacuum (\(c\)) is the same in all inertial frames of reference.

If we have a frame of reference \(S^{\prime}\) moving at a velocity \(\mathbf{v} = v \mathbf{\hat{x}}\) relative to another frame of reference \(S\) (commonly called the lab frame), then the Lorentz transformation equations for a 4-vector \(A^{\mu} = (A^0, \mathbf{A})\) are:

\[\begin{align*} A^{\prime 0} &= \gamma (A^0 - v A^1) \, , \\ A^{\prime 1} &= \gamma (A^1 - v A^0) \, , \\ A^{\prime 2} &= A^2 \, , \\ A^{\prime 3} &= A^3 \, , \end{align*}\]

where \(\gamma = 1/\sqrt{1 - v^2}\) is the Lorentz factor. This can be written more compactly as

\[A^{\prime \mu} = \Lambda^{\mu}_{\phantom{\mu} \nu} A^{\nu} \, ,\]

where \(\Lambda^{\mu}_{\phantom{\mu} \nu}\) is the Lorentz transformation matrix, and can be written in two forms:

\[\begin{pmatrix} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \quad \text{or} \quad \begin{pmatrix} \cosh(\eta) & -\sinh(\eta) & 0 & 0 \\ -\sinh(\eta) & \cosh(\eta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \, ,\]

where \(\eta = \tanh^{-1}(v)\) is the rapidity. For successive Lorentz boosts, the rapidities add (just like for successive 2D rotations the angles add). For the inverse transformation, we use \(v \rightarrow -v\).

The inner product of two 4-vectors \(A^{\mu}\) and \(B^{\mu}\) is given by

$$ A \cdot B = A^{\mu} B_{\mu} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 \, ,

Kinematics

In kinematics, we use the position 4-vector \(x^{\mu} = (t, \mathbf{x})\), where \(t\) is the time and \(\mathbf{x}\) is the position vector. In practice, it is easier to use \(\Delta x^{\mu} = (\Delta t, \Delta \mathbf{x})\), where \(\Delta t\) is the time interval and \(\Delta \mathbf{x}\) is the displacement vector.

Applying the Lorentz transformation to the position 4-vector, we find

\[\begin{align*} \Delta t^{\prime} &= \gamma (\Delta t - v \Delta x) \, , \\ \Delta x^{\prime} &= \gamma (\Delta x - v \Delta t) \, , \\ \Delta y^{\prime} &= \Delta y \, , \\ \Delta z^{\prime} &= \Delta z \, . \end{align*}\]

The invariant spacetime interval between two events is given by

\[x^{\mu} x_{\mu} = t^2 - \mathbf{x}^2 \, .\]

The velocity addition formula in special relativity is

\[u = \frac{\text{d}x}{\text{d}t} = \frac{\gamma (\text{d} x^{\prime} + v \text{d} t^{\prime})}{\gamma (\text{d} t^{\prime} + v \text{d} x^{\prime})} = \frac{\frac{\text{d} x^{\prime}}{\text{d} t^{\prime}} + v}{1 + v \frac{\text{d} x^{\prime}}{\text{d} t^{\prime}}} = \frac{u^{\prime} + v}{1 + u^{\prime} v} \, ,\]

where \(u = \text{d}x/\text{d}t\) is the velocity of the object in the lab frame \(S\), and \(u^{\prime} = \text{d}x^{\prime}/\text{d}t^{\prime}\) is the velocity of the object in frame \(S^{\prime}\) moving at velocity \(v\) relative to \(S\).