Definition: Energy and Momentum

The energy $E$ and momentum $\mathbf p$ respectively of a particle moving at speed $v$ with $\gamma=\sqrt{1-v^2}$ are $$E=\gamma m\qquad \mathbf p=\gamma m\mathbf v$$It can be seen that energy and momentum follow two important relations: $$E^2-p^2=m^2\qquad v=\frac{p}{E}$$If the primed frame moves at a speed $+v$ in the lab frame, the Lorentz transformations of $E$ and $p$ are \begin{gather*} E=\gamma (E'+vp')
p=\gamma(p'+vE') \end{gather*}

Definition: 4-momentum

The energy-momentum 4-vector (or 4-momentum) is defined as $$P=(E,\mathbf p)=(E,p_x,p_y,p_z)$$ The dot product of two 4-vectors $P_1=(a_1,\mathbf b_1)$ and $P_2=(a_2,\mathbf b_2)$ is defined as $$P_1P_2=a_1a_2-\mathbf b_1\cdot \mathbf b_2$$

Theorem: Properties of 4-momentum

Three key properties of 4-momentum you should remember:

1. $P^2=m^2$
2. The inner product of two 4-momenta $P_1P_2$ in any given frame is invariant under the Lorentz transformations.
3. During collisions and decays, the total 4-momentum is conserved.

Definition: Photons

Photons are massless light particles. In units of $c=1$, the energy and momentum of a photon are equal: $E=p$. The energy of a photon can be related to its frequency $f$ as $E=hf$, and its momentum can be related to its wavelength $\lambda$ as $p=h/\lambda$. It can be seen that the relation $f\lambda=c$ holds.

Definition: Force

Force is defined as $$\mathbf F=\frac{d\mathbf p}{dt}$$It can be shown via taking the derivative that a force applied in the direction of motion is related to acceleration by $$F=\gamma^3 ma$$The transformations of $F$ are \begin{gather*}F_\parallel=F_\parallel'
F_\perp = \frac{F_\perp'}{\gamma}\end{gather*}