# Why does four-momentum have the same transformation matrix as spacetime coordinates?

I will outline my question in 1+1D for brevity. We can passively transform our coordinate system using a Lorentz boost; $$\Lambda^{\bar{\nu}}_{\mu}x^{\mu}=x^{\bar{\nu}}$$. I've seen that, by stipulating that the speed of light is the same to all observers, rather than time, we can describe this transformation using the matrix:

$$\begin{bmatrix} \bar{ct} \\ \bar{x} \\ \end{bmatrix}=\begin{bmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & \gamma\\ \end{bmatrix}\begin{bmatrix} ct \\ x \\ \end{bmatrix}$$

However, what's not obvious to me is why transformations $$\Lambda^{\bar{\nu}}_{\mu}p^{\mu}=p^{\bar{\nu}}$$ have the same form.

$$\begin{bmatrix} \bar{E} \\ \bar{p_{x}} \\ \end{bmatrix}=\begin{bmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & \gamma\\ \end{bmatrix}\begin{bmatrix} E \\ p_{x} \\ \end{bmatrix}$$ I understand that all four-vectors transform under Lorentz transformations. The part I don't understand is why the Lorentz matrix has the same form, even though we're transforming a different basis. Intuitively it doesn't make sense to me why energy and momentum are transforming as though they were spacetime coordinates.

• In my answer here : Transformation of 4− velocity you will see another proof that 4-velocity is a Lorentz 4-vector under a Lorentz boost along an arbitrary direction. Lengthy but with useful relations (transformation of velocity 3-vectors, relations between $\gamma-$factors etc). May 21, 2022 at 7:29

## 4 Answers

four-momentum is literally the first (proper) time derivative of position, multiplied by mass, so it is a vector with the same transformation rule.

• Yeh makes sense, cheers May 20, 2022 at 19:07

Matematically, it is only the derivative at both sides with respect to the invariant parameter $$\tau$$, and multiplication by another scalar invariant $$m$$. As the Lorentz matrix is a constant, that is, it doesn't depend on $$\tau$$, the first equation leads to the second one.

• Yeh makes sense, cheers May 20, 2022 at 19:07

$$p^\mu$$ has to transform as $$p^\mu \to \Lambda^\mu_{~~\nu}p^\nu$$ if you want quantities like $$x^\mu p_\mu$$ to be lorentz invariant. For example if you want to measure the mass of a particle you might choose $$x^\mu = (1,\mathbf{0})$$ and $$p^\mu = (m,\mathbf{0}),$$ so$$x^\mu p_\mu = m$$ If this wasn't boost invariant then you'd measure a different mass for the particle in different frames.

• Ah yes that adds to it too. thanks! May 21, 2022 at 11:37

As a supplement to the earlier answers, one can form a scalar $$S=(tE-\vec x\cdot \vec p)$$ which has the units of action. Regarding this as the [Lorentz-metric] dot-product of two 4-vectors, this implies $$(E,\vec p)$$ also transforms a 4-vector.