How to solve problems in special relativity using four-dimensional vectors (basic transformations of spacetime and collisions)

During my study of physics competitions, I found that there is a severe lack of articles and even books in Chinese regarding four-dimensional vectors in special relativity (STR). In light of this, I hope to combine what I have learned to write a related article, aiming to provide a learning reference for students interested in this topic.

Why Use Four-Dimensional Vectors#

When studying STR, I often found myself troubled by the "eigen" states of various scenarios. Many exercises and even the problems themselves contained certain errors, which led to a chaotic understanding of the scenarios. But why does this "classical" transformation feel so counterintuitive? At its core, it is likely because our understanding of time and space is based on classical Galilean spacetime, which does not hold in Einstein's theory of special relativity, as its spacetime is not independent. This is based on the two postulates of special relativity:

The principle of constancy of the speed of light: The speed of light in a vacuum is the same constant in all inertial reference frames, regardless of the motion of the light source and the observer.
The principle of relativity: The form of physical laws is the same in all inertial reference frames; there is no "absolute rest" inertial frame.

What conclusions can we draw from this?

From the constancy of the speed of light, we can derive\ x^2+y^2+z^2-c^2t^2=0

This conclusion can be derived simply by imagining the equations of wavefronts of two beams of light at different time intervals (this will not be elaborated here, as it is not an introductory article on STR).

Having seen this, since the old spacetime system leads to complexity in understanding, is there a way to resolve this? The answer is affirmative; Minkowski proposed such a space because special relativity is essentially a theory of invariants of the Lorentz group. Therefore, as long as we construct such a spacetime, we can elegantly solve this problem, which is the Minkowski spacetime.

Basic Four-Dimensional Vectors in Minkowski Spacetime#

From the aforementioned invariant

x^2+y^2+z^2-c^2t^2=0

we can obtain the first set of four-dimensional vectors

(x,y,z,ict)

So how do we derive the basic Lorentz transformation from it? That is, the relationship between (x,y,z,ict) and (x',y',z',ict').

\text{Starting from the general transformation}

\begin{pmatrix} x' \\ y' \\ z' \\ ict' \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ % Here a_31 and a_32 have been corrected; if the original intention was a_31,a_31, it can be changed back a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ ict \end{pmatrix}

\text{Next, we consider the motion equations of the coordinate origin $O$ in the $S'$ system}

x' = a_{14} t \ , \quad y' = a_{24} t \ , \ z'=a_{34} t \ , \ t'=a_{44} t

\text{Thus we have}

v_x'=\frac{dx'}{dt'}=-v \ , \ v_y'=\frac{dy'}{dt'}=0 \ , \ v_z'=\frac{dz'}{dt'}=0

\text{For the coordinate origin O', there should be a velocity relationship}

\begin{gathered} % These closely related formulas can be placed in a gathered environment a_{11}v_x+a_{12}v_y+a_{13}v_z+a_{14}=0 \\ a_{21}v_x+a_{22}v_y+a_{23}v_z+a_{24}=0 \\ a_{31}v_x+a_{32}v_y+a_{33}v_z+a_{34}=0 \end{gathered}

\text{Also, there is velocity}

v_x=v \ , \ v_y=0 \ , \ v_z=0

\text{Let us set}

a_{44}=\gamma

\text{Then from the above equations, we can obtain}

\begin{gathered} % These closely related formulas can be placed in a gathered environment x'=\gamma x+a_{12}y+a_{13}z-\gamma vt \\ y'=a_{22}y+a_{23}z \\ z'=a_{32}y+a_{33}z \\ t'=a_{41}x+a_{12}y+a_{13}z+\gamma t \end{gathered}

\text{Substituting the above into}

x'^2+y'^2+z'^2-c^2t'^2=0

\text{Adding the anisotropy of spatial position gives}

x'^2+y'^2+z'^2-c^2t'^2=x^2+y^2+z^2-c^2t^2

\text{From coefficient comparison, we can conclude}

\begin{gathered} % These closely related formulas can be placed in a gathered environment \gamma = \pm \frac{1}{\sqrt{1-\beta^2}} \ , \ \beta=\frac{v}{c} \\ a_{41}=\frac{\gamma v}{c^2} \ , \ a_{12}=a_{13}=a_{42}=a_{43}=0 \\ a_{22}^2+a_{23}^2=a_{32}^2+a_{33}^2=1 \ , \ a_{22}a_{23}+a_{32}a_{33}=0 \end{gathered}

\text{Also, since the transformation of the y-z space is an identity transformation, we should have}

a_{22}=a_{33}=1 \ , \ a_{23}=a_{32}=0

\text{From the transformation with } v=0 \text{ being an identity transformation, we can conclude that } \gamma \text{ is positive}

\text{In summary, we can obtain}

\begin{pmatrix} x' \\ y' \\ z' \\ ict' \end{pmatrix} = \begin{pmatrix} \gamma & 0 & 0 & i \gamma \beta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -i \gamma \beta & 0 & 0 & \gamma \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ ict \end{pmatrix}

This is the most basic four-dimensional vector of spacetime coordinates. What other basic four-dimensional vectors are there?

Four-dimensional wave vector
$k_p=(k_x \ ,\ k_y \ , \ k_z \ , \ i\frac{\omega}{c})$
This can be understood through
$kr-wt=0$
Four-dimensional current density vector
$j_p=(j_x \ ,\ j_y \ , \ j_z \ , \ ic\rho)$
This can be understood through
$\nabla \cdot \mathbf{j} + \frac{\partial \rho}{\partial t}=0$
Four-dimensional momentum vector
$p_\mu = (p_x \ ,\ p_y \ , \ p_z \ , \ i\frac{W}{c})$

In fact, we can use these quantities to derive many four-dimensional vectors.
For example, the four-dimensional velocity is simply the differentiation of the spacetime coordinates, which leads to
$U_\mu = \frac{dx_\mu}{d\tau}=\gamma(u,ic)$
It is not difficult to see that
$p_\mu = mU_\mu \ , \ j_p=\rho U_p$
As for dynamics, differentiating P can yield the four-dimensional force vector K; this will not be elaborated here (by the way, it can also be derived using four-dimensional acceleration; interested readers can derive it themselves).

Properties of Four-Dimensional Vectors#

Having discussed so much, we still haven't properly introduced the properties of four-dimensional vectors, but these are key to solving practical problems later. Below, I will introduce several properties of four-dimensional vectors.

Lorentz Covariance#

This is an important property of four-dimensional vectors, and let's derive it.

\begin{gather*} \text{This is an important property of four-dimensional vectors.} \\[1em] \text{In special relativity, the form of physical laws is the same in all inertial reference frames.} \\[1em] \text{This means that physical quantities should maintain some invariance under Lorentz transformations.} \\[1em] \text{The introduction of four-dimensional vectors is precisely to mathematically realize this invariance.} \\[1em] \text{For a four-dimensional vector } X=(x, y, z, ict) \text{ and another four-dimensional vector } Y=(x', y', z', ict'). \\[1em] \text{Their inner product is defined as:} \\[1em] X \circ Y = x x' + y y' + z z' + (ict)(ict') = x x' + y y' + z z' - c^2 tt' \\[1em] \text{This inner product is a scalar.} \\[1em] \text{It has the same value in all inertial reference frames.} \\[1em] \text{Thus, it remains invariant under Lorentz transformations.} \\[1em] \text{Let's derive this.} \\[1em] \text{Let the four-dimensional vector } X \text{ be represented as a column vector in the } S \text{ frame.} \\[1em] \text{In } S' \text{ it is represented as } X'. \\[1em] \text{And they are related by the Lorentz transformation matrix } L \text{ such that } X' = L X\text{.} \\[1em] \text{Similarly, } Y' = LY\text{.} \\[1em] \text{(It is worth noting that in the coordinate system using the imaginary time component } ict \text{, the Lorentz transformation matrix } L \text{ is an orthogonal matrix.)} \\[1em] \text{That is, it satisfies } L^T L = I\text{, where } I \text{ is the identity matrix.} \\[1em] \text{This property ensures that the numerical value of the inner product remains unchanged after transformation.)} \\[1em] \begin{gathered} \text{Then, the inner product of the transformed four-dimensional vectors } X' \text{ and } Y' \text{ is:} \\[8pt] X' \circ Y' = (X')^T Y' = (L X)^T (L Y) = X^T L^T L Y \end{gathered} \\[1em] \begin{gathered} \text{Since the Lorentz transformation matrix } L \text{ satisfies the orthogonality condition in the imaginary time coordinate system } L^T L = I\text{,} \\[8pt] \text{the above expression becomes:} \\ X^T L^T L Y = X^T I Y = X^T Y = X \circ Y \end{gathered} \\[1em] \text{This proves Lorentz covariance:} \\[1em] \text{After a Lorentz transformation, the numerical value of the inner product of four-dimensional vectors remains unchanged.} \\[1em] \text{This property allows us to simplify the treatment of physical quantities by constructing Lorentz invariants (such as the magnitude of four-dimensional vectors).} \\[1em] \text{We need not worry that choosing different inertial reference frames will lead to numerical changes.} \\[1em] \text{For example, the magnitude of the four-dimensional vector } X \circ X = x^2+y^2+z^2-c^2t^2 \text{ is invariant in any reference frame.} \end{gather*}

Inner Products and Conservation of Some Special Four-Dimensional Vectors#

Inner product of four-dimensional velocity
$U \circ U =U \circ U =-c^2$
Inner product of four-dimensional momentum

P \circ U = m(U \circ U)=-mc^2=-E_0 \ , \ P \circ P = -m^2c^2

Four-dimensional momentum is conserved.

Have you noticed that due to Lorentz covariance, we can obtain

P \circ P = -(\frac{E_{tot}}{c})^2 + p^2 \ , \ P' \circ P' = -(\frac{E_0}{c})^2

This is the famous relativistic energy-momentum relation
$E_{tot}^2 = E_0^2 + p^2c^2$

In this case, for photons, we have

E_{tot}=-p \cdot c \ , \ P \circ P = 0

The Wonderful Use of Four-Dimensional Vectors in Solving Various Spacetime and Collision Problems#

Doppler Effect in Relativistic Situations#

From the four-dimensional wave vector, we can obtain

k'_x = \gamma\left(k_x - \frac{v}{c^2}\omega\right) \ , \ k'_y = k_y \ , \ k'_z = k_z \ , \ \omega' = \gamma(\omega - v k_x)

Thus we have
$\omega' = \gamma\omega(1 - \frac{v}{c}cos\theta)$
This is the Doppler effect in relativistic situations.

Particle Collision Annihilation Producing Photons#

pair_annihilation_hd

For each particle, the following four-dimensional momentum will apply:

Incident particle:
$P_1=\gamma m \cdot (v \ , \ 0 \ , \ 0 , \ ic)$
Stationary particle:
$P_2=M \cdot (0 \ , \ 0 \ , \ 0 , \ ic)$
Perpendicular emitted photon beam:
$P_3=\frac{hf_1}{c} \cdot (0 \ , \ 1 \ , \ 0 \ , \ i)$
Diagonally emitted photon beam:
$P_4=\frac{hf_2}{c} \cdot (cos(\theta) \ , \ -sin(\theta) \ , \ 0 \ , \ i)$
From the conservation of four-dimensional momentum, we have
$P_1 + P_2 = P_3 + P_4$
We can obtain:

E_1 + E_2 = E_3 + E_4\\ E_1 \cdot \frac{v}{c} = E_4 \cdot cos(\theta) \\ E_3 = -E_4 \cdot sin(\theta) \end{gathered}

Thus we can have

\begin{gathered} E_4^2 - E_3^2=E_1^2 \cdot \frac{v^2}{c^2} \\ (E_1 + E_2) \cdot (E_4 - E_3)=E_4^2 - E_3^2 \\ E_4 - E_3 = \frac{E_1^2 \cdot \frac{v^2}{c^2}}{E_1 + E_2} \end{gathered}

We also have
$\frac{v^2}{c^2} = 1 - \frac{1}{\gamma^2} = 1 - \frac{E_2^2}{E_1^2} = \frac{E_1^2 - E_2^2}{E_1^2}$
Thus we obtain
$2 \cdot E_4 = E_1 + E_2 + \frac{E_1^2}{E_1+E_2} \cdot \frac{E_1^2-E_2^2}{E_1^2} = E_1 + E_2 + E_1 - E_2 = 2 \cdot E_1$
Thus

E_1 = E_4 \\ E_2 = E_3 \end{gathered}

$cos(\theta) = \frac{v}{c} = \sqrt{1 - \frac{E_2^2}{E_1^2}}$

Light Illuminating Particles to Produce New Particles#

pair_production_hd

For each particle, the following four-dimensional momentum will apply:

Incident light:
$P_1=\frac{hf}{c} \cdot (1 \ , \ 0 \ , \ 0 \ , \ i)$
Stationary particle:
$P_2=M \cdot (0 \ , \ 0 \ , \ 0 \ , \ ic)$
New particle:
$P_3$
From the conservation of four-dimensional momentum, we can obtain
$P_1 + P_2 = P_3$
Squaring gives
$P_1 \circ P_1+2 P_1 \circ P_2+P_2 \circ P_2=P_3 \circ P_3$
If the original particles are much greater than the new particles, we have:
$0 + 2 \cdot h \cdot f \cdot M + (M \cdot c)^2 \approx (M + 2 \cdot m)^2 \cdot c^2$
Simplifying gives
$h \cdot f \approx 2 \cdot m \cdot c^2 + 2 \cdot m^2 \cdot c^2 / M = 2 \cdot m \cdot c^2 \cdot \left(1 + \frac{m_0}{M}\right)$

From this, we can see: if the particles involved are electrons, then the energy of the incident photons must be at least twice the rest energy of the produced particles.

Complete Inelastic Collision of Particles Generating a New Particle#

relativistic_collision_hd

For each particle, the following four-dimensional vectors will apply:

Incident particle:
$P_1$
Target particle:
$P_2$
Generated particle:
$P_3$
From the conservation of four-dimensional momentum, we have:
$P_1 + P_2 = P_3$
Squaring gives
$P_1 \circ P_1 +P_2 \circ P_2 + 2P_1 \circ P_2 = P_3 \circ P_3$
This leads to
$m_1^2 + 2 \gamma_1 \gamma_2m_1m_2 \cdot \frac{(c^2-v_1v_2)}{c^2} + m_2^2 = m_3^2$
This allows us to obtain
$m_3$
From the conservation of four-dimensional momentum, we can also obtain:
$v_3 = \frac{\gamma_1 \cdot m_1 \cdot v_1 + \gamma_2 \cdot m_2 \cdot v_2}{\gamma_1 \cdot m_1 + \gamma_2 \cdot m_2}$

Complete Elastic Collision of Particles#

elastic_collision_theta_alpha_hd

For each particle, the following four-dimensional vectors will apply:

Incident particle:
$P_1=\gamma_1 m \cdot (v \ , \ 0 \ , \ 0 , \ ic)$
Stationary particle:
$P_2=M \cdot (0 \ , \ 0 \ , \ 0 , \ ic)$
Generated particle 1:
$P_3 = \gamma_3 M \cdot (wcos(\theta) \ , \ wsin(\theta) \ , \ 0 \ , \ ic)$
Generated particle 2:
$P_4 = \gamma_4 m \cdot (rcos(\alpha) \ , \ rsin(\alpha) \ , \ 0 \ , \ ic)$
From the conservation of four-dimensional momentum, we have
$P_1 + P_2 = P_3 + P_4$
Squaring gives
$P_1 \circ P_1 +P_2 \circ P_2 + 2P_1 \circ P_2 = P_3 \circ P_3 +P_4 \circ P_4 + 2P_3 \circ P_4$
Also, since
$P_1 \circ P_3 + P_2 \circ P_3 = P_3 \circ P_3 + P_4 \circ P_3 = P_3 \circ P_3 + P_1 \circ P_2$
We can obtain

\begin{gathered} P_1 \circ P_3 = \gamma_1 \cdot \gamma_3 \cdot m \cdot M \cdot (c^2 - v \cdot w \cdot \cos(\alpha))\\ P_2 \circ P_3 = \gamma_3 \cdot M^2 \cdot c^2, P_3 \circ P_3 = M^2 \cdot c^2 \\ P_1 \circ P_2 = \gamma_1 \cdot m \cdot M \cdot c^2 \end{gathered}

We can solve for

From energy conservation and momentum conservation, we can solve for

\begin{gathered} \gamma_4 = \gamma_1 + \frac{M}{m} - \frac{M}{m} \cdot \gamma_3 \\ r_x = \left(\gamma_1 \cdot v - \gamma_w \cdot \frac{M}{m} \cdot w_x\right)/\gamma_r \\ w_x = w \cdot cos(\alpha) \\ \beta = \sin^{-1}(r_x/r) \end{gathered}

Compton Scattering#

compton_scattering_hd

For each particle, the following four-dimensional vectors will apply:

Incident light:
$P_1=\frac{hf}{c} \cdot (1 \ , \ 0 \ , \ 0 \ , \ i)$
Stationary particle:
$P_2=m \cdot (0 \ , \ 0 \ , \ 0 \ , \ ic)$
New particle 1:
$P_3 = \frac{hf'}{c} \cdot (cos(\theta) \ , \ sin(\theta) \ , \ 0 \ , \ i)$
New particle 2:
$P_4 = \gamma m \cdot ( \vec{v}\ , \ ic)$
From the conservation of four-dimensional momentum, we have
$P_1 + P_2 = P_3 + P_4$
Squaring gives
$P_1 \circ P_1 +P_2 \circ P_2 + 2P_1 \circ P_2 = P_3 \circ P_3 +P_4 \circ P_4 + 2P_3 \circ P_4$
This leads to
$P_1 \circ P_2 = P_3 \circ P_4$
Also, since
$P_1 \circ P_3 + P_2 \circ P_3 = P_3 \circ P_3 + P_4 \circ P_3 = P_1 \circ P_2$
This gives
$\frac{h}{\lambda} \cdot \frac{h}{\lambda'}(1-\cos \theta)+m_0 \cdot c \cdot \frac{h}{\lambda'}=m_0 \cdot c \cdot \frac{h}{\lambda}$
We obtain
$\lambda'-\lambda = \frac{h}{m_0\cdot c} \cdot (1-\cos \theta)$
This is the classic conclusion of Compton scattering!

Inverse Compton Scattering#

inverse_compton_scattering_hd

For each particle, the following four-dimensional vectors will apply:

Incident light:
$P_1=\frac{hf}{c} \cdot (1 \ , \ 0 \ , \ 0 \ , \ i)$
High-speed particle:
$P_2 = \gamma_2 m \cdot (v \ , \ 0 \ , \ 0 \ , \ ic)$
New particle 1:
$P_3 = \frac{hf'}{c} \cdot (1 \ , \ 0 \ , \ 0 \ , \ i)$
New particle 2:
$P_4 = \gamma_4 m \cdot (v' \ , \ 0 \ , \ 0 \ , \ ic)$
From the conservation of four-dimensional momentum, we have
$P_1 + P_2 = P_3 + P_4$
Squaring gives
$P_1 \circ P_1 +P_2 \circ P_2 + 2P_1 \circ P_2 = P_3 \circ P_3 +P_4 \circ P_4 + 2P_3 \circ P_4$
This leads to
$P_1 \circ P_2 = P_3 \circ P_4$
Also, since
$P_1 \circ P_3 + P_2 \circ P_3 = P_3 \circ P_3 + P_4 \circ P_3 = P_1 \circ P_2$
This gives
$\frac{2}{\gamma_2 m c^2} \cdot h \cdot f \cdot h \cdot f' + h \cdot f' \cdot \left(1-\frac{v}{c}\right) = h \cdot f \cdot \left(1+\frac{v}{c}\right) \approx 2 \cdot h \cdot f$
If v approaches c, we have
$h \cdot f' \cdot \left(\frac{1}{\gamma_2 m_0 c^2} \cdot h \cdot f + \frac{1}{2} \cdot \left(1-\frac{v}{c}\right)\right) = h \cdot f$
This gives us the relationship between f and f‘!

Thoughts on Four-Dimensional Vectors#

I hope this introductory article can provide you with some insights, but in fact, the wonderful uses of four-dimensional vectors go far beyond this. Due to space constraints, I would like to conclude this article here (perhaps later I can start a series to document this: P). We see that four-dimensional vectors provide a unified and elegant mathematical framework for understanding relativistic phenomena. I hope this can inspire you and provoke thought; sometimes abstract mathematical tools can greatly advance our understanding of physics.

PS: Regarding whether four-dimensional vectors can be used in middle school physics competitions, my current experience is that they can be attempted for problems that one is very confident about, but once a mistake is made, do not expect to receive partial credit for the process. Unfortunately, physics competitions are also exams, and the most basic and classical methods are often the most favored (but using four-dimensional vectors for verification is still very good: D).