Elastic Collisions in Minkowski Momentum Space with Lorentz Transformations

We reexamined the elastic collision problems in the special relativity for both one and two dimensions from a different point of view. In order to obtain the final states in the laboratory system of the collision problems, almost all textbooks in the special relativity calculated the simultaneous equations. In contrast to this, we make a detour through the center-of-mass system. The two frames of references are connected by the Lorentz transformation with the velocity of the center-of-mass. This route for obtaining the final states is easy for students to understand the collision problems. For one dimensional case, we also give an example for illustrating the states of the particles in the Minkowski momentum space, which shows the whole story of the collision.


Introduction
Collisions of the interacting particles have fundamental importance in both classical mechanics and special relativity. Illustrating the collision problems is rewarding to understand them clearly and quickly.
For one dimensional collision in classical mechanics, mass-momentum diagram plays a key role [1] [2]. We can see the whole story of the collision in the single diagram for both the center-of-mass and the laboratory systems. For two dimensional collision in classical mechanics, two-dimensional momentum space describes the collision clearly in the textbook [3]. We also see the slightly different illustration which lays emphasis on the transformation of the two systems [4]. For one dimensional collision in the special relativity, Saletan [5] proposed to understand the collision problems in the Minkowski momentum space, with along the horizontal axis. The states of the particles are expressed by the arrow in the space. The quantitative application of it is stated by [6]. We do not need any calculation for obtaining the whole story of the collision. For two dimensional collision in the special relativity, illustration is clearly stated in the literature [6] [7]. We also see the slightly different illustration which lays emphasis on the transformation of the two systems [8].
In this article, we propose a different point of view for the elastic collision problems in the special relativity. We make a detour through the center-of-mass system for obtaining the final states in the laboratory system. It is applicable to both one and two dimensional collisions. This method shows the unified way to think about collision problems. Now, consider two reference frames K and K'. We assume that the frame K' moves in the x-direction at speed V with respect to the frame K. And let us assume the origins O and O' of the two reference frames coincide at time . An event that occurs at some point is observed from both frames and is charac- where β = V c and 2 1 1 γ β = − . In the following paper, we designate the frame K as the laboratory system, while K' as the center-of-mass system. Accordingly, the velocity V describes the velocity of the center-of-mass. The inverse transformation is given by just putting β − to β in Equation (1). Our strategy is pictorially stated in Figure 1. In the textbooks of physics, we have to calculate the simultaneous equations of momentum-and energy-conservation in order to obtain the final states in the laboratory system. See the dashed arrow in Figure 1. Our strategy is as follows.
1) By the Lorentz inverse transformation, we obtain the velocity V of the center-of-mass in terms of energies ( A E , B E ) and momenta ( A p , B p ) in the laboratory system before the collision. The velocity V does not change throughout the collision.
2) By the Lorentz transformation, we obtain the momenta ( * A p , * B p ) in the center-of-mass system before the collision. See the strategy 2 in Figure 1. In this frame, two particles make a head on collision with the same magnitude of the momentum * p . 3) We determine the momenta ( * ′ A p , * ′ B p ) in the center-of-mass system after Figure 1. The usual approach to the collision problems is along the dashed arrow. The strategy in this article is on the detour of the solid arrows.

4) By the Lorentz inverse transformation, we obtain the momenta ( ′
in the laboratory system after the collision. See the strategy 4 in Figure 1. Finally, we reach the final states. We never solve the simultaneous equations in contrast with the usual treatment of the collision problems. 5) Let us consider the two special cases. One is that the target particle is at rest ) in the laboratory system before the collision. The other is that, in addition to the condition above, two particles have equal masses. 6) We check the limit → ∞ c and see whether these strategies recover the Newtonian mechanics.
This paper is organized in the following way. In Section 2, we discuss one dimensional collisions, according to the strategy stated above. We also show the illustration of these collisions in Minkowski momentum space. This diagram shows the whole story of the one dimensional collision in the special relativity. In Section 3, we turn to the two dimensional collision case. We introduce the collision angle θ * of the incident particle in the center-of-mass system. We show the theoretical background for the diagrammatic approach [6] [7] [8]. Section 4 is devoted to a summary.

Elastic Collisions in One Dimension
Let us discuss the one dimensional elastic collisions. The motions of the particles are restricted in the x-direction. Therefore, the yand z-components of the momentum are zero. Although the illustrations of the contents of this section are already done by [5] [6], we reexamined how we draw the collision problems in the Minkowski momentum space.
transformation with the whole two body system, E E m c m c p p c c (7) where A m and B m are the masses of the colliding particles. We used the relation ( ) ( ) 2 2 2 − = E c p mc , which is satisfied by the relativistic particle. When we define W as the total energy in the center-of-mass system, then W is written in terms of s as follows: We also calculate the following quantities from Equation (5), which are frequently used in the following sections. Figure 2 depicts the states of two particles before the collision in the laboratory system: According to the parallelogram law, we obtain the vector = + OC OA OB , which indicates the state of the center-of-mass. The β in Equation (5) c. The tips A and B show the states of the particles before the collision in the laboratory system. The tip C is determined from A and B by the parallelogram law.
of the perfect inelastic collision in the special relativity, i.e., the two particles are combined and move with the velocity β after the collision. Contrary to this, this diagram is also interpreted as the decay process. The parent particle OC decays into two daughter particles OA and OB .

Momenta and Energies in the Center-of-Mass System before the Collision
We discuss the strategy 2 in the Introduction. Concerning the Lorentz transformation for each particle, we obtain the momenta in the center-of-mass system before the collision; where we used Equations (9). It is natural that 0 * * + = for later use. The energies of the particles in the system are also given by Equa- where we used Equations (7) and (9). These energies are also derived by ( ) (12) and (13). Summing up these energies, we can easily see Equation (8).
We obtain these results from Figure 3. We draw a new * p -axis which has the slope tanθ with respect to the horizontal p-axis. Drawing  E c in the center-of-mass system before the collision.

Momenta and Energies in the Center-of-Mass System after the Collision
We discuss the strategy 3 in the Introduction. We determine the momenta in the center-of-mass system after the collision. In this frame, the particles move in the opposite direction after the collision with the same magnitude of * p in Equation (14). We write down the momenta in the center-of-mass system after the collision , .
Since the magnitudes of the momenta do not change, the energies of the particles , * * * *

Momenta and Energies in the Laboratory System after the Collision
We discuss the strategy 4 in the Introduction. Consider the Lorentz inverse transformation for each particle, we obtain the momenta in the laboratory system after the collision. From the second row of these matrices, we obtain 2 2 , βγ γ * * where we used Equations (9), (17) and (18 show the states after the collision. Namely, the tip A(B) can slide on the hyper- Once the momenta and energies of the particles before the collision are given, we obtain the final states as shown in Figure 4 without any calculations.    (20) and (21), we obtain the momenta in the laboratory system after the collision 2 2 2 2 2 ,

In
The second term of the right hand side of Equation (27) is a momentum lost by the particle A, and this transfers to the momentum gained by the particle B in Equation (28). This is the impulse in the special relativity. We obviously understand the conservation of the momentum: In addition, we obtain the energies from Equations (22) and (23) The second terms of the right hand side of both equations are the same and it transfers from the particle A to B. This is the work in the special relativity. The sum of these energies shows the conservation law of energy. , and we obtain the following relations  , , in the laboratory system. After the collision, the incident particle A stops and the initially rest particle B moves with the momentum of which the particle A had before the collision.

In the Limit c → ∞
In the limit → ∞ c , the relativistic energy E is replaced by 2 mc . Equation (5) becomes .
This is the velocity of the center-of-mass in Newtonian mechanics. Equation Using these equations, we obtain the momenta Equations (20) and (21), after the collision in the laboratory system, which are recovered the case in Newtonian mechanics.

Elastic Collisions in Two Dimensions
Let us turn our discussion to the case of the two dimensional elastic collisions.
We suppose that the motions of the particles are restricted in the x-y plain, so that the z-component of the momentum is zero. Since the motions of the particles are supposed along the x-direction before the collision, we repeat the same discussion of Subsections 2.1 and 2.2. The illustration of this section is already done by [7] [8].

Momenta and Energies in the Center-of-Mass System after the Collision
Let us start our discussion from the strategy 3 in the Introduction. In the center-of-mass system, the magnitudes of the momenta do not change before and after the collision. Thus, we write down the momenta in the same way with Equation (14), where s is defined by Equation (7).
However, the direction of the momenta changes after the collision in two dimensions. As shown in Figure 6, we define the sense of the momentum * ′ p A as ( ) cos ,sin , 0 θ θ * * * = n , where θ * is the scattering angle of the particle A in the center-of-mass system. In other words, the momenta after the collisions are denoted by the vector-form: Since the magnitudes of the momenta do not change in this frame throughout the collision, the energies of each particle do not change either:

Momenta and Energies in the Laboratory System after the Collision
We discuss the strategy 4 in the Introduction. The motion of the particles after the collision is supposed to occur in the x-y plain. Thus, the momenta are These equations show the ellipse with the following parameters: major semiaxis , where Equations (7), (9), (14), (15) and (16)  The second terms of the right hand side in Equations (56) and (57) show the energy lost by the particle A and the energy gained by the particle B. This is the work in the special relativity. From these energies, we clearly see the conservation law of the energy: is illustrated in the solid line.
The points E' and E'' are the foci of this ellipse and the midpoint of them is depicted by E. The dashed circle shows the collision in the center-of-mass system [8].
Because of the relation of the tangent