Question

A famous result in Newtonian dynamics is that if a particle in motion collides elastically with an identical particle at rest, the two particles emerge from the collision on perpendicular trajectories. Does the same hold in the special theory of relativity? Suppose a particle of rest mass \(m\) and total energy \(E\) collides with an identical particle at rest, and the two particles emerge from the collision with new velocities. Are those velocities necessarily perpendicular? Explain.

Short answer

Answer: No, in the special theory of relativity, the final velocities of the particles after an elastic collision are not necessarily perpendicular. Although similar outcomes may arise under specific conditions, such a general conclusion as in Newtonian dynamics does not hold universally for relativistic cases.

Step-by-step solution

Conservation of momentum and energy in the collision

In an elastic collision, both momentum and energy are conserved. Let's label the moving particle as A and resting particle as B. Before the collision, particle A has a momentum pA and energy E, while particle B has momentum pB=0 and energy E=mc^2 (rest energy). After the collision, let the momenta of particles A and B be p'A and p'B, and energies E_A' and E_B', respectively. Conservation of momentum: \(p_A + p_B = p'_A + p'_B\) Conservation of energy: \(E + mc^2 = E_A' + E_B'\)

Determine component-wise momentum

We can consider the x and y components of momenta of the particles separately. Let the initial momentum of A be in the x direction: Before collision: \(p_{Ax} = p_A\) \(p_{Ay} = 0\) After collision: \(p'_{Ax} = p'_A \cos\theta\) \(p'_{Ay} = p'_A \sin\theta\) \(p'_{Bx} = p'_B \cos\phi\) \(p'_{By} = p'_B \sin\phi\) Now we apply the conservation of momentum component-wise.

Conservation of momentum in the x-direction

From the conservation of momentum, we have: \(p_A = p'_A\cos\theta + p'_B\cos\phi\)

Conservation of momentum in the y-direction

As particle A initially has no momentum in the y-direction, we have: \(0 = p'_A\sin\theta - p'_B\sin\phi\)

Determine the final energies

Using the conservation of energy and the energy-momentum relation (E^2 = (pc)^2 + (mc^2)^2), we can express the final energies as: \(E_A' = \sqrt{(p'_A c)^2 + (mc^2)^2}\) \(E_B' = \sqrt{(p'_B c)^2 + (mc^2)^2}\)

Conservation of energy

Now we apply the conservation of energy: \(E + mc^2 = E_A' + E_B'\) \(E + mc^2 = \sqrt{(p'_A c)^2 + (mc^2)^2} + \sqrt{(p'_B c)^2 + (mc^2)^2}\)

Check for perpendicular velocities

In order for the particles' final velocities to be perpendicular, the angle between their velocities should be 90 degrees. Therefore, their scalar product should be zero: \(v'_A v'_B \cos(\theta - \phi) = 0\)

Conclusion

Given the conservation of momentum and energy Equations and the scalar product condition for perpendicular velocities, the final velocities are not necessarily perpendicular in the special theory of relativity as we cannot simply determine the resulting angles. Similar outcomes may arise under specific conditions, but such a general conclusion as in Newtonian dynamics does not hold universally for relativistic cases.

Key concepts

Conservation of Momentum

In physics, the principle of conservation of momentum states that the total momentum of a closed system remains constant over time if it is not affected by external forces.

Imagine two ice skaters pushing away from one another: the initial combined momentum of the skaters is zero, but as they glide apart, each skater gains momentum in opposite directions. Importantly, the sum of their momenta is still zero, showcasing momentum conservation.

In the context of the special theory of relativity, this principle still holds. However, we need to account for the fact that the momentum of an object increases rapidly as its speed approaches the speed of light, behaving differently from classical expectations. The concept is crucial when considering collisions at high speeds, ensuring that the total momentum before an event, like a collision, is the same as after, as illustrated in our example with two particles.

Elastic Collision

An elastic collision is a fundamental interaction where the total kinetic energy and momentum are conserved. Such collisions are often idealized because in the real world most collisions are not perfectly elastic - some energy is inevitably transformed into other forms, such as heat or sound.

When dealing with subatomic particles at high velocities, as is common in special relativity, the idealization of elastic collisions is useful. By assuming no kinetic energy is lost, we can focus on how other relativistic effects come into play. For instance, understanding what happens when a moving particle strikes a stationary one helps us to grasp more about the underlying physics than if we also had to consider energy losses.

Energy-Momentum Relation

The energy-momentum relation is one of the key outcomes of Einstein's special theory of relativity. It links an object's energy with its momentum and rest mass through the famous equation:
\[E^2 = (pc)^2 + (m_0c^2)^2\]
where E represents the total energy, p is the momentum, m_0 is the rest mass, and c is the speed of light.

This relation shows that energy and momentum are not independent; rather they are closely tied together. When particles collide at high speeds, as in our textbook problem, analyzing the situation requires using this relation to ensure that both energy and momentum conservation can be satisfied in the outcome of the collision.

Relativistic Dynamics

The realm of relativistic dynamics extends the familiar Newtonian mechanics to situations where objects move at speeds comparable to the speed of light. In this regime, we cannot assume that time and space are the same for all observers, and factors like time dilation and length contraction come into play.

Hence, when two particles collide at relativistic speeds, the dynamics of their interaction are not intuitive. Unlike Newtonian predictions, relativistic outcomes depend on the reference frame of the observer. This complicates the analysis of motion and forces involved in high-speed collisions and requires a careful application of relativistic principles to understand phenomena such as particle interactions in particle accelerators or cosmic rays striking the atmosphere.