Understanding the Conservation of Momentum in Special Relativity Collisions

In summary, there are two main collision problems in special relativity: elastic collisions, where total kinetic energy is conserved, and inelastic collisions, where it is not. Momentum is conserved in collisions in special relativity, just as in classical mechanics. Head-on collisions, where objects have equal and opposite initial velocities, and off-center collisions, where they collide at an angle, are different. Relativistic effects, like time dilation and length contraction, can impact collisions by affecting energy and momentum conservation and final velocities. The Lorentz transformation is a crucial tool for solving collision problems in special relativity, allowing for accurate descriptions of motion from different observer perspectives.
  • #1
Haorong Wu
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Homework Statement
The problem is from Problem book in relativity and gravitation by Alan P. Lightman et al.

2.2 When a photon scatters off a charged particle which is moving with a speed very nearly that of light, the photon is said to have undergone an inverse Compoton scattering. Consider an inverse Compton scattering in which a charged particle of rest mass ##m## and total mass-energy (as seen in the lab frame) ##E\gg m##, collides head-on with a photon of freqency ##\nu (h\nu \ll m)##. What is the maximum energy the particle can transfer to the photon?

2.12 Consider the elastic collision of a particle of mass ##m_1## with a stationary particle of mass ##m_2<m_1##. Let ##\theta_{max}## be the maximum scattering angle of ##m_1##. In nonrelativistic calculations, ##\sin \theta_{max} =m_2/m_1##. Prove that this result also holds relativistically.
Relevant Equations
The conservation of 4-momentum
Solutions are given in the book, but I could not understand some part of them.

For problem 2.2, denote the 4-momentum of the photon by ##\mathbf P_\gamma##, that of the particle by ##\mathbf P## and the values after scattering by primes.
Then by the conservation of momentum, we have ## \mathbf P_\gamma+\mathbf P=\mathbf P_\gamma^{'}+\mathbf P^{'}##. Also, ##\mathbf P^2=-m^2##. But then the solution gives a equation that $$\mathbf P_\gamma \cdot \mathbf P_\gamma^{'}=\mathbf P \cdot (\mathbf P_\gamma-\mathbf P_\gamma^{'}) .$$
I could not figure out what laws this equation stands for. Other than this equation, I could understand the solution.

For problem 2.12, the solution first consider the C.M. frame. It reads, since the collision is elastic, ##E_1^{CM}=E_{1'}^{CM}##, i.e., $$ \mathbf P_1 \cdot \mathbf u_{CM}= \mathbf P_{1'} \cdot \mathbf u_{CM}$$ which can be evaluated in the lab frame yielding $$-E_1 +\mathbf p_1 \cdot \mathbf v_{CM}=-E_{1'}+\mathbf p_{1'} \cdot \mathbf v_{CM} $$ where ##\mathbf v_{CM}=\mathbf p_1/(E_1+m_2)##. The last equation is my problem.
In classical mechanics, I know ##\mathbf v_{CM}=(\mathbf p_1+\mathbf p_2)/(m_1+m_2)##. But I have difficulties to generalize this result to SR.

PS. I am not sure what the convention to type 4-vectors, the spatial components of them, and ordinary vectors. So the variables may look scrambled. I apologize for that.

Thank you for your time.
 
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  • #2
Haorong Wu said:
For problem 2.2, denote the 4-momentum of the photon by ##\mathbf P_\gamma##, that of the particle by ##\mathbf P## and the values after scattering by primes.
Then by the conservation of momentum, we have ## \mathbf P_\gamma+\mathbf P=\mathbf P_\gamma^{'}+\mathbf P^{'}##. Also, ##\mathbf P^2=-m^2##. But then the solution gives a equation that $$\mathbf P_\gamma \cdot \mathbf P_\gamma^{'}=\mathbf P \cdot (\mathbf P_\gamma-\mathbf P_\gamma^{'}) .$$
I could not figure out what laws this equation stands for. Other than this equation, I could understand the solution.
Try squaring ##(\mathbf P_\gamma - \mathbf P_\gamma^{'}) +\mathbf P = \mathbf P^{'}##.
 
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  • #3
Haorong Wu said:
For problem 2.12, the solution first consider the C.M. frame. It reads, since the collision is elastic, ##E_1^{CM}=E_{1'}^{CM}##, i.e., $$ \mathbf P_1 \cdot \mathbf u_{CM}= \mathbf P_{1'} \cdot \mathbf u_{CM}$$ which can be evaluated in the lab frame yielding $$-E_1 +\mathbf p_1 \cdot \mathbf v_{CM}=-E_{1'}+\mathbf p_{1'} \cdot \mathbf v_{CM} $$ where ##\mathbf v_{CM}=\mathbf p_1/(E_1+m_2)##. The last equation is my problem.
In classical mechanics, I know ##\mathbf v_{CM}=(\mathbf p_1+\mathbf p_2)/(m_1+m_2)##. But I have difficulties to generalize this result to SR.
For an object of mass ##m##, the momentum and energy are given by ##p=\gamma m v## and ##E = \gamma m##, and it's easy to see that ##v = p/E##. This same formula is being applied to the two-particle system.

Alternatively, you know that in the center-of-mass frame, ##p_1 = -p_2##. You know the momentum and energy of both masses in the lab frame. You can use the Lorentz transformation to solve for the velocity required so that the masses have momenta equal in magnitude but opposite in direction.
 
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FAQ: Understanding the Conservation of Momentum in Special Relativity Collisions

What is the theory of special relativity (SR)?

The theory of special relativity (SR) is a fundamental theory in physics that describes the relationship between space and time. It was developed by Albert Einstein in 1905 and is based on two main principles: the laws of physics are the same for all observers in uniform motion, and the speed of light in a vacuum is constant for all observers regardless of their relative motion.

What is the concept of time dilation in SR?

Time dilation is a phenomenon predicted by the theory of special relativity, which states that time moves slower for an observer who is moving at a high speed relative to another observer. This means that time is not absolute, but rather relative to the observer's frame of reference. This effect has been confirmed through experiments and has important implications for space travel and the behavior of particles at high speeds.

How does SR explain the concept of length contraction?

Length contraction is another consequence of the theory of special relativity, which states that an object's length appears shorter to an observer who is moving at a high speed relative to the object. This is because, according to SR, space and time are intertwined and change depending on the observer's frame of reference. This effect has also been confirmed through experiments and is an important concept in understanding the behavior of objects at high speeds.

What are the two collision problems in SR?

The two collision problems in SR refer to the elastic and inelastic collisions between two objects moving at high speeds. These problems arise because the traditional equations for momentum and kinetic energy do not hold true in the theory of special relativity. Instead, special relativity introduces new equations that take into account the effects of time dilation and length contraction on the objects' velocities and masses.

What are some practical applications of SR?

The theory of special relativity has numerous practical applications in modern technology, including GPS systems, particle accelerators, and nuclear power plants. It also plays a crucial role in our understanding of the universe, particularly in the fields of cosmology and astrophysics. Additionally, the principles of SR have been incorporated into many other areas of physics, such as quantum mechanics and general relativity, leading to a deeper understanding of the fundamental laws of the universe.

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