Chapter 15: Problem 88
Two particles \(a\) and \(b\) with masses \(m_{a}=0\) and \(m_{b}>0\) approach one another. Prove that they have a CM frame (that is, a frame in which their total three-momentum is zero). [Hint: As you should explain, this is equivalent to showing that the sum of two four-vectors, one of which is forward light-like and one forward time-like, is itself forward time-like.]
Short Answer
Step by step solution
Understanding the Problem
Four-Vector Definitions
Summing the Four-Vectors
Condition for Forward Time-like Vector
Conclusion: Existence of CM Frame
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Four-momentum Vector
The energy in this context is often linked to the mass and velocity of the particle. For massive particles, energy is given by the relativistic energy-momentum relationship \( E^2 = (pc)^2 + (m_0c^2)^2 \), where \( m_0 \) is the rest mass of the particle and \( c \) is the speed of light. For a massless particle, such as a photon, the relation simplifies to \( E = pc \).
In exercises involving four-momentum vectors, like the current one, understanding how to add and interpret these vectors is crucial. Summing four-momentum vectors can indicate properties like collisions or transformations between different frames of reference.
Special Relativity
This theory introduces important consequences such as time dilation and length contraction. Time dilation implies that a moving clock ticks slower compared to a stationary one, while length contraction indicates that an object's length shortens along the direction of motion as its speed approaches the speed of light.
Special relativity also provides the framework for analyzing four-momentum vectors, which are essential for understanding particle physics phenomena. These vectors appropriately account for the relativistic effects acting on moving particles, enabling precise calculations and predictions.
Light-like and Time-like Vectors
Time-like vectors, on the other hand, pertain to particles that have a rest mass. These vectors satisfy the condition \( E > |\vec{p}| \). For such particles, there exists a reference frame where the particle is at rest, and the energy component of the four-vector is associated with the rest mass.
In the context of this exercise, different combinations of these vectors help determine if a center-of-mass frame exists. Adding a light-like vector and a time-like vector can result in a forward time-like vector, allowing for a zero total three-momentum in some frame, confirming the presence of a CM frame.
Reference Frame
Reference frames can be inertial, meaning they are either at rest or in uniform motion, and non-inertial, which involves acceleration. In an inertial reference frame, Newton's laws of motion hold true. The center-of-mass (CM) frame, often used in particle physics, is one such inertial frame where the total momentum is zero.
Finding the CM frame for two particles involves demonstrating that their combined momentum can be zero in this perspective, as seen in the current task. Such frames simplify calculations, particularly in collision problems, by transforming the complexity of motion into a manageable form.