Chapter 15: Problem 66
In non relativistic mechanics, the energy contains an arbitrary additive constant \(-\) no physics is changed by the replacement \(E \rightarrow E+\) constant. Show that this is not the case in relativistic mechanics. [Hint: Remember that the four-momentum \(p\) is supposed to transform like a four- vector.]
Short Answer
Step by step solution
Understand Non-Relativistic Mechanics
Consider Relativistic Energy-Momentum Relation
Transform Four-Momentum under Lorentz Transformations
Analyze the Impact of Adding a Constant to Energy
Conclusion based on Scalar Invariance
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Relativistic Energy-Momentum Relation
Why is this link crucial? Because it retains the consistency of physical laws for observers in different inertial frames. The relationship is determined by the relativistic equation:
\[ E^2 = (pc)^2 + (m_0 c^2)^2 \] Here, \( m_0 \) is the invariant mass, a term we'll discuss further. This equation is a cornerstone of relativity, illustrating that changes to energy affect momentum and vice versa.
- The energy term encodes not just kinetic energy but also rest mass energy.
- Momentum remains coupled with energy, contrary to classical physics.
Lorentz Transformation
A four-vector, such as four-momentum \( p^\mu = ( \frac{E}{c}, \mathbf{p} ) \), transforms according to the Lorentz transformation equations:\[ p'^\mu = \Lambda^\mu_{\ u} p^u \] Where \( \Lambda^\mu_{\ u} \) represents the transformation matrix, tailored to the velocity between the frames.
- The transformation ensures that the laws of physics remain the same in all inertial frames.
- It binds spatial and temporal coordinates, illustrating how they can't be independently changed without affecting the other.
- In turn, the concepts of energy and momentum transform as parts of a single entity — the four-momentum.
Invariant Mass
Mathematically, the invariant mass is part of the four-momentum scalar product:
\[ (p^\mu p_\mu) = \left( \frac{E}{c} \right)^2 - \mathbf{p}^2 = (m_0 c^2)^2 \] From this relation, it is apparent that altering energy by some constant changes the scalar product and thereby the invariant mass as well.
- Invariant mass represents the 'true' mass as it is experienced in any frame of reference.
- This mass is a key measure independent of the body's speed or energy.
- In a broader sense, invariant mass provides a single-valued measure of a body's energy content when it is at rest.