Question

If one defines a variable mass \(m_{\text {var }}=\gamma m\), then the relativistic momentum \(\mathbf{p}=\gamma m \mathbf{v}\) becomes \(m_{\text {var }} \mathbf{v}\) which looks more like the classical definition. Show, however, that the relativistic kinetic energy is not equal to \(\frac{1}{2} m_{\mathrm{var}} v^{2}\)

Short answer

Relativistic and classical kinetic energies differ due to relativistic effects and the Lorentz factor \(\gamma\).

Step-by-step solution

Define Variables and Formulas

Let's begin by defining the variables and important concepts. We have the relativistic mass given by the variable \(m_{\text{var}} = \gamma m\) where \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) is the Lorentz factor. The classical momentum is \(m_{\text{var}} \mathbf{v}\). We need to analyze the kinetic energy expression \(\frac{1}{2} m_{\text{var}} v^2\) and see how it compares to the relativistic kinetic energy.

Relativistic Kinetic Energy Formula

The relativistic kinetic energy is given by the formula \(T = (\gamma - 1)mc^2\). This formula results from the difference between the total energy \(E = \gamma mc^2\) and the rest energy \(E_0 = mc^2\).

Compare Classical and Relativistic Kinetic Energies

The classical kinetic energy formula is \(T_{\text{classical}} = \frac{1}{2}m_{\text{var}} v^2\). Substituting \(m_{\text{var}} = \gamma m\), we have:\[T_{\text{classical}} = \frac{1}{2}\gamma m v^2\]Compare this expression with the relativistic kinetic energy \((\gamma - 1)mc^2\). Note that they are fundamentally different due to the presence of the Lorentz factor \(\gamma\) and the \(c^2\) term in the relativistic formula.

Analyze Differences in Formulas

The classical kinetic energy \(\frac{1}{2} \gamma mv^2\) does not account for the relativistic effects encapsulated in \(\gamma\). As the speed \(v\) approaches the speed of light \(c\), \(\gamma\) becomes significantly larger, leading to discrepancies between both kinetic energy expressions. Classically, \(\gamma\) is not considered, leading to fundamentally different results as \(v\) increases.

Key concepts

Lorentz Factor

The Lorentz factor, often denoted as \( \gamma \), is a crucial concept in the realm of relativity. It is defined as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). This factor becomes significant especially when dealing with objects moving at speeds comparable to the speed of light, \( c \). The key idea behind the Lorentz factor is that it accounts for how time and space measurements change for objects moving at high velocities.

• When an object moves much slower than the speed of light, \( \gamma \approx 1 \) and relativistic effects are negligible.
• As the object's speed \( v \) increases towards \( c \), \( \gamma \) increases, leading to more noticeable relativistic effects.

This factor appears in several relativistic equations, like in the expression of relativistic momentum \( \mathbf{p} = \gamma m \mathbf{v} \) and relativistic energy \( E = \gamma mc^2 \). It shows how these physical quantities differ from their classical counterparts as speed increases.

Relativistic Mass

Relativistic mass is a concept that takes into account the increase in mass that a body experiences as its velocity approaches the speed of light. It is expressed as \( m_\text{var} = \gamma m \), where \( m \) is the rest mass and \( \gamma \) is the Lorentz factor.

This term \( m_\text{var} \) emphasizes that the effective inertial mass of an object is velocity-dependent. As a result:

• The faster an object moves, the more massive it seems, due to \( \gamma \).
• At speeds approaching \( c \), \( \gamma \) increases and so does the relativistic mass.

This concept helps in explaining why accelerating objects to the speed of light is energetically prohibitive since their mass would effectively become infinite, requiring infinite energy to continue acceleration.

Classical Mechanics

Classical mechanics, established by Isaac Newton, describes the motion of objects using laws that assume time and space are absolute. This field is effective at low velocities compared to the speed of light.

In classical mechanics, concepts like momentum and kinetic energy have simpler forms:

• Momentum \( \mathbf{p} = m \mathbf{v} \)
• Kinetic energy \( T_\text{classical} = \frac{1}{2} m v^2 \)

These formulas are adequate when speeds are much lower than \( c \). However, as speeds increase, these expressions diverge from reality, requiring relativistic mechanics to account for effects such as time dilation and length contraction. Consequently, formulas incorporating the Lorentz factor are essential for accurate predictions at high velocities.

Speed of Light

The speed of light, denoted by \( c \), is a constant with a value of approximately 299,792,458 meters per second. It is fundamental in Einstein's theory of relativity, serving as the ultimate speed limit in the universe.

Several key points concerning the speed of light include:

• Nothing with mass can travel at or faster than the speed of light, as this would require infinite energy.
• Light travels at \( c \) in a vacuum, and this speed is invariant—meaning all observers, regardless of their relative motion, measure the speed of light as \( c \).

The speed of light plays a pivotal role in relativistic equations, notably those involving the Lorentz factor and relativistic energy, illustrating how \( c \) impacts time and space perception at high velocities. Understanding \( c \) enables a deeper grasp of how the universe operates at a fundamental level.