Problem 56 - University Physics with Modern Physics

Question

Show that momentum and energy transform from one inertial frame to another as \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right) ; p_{y}^{\prime}=p_{y}\) \(p_{z}^{\prime}=p_{p} ; E^{\prime}=\gamma\left(E-v p_{x}\right) .\) Hint: Look at the derivation for the space-time Lorentz transformation.

Step-by-Step Solution

Answer

Based on our analysis and solution, we have demonstrated that the momentum and energy transform from one inertial frame to another according to the given equations by applying the space-time Lorentz transformation. We defined the four-momentum vector, applied the Lorentz transformation matrix, and compared our results to the given equations to verify the relationships between the inertial frames.

Step 1: Review the Lorentz Transformation

The Lorentz transformation is a set of equations that relates the space-time coordinates \((t, x, y, z)\) of an event in one inertial frame (S) to the space-time coordinates \((t', x', y', z')\) in another inertial frame (S'). The frames are moving relative to each other with a constant velocity v along the x-axis. The transformation equations are: 1. \(t^{\prime} = \gamma\left(t - \frac{v x}{c^2}\right)\) 2. \(x^{\prime} = \gamma\left(x - vt\right)\) 3. \(y^{\prime} = y\) 4. \(z^{\prime} = z\) where \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) is the Lorentz factor.

Step 2: Define the Four-Momentum Vector

In special relativity, the four-momentum vector is defined as a four-dimensional vector with components: $$ P = \begin{pmatrix} E/c \\ p_x \\ p_y \\ p_z \end{pmatrix} $$ with \(E = \gamma m c^2\) being the relativistic energy and \((p_x, p_y, p_z)\) the relativistic momentum. The four-momentum transforms according to the Lorentz transformation for contravariant vectors in Minkowski space: $$ P^{\mu\prime} = \Lambda^{\mu}{\underline{\phantom{xx}}}_{\nu} P^\nu \\ P^\prime = \begin{pmatrix} E^{\prime}/c \\ p_x^{\prime} \\ p_y^{\prime} \\ p_z^{\prime} \end{pmatrix} $$ where \(\Lambda\) is the Lorentz transformation matrix.

Step 3: Evaluate the Lorentz Transformation for the Four-Momentum Vector

Now, we evaluate the Lorentz transformation applied to the four-momentum vector: $$ \begin{pmatrix} E^{\prime}/c \\ p_x^{\prime} \\ p_y^{\prime} \\ p_z^{\prime} \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma v/c & 0 & 0 \\ -\gamma v/c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} E/c \\ p_x \\ p_y \\ p_z \end{pmatrix} $$ By multiplying the matrices we obtain: 1. \(E^{\prime}/c = \gamma(E/c - v p_x/c^2)\) 2. \(p_x^{\prime} = \gamma(p_x - vp_z)\) 3. \(p_y^{\prime} = p_y\) 4. \(p_z^{\prime} = p_z\)

Step 4: Simplify the Transformations and Compare with the Given Equations

Now we need to simplify the results and compare them with the given equations for momentum and energy. 1. Multiplying both sides of the first equation by c gives \(E^{\prime} = \gamma\left(E-v p_x\right)\) 2. The second equation is \(p_x^{\prime} = \gamma\left(p_x - v E / c^2\right)\) 3. The third equation is \(p_y^{\prime} = p_y\) 4. The fourth equation is \(p_z^{\prime} = p_z\) Comparing these equations with the given ones, we see that they match. Thus, we have successfully shown that momentum and energy transform from one inertial frame to another according to the given equations by analyzing the space-time Lorentz transformation.

Key Concepts

Special Relativity

Special Relativity is a theory proposed by Albert Einstein that revolutionized our understanding of physics. It addresses the relationship between space and time, asserting that the laws of physics are the same for all non-accelerating observers. This theory produces several surprising results:

• Time dilation – time moves slower for an object moving relative to a stationary observer.
• Length contraction – moving objects are measured to be shorter in the direction of motion.
• Relative simultaneity – events that are simultaneous in one frame may not be in another.

This framework leads us to rethink concepts like mass and energy, encapsulated in the famous equation: \(E=mc^2\). This relationship indicates that mass can be converted into energy, and vice versa, highlighting the interconnectedness of mass and energy in relativistic contexts. Special Relativity is crucial for understanding the behavior of objects at relativistic speeds, which are close to the speed of light \(c\).

Four-Momentum

Four-momentum is a fundamental concept in the theory of Special Relativity. It extends the classical idea of momentum into four-dimensional spacetime. Unlike classical momentum, which only involves spatial components, four-momentum adds a temporal component. Defined as a four-dimensional vector:\[P = \begin{pmatrix} E/c \ p_x \ p_y \ p_z \end{pmatrix} \]where:

• \(E\) is the relativistic energy of the particle.
• \(c\) is the speed of light.
• \(p_x, p_y, p_z\) are the components of momentum in the respective spatial directions.

The beauty of four-momentum lies in its invariance; while individual components may change between reference frames, the overall four-momentum does not, because it transforms covariantly with the Lorentz transformation. This invariance is analogous to how the magnitude of a three-vector remains the same under rotations in three-dimensional space.

Relativistic Energy

Relativistic Energy is an essential concept when considering objects moving at speeds close to the speed of light. Unlike classical physics, where energy is linear with respect to mass and speed, relativistic energy incorporates mass, velocity, and the restriction imposed by the speed of light:\[E = \gamma m c^2\]Here, \(\gamma\) is the Lorentz factor: \[\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\]

• \(m\) is the rest mass of the object.
• \(v\) is the velocity of the object.
• \(c\) is the speed of light.

This formula shows that as an object's speed approaches the speed of light, its energy sharply increases. Due to this, no object with mass can reach the speed of light, as doing so would require infinite energy. The relation also bridges energy and mass, reinforcing the concept that they are interchangeable, as seen in nuclear reactions.

Minkowski Space

Minkowski Space is a geometric framework that profoundly changes our comprehension of the universe's fabric. It merges the three-dimensional space with time into a single four-dimensional continuum. Developed by Hermann Minkowski, it forms the backbone of Special Relativity, which can be visualized in terms of events rather than just positions and times.

• The coordinates are given as \((ct, x, y, z)\) where \(ct\) incorporates time into the space dimensions.
• This leads to a vector space that combines dimensions of space and time.

Within Minkowski Space, the Lorentz transformation naturally arises, illustrating how different observers perceive space and time differently. This space-time model comprehensively represents how intervals between events are invariant, though these coordinates themselves change. Minkowski Space thus provides an essential language through which we understand phenomena at relativistic speeds, maintaining consistency across various inertial frames.