Question

A \(1 \mathrm{~kg}\) object moves at \(0.8 \mathrm{c}\) relative to Earth. (a) Calculate the momentum and energy of the object. (b) Determine the Lorentz transformation matrix from Earth's frame to the object's frame. (c) Find the momentum and energy of the object in the new frame via matrix multiplication.

Short answer

The relativistic momentum and energy of the object in Earth frame are calculated first, followed by the construction of the Lorentz transformation matrix for the given velocity. The transformed momentum and energy are found by multiplying the Lorentz transformation matrix with the energy-momentum 4-vector.

Step-by-step solution

Calculate Momentum and Energy

The first step is to calculate the momentum and energy of the object relative to the Earth. The relativistic momentum is given by the formula \(p = \gamma mv\), where \(m\) is the object's mass, \(v\) is the object's velocity, and \(\gamma\) is the Lorentz factor, which is given by \(\gamma = \frac{1}{\sqrt{1 - v^{2}/c^{2}}}\). The relativistic energy is given by \(E = mc^{2}\gamma\). Substitute the given values into the formulas to find the momentum and energy.

Lorentz Transformation Matrix

Now, construct the Lorentz transformation matrix for the velocity \(v = 0.8c\). The Lorentz transformation matrix for a boost in the x-direction is \(\begin{pmatrix} \gamma & -\gamma \beta \ -\gamma \beta & \gamma \end{pmatrix}\), where \(\beta = v / c\). Substitute the value \(\beta = 0.8\) into the matrix to obtain the Lorentz transformation matrix.

Transformation of Energy and Momentum

To find the transformed energy and momentum, multiply the Lorentz transformation matrix by the energy-momentum 4-vector \(\begin{pmatrix} E/c \ p \end{pmatrix}\). This will result in a new 4-vector expressing the energy and momentum in the object's rest frame. Decompose this 4-vector into energy and momentum components.

Key concepts

Lorentz Transformation Matrix

To understand the effects of high-speed travel relative to an observer, the Lorentz transformation matrix becomes a cornerstone concept in relativistic physics. What this matrix does is decidedly fascinating; it enables the transition of measurements of space and time from one inertial frame to another, especially when these frames are in relative motion close to the speed of light, denoted as 'c'.

Picture two observers: one standing on Earth and another traveling in a spaceship at a significant fraction of light speed. The Lorentz transformation expresses how the second observer's measurements will differ from the first's. This matrix is not merely a shift but a squeeze-and-stretch transformation that couples time and space dimensions in ways that are not intuitive at low speeds.

For an object moving in the x-direction, like in our exercise, the transformation matrix takes the form of a 2x2 matrix where \( \beta = v/c \) and \( \gamma = 1/\sqrt{1-\beta^2} \). This mathematical depiction elegantly quantifies the relativistic effects of time dilation and length contraction that occur due to the object's high speed.

Applying it in our scenario, with \( v = 0.8c \), requires us to substitute the appropriate values and work out the matrix's elements, which becomes an essential step towards analyzing how the object's velocity alters its energy and momentum from different observational frames.

Relativistic Momentum

In contexts where velocities approach that of light, Newtonian mechanics starts to bend under the weight of its assumptions. At such relativistic speeds, we must turn to the concept of relativistic momentum to accurately describe motion. It accounts for the increase in mass as an object moves faster, reflecting in the equation \( p = \gamma mv \), with \( \gamma \) as the Lorentz factor.

Unlike classical momentum, which multiplies just mass by velocity, relativistic momentum incorporates the Lorentz factor, a term that grows significantly as velocity approaches the speed of light. This adjustment ensures that as the object's speed increases, so too does its momentum, but in a way that aligns with the constraints imposed by the speed of light.

In solving the textbook problem, we calculated the relativistic momentum by considering a mass of \(1 \, \text{kg}\) moving at \(0.8c\) relative to the Earth. The result isn't just a larger number than what classical mechanics would predict; it reflects the profound reality that, at high speeds, momentum becomes a far more complex beast, deeply intertwined with the fabric of spacetime itself.

Energy-Momentum 4-Vector

As we venture further into the realm of relativistic physics, we encounter the energy-momentum 4-vector, a crucial entity that unifies energy and momentum into a single four-dimensional vector. This vector is a staple in the special theory of relativity, providing an invariant quantity that remains constant across all inertial reference frames.

The energy-momentum 4-vector for an object contains its energy and three components of momentum, all converted into compatible units (energy often divided by the speed of light 'c' for this purpose). Mathematically, it appears as \( \begin{pmatrix} E/c \ p_x \ p_y \ p_z \end{pmatrix} \), where \( p_x \) is the relativistic momentum in the x-direction, and similarly for \( p_y \) and \( p_z \) in their respective directions.

In the given exercise, we've used this 4-vector concept alongside the Lorentz transformation matrix to discern the object's energy and momentum from a different frame. By performing matrix multiplication, we can translate these quantities from Earth's frame, where the object moves at \(0.8c\), to the object's rest frame. This translation gives us a new perspective on the object's energetic properties and highlights the transformative power of the Lorentz transformations in our understanding of physics at near-light speeds.