Question

(a) Determine the Lorentz transformation matrix giving position and time in frame \(S^{\prime}\) from those in frame \(S\) for the case \(v=0.5 c\) ( (b) If frame \(S^{\prime \prime}\) moves at \(0.5 c\) relative to frame \(S^{\prime}\), the Lorentz transformation matrix is the same as the previous one. Find the product of the two matrices, which gives \(x^{\prime \prime}\) and \(t^{\prime \prime}\) from \(x\) and \(t\). (c) To what single speed does the transformation correspond? Explain this result.

Short answer

The Lorentz Transformation Matrix for v=0.5c is \[\begin{pmatrix}\sqrt{4/3} & -\sqrt{4/3}/2 & 0 & 0 \-\sqrt{4/3}/2 & \sqrt{4/3} & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{pmatrix}\] The product of two such transformations is \[\begin{pmatrix}4/3 & -2/3 & 0 & 0 \-2/3 & 4/3 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{pmatrix}\] These transformations correspond to a speed \(v = c / \sqrt{3}\).

Step-by-step solution

Determine the Lorentz Transformation Matrix

A Lorentz transformation matrix for a boost along the x-axis has general form: \[\begin{pmatrix}\gamma & -\beta\gamma & 0 & 0 \-\beta\gamma & \gamma & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{pmatrix}\] where \(\gamma = 1/ \sqrt{1 - v^2/c^2}\) and \(\beta = v/c\). Given \(v = 0.5c\), this gives:\[\begin{pmatrix}\sqrt{4/3} & -\sqrt{4/3}/2 & 0 & 0 \-\sqrt{4/3}/2 & \sqrt{4/3} & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{pmatrix}\]

Find the Product of Two Matrices

If frame \(S^{\prime \prime}\) moves at \(0.5 c\) relative to frame \(S^{\prime}\), the Lorentz transformation matrix is the same as before. To find \(x^{\prime \prime}\) and \(t^{\prime \prime}\) from \(x\) and \(t\), find the product of these two Lorentz transformations. This requires matrix multiplication, and the result is:\[\begin{pmatrix}4/3 & -2/3 & 0 & 0 \-2/3 & 4/3 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 1\end{pmatrix}\]

Determine Corresponding Speed

To find the speed the transformation corresponds to, observe the relevant coordinates of the matrix product. \(t^{\prime \prime}/t\) and \(x^{\prime \prime}/x\). These correspond to \(\gamma\) and \(-\beta\gamma\), respectively. Solve these equations for \(v\), which gives \(v = c / \sqrt{3}\). Thus, the two boosts with \(v = 0.5c\) are equivalent to a single boost with \(v = c / \sqrt{3}\).

Key concepts

Special Relativity

Special relativity is a theory proposed by Albert Einstein that revolutionized our understanding of space, time, and motion. This theory posits that the laws of physics are the same for all observers, regardless of their constant velocity relative to each other. A key insight of special relativity is that the speed of light is constant and does not change even if the observer or the source of light is moving.
This leads to some fascinating consequences:

• Time Dilation: Moving clocks tick slower compared to stationary ones as experienced by a moving observer.
• Length Contraction: Objects contract in length along the direction of motion as they move close to the speed of light.
• Simultaneity: Events that are simultaneous in one frame may not be simultaneous in another that is moving relative to the first.

The Lorentz transformation equations are mathematical tools in special relativity that allow us to transition between the coordinates of two such inertially moving frames.

Matrix Multiplication

Matrix multiplication is an essential arithmetic process in mathematics used to determine the combined effect of successive transformations. In the context of physics, particularly special relativity, we use matrix multiplication to apply multiple Lorentz transformations to coordinates.
The Lorentz transformation matrix in our problem helps us understand how one reference frame transforms into another. Given two frames, say \(S\) and \(S'\), moving relatively, you use matrix multiplication to find out the new coordinates. Specifically,

• The product of two Lorentz matrices combines the transformations into one.
• In our scenario, frame \(S\) transforms to \(S'\) and then from \(S'\) to \(S''\).

The result tells us the direct transformation from \(S\) to \(S''\), giving us the net effect of these two transformations.

Velocity Addition

Velocity addition in special relativity is different from classical physics. Unlike simple addition in everyday math, adding velocities in special relativity considers the effects of approaching the speed of light.

• Normal addition would suggest that speeds directly add up. For example, if one object is moving at 0.5c relative to another, and this second object is moving at 0.5c, classical physics would predict a total speed of 1c.
• However, in relativistic conditions, velocities add according to the formula: \[u' = \frac{u + v}{1 + \frac{uv}{c^2}}\] where \(u\) and \(v\) are the velocities of the two objects, and \(c\) is the speed of light.
• This ensures that no object surpasses the universal speed limit of light.

The exercise demonstrates this by showing how two subsequent moves with 0.5c result in a final speed less than 1c.

Relativistic Speed

Relativistic speeds are those that approach the speed of light, causing the relativistic effects described by Einstein’s special relativity.

• As an object's speed nears that of light, relativistic effects such as time dilation and length contraction become significant.
• Measurements of time and space begin to differ significantly when different observers are in motion relative to one another.
• For example, the product of two Lorentz matrices transforming a point from frame \(S\) to \(S''\) results in a single relativistic speed \(v = c / \sqrt{3}\) from the original speeds of 0.5c.

This outcome underlines how speeds combine non-linearly in relativity, elucidating the counter-intuitive nature of relativistic phenomena.

Gamma Factor

The gamma factor, denoted by \(\gamma\), is crucial in special relativity, representing the factor by which time, length, and relativistic mass change for an object moving at a relativistic speed.

• The formula for gamma is \(\gamma = \frac{1}{\sqrt{1 - \left(\frac{v^2}{c^2}\right)}}\).
• It appears in the Lorentz transformation equations, scaling time and space coordinates.
• As speed \(v\) approaches the speed of light \(c\), \(\gamma\) increases towards infinity, explaining the drastic changes in relativistic measurements.

In the problem, choosing \(v = 0.5c\) gives a specific gamma factor of \( \sqrt{\frac{4}{3}} \). This highlights how handy the gamma factor is in calculating the effects of relativity depending on velocity.