Question

Observer \(S\) reports that an event occurred on the \(x\) axis at \(x=\) \(3.20 \times 10^{\mathrm{s}} \mathrm{m}\) at a time \(t=2.50 \mathrm{~s}\). (a) Observer \(S^{\prime}\) is moving in the direction of increasing \(x\) at a speed of \(0.380 c\). What coordinates would \(S^{\prime}\) report for the event? (b) What coordinates would \(S^{\prime \prime}\) report if \(S^{\prime \prime}\) were moving in the direction of decreasing \(x\) at this same speed?

Short answer

The final solution comes from performing the mathematical operations prescribed in the Lorentz transformations for each observer. Depending on the direction of travel, we substitute \(v = 0.380c\) or \(-v = -0.380c\) into the Lorentz equations, along with the given event coordinates \(x = 3.20 \times 10^8 m\) and \(t = 2.50 s\).

Step-by-step solution

Apply Lorentz transformations for observer \(S'\)

The Lorentz transformations are given by \(x' = \gamma (x - vt)\) and \(t' = \gamma (t - vx/c^2)\), where \(v\) is the relative speed between the observers, \(x\) and \(t\) are the space and time coordinates in observer \(S\)'s frame, and \(x'\) and \(t'\) are the space and time coordinates in observer \(S'\)'s frame. Here, we're given \(x = 3.20 \times 10^8 m, t = 2.50 s\) and \(v = 0.380c\). Solving these equations will provide the coordinates that \(S'\) reports for the event.

Apply Lorentz transformations for observer \(S''\)

For observer \(S''\), moving in the opposite direction, the relative speed is \(-v\). Thus, the Lorentz transformations become \(x'' = \gamma (x + vt)\) and \(t'' = \gamma (t + vx/c^2)\). Solving these equations will provide the coordinates that \(S''\) would report for the event.

Compute the Lorentz factor \(\gamma\)

To perform the Lorentz transformation, we must first compute the Lorentz factor, which is given by \(\gamma = 1/ \sqrt{1 - v^2 / c^2}\). Note that \(v\) must be expressed as a fraction of the speed of light \(c\), then this equation can be evaluated.

Key concepts

Special Relativity

Special Relativity is a theory developed by Albert Einstein in 1905. It introduced groundbreaking insights into the nature of space and time. This theory is based on two fundamental principles: the invariance of the laws of physics for all inertial frames, and the constancy of the speed of light for all observers, regardless of their relative motion.

Imagine you and a friend are both on spaceships moving at different speeds. Despite moving quickly, light will always pass both of you at the same speed. This is fascinating because it defies our everyday experiences where speeds simply add up. In Special Relativity:

• Neither time nor space is absolute; they both change with different frames of motion.
• Each observer might disagree on measurements of distance and time, but physical laws remain consistent.

These surprising results arise from the Lorentz Transformations, mathematical equations that relate time and space coordinates from one inertial frame to another. This necessity for transformation comes into play when observers are moving at significant fractions of the speed of light, as in the original exercise provided.

Time Dilation

Time Dilation is one of the astonishing predictions of Special Relativity. It explains how time measured by an observer in a stationary frame compares to time measured in a moving frame. An outsider will find that time ticks slower in the moving frame.

Let's consider the Lorentz transformation equations: For time, we use \(t' = \gamma (t - vx/c^2)\). Here, \(\gamma\) is the Lorentz factor, which always has a value greater than 1 when there is relative motion.

• \(\gamma = 1/\sqrt{1 - v^2/c^2}\) increases with velocity.
• The greater the velocity \(v\), the slower time passes in the moving frame relative to the stationary observer.

As an intuitive example, consider two clocks: one stationary and one on a high-speed rocket. To a stationary observer, the rocket’s clock appears to run slower. This effect was the key consideration in solving the given problem, where observers in different relative motions measured the same event at different times.

Length Contraction

Length Contraction is the counterpart to Time Dilation in the realm of Special Relativity. As an object moves relative to an observer, its length parallel to its motion appears shorter. This occurs due to the Lorentz transformation for space: \(x' = \gamma (x - vt)\).

To better understand, imagine observing a train passing by at a speed close to the speed of light.

• The train's length appears contracted from the observer's point of view.
• The equation \(L = L_0/\gamma\) relates the observed length \(L\) to the proper length \(L_0\), which is the length measured in the train’s own rest frame.

In the given exercise, as observers \(S'\) and \(S''\) have different velocities relative to the event on the \(x\) axis, they would report different spatial coordinates due to length contraction. This demonstrates how crucial it is to consider both time dilation and length contraction when dealing with high-speed frames in Special Relativity.