Problem 65 - University Physics with Modern Physics

Question

Two events observed in a frame of reference Shave positions and times given by \(\left(x_{1}, t_{1}\right)\) and \(\left(x_{2}, t_{2}\right),\) respectively. (a) Frame \(S^{\prime}\) moves along the \(x\) -axis just fast enough that the two events occur at the same position in \(S^{\prime} .\) Show that in \(S^{\prime},\) the time interval \(\Delta t^{\prime}\) between the two events is given by $$\Delta t^{\prime}=\sqrt{(\Delta t)^{2}-\left(\frac{\Delta x}{c}\right)^{2}}$$ where \(\Delta x=x_{2}-x_{1}\) and \(\Delta t=t_{2}-t_{1}\) . Hence show that if \(\Delta x>c \Delta t,\) there is \(n o\) frame \(S^{\prime}\) in which the two events occur at the same point. The interval \(\Delta t^{\prime}\) is sometimes called the proper time interval for the events. Is this term appropriate? (b) Show that if \(\Delta x>c \Delta t,\) there is a different frame of reference \(S\) in which the two events occur simultaneously. Find the distance between the two events in \(S^{\prime} ;\) express your answer in terms of \(\Delta x, \Delta t,\) and \(c\). This distance is sometimes called a proper length. Is this term appropriate? (c) Two events are observed in a frame of reference \(S^{\prime}\) to occur simultancously at points separated by a distance of 2.50 \(\mathrm{m}\) . In a second frame \(S\) moving relative to \(S^{\prime}\) along the line joining the two points in \(S^{\prime},\) the two events appear to be separated by 5.00 \(\mathrm{m}\) . What is the time interval between the events as measured in \(S ?[\text { Hint: Apply the result obtained in part (b).1 }\)

Step-by-Step Solution

Answer

Events cannot occur at the same position in any frame if \(\Delta x > c \Delta t\). In another frame, events can be simultaneous and separated by the calculated distance based on transformations.

Step 1: Understand the Problem Statement

We are tasked to show that there exists a frame \(S'\) moving such that two events occur at the same position in \(S'\), and derive the time interval \(\Delta t'\). Additionally, we need to explore the conditions and implications if \(\Delta x > c \Delta t\), and finally calculate the time interval for simultaneous events observed in another frame.

Step 2: Use Lorentz Transformation

To find the frame \(S'\) where both events occur at the same position, we use the Lorentz transformation equations which provide:\[ x' = \gamma(x - vt) \] and \[ t' = \gamma(t - vx/c^2) \] where \(\gamma\) is the Lorentz factor \(\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\). We apply the condition that in \(S'\), \(x'_1 = x'_2\).

Step 3: Derive Time Interval for Same Position

Given \(x'_1 = x'_2\), we set \(\gamma(x_2 - vt_2) = \gamma(x_1 - vt_1)\). Solving for \(v\), we find \(v = \frac{\Delta x}{\Delta t}\). Consequently, the time interval \(\Delta t' = t'_2 - t'_1\) becomes:\[ \Delta t' = \gamma \left( \Delta t - \frac{v \Delta x}{c^2} \right) \] Substituting \(v = \frac{\Delta x}{\Delta t}\) into the equation yields \(\Delta t' = \sqrt{(\Delta t)^2 - \left(\frac{\Delta x}{c}\right)^2}\).

Step 4: Analyze Condition \(\Delta x > c \Delta t\)

If \(\Delta x > c \Delta t\), the expression under the square root becomes negative, indicating no real solution exists for \(\Delta t'\). Therefore, no frame \(S'\) exists where both events can occur at the same position. The term 'proper time interval' refers to the minimum time separation that can occur between two events when observed in a frame where these events are co-located.

Step 5: Simultaneous Events in Another Frame

For events to occur simultaneously in a frame \(S\), we apply the inverse Lorentz transformation, ensuring \(t_2' = t_1'\) in this new frame. The separation in \(S\) is related by:\[ \Delta x' = \gamma(\Delta x - v\Delta t) \] When \(\Delta x > c\Delta t\), \(v = \frac{c^2 \Delta t}{\Delta x}\), leading to \(\gamma = \frac{1}{\sqrt{1 - (c \Delta t/\Delta x)^2}}\).

Step 6: Application to Measured Distance

From part (b)'s result, with \(\Delta x' = 2.50\, \text{m}\) and \(\Delta x = 5.00\, \text{m}\), we use the relation:\[ v = c\sqrt{1 - (\Delta x/\Delta x')^2} \] to find the velocity, then calculate the time interval using the Lorentz transformation for time. The time interval is \(t = \gamma \frac{\Delta x'}{v}\).

Step 7: Calculate Resultant Time Interval

By plugging the values calculated before into the formula, the result is:\[ \Delta t = \frac{\Delta x}{c} \] or equivalent using given distances to verify exact time interval as it accounts for discrepancies between rest lengths.

Key Concepts

Special Relativity

Special Relativity is a core area of physics developed by Albert Einstein, focusing on how the laws of physics are invariant across all inertial frames of reference. A key aspect of Special Relativity is the notion that the speed of light, denoted by \( c \), remains constant at approximately \( 299,792,458 \rac{m}{s} \) regardless of the observer's state of motion.

Special Relativity redefines conventional concepts of time and space.

• It introduces time dilation, where a moving clock appears to tick slower relative to a stationary observer.
• It describes length contraction, meaning an object's length appears shorter in the direction of motion when observed from a moving reference frame.

These effects become significant at speeds approaching the speed of light. The transformations that allow us to calculate these effects between different reference frames are the Lorentz Transformations.

Ultimately, Special Relativity shows us that our understanding of time and space is relative to our frame of reference and relative velocity.

Proper Time

Proper Time refers to the time interval between two events as measured in a frame where the events occur at the same spatial location. It's the shortest possible time interval for those events, providing a standard measure across different reference frames.

In the context of Special Relativity, Proper Time is often derivable using the formula:\[ \\Delta t' = \sqrt{(\Delta t)^2 - \left(\frac{\Delta x}{c}\right)^2} \\]where \( \Delta t \) is the time interval between two events in one frame, and \( \Delta x \) is the spatial separation between these events in the same frame.

• If \( \Delta x > c \Delta t \), the Proper Time becomes imaginary — indicating that no such frame exists where the events can occur at the same location simultaneously.
• Proper Time is used to objectively compare time intervals from different frames of reference.

By focusing on co-located events, Proper Time provides clarity into the nature of time across different observations.

Frame of Reference

A Frame of Reference is essentially a backdrop or perspective from which observations are made. It's critical in physics to describe the motion and interaction of objects consistently.

In Special Relativity, understanding different Frames of Reference is crucial because:

• They define spatial positions and times differently, thereby affecting the perception of events.
• The relative velocity between frames affects time dilation and length contraction.

Each observer's measurements might differ, but the laws of physics remain consistent across all inertial frames thanks to Lorentz Transformations.

Switching between different Frames of Reference, especially at relativistic speeds, can demonstrate how relative motion influences our perceptions of time and space. It reminds us how our understanding of "movement" and "rest" are relative, grounded in the speed and perspective from which we observe events.

Simultaneous Events

Simultaneous Events occur when two events happen at the same time within a given frame of reference. However, simultaneity is a relative concept in Special Relativity.

Events that are simultaneous in one frame may not be in another due to the finite speed of light:

• The notion of simultaneity is altered because observers moving at different velocities may disagree on whether events are synchronous.
• This leads to the realization that time can be experienced differently depending on the observer’s motion relative to the events occurring.

By applying Lorentz Transformations, one can determine the conditions under which events are simultaneous in a specific reference frame, while perhaps separated in space and time in another.

Understanding simultaneous events challenges our classical intuitions and highlights the novelty and counterintuitive nature of Special Relativity.