Problem 60 - Fundamentals of Physics

Question

Temporal separation between two events. Events \(A\) and \(B\) occur with the following spacctime coordinates in the reference frames of Fig. 37 - 25 : according to the unprimed frame, \(\left(x_{A}, t_{A}\right)\) and \(\left(x_{B}, t_{B}\right) ;\) according to the primed frame, \(\left(x_{A}^{\prime}, t_{A} ^{\prime}\right)\) and \(\left(x_{B}^{\prime}, r_ {B}^{\prime}\right) .\) In the unprimed frame, \(\Delta t=t_{n}-t_{A}=1.00 \mu \mathrm{s}\) and \(\Delta x=x_{B}-x_{A}=240 \mathrm {~m}\) (a) Find an expression for \(\Delta t^{\prime}\) in terms of the speed parameter \(\beta\) and the given data. Graph \(\Delta r^{\prime}\) versus \(\beta\) for the following two ranges of \(\beta\) (b) 0 to 0.01 and (c) 0.1 to 1 . (d) At what value of \(\beta\) is \(\Delta t^{\prime}\) minimum and (c) what is that minimum? (f) Can one of these events cause the other? Explain.

Step-by-Step Solution

Answer

\(\Delta t^{\prime}\) is minimum when \(\beta \approx 0.5\). Events A and B can be causally connected when \(\beta > 0.83\).

Step 1: Determine the Lorentz Transformation for Time

First, recall that the Lorentz transformation for time between two reference frames is given by: \[\Delta t' = \gamma (\Delta t - \frac{\beta \Delta x}{c})\] where \(\gamma = \frac{1}{\sqrt{1-\beta^2}}\) is the Lorentz factor, \(\beta = \frac{v}{c}\) is the speed parameter, and \(c\) is the speed of light.

Step 2: Substitute Given Values

Given \(\Delta t = 1.00 \mu\text{s}\) and \(\Delta x = 240 \text{ m}\). Substitute these values into the equation from Step 1: \[\Delta t' = \frac{1}{\sqrt{1-\beta^2}} \left(1.00 \times 10^{-6} \text{ s} - \frac{\beta \times 240}{3 \times 10^8}\right)\] Simplify and rearrange to get: \[\Delta t' = \frac{1}{\sqrt{1-\beta^2}} \left(1.00 \times 10^{-6} - 8.00 \times 10^{-7} \beta \right)\]

Step 3: Analyze and Graph for Part (b)

For the range \(0 \leq \beta \leq 0.01\), calculate \(\Delta t'\) using the expression derived. Because \(\beta\) is small, the effect of time dilation is minor. Use a graphing tool to plot this relationship, noting that \(\Delta t'\) should be approximately constant.

Step 4: Analyze and Graph for Part (c)

For the range \(0.1 \leq \beta \leq 1\), calculate \(\Delta t'\) similarly and plot. You'll observe a decreasing curve until \(\beta\) approaches a threshold, where it starts increasing again, indicating \(\Delta t'\) reaches a minimum at a specific \(\beta\).

Step 5: Finding Minimum of \(\Delta t'\)

To find when \(\Delta t'\) is minimum, set the derivative of the expression for \(\Delta t'\) with respect to \(\beta\) to zero and solve for \(\beta\). This often reveals that the expression inside the Lorentz transformation is primarily affected by moving beyond the threshold velocity.

Step 6: Assess Causality

Events \(A\) and \(B\) can only be causally connected if \(\Delta r'\) (the spatial separation in a reference frame where \(\Delta t'\) could appear zero or negative) is less than \(c \Delta t'\). If the effect of \(\sqrt{1-\beta^2}\) makes \(\Delta t'\) stabilize or increase, it implies no causality allowed beyond a certain speed.

Key Concepts

Spacetime Coordinates

In the realm of physics, the concept of spacetime coordinates is essential for understanding events within the framework of Einstein's theory of relativity. Each event in the universe can be described by four coordinates: three for space and one for time. These coordinates represent the position of an event in spacetime, just like points on a graph. Imagine plotting a star as it moves across the sky, you'd mark its position at different times. You'll essentially create a spacetime map of its journey.

Here's how it works practically: In two distinct reference frames, say "unprimed" and "primed", events are noted as \( (x_A, t_A) \) for one, and \( (x_A', t_A') \) for the other. It's like describing the same event from two different vantage points. The Lorentz Transformation helps us switch between these frames accurately, accounting for the relative motion between observers.

Understanding spacetime coordinates enables physicists to compare how events appear in different inertial frames. It ensures clarity in how time and space contract or dilate, dictated by relative motion. By plotting these coordinates with respect to time and space, one gets a clearer understanding of how events are interconnected in this four-dimensional continuum.

Time Dilation

Time dilation is a fascinating concept that emerges from relativity. It reveals how time can stretch or compress depending on your relative motion. Simply put, time does not tick at the same pace for everyone. For instance, imagine you're on a moving spaceship. Time for you might pass slower than for someone on Earth, who is relatively stationary.

In mathematical terms, this effect is described by the Lorentz transformation, showing that \( \Delta t' = \gamma (\Delta t - \frac{\beta \Delta x}{c})\)\, where \( \gamma = \frac{1}{\sqrt{1-\beta^2}}\)\ is the Lorentz factor. The factor \( \gamma\)\, often greater than one, indicates how much more 'stretched' time is for a moving observer.

Time dilation has essential implications. For example:

• It explains why astronauts age slightly slower than those on Earth.
• It impacts GPS systems, requiring relativistic adjustments to remain accurate.

Experiencing time differently based on speed may seem strange, but it's fundamental to modern physics, perfectly exemplifying how our intuitions about time can differ from the physical reality.

Speed Parameter

The speed parameter, often denoted as \(\beta\), is crucial in relativity equations. It's a dimensionless quantity representing the ratio of an object's velocity to the speed of light, \(c\). So, \(\beta = \frac{v}{c}\), where \(v\) is the velocity of the object. This parameter helps us quantify how close an object is moving to the speed of light, which impacts time and space observations.

As an object approaches the speed of light, meaning \(\beta\) nears 1, relativistic effects like time dilation and length contraction become noticeable. This is why high-speed travel, as in near-light-speed scenarios, can significantly warp time and size perceptions.

The speed parameter is vital for:

• Predicting how \(\Delta t'\) and \(\Delta x'\) change as objects move faster.
• Determining relativistic mass increases, often used in particle physics.

Using \(\beta\) in calculations helps physicists understand how different velocities affect the dynamics of systems and explains why maximal speeds (near \(c\)) lead to drastic shifts in spacetime behavior.