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Chapter 12: Problem 20

(a) Event \(A\) happens at point \(\left(x_{A}=5, y_{A}=3, z_{A}=0\right)\) and at time \(t_{A}\) given by \(c t_{A}=15\); event \(B\) occurs at \((10,8,0)\) and \(c t_{B}=5\), both in system \(S\). (i) What is the invariant interval between \(A\) and \(B\) ? (ii) Is there an inertial system in which they occur simultaneously? If so, find its velocity (magnitude and direction) relative to \(\mathcal{S}\). (iii) Is there an inertial system in which they occur at the same point? If so, find its velocity relative to \(\mathcal{S}\). (b) Repeat part (a) for \(A=(2,0,0), c t=1 ;\) and \(B=(5,0,0), c t=3 .\)

Short Answer

Expert verified
(a) Invariant interval for A and B: \(\sqrt{50}\). No inertial system for simultaneity or same point. (b) Invariant interval: \(\sqrt{-5}\). Simultaneity possible with velocity \(v = \frac{2c^2}{3}\) along x-axis; same point possible with complex calculations.

Step by step solution

01

Calculate Invariant Interval for Part (a)

The invariant interval between two events in spacetime is given by the formula:\[ s^2 = (c^2 (t_B - t_A)^2) - ((x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2) \]Substitute the given values:- Coordinates of A: \(x_A = 5, y_A = 3, z_A = 0, c t_A = 15\)- Coordinates of B: \(x_B = 10, y_B = 8, z_B = 0, c t_B = 5\)\[ s^2 = (15 - 5)^2 - ((10 - 5)^2 + (8 - 3)^2 + (0 - 0)^2) \]\[ s^2 = 10^2 - (5^2 + 5^2) \]\[ s^2 = 100 - 50 = 50 \]So, the invariant interval is \(\sqrt{50}\).
02

Determine if Simultaneity is Possible for Part (a)

Events A and B cannot be simultaneous in any inertial reference frame if their invariant interval \(s^2\) is non-zero and positive. Since we found \(s^2 = 50\), these events cannot be made to occur simultaneously in any inertial frame. Therefore, there is no inertial system where they occur simultaneously.
03

Check if A and B Occur at the Same Point for Part (a)

For two events to occur at the same point, their spatial coordinates must be transformable to coincide and \(s^2\) must be negative. Since \(s^2 = 50 > 0\), these events are spatially separated and cannot occur at the same point in any inertial reference frame. Thus, there is no inertial system where they occur at the same point.
04

Calculate Invariant Interval for Part (b)

Use the invariant interval formula for the second set of data:- Coordinates of A: \(x_A = 2, y_A = 0, z_A = 0, c t_A = 1\)- Coordinates of B: \(x_B = 5, y_B = 0, z_B = 0, c t_B = 3\)\[ s^2 = (3 - 1)^2 - ((5 - 2)^2 + (0 - 0)^2 + (0 - 0)^2) \]\[ s^2 = 2^2 - (3^2) \]\[ s^2 = 4 - 9 = -5 \]Thus, the invariant interval is imaginary, \(\sqrt{-5}\), indicating events are time-like separated.
05

Determine if Simultaneity is Possible for Part (b)

Since \(s^2 = -5\) (negative), the events can be transformed to occur simultaneously in some inertial frame. Therefore, there is an inertial system where they could occur simultaneously, but we need to find the velocity ensuring this condition.
06

Find Velocity for Simultaneity in Part (b)

For simultaneity, time difference in another frame must be zero. We use Lorentz transformations for time:\[ t' = \gamma (t_B - t_A - \frac{v(x_B - x_A)}{c^2}) = 0 \]Solve for \(v\) using: \( t_B - t_A = \frac{v(x_B - x_A)}{c^2} \)\[ 2 = \frac{v(3)}{c^2} \Rightarrow v = \frac{2c^2}{3} \]Direction is along the x-axis with magnitude related to the ratio of spatial to temporal differences.
07

Check if A and B Occur at the Same Point for Part (b)

Since \(s^2 = -5 < 0\), there exists an inertial frame where both events can occur at the same spatial location. Calculate velocity:Using invariant interval as time-like, use similarity to step above but solve spatial transformation to equate spatial coordinates. Velocity magnitude will exceed \(c \), non-physical, indicating financial in initial frame.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Invariant Interval
In the realm of Special Relativity, the "Invariant Interval" is a specific measurement that remains constant between two events, irrespective of the chosen inertial reference frame. This interval helps us understand the nature of the separation between events in spacetime.
The formula used to find the invariant interval is: \[ s^2 = (c^2 (t_B - t_A)^2) - ((x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2) \]
  • When \(s^2 > 0\), the interval is space-like, meaning the events are separated by more space than time, and cannot occur simultaneously in any frame.
  • If \(s^2 < 0\), the interval is time-like, indicating the events can be perceived as simultaneous in a specific inertial frame.
  • For \(s^2 = 0\), the interval is light-like, which means the events are connected by the path of light.
This concept forms the backbone of how we understand separations in spacetime under relativity theory.
Simultaneity
Simultaneity is an important concept in Special Relativity, challenging the classical idea that events can be seen as occurring at the same time in all frames of reference. Einstein鈥檚 theory suggests that simultaneity is dependent on the observer's inertial frame.
  • Two events are simultaneous in one inertial frame, but they might not be in another frame moving relative to the first.
  • The Lorentz transformation equations assist in analyzing when and if events occur simultaneously in different frames.
  • Simultaneity is also related to the invariant interval; for events to be simultaneous, a specific velocity exists where the time difference becomes zero between events.
This variability in the perception of time underscores the non-absolute nature of simultaneity in relativistic physics.
Inertial Reference Frame
An "Inertial Reference Frame" is a fundamental idea in physics, particularly in the context of Special Relativity. It refers to a frame that is either at rest or moves with constant velocity, meaning no net external forces act upon it.
  • Physical laws remain consistent in all inertial frames, as articulated by Einstein's postulates.
  • Observers in different inertial frames may measure different times and distances between events, but the invariant interval remains the same for all.
  • Analyzing events in these frames helps determine if they can occur simultaneously or in the same location, which involves transformations like the Lorentz transformation.
Understanding inertial reference frames is crucial for explaining how observations change with relative motion.
Lorentz Transformation
The "Lorentz Transformation" is a set of equations that relate the space and time coordinates of two events as observed from different inertial frames. It is crucial in understanding how different observers perceive events relative to their motion.The transformation equations are given by:
  • For time: \( t' = \gamma (t - \frac{vx}{c^2}) \)
  • For position: \( x' = \gamma (x - vt) \)
Here, \( \gamma \) is the Lorentz factor, defined as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \).
  • The transformations ensure that the speed of light is constant in all inertial frames.
  • These equations are instrumental in calculating how time dilation and length contraction occur.
  • They are used to find velocities for specific conditions like simultaneity or colocation of events in different frames.
Grasping the Lorentz transformation is key to handling any problems involving different viewpoints of spacetime events.

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Most popular questions from this chapter

In the past, most experiments in particle physics involved stationary targets: one particle (usually a proton or an electron) was accelerated to a high energy \(E\), and collided with a target particle at rest (Fig. 12.29a). Far higher relative energies are obtainable (with the same accelerator) if you accelerate both particles to energy \(E\), and fire them at each other (Fig. 12.29b). Classically, the energy \(\hat{E}\) of one particle, relative to the other, is just \(4 \mathrm{E}\) (why?) ... not much of a gain (only a factor of 4). But relativistically the gain can be enormous. Assuming the two particles have the same mass, \(m\), show that $$ \bar{E}=\frac{2 E^{2}}{m c^{2}}-m c^{2} $$

A neutral pion of (rest) mass \(m\) and (relativistic) momentum \(p=\) \(\frac{3}{4} m_{c}\) decays into two photons. One of the photons is emitted in the same direction as the original pion, and the other in the opposite direction. Find the (relativistic) energy of each photon.

A Lincoln Continental is twice as long as a VW Beetle, when they are at rest. As the Continental overtakes the VW, going through a speed trap, a (stationary) policeman observes that they both have the same length. The VW is going at half the speed of light. How fast is the Lincoln going? (Leave your answer as a multiple of c.)

Suppose you have a collection of particles, all moving in the \(x\) direction, with energies \(E_{1}, E_{2}, E_{3} \ldots\) and momenta \(p_{1}, p_{2}, p_{3}, \ldots\). Find the velocity of the center of momentum frame, in which the total momentum is zero.

Problem 12.6 Every 2 years, more or less, The New York Times publishes an article in which some astronomer claims to have found an object traveling faster than the speed of light. Many of these reports result from a failure to distinguish what is seen from what is observed - that is, from a failure to account for light travel time. Here's an example: A star is traveling with speed \(v\) at an angle \(\theta\) to the line of sight (Fig. 12.6). What is its apparent speed across the sky? (Suppose the light signal from \(b\) reaches the earth at a time \(\Delta t\) after the signal from \(a\), and the star has meanwhile advanced a distance \(\Delta s\) across the celestial sphere; by "apparent speed," I mean \(\Delta s / \Delta t\).) What angle \(\theta\) gives the maximum apparent speed? Show that the apparent speed can be much greater than \(c\), even if \(v\) itself is less than \(c\).
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