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Chapter 12: Problem 21

The coordinates of event \(A\) are \(\left(x_{A}, 0,0\right), t_{A}\), and the coordinates of event \(B\) are \(\left(x_{B}, 0,0\right), t_{B}\). Assuming the displacement between them is spacelike, find the velocity of the system in which they are simultaneous.

Short Answer

Expert verified
The velocity is \(v = c^2(t_B - t_A)/(x_B - x_A)\).

Step by step solution

01

Understanding Spacelike Interval

For the displacement to be spacelike, the spacetime interval between events \(A\) and \(B\) must satisfy: \((t_B - t_A)^2 < (x_B - x_A)^2\). This means that you cannot reach one event from the other by traveling at or below the speed of light.
02

Define the Velocity for Simultaneity

To find the velocity \(v\) of the system where events \(A\) and \(B\) are simultaneous, we set \(t'_A = t'_B\). In the new frame, the time difference should be zero: \(\gamma (t_B - t_A - v(x_B - x_A)/c^2) = 0\), where \(\gamma\) is the Lorentz factor, \(\gamma = 1 / \sqrt{1 - v^2/c^2}\).
03

Solve for Velocity

Solving the equation \(\gamma (t_B - t_A - v(x_B - x_A)/c^2) = 0\) allows us to find the velocity \(v\). Since \(\gamma\) is never zero (it's infinite when \(v = c\), otherwise finite), this simplifies to: \(t_B - t_A = v(x_B - x_A)/c^2\). Thus, \(v = c^2(t_B - t_A)/(x_B - x_A)\).
04

Interpret the Result

The velocity \(v\) is the velocity of the reference frame in which both events are perceived as happening simultaneously. It depends on the ratio of the time difference to the spatial separation between the events, weighted by \(c^2\).

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Key Concepts

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

Spacelike Interval
In the realm of special relativity, understanding the nature of the spacetime interval is crucial. A spacetime interval refers to the separation between two events, and it can have different characteristics depending on their coordinates. When we say an interval is "spacelike," we mean that the spatial separation between the events dominates over the time difference between them. Mathematically, this is expressed by the inequality:
  • \((t_B - t_A)^2 < (x_B - x_A)^2\)
This indicates that no signal, even light, can travel fast enough to connect the two events directly.
For intervals that satisfy this condition, it is impossible to adjust the reference frame in such a way that one event is causally connected to the other.
In simpler terms, they happen "too far apart," in space, for the time difference to matter noticeably.
Simultaneity
Simultaneity is a concept that becomes fascinatingly complex once we dive into the relativity theory sea. Normally, we think of two events happening at the same time as simultaneous. However, in different frames of reference, events can lose this harmonious synchronization.
This is where the velocity adjustment comes into play. By finding the right velocity for a specific frame of reference, you can make two spacelike-separated events appear simultaneous in that particular frame.
This is achieved by ensuring their time difference in a moving frame is zero:
  • The formula employed is \(\gamma (t_B - t_A - v(x_B - x_A)/c^2) = 0\)
Here, \(v\) represents the velocity needed to achieve simultaneity. What makes simultaneity in relativity so intriguing is that it is not an absolute fact, but a perspective that hinges entirely on the observer鈥檚 state of motion.
Lorentz Factor
The Lorentz Factor, denoted as \(\gamma\), plays a pivotal role in making adjustments to time and space measures when changing frames of reference. It is defined by the equation:
  • \(\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\)
This factor influences time dilation and length contraction, key elements of relativity.
  • Time dilation describes how time differs for observers in different states of motion
  • Length contraction concerns how an object's length is observed to shrink when moving at high speeds
In the context of simultaneity, \(\gamma\) ensures that when calculating the time difference for events across moving frames, relative speeds determine how time aligns.
Essentially, the Lorentz Factor accounts for the discrepancies that arise when merging Newtonian mechanics' classical world with the world of relativity.

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Most popular questions from this chapter

If a particle's kinetic energy is \(n\) times its rest energy, what is its speed?

An electromagnetic plane wave of (angular) frequency \(\omega\) is traveling in the \(x\) direction through the vacuum. It is polarized in the \(y\) direction, and the amplitude of the electric field is \(E_{0}\). (a) Write down the electric and magnetic fields, \(\mathbf{E}(x, y, z, t)\) and \(\mathbf{B}(x, y, z, t)\). [Be sure to define any auxiliary quantities you introduce, in terms of \(\omega, E_{0}\), and the constants of nature.] (b) This same wave is observed from an inertial system \(\mathcal{S}\) moving in the \(x\) direction with speed \(v\) relative to the original system \(S\). Find the electric and magnetic fields in \(\mathcal{S}\), and express them in terms of the \(\mathcal{S}\) coordinates: \(\overline{\mathbf{E}}(\tilde{x}, \tilde{\mathbf{y}}, \bar{z}, \bar{t})\) and \(\overline{\mathbf{B}}(\bar{x}, \bar{y}, \bar{z}, \bar{l})\). [Again, be sure to define any auxiliary quantities you introduce.] (c) What is the frequency \(\tilde{\omega}\) of the wave in \(\mathcal{S}\) ? Interpret this result. What is the wavelength \(\bar{\lambda}\) of the wave in \(\overline{\mathcal{S}}\) ? From \(\bar{\omega}\) and \(\bar{\lambda}\), determine the speed of the waves in \(\overline{\mathcal{S}}\). Is it what you expected? (d) What is the ratio of the intensity in \(\overline{\mathcal{S}}\) to the intensity in \(\mathcal{S}\) ? As a youth, Einstein wondered what an electromagnetic wave would look like if you could run along beside it at the speed of light. What can you tell him about the amplitude, frequency, and intensity of the wave, as \(v\) approaches \(c\) ?

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.)

You may have noticed that the four-dimensional gradient operator \(\partial / \partial x^{\mu}\) functions like a covariant 4 -vector-in fact, it is often written \(\partial_{\mu}\), for short. For instance, the continuity equation, \(\partial_{\mu} J^{\mu}=0\), has the form of an invariant product of two vectors. The corresponding contravariant gradient would be \(\partial^{\mu}=\partial / \partial x_{\mu}\). Prove that \(\partial^{\mu} \phi\) is a (contravariant) 4 -vector, if \(\phi\) is a scalar function, by working out its transformation law, using the chain rule.

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