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Chapter 1: Problem 3

Draw the \(t\) and \(x\) axes of the spacetime coordinates of an observer \(\mathcal{O}\) and then draw: (a) The world line of \(\mathcal{O}\) 's clock at \(x=1 \mathrm{~m}\). (b) The world line of a particle moving with velocity \(\mathrm{d} x / \mathrm{d} t=0.1\), and which is at \(x=0.5 \mathrm{~m}\) when \(t=0\) (c) The \(\bar{t}\) and \(\bar{x}\) axes of an observer \(\overline{\mathcal{O}}\) who moves with velocity \(v=0.5\) in the positive \(x\) direction relative to \(\mathcal{O}\) and whose origin \((\bar{x}=\bar{t}=0)\) coincides with that of \(\mathcal{O}\). (d) The locus of events whose interval \(\Delta s^{2}\) from the origin is \(-1 \mathrm{~m}^{2}\). (e) The locus of events whose interval \(\Delta s^{2}\) from the origin is \(+1 \mathrm{~m}^{2}\). (f) The calibration ticks at one meter intervals along the \(\bar{x}\) and \(\bar{t}\) axes. (g) The locus of events whose interval \(\Delta s^{2}\) from the origin is \(0 .\) (h) The locus of events, all of which occur at the time \(t=2 \mathrm{~m}\) (simultaneous as seen by \(\mathcal{O}\) ). (i) The locus of events, all of which occur at the time \(\bar{t}=2 \mathrm{~m}\) (simultaneous as seen by \(\overline{\mathcal{O}}\) ). (j) The event which occurs at \(\bar{t}=0\) and \(\bar{x}=0.5 \mathrm{~m}\). (k) The locus of events \(\bar{x}=1 \mathrm{~m}\). (1) The world line of a photon which is emitted from the event \(t=-1 \mathrm{~m}, x=0\), travels in the negative \(x\) direction, is reflected when it encounters a mirror located at \(\bar{x}=-1 \mathrm{~m}\), and is absorbed when it encounters a detector located at \(x=0.75 \mathrm{~m}\).

Short Answer

Expert verified
Plot lines and points for axes, world lines, calibration ticks, and intervals according to spacetime transformations and observer velocities.

Step by step solution

01

Draw Axes for Observer O

Start by drawing the two main axes of the spacetime diagram: the time axis (t) which is vertical, and the space axis (x) which is horizontal. These will represent the stationary observer O's frame of reference.
02

World Line of Observer's Clock

Plot a vertical line at x = 1 m, parallel to the t-axis. This line represents the world line of observer O's clock moving through time at x = 1 m.
03

World Line of a Moving Particle

To draw the world line of a particle with velocity dx/dt = 0.1, starting at x = 0.5 m when t = 0, plot points at intervals that form a straight line from the point (0, 0.5) with a slope of 0.1. This means for each unit increase in t, x increases by 0.1.
04

t and x Axes for Moving Observer

For observer O虆, moving with velocity v = 0.5, draw lines at angles dictated by the Lorentz transformation. The x虆 axis will tilt towards the t-axis, and the t虆 axis will tilt towards the x-axis, intersecting the origin (0,0) with appropriate scaling.
05

Locus of Events with Negative Interval

Plot the hyperbola defined by 螖s虏 = -1 m虏. This implies a timelike separation. The hyperbola opening vertically (aligned with the t-axis) represents events accessible to a subluminal (less than light-speed) observer, centered at the origin.
06

Locus of Events with Positive Interval

For 螖s虏 = +1 m虏, the locus is a hyperbola symmetric around the x-axis, consistent with spacelike separation, implying these events cannot causally affect each other.
07

Calibration Ticks for Observer a虆

To calibrate O虆's axes, mark points at one meter intervals along the x虆 and t虆 axes, similar to tick marks on the standard coordinate axes, adjusted by the Lorentz transformation.
08

Locus with Zero Interval

This locus is defined by 螖s虏 = 0, which represents the light cone of the origin. Plot a 45-degree line through the origin to show the path a photon would take, thereby signifying causally connected events.
09

Line of Simultaneous Events in O's Frame

For events occurring simultaneously when t = 2 m, plot a horizontal line at this t value from the x-axis, parallel to the x-axis.
10

Line of Simultaneous Events in O虆's Frame

Using the velocity v = 0.5 and applying Lorentz transformation, plot the line parallel to x虆 corresponding to 虆t = 2 m as it will appear tilted in O's frame due to time dilation.
11

Event at 虆t=0, 虆x=0.5 m

Mark point on the diagram that corresponds to these coordinates in observer O虆's reference frame, applying the transformation to find its place in O's diagram.
12

Line for 虆x=1 m

Translate x虆 = 1 m to observer O鈥檚 coordinates using the Lorentz transformation and plot as a line representing all events with this condition.
13

World Line of Reflected Photon

Draw the path of a photon starting at (t=-1 m, x=0) moving linearly with slope -1, reflecting off a boundary set at x虆 = -1 m, then adjusting its path to finish at (x=0.75 m).

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Key Concepts

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

World Line
In the context of spacetime diagrams, a world line is a crucial concept that helps us understand the path an object takes through both time and space. Let鈥檚 break this down into simpler terms. Imagine you are tracking the journey of a particle or an observer's clock.
  • The world line for observer \( \mathcal{O} \) is essentially the path this clock takes as it moves through time at a constant position \( x = 1 \text{ m} \). It is visualized as a vertical line parallel to the t-axis, representing a stationary position in space.
  • For a moving particle, like in part (b) of the exercise, the world line will be a diagonal line, showcasing movement with time. If the particle starts at \( x = 0.5 \text{ m} \) and has a velocity of 0.1, its world line will be less steep than the clock鈥檚 vertical line, indicating its gradual movement through space.
      • Understanding world lines helps us visualize how various objects experience and consume space and time differently, depending on their velocities and starting points.
Lorentz Transformation
The Lorentz transformation is at the heart of understanding how observations by different observers in relative motion must be adjusted. It鈥檚 fundamentally about translating coordinates between frames that move relative to each other at a constant velocity.
  • For observer \( \overline{\mathcal{O}} \), who moves with velocity \( v = 0.5 \) relative to \( \mathcal{O} \), the axes \( \bar{t} \) and \( \bar{x} \) become skewed compared to \( t \) and \( x \) axes of \( \mathcal{O} \).
  • The transformation ensures that the speed of light remains constant across frames, a core aspect of the theory of relativity.
      • This means observers will disagree on measurements of distances and times unless these transformations are used to align their observations. Thus, it is pivotal in accurately predicting how events unfold in different frames of reference.
Intervals in Spacetime
In special relativity, intervals in spacetime play a fundamental role, which can be timelike, spacelike, or lightlike, and they dictate how different events are related.
  • A timelike interval implies that two events can influence one another causally. In the diagram, this is shown by curves like hyperbolas extending from the origin, focusing more along the time dimension.
  • A spacelike interval suggests no causal connection between events; they鈥檙e so far apart space-wise that effects can鈥檛 travel between them at the speed of light or slower. In spacetime diagrams, these are hyperbolas that open along the space axis.
  • The lightlike interval, characterized by \( \Delta s^2 = 0 \), represents the light cone and is key for understanding signals or particles traveling at light-speed. This interval captures the boundary between causally connected and disconnected events.
      • These intervals help illustrate the geometry of spacetime, showing how time and space are intertwined and how they affect the potential for events to be connected.
Simultaneity in Relativity
Understanding simultaneity in relativity challenges our everyday perceptions of what it means for events to occur at the 'same time.' In Einstein鈥檚 relativity, two observers might not agree on whether two events are simultaneous, due to how time is affected by speed.
  • For \( \mathcal{O} \), events simultaneous at \( t = 2 \text{ m} \) form a horizontal line across the spacetime diagram. For observer \( \overline{\mathcal{O}} \), however, simultaneity is altered due to their motion, creating a line at an angle relative to the x-axis.
  • This effect is due to what is called relativity of simultaneity, where timelines skew relative to each other. This means that events seen as simultaneous in one frame may occur at different times in another.
      • These differences emphasize why it is crucial to use transformations like those by Lorentz to relate observations accurately across different frames of reference. This core idea rewires our understanding of time, emphasizing that simultaneity is not absolute but deeply relative.

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Most popular questions from this chapter

Show that the hyperbolae \(-t^{2}+x^{2}=a^{2}\) and \(-t^{2}+x^{2}=-b^{2}\) are asymptotic to the lines \(t=\pm x\), regardless of \(a\) and \(b\).

Suppose that the velocity \(v\) of \(\overline{\mathcal{O}}\) relative to \(\mathcal{O}\) is small, \(|v| \ll 1\). Show that the time dilation, Lorentz contraction, and velocity-addition formulae can be approximated by, respectively: (a) \(\Delta t \approx\left(1+\frac{1}{2} v^{2}\right) \Delta \bar{t}\), (b) \(\Delta x \approx\left(1-\frac{1}{2} v^{2}\right) \Delta \bar{x}\), (c) \(w^{\prime} \approx w+v-w v(w+v)\) (with \(|w| \ll 1\) as well). What are the relative errors in these approximations when \(|v|=w=0.1\) ?

(a) Show that if two events are timelike separated, there is a Lorentz frame in which they occur at the same point, i.e. at the same spatial coordinate values. (b) Similarly, show that if two events are spacelike separated, there is a Lorentz frame in which they are simultaneous.

Write out all the terms of the following sums, substituting the coordinate names \((t, x, y, z)\) for \(\left(x^{0}, x^{1}, x^{2}, x^{3}\right)\) (a) \(\sum_{\alpha=0}^{3} V_{\alpha} \Delta x^{\alpha}\), where \(\left\\{V_{\alpha}, \alpha=0, \ldots, 3\right\\}\) is a collection of four arbitrary numbers. (b) \(\sum_{i=1}^{3}\left(\Delta x^{i}\right)^{2}\).

The half-life of the elementary particle called the pi meson (or pion) is \(2.5 \times 10^{-8} \mathrm{~s}\) when the pion is at rest relative to the observer measuring its decay time. Show, by the principle of relativity, that pions moving at speed \(v=0.999\) must have a half-life of \(5.6 \times 10^{-7} \mathrm{~s}\), as measured by an observer at rest.
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