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Chapter 15: Problem 43

(a) Show that if a body has speed \(v < c\) in one inertial frame, then \(v < c\) in all frames. [Hint: Consider the displacement four-vector \(d x=(d \mathbf{x}, c d t),\) where \(d \mathbf{x}\) is the three-dimensional displacement in a short time \(d t .]\) (b) Show similarly that if a signal (such as a pulse of light) has speed \(c\) in one frame, its speed is \(c\) in all frames.

Short Answer

Expert verified
By Lorentz invariance, if \(v < c\) or \(v = c\) in one frame, it holds in all frames.

Step by step solution

01

Understanding the Four-Vector

The displacement four-vector is given by \( dx = (d \mathbf{x}, c dt) \), where \( d \mathbf{x} \) is the spatial displacement and \( c dt \) represents displacement in time scaled by the speed of light. This vector is used to analyze transformations between inertial frames.
02

Consideration in One Frame

In a given inertial frame, the speed of a body is expressed as \( v = \frac{d\mathbf{x}}{dt} \). A body has speed \( v < c \) which implies that its time-like interval satisfies \( (c dt)^2 > (d\mathbf{x})^2 \).
03

Lorentz Transformation

Under Lorentz transformation, the interval \( (c dt)^2 - (d\mathbf{x})^2 \) is invariant. This means that for any other frame, the inequality \( (c dt)^2 > (d\mathbf{x})^2 \) still holds, ensuring that \( v < c \) in all frames.
04

Verify Speed of Light Scenario

For a signal traveling at speed \( c \) in one frame, we have \( v = c \rightarrow (c dt)^2 = (d\mathbf{x})^2 \). This makes the interval space-like or null. The Lorentz invariance ensures that \( (c dt)^2 = (d\mathbf{x})^2 \) in any frame, hence \( v = c \) in all frames.

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Key Concepts

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

Four-Vector
The concept of four-vector is crucial in understanding relativity. A four-vector combines quantities from both space and time into a single framework. Imagine you have three dimensions of space, much like a typical graph with width, length, and height. Now, add time as an additional dimension. This makes up a four-dimensional structure.

An example is the displacement four-vector, expressed as \( dx = (d \mathbf{x}, c dt) \). Here:
  • \( d \mathbf{x} \) represents the displacement in three-dimensional space, and
  • \( c dt \) scales time with the speed of light \( c \). It effectively unites space and time for analysis across different perspectives or frames.
The four-vector concept is fundamental when analyzing how physical quantities change (or don't change) under transformations like those described in relativity.
Inertial Frame
An inertial frame is a reference point where observers see systems at rest or moving at a constant speed in a straight line. There are no external forces acting on these systems.

Imagine you're sitting in a car moving at a steady speed on a freeway. Inside the car, everything seems normal—items are stationary unless you move them. This is akin to being in an inertial frame.
Different inertial frames may exist, each offering a unique viewpoint. For instance, an observer on the sidewalk sees the car moving, while to you, inside the car, it might feel like you're still.

Relativity tells us that the laws of physics remain consistent across these frames. This means if you measure the speed of light inside or outside the car (assuming windows don’t distort anything), you'll still find the same result. This fundamental insight from Einstein’s relativity informs us that no preferential reference frame dictates the laws of nature.
Speed of Light
The speed of light, denoted as \( c \), is a universal constant approximately equal to 299,792,458 meters per second in a vacuum. This speed is special and acts as the ultimate speed limit in the universe according to relativity.

Never increasing or decreasing, the speed of light ensures that light travels uniformly across different inertial frames. Hence, if light moves at speed \( c \) in one relative viewpoint, it maintains that speed everywhere.
To comprehend relativity, it's pivotal to grasp how speeds combine: In classical thought, speeds would simply add up, but in relativity, they align so that nothing surpasses \( c \).

That's why even if you're in a fast-moving vehicle—and turn on your headlights—the light continues to travel at the same speed, \( c \), not \( c \) plus your speed. This constancy leads to numerous fascinating outcomes, such as time dilation and length contraction.
Relativity
Relativity fundamentally alters how we understand motion and time. Developed by Albert Einstein, it reveals that space and time are interwoven into a single entity known as spacetime. This alters how we perceive the universe when we or objects approach the speed of light.

Under relativity, common sense ideas about speed and duration shift. Time, for instance, dilates or stretches out for objects nearing the speed of light. Similarly, lengths contract along the direction of movement.
These principles mean that observations can vary dramatically between different inertial frames, yet the ultimate rule of relativity ensures all frames adhere to the same physical laws—particularly the unchanging speed of light.
  • Relativity clarifies why a body's speed remains less than \( c \) across all inertial frames if it remains so in one frame.
  • It also shows why signals traveling at \( c \) maintain that speed universally, unaffected by shifts from one frame to another.
Relativity not only reshapes physics but enriches how we contemplate the very fabric of our universe.

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Most popular questions from this chapter

One way to create exotic heavy particles is to arrange a collision between two lighter particles $$a+b \rightarrow d+e+\cdots+g$$ where \(d\) is the heavy particle of interest and \(e, \cdots, g\) are other possible particles produced in the reaction. (A good example of such a process is the production of the \(\psi\) particle in the process \(\left.e^{+}+e^{-} \rightarrow \psi, \text { in which there are no other particles } e, \cdots, g .\right)\) (a) Assuming that \(m_{d}\) is much heavier that any of the other particles, show that the minimum (or threshold) energy to produce this reaction in the CM frame is \(E_{\mathrm{cm}} \approx m_{d} c^{2} .\) (b) Show that the threshold energy to produce the same reaction in the lab frame, where the particle \(b\) is initially at rest, is \(E_{\mathrm{lab}} \approx m_{d}^{2} c^{2} / 2 m_{b} .\) (c) Calculate these two energies for the process \(e^{+}+e^{-} \rightarrow \psi,\) with \(m_{e} \approx 0.5 \mathrm{MeV} / c^{2}\) and \(m_{\psi} \approx 3100 \mathrm{MeV} / c^{2} .\) Your answers should explain why particle physicists go to the trouble and expense of building colliding-beam experiments.

The pion \(\left(\pi^{+} \text {or } \pi^{-}\right)\) is an unstable particle that decays with a proper half-life of \(1.8 \times 10^{-8}\) s. (This is the half-life measured in the pion's rest frame.) (a) What is the pion's half-life measured in a frame \(\mathcal{S}\) where it is traveling at \(0.8 c ?\) (b). If 32,000 pions are created at the same place, all traveling at this same speed, how many will remain after they have traveled down an evacuated pipe of length \(d=36 \mathrm{m} ?\) Remember that after \(n\) half-lives, \(2^{-n}\) of the original particles survive. (c) What would the answer have been if you had ignored time dilation? (Naturally it is the answer (b) that agrees with experiment.)

The relativistic kinetic energy of a particle is \(T=(\gamma-1) m c^{2} .\) Use the binomial series to express \(T\) as a series in powers of \(\beta=v / c .\) (a) Verify that the first term is just the non relativistic kinetic energy, and show that to lowest order in \(\beta\) the difference between the relativistic and non relativistic kinetic energies is \(3 \beta^{4} m c^{2} / 8 .\) (b) Use this result to find the maximum speed at which the non relativistic value is within \(1 \%\) of the correct relativistic value.

Prove the following useful result, called the zero-component theorem: Let \(q\) be a four-vector, and suppose that one component of \(q\) is found to be zero in all inertial frames. (For example, \(q_{4}=0\) in all frames.) Then all four components of \(q\) are zero in all frames.

The neutral pion \(\pi^{0}\) is an unstable particle (mass \(m=135\ \mathrm{MeV} / c^{2}\) ) that can decay into two photons, \(\pi^{0} \rightarrow \gamma+\gamma .\) (a) If the pion is at rest, what is the energy of each photon? (b) Suppose instead that the pion is traveling along the \(x\) axis and that the photons are observed also traveling along the \(x\) axis, one forward and one backward. If the first photon has three times the energy of the second, what was the pion's original speed \(v ?\)
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