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Chapter 2: Problem 6

In the twin paradox situation, a fellow student objects to the argument that Anna's acceleration is the root of the asymmetry. "All motion is relative! Anna is accelerating relative to Bob. but Bob is accelerating relative to Anna." Answer this objection.

Short Answer

Expert verified
The objection is based on a misinterpretation. While it's true that uniform motion is relative, acceleration isn't. Anna鈥檚 acceleration, inevitable during her turn-around in the voyage, breaks the symmetry between her and Bob鈥檚 experiences. So, despite Bob appearing to accelerate from Anna's perspective, the physical reality of her own accelerated motion is undeniable and causes her to age less upon her return to Earth.

Step by step solution

01

Understand the Twin Paradox

The twin paradox is a thought experiment rooted in Einstein's special theory of relativity. Let's say Anna and Bob are twins, and Anna embarks onto a space voyage near light speed while Bob stays on Earth. According to special relativity, both Anna and Bob would deem the other's clock to be ticking slower (time dilation). However, when Anna returns to Earth, it will be seen that she aged less than Bob, contrary to the symmetry one can expect from the principle of relativity. This is the paradox.
02

Understand the Role of Acceleration in Special Relativity

In Einstein's special relativity, the laws of physics are identical in all inertial frames of reference - those moving at constant velocity. However, acceleration breaks this symmetry as it isn't relative, unlike constant velocity motion. In the twin paradox scenario, Anna's journey involves heavy acceleration periods when she turns around to return to Earth. This is where she breaks the symmetry, and her clock runs slower than Bob's as viewed from an inertial frame of reference (i.e., Bob's frame).
03

Answering the Objection

The false assumption in the student's objection is that all forms of motion, including acceleration, are relative. In special relativity, only uniform motion is relative. An observer can be said to be at rest or moving at a constant velocity simply depending on one's frame of reference. But an accelerating observer is accelerating in all frames. So in the twin paradox, while Anna does see Bob to be accelerating and herself at rest during her turn-around, due to their different circumstances - one actually undergoing the physical stresses of acceleration, the other not - we should not equate their experiences. Therefore, it's only Anna's acceleration that leads to the asymmetry and causes her to age less than Bob.

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Key Concepts

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

Special Relativity
Special relativity, introduced by Albert Einstein in 1905, revolutionized our understanding of space and time. One of its central ideas is that the laws of physics are the same for all observers moving at constant speed鈥攖hese are called "inertial frames." So, if you're in a train moving smoothly at a constant speed, the laws of physics you experience are the same as someone standing still on the ground.

A key principle of special relativity is that there is no absolute state of rest. Everything is relative鈥攚hich means motion is always described with respect to something else. This is why Anna sees Bob's clock moving slower and vice versa when they are moving relative to each other. However, special relativity doesn't apply directly when acceleration comes into play, because Einstein's theory originally dealt only with inertial frames, where objects move at a constant velocity.
  • Laws of physics are uniform in all inertial frames.
  • No absolute rest state鈥攎otion is relative.
  • Acceleration introduces complexity into the theory.
Time Dilation
Time dilation is a fascinating phenomenon predicted by special relativity. It suggests that time as observed by different observers depends on their relative motion. In the twin paradox, both Anna and Bob observe each other's time ticking slower when moving at nearly the speed of light. This feature arises because the speed of light is always the same for all observers, no matter how fast they are moving relative to each other.

Time dilation is more pronounced as speeds approach that of light. This means that, theoretically, if Anna were to travel through space close to light-speed, she could age only a few hours, while years might pass for Bob on Earth.
  • Time appears to slow down for objects moving close to light speed.
  • Simultaneous experiences can vary greatly for different observers.
  • Crucial for understanding scenarios like the twin paradox.
Acceleration in Physics
Acceleration plays a significant role in distinguishing between different types of motion. Unlike constant velocity and uniform motion, which special relativity explains, acceleration requires a deeper look. When an object accelerates, its position changes non-uniformly in comparison to other objects.

In the context of the twin paradox, Anna's acceleration is not relative. Unlike time dilation which is purely relative, acceleration affects the very framework of space and time around the accelerating object. Consequently, Anna's experience is physically different鈥攕he undergoes changes in velocity that Bob does not. This breaks the symmetry that special relativity assumes for relative motion.
  • Acceleration is a non-relative form of motion.
  • It affects spacetime directly, unlike constant velocity.
  • Accelerated frames can detect acceleration through techniques like feeling a force.

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Most popular questions from this chapter

A point charge \(+q\) rests halfway between two steady streams of positive charge of equal charge per unit length \(\lambda\), moving opposite directions and each at \(c / 3\) relative to point charge. With equal electric forces on the point charge, it would remain at rest. Consider the situation from a frame moving right at \(c / 3\). (a) Find the charge per unit length of each stream in this frame. (b) Calculate the electric force and the magnetic force on the point charge in this frame, and explain why they must be related the way they are. (Recall that the electric field of a line of charge is \(\lambda / 2 \pi \varepsilon_{0} r\), that the magnetic field of a long wire is \(\mu_{0} I / 2 \pi r\), and that the magnetic force is \(q \mathbf{v} \times \mathbf{B}\). You will also need to relate \(\lambda\) and the current \(L\).)

With reckless disregard for safety and the law. you set your high-perfomance rocket cycle on course to streak through an intersection at top speed. Approaching the intersection, you observe grcen \((540 \mathrm{nm})\) light from the traffic signal. After passing through, you look back to observe red \((650 \mathrm{nm})\) light. Actually, the traffic signal never changed color wit didn't have time! What is the top speed of your rocket cycle, and what was the color of the traffic signal (according to an appalled bystander)?

You fire a light signal at \(60^{\circ}\) norh of west. (a) Find the velocity components of this signal according to an observer moving eastward relative to you at half the speed of light. From them, determine the magnitude and direction of the light signal's velocity according to this other observer. (b) Find the components according to a different observer, moving westward relative to you at half the speed of light.

In the collision shown, energy is conserved. because both objects have the same speed and mass after as before the collision. Since the collision merely reverses the velocities, the final (total) momentum is opposite the initial. Thus, momentum can be conserved ooly if it is zero. (a) Using the relativistically correct expression for momentum, show that the total momentum is zero that momentum is conserved. (Masses are in arbitrary units.) (b) Using the relativistic velocity transformation, find the four velocities in a frame moving to the right at \(0.6 c\) (c) Verify that momentum is conserved in che new frame.

Planet \(W\) is 12 ly from Earth. Anna and Bob are both 20 yr old. Anna travels to Planet \(W\) at \(0.6 c\), quickly turns around, and returns to Earth at \(0.6 c\). How old will Anna and Bob be when Anna gets back?
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