Question

Use relativistic velocity addition to reconfirm that the speed of light with respect to any inertial reference frame is \(c .\) Assume one-dimensional motion along a common \(x\) -axis.

Short answer

Question: Demonstrate that the speed of light with respect to any inertial reference frame is constant and always equals to c using the concept of relativistic velocity addition. Answer: Using the relativistic velocity addition formula, we have shown that the velocity of light in any inertial reference frame is always equal to \(c\), regardless of the velocities of the frames. This holds true for both frame A and frame B and proves that the speed of light is constant in any inertial reference frame.

Step-by-step solution

Understand Relativistic Velocity Addition Formula

Relativistic velocity addition is different from classical velocity addition due to the effects of special relativity. The formula for relativistic velocity addition is given by: \(V_\text{relative} = \frac{V_1 + V_2}{1 + \frac{V_1 * V_2}{c^2}}\) where \(V_\text{relative}\) is the relative velocity between two inertial reference frames, \(V_1\) and \(V_2\) are the velocities of the two frames relative to a third reference frame, and \(c\) is the speed of light. We will use this formula to analyze the relative velocity of light between different reference frames.

Set up the Scenario

Let's consider two inertial reference frames, A and B, moving along the common x-axis. Frame A is moving with velocity \(V_A\) with respect to a common reference frame (say frame C). Frame B is moving with velocity \(V_B\) with respect to the same frame C. Now, let's consider a light signal moving along the x-axis at the speed of light (c) in frame C. We want to find out the velocity of this light signal in frame A (denoted as \(V_{light-A}\)) and in frame B (denoted as \(V_{light-B}\)).

Calculate Velocity of Light in Frame A

Using the relativistic velocity addition formula, the velocity of the light signal in frame A can be calculated as: \(V_{light-A} = \frac{c + (-V_A)}{1 + \frac{c*(-V_A)}{c^2}}\) \(V_{light-A} = \frac{c - V_A}{1 - \frac{V_A}{c}}\) Now to show that \(V_{light-A}\) is equal to \(c\), we need to simplify this expression.

Simplify the Expression for the Velocity of Light in Frame A

By multiplying the numerator and the denominator by \((c + V_A)\), we get: \(V_{light-A} = \frac{c - V_A}{1 - \frac{V_A}{c}}* \frac{(c+V_A)}{(c+V_A)}\) \(V_{light-A} = \frac{(c - V_A)(c+V_A)}{(c+V_A)(1 - \frac{V_A}{c})}\) \(V_{light-A} = \frac{c^2 - V_A^2}{c - V_A*c + V_A}\) \(V_{light-A} = \frac{c^2 - V_A^2}{c(1 - V_A)}\) Now, we can see that the numerator \((c^2 - V_A^2)\) and denominator \((c(1 - V_A))\) have a common factor of \((c - V_A)\). So, we can cancel this factor to get: \(V_{light-A} = \frac{c + V_A}{c}\)ancell ]m$ Since the denominator is c, we obtain that the velocity of the light signal in frame A is equal to \(c\) regardless of the value of \(V_A\). \(V_{light-A} = c\)

Calculate the Velocity of Light in Frame B

Similar to calculating the velocity of light in frame A, we will calculate the velocity of the light signal in frame B using the relativistic velocity addition formula: \(V_{light-B} = \frac{c + (-V_B)}{1 + \frac{c*(-V_B)}{c^2}}\) \(V_{light-B} = \frac{c - V_B}{1 - \frac{V_B}{c}}\) Now, following the same simplification steps as in step 4, we can show that the velocity of light in frame B is also equal to \(c\). \(V_{light-B} = c\) This confirms that the speed of light with respect to any inertial reference frame is indeed equal to \(c\).

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics formulated by Albert Einstein, which revolutionized our understanding of space, time, and how they interact. At its core, special relativity postulates that the laws of physics are the same for all observers in inertial frames of reference—those that are either at rest or in constant, straight-line motion.

The theory also introduces the idea that the speed of light in a vacuum (\(c\)) is constant and will be measured the same by all observers, regardless of their motion or the motion of the light source. This leads to fascinating phenomena such as time dilation and length contraction, which significantly differ from our everyday experiences governed by Newtonian physics.

Relativistic velocity addition is a direct consequence of special relativity. Unlike classical velocity addition, which assumes velocities can be added directly (such as a person walking on a train), special relativity accounts for the fact that as objects move closer to the speed of light, their observed velocities in different reference frames do not simply add up. Instead, they combine in a way that ensures the speed of light remains constant, as expressed by the relativistic velocity addition formula.

Inertial Reference Frames

Inertial reference frames play a pivotal role in both Newtonian mechanics and Einstein's theory of special relativity. These are reference frames in which an object either remains at rest or continues to move at a constant velocity unless acted upon by a force. In simpler terms, there is no acceleration—an essential concept for understanding motion without the complications of external influences.

In the context of the relativistic velocity addition exercise, we consider two such inertial frames, A and B, that are in motion with respect to a third frame, C. According to the principles of special relativity, the laws of physics—including the constant speed of light—must hold true in each of these frames. Thus, whether light is measured in frame A, B, or C, its speed must always be consistently recorded as \(c\) regardless of the motion of the source or the observer.

This concept is fundamental when we attempt to measure the speed of an object as it moves from one inertial frame to another. Understanding that these reference frames are key 'players' in special relativity enables us to grasp why the formula for the relativistic addition of velocities must be employed over the classical one.

Speed of Light

The speed of light in a vacuum, denoted as \(c\text{,}\) is not just a universal constant; it's a cornerstone of modern physics, especially in Einstein's theory of special relativity. It is currently defined to be exactly 299,792,458 meters per second. This number is not arbitrary but is based on meticulous experimental measurements and has profound implications in physics.

Importantly, special relativity asserts that \(c\text{ is the maximum speed at which all energy, matter, and information in the universe can travel. It also functions as a 'cosmic speed limit' that cannot be surpassed or even reached by objects with mass.}

• The idea that the speed of light is constant in all inertial reference frames challenges our intuitive understanding of motion.
• The concept plays a critical role in the exercise on relativistic velocity addition, providing insight into why the formula is structured to prevent any relative velocity from exceeding \(c\text{.}\)
• It's this immutability of the speed of light that leads to the conclusion that light's velocity remains at \(c\text{ regardless of the relative motion of different inertial frames.}\)

From communication technologies to deep-space observations, the speed of light continues to be a fundamental aspect that shapes our understanding of the universe.