91Ó°ÊÓ

(a) Consider the Galilean transformation along the \(x\) -direction: \(x^{\prime}=x-\) vt and \(t^{\prime}=t .\) In frame \(S\) the wave equation for electromagnetic waves in a vacuum is $$\frac{\partial^{2} E(x, t)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}}=0$$ where \(E\) represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame \(S^{\prime}\) is found to be $$\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime 2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime} \partial t^{\prime}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0$$ This has a different form than the wave equation in \(S .\) Hence the Galilean transformation violates the first relativity postulate that all physical laws have the same form in all inertial reference frames. (Hint: Express the derivatives \(\partial / \partial x\) and \(\partial / \partial t\) in terms of \(\partial / \partial x^{\prime}\) and \(\partial / \partial t^{\prime}\) by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Egs. \((37.21),\) and show that in frame \(S^{\prime}\) the wave equation has the same form as in frame \(S :\) $$\frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{2}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{\prime 2}}=0$$ Explain why this shows that the speed of light in vacuum is \(c\) in both frames \(S\) and \(S^{\prime} .\)

Step by step solution

01

Understand the Given Equation in Frame S

The wave equation for electromagnetic waves in a vacuum in frame \(S\) is given as \(abla^{2} E(x, t) - \frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}} = 0\). This equation describes the behavior of electromagnetic waves in a vacuum.

02

Apply Galilean Transformation

Using the Galilean transformation along the \(x\)-direction, where \(x' = x - vt\) and \(t' = t\), we will express the spatial and temporal partial derivatives in terms of the new variables \(x'\) and \(t'\) using the chain rule.

03

Change of Variables Using Chain Rule

Using the chain rule for partial derivatives: \[\frac{\partial}{\partial x} = \frac{\partial x'}{\partial x} \frac{\partial}{\partial x'} = \frac{\partial}{\partial x'}\]\[\frac{\partial}{\partial t} = \frac{\partial x'}{\partial t} \frac{\partial}{\partial x'} + \frac{\partial t'}{\partial t} \frac{\partial}{\partial t'} = -v \frac{\partial}{\partial x'} + \frac{\partial}{\partial t'}\]

04

Apply Derivatives to the Wave Equation

Substitute these results into the wave equation: \[\frac{\partial^{2} E}{\partial x^{2}} = \frac{\partial^{2} E}{\partial x'^2}\]\[\frac{\partial^{2} E}{\partial t^{2}} = \left(-v \frac{\partial}{\partial x'} + \frac{\partial}{\partial t'}\right)^2 E\]

05

Simplify the Equation

Expanding the time derivative term using binomial expansion:\[\frac{\partial^{2} E}{\partial t^{2}} = \left(v^2 \frac{\partial^2}{\partial x'^2} - 2v \frac{\partial^2}{\partial x' \partial t'} + \frac{\partial^2}{\partial t'^2}\right) E\]

06

Substitute Back into Original Equation

Place these into the original wave equation:\[\frac{\partial^{2} E}{\partial x'^2} - \frac{1}{c^2} \left(v^2 \frac{\partial^2 E}{\partial x'^2} - 2v \frac{\partial^2 E}{\partial x' \partial t'} + \frac{\partial^2 E}{\partial t'^2}\right) = 0\]

07

Rearrange Terms to Get the Galilean Equation

Organizing the terms, we find:\[\left(1 - \frac{v^2}{c^2}\right) \frac{\partial^2 E}{\partial x'^2} + \frac{2v}{c^2} \frac{\partial^2 E}{\partial x' \partial t'} - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} = 0\]

08

Explain Violation of Relativity Postulate

In this form, the wave equation in frame \(S'\) is different from that in frame \(S\), showing that the Galilean transformation does not preserve the form of the wave equation and thus violates the first relativity postulate.

09

Lorentz Transformation Analysis

Apply Lorentz transformations for \(x\) and \(t\): \[x' = \gamma (x - vt)\quad \text{and} \quad t' = \gamma \left(t - \frac{vx}{c^2}\right)\]where \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).

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Key Concepts

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

Wave Equation

The wave equation is a fundamental concept in physics. It describes how waves, such as electromagnetic waves, travel through a medium or space. It is particularly important when studying light and other forms of radiation. In a vacuum, the wave equation for electromagnetic waves is given by:

• \( \frac{\partial^{2} E(x, t)}{\partial x^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}} = 0 \)

where \( E(x, t) \) represents the electric field and \( c \) is the speed of light in a vacuum. The equation states that the second spatial derivative of the electric field minus the second time derivative of the electric field, scaled by the speed of light squared, is zero. This means that the changes in the electric field over space and time must be balanced.
To convert this equation to another frame of reference, transformations like the Galilean or Lorentz transformations can be used. However, only certain transformations manage to preserve the original form of this equation, maintaining the uniform speed of light across different reference frames.

Lorentz Transformation

The Lorentz transformation is crucial when dealing with high velocities, close to the speed of light. It offers a means to convert physical laws between different inertial frames without altering their form. The transformations for space and time are:

• \( x' = \gamma (x - vt) \)
• \( t' = \gamma \left(t - \frac{vx}{c^2}\right) \)

Here, \( \gamma \) is the Lorentz factor, defined as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). It accounts for time dilation and length contraction, the phenomena predicted by Einstein's theory of relativity.
When these transformations are applied, they ensure that the wave equation retains its original form. This means that the laws of physics, such as the constant speed of light, hold true regardless of how fast an observer is moving. Therefore, unlike the Galilean transformation, the Lorentz transformation adheres to the principle of relativity.

Principle of Relativity

The principle of relativity is foundational in modern physics. It postulates that the laws of physics are the same for all observers, regardless of their relative motion. This principle is a cornerstone of Einstein's theory of relativity and ensures that scientific laws have a consistent form in all inertial frames of reference.
In the context of the wave equation and transformations, this principle implies that the speed of light should be constant across all reference frames. The Galilean transformation fails this test, as it results in a distorted wave equation when moving to a new frame. In contrast, the Lorentz transformation preserves the wave equation's structure, thereby maintaining the consistency of the speed of light.
This consistency reinforces our understanding of how light behaves and supports the concept that the speed of light in a vacuum is the ultimate speed limit of the universe.