Problem 76 - University Physics with Modern Physics

Question

(a) Consider the Galilean transformation along the \(x\) -direction: \(x^{\prime}=x-v t\) 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 $$\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime 2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial x^{\prime} \partial t^{\prime}}-\frac{1}{c^{2}} \frac{\partial^{2} E\left(x^{\prime}, t^{\prime}\right)}{\partial t^{2}}=0$$ This has a different form than the wave equation in \(S\) . Hence the Galiean 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 part (a), but use the Lorentz coordinate transformations, Eqs. (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

Answer

Galilean transformation changes the wave equation form, violating relativity. Lorentz transformation preserves form, confirming constant speed of light across frames.

Step 1: Understand the Galilean Transformation

The Galilean transformation is a set of equations used to transform between two reference frames: \(x^{\prime} = x - vt\) and \(t^{\prime} = t\). It assumes that the addition of velocities is valid, which does not hold true for speeds close to the speed of light.

Step 2: Apply Transformation to the Wave Equation

Starting with the wave equation in frame \(S\):\[ \frac{\partial^{2} E(x, t)}{\partial x^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} E(x, t)}{\partial t^{2}} = 0 \]Substitute \(x = x^{\prime} + vt\) into the derivatives for \(x\):\[ \frac{\partial}{\partial x} = \frac{\partial x^{\prime}}{\partial x} \cdot \frac{\partial}{\partial x^{\prime}} + \frac{\partial t^{\prime}}{\partial x} \cdot \frac{\partial}{\partial t^{\prime}} = \frac{\partial}{\partial x^{\prime}} \]Since \(t^{\prime} = t\), the derivatives with respect to \(t\) remain unchanged.Substituting these into the wave equation gives the new form as:\[ \left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} E(x^{\prime}, t^{\prime})}{\partial x^{\prime 2}} + \frac{2v}{c^{2}} \frac{\partial^{2} E(x^{\prime}, t^{\prime})}{\partial x^{\prime} \partial t^{\prime}} - \frac{1}{c^{2}} \frac{\partial^{2} E(x^{\prime}, t^{\prime})}{\partial t^{2}} = 0 \]

Step 3: Notice the Form Change in Frame S'

The resulting equation has extra terms that are not in the original wave equation, indicating a change in its form. This is because the Galilean transformation does not take into account relativistic effects, especially for electromagnetic phenomena which require the Lorentz transformation for preservation of the wave equation form.

Step 4: Use Lorentz Transformation to Preserve Form

To preserve the form of the wave equation across frames, use the Lorentz transformation:\[ x^{\prime} = \gamma(x - vt) \] and \[ t^{\prime} = \gamma(t - \frac{vx}{c^{2}}) \], where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \).These transformations ensure that the form of physical laws is identical in all inertial frames. Once substituted correctly, the wave equation in frame \(S^{\prime}\) becomes:\[ \frac{\partial^{2} E(x^{\prime}, t^{\prime})}{\partial x^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} E(x^{\prime}, t^{\prime})}{\partial t^{\prime 2}} = 0 \], same as frame \(S\).

Step 5: Conclude on Speed of Light Constancy

Since the Lorentz transformations preserve the form of the wave equation between frames \(S\) and \(S^{\prime}\), this demonstrates that the speed of light is constant in all inertial reference frames, aligned with Einstein's postulate that the speed of light in a vacuum remains \(c\) irrespective of the motion of the source or observer.

Key Concepts

Galilean Transformation

The Galilean transformation is a mathematical tool used to shift between two reference frames that move at a constant velocity relative to each other. It comes into play when both frames move along a single direction, typically the x-axis, with one moving with a velocity \( v \). In its simplest form, it is expressed as:

• \( x' = x - vt \)
• \( t' = t \)

These formulas suggest a straightforward relationship between the coordinates and time of two observers in different frames. Here, \( t' = t \) assumes time is the same in both frames, a key assumption of the Galilean transformation.
However, this approach works well when dealing with low speeds, much slower than the speed of light. At these speeds, the addition of velocities seems simple, and the temporal aspect requires no alteration. But when speeds approach the speed of light, such straightforward assumptions falter. The transformation does not accommodate the relativistic effects necessary for light-speed scenarios, leading to discrepancies in equations such as the wave equation for electromagnetic waves.

Lorentz Transformation

The Lorentz transformation steps in where the Galilean transformation is limited. It adjusts for the peculiarities of relativistic speeds, those nearing the speed of light \( c \). This transformation, unlike the Galilean, properly accounts for alterations in space and time as per relativity. Based on Einstein's theory of relativity, its equations are:

• \( x' = \gamma(x - vt) \)
• \( t' = \gamma(t - \frac{vx}{c^2}) \)
• where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

The \( \gamma \) factor, or Lorentz factor, adjusts for the effects of relative motion on measurements of time and space between two frames. It ensures that at high velocities, near the speed of light, the distortions described by relativity are accounted for properly.
Using Lorentz transformations, the wave equation maintains its form across frames, proving the particle's behavior remains consistent. This confirms that laws of physics, such as the wave equation, sustain uniformity across all inertial frames when critical speed thresholds are approached.

Speed of Light

The speed of light, denoted by \( c \), is a fundamental constant in physics, approximately equal to \( 3 \times 10^8 \) meters per second. It is a central tenet in the theory of relativity, serving as the upper limit for how fast information or matter can travel. Importantly, no matter the motion of the source or the observer, light's speed in a vacuum remains \( c \).
This invariance is a key part of Einstein's postulates of special relativity. In our context, properly applying Lorentz transformations rather than Galilean ensures the wave equation remains unchanged across inertial frames. The consistent form of the wave equation vindicates the constancy of \( c \) in every frame.
This constancy elucidated by the wave equation's form implies that in both frame \( S \) and \( S' \), light maintains its speed \( c \). Therefore, the universe's "speed limit" applies everywhere and every time, highlighting how relativity reshapes our understanding of speed and time.

Inertial Frames

Inertial frames are a concept essential to classical and relativistic physics. These are frames of reference where an object in motion remains in motion at constant velocity unless an external force acts upon it. In such frames, the laws of physics, including mechanics, always sustain the same form.
The need for consistent laws across inertial frames leads us to inspect transformations like Galilean and Lorentz more closely. Especially at speeds closer to light, discrepancies arise if purely Galilean approaches are used. Lorentz transformations emerge necessary to preserve the law's nature across various inertial frames.

• All inertial frames are equivalent in their description of physical processes.
• The Lorentz transformation ensures identical physical laws across frames moving at constant, high velocities.

This reader-friendly notion aligns with the principle of relativity, stating all physics' laws must be equally valid and take the same form across all inertial frames, no matter their relative velocity.