Question

(a) Consider the Galilean transformation along the \(x\)-direction : \(x' = x - vt\) and \(t' = t\). In frame \(S\) the wave equation for electromagnetic waves in a vacuum is $${\partial^2E(x, t)\over \partial x^2} - {1 \over c^2} {\partial^2E(x, t)\over \partial t^2} = 0$$ where \(E\) represents the electric field in the wave. Show that by using the Galilean transformation the wave equation in frame \(S'\) is found to be $$(1 - {v^2 \over c^2} ) {\partial^2E(x', t') \over \partial{x'^2}} + {2v \over c^2} {\partial^2E(x', t') \over \partial{x'} \partial{t'}} - {1 \over c^2} {\partial^2E(x', t') \over \partial{t'^2}} = 0$$ 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 \(\partial/\partial{x'}\) and \(\partial/\partial{t'}\) by use of the chain rule.) (b) Repeat the analysis of part (a), but use the Lorentz coordinate transformations, Eqs. (37.21), and show that in frame \(S'\) the wave equation has the same form as in frame \(S\): $${\partial^2E(x', t') \over \partial{x'}^2} - {1 \over c^2} {\partial^2E(x', t') \over \partial{t'}^2} = 0$$ Explain why this shows that the speed of light in vacuum is c in both frames \(S\) and \(S'\).

Short answer

The Galilean transformation alters the wave equation, violating relativity; the Lorentz transformation preserves its form, showing light speed is constant.

Step-by-step solution

Express Derivatives Using Galilean Transformation

The Galilean transformation is given by \(x' = x - vt\) and \(t' = t\). To express derivatives in terms of \(x'\) and \(t'\), use the chain rule.\[\frac{\partial}{\partial x} = \frac{\partial x'}{\partial x} \frac{\partial}{\partial x'} + \frac{\partial t'}{\partial x} \frac{\partial}{\partial t'} = \frac{\partial}{\partial x'},\]\[\frac{\partial}{\partial t} = \frac{\partial x'}{\partial t} \frac{\partial}{\partial x'} + \frac{\partial t'}{\partial t} \frac{\partial}{\partial t'} = -v \frac{\partial}{\partial x'} + \frac{\partial}{\partial t'}.\]

Apply the Chain Rule to Second Derivatives

Compute the second derivatives using the expressions obtained in Step 1:\[\frac{\partial^2}{\partial x^2} = \left(\frac{\partial}{\partial x'}\right)^2 = \frac{\partial^2}{\partial x'^2},\]\[\frac{\partial^2}{\partial t^2} = \left(\frac{\partial}{\partial t}\right)^2 = \left(-v \frac{\partial}{\partial x'} + \frac{\partial}{\partial t'}\right)^2 = v^2 \frac{\partial^2}{\partial x'^2} - 2v\frac{\partial^2}{\partial x' \partial t'} + \frac{\partial^2}{\partial t'^2}.\]

Substitute Back into Wave Equation

Substitute the expressions for the second derivatives into the wave equation:\[\frac{\partial^2 E}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2} = \frac{\partial^2 E}{\partial x'^2} - \frac{1}{c^2}(v^2 \frac{\partial^2 E}{\partial x'^2} - 2v \frac{\partial^2 E}{\partial x' \partial t'} + \frac{\partial^2 E}{\partial t'^2})\]Simplify to obtain:\[(1 - \frac{v^2}{c^2}) \frac{\partial^2 E}{\partial x'^2} + \frac{2v}{c^2} \frac{\partial^2 E}{\partial x' \partial t'} - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} = 0.\]

Use Lorentz Transformation for Part (b)

The Lorentz transformation is given by:\[x' = \gamma (x - vt), \quad t' = \gamma (t - \frac{vx}{c^2}),\]where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \). Use the chain rule to express derivatives in terms of \(x'\) and \(t'\):\[\frac{\partial}{\partial x} = \gamma \frac{\partial}{\partial x'} - \frac{v\gamma}{c^2} \frac{\partial}{\partial t'}, \quad \frac{\partial}{\partial t} = -v\gamma \frac{\partial}{\partial x'} + \gamma \frac{\partial}{\partial t'}.\]With these expressions, you can show that the wave equation retains its form:\[\frac{\partial^2 E}{\partial x'^2} - \frac{1}{c^2} \frac{\partial^2 E}{\partial t'^2} = 0.\]

Key concepts

Wave Equation

The wave equation describes how waves, such as electromagnetic waves, propagate through space. In a vacuum, the wave equation for an electric field \( E(x, t) \) is given by:

\[ \frac{\partial^2 E(x, t)}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 E(x, t)}{\partial t^2} = 0 \]
This equation essentially states that the second spatial derivative of the electric field minus the second temporal derivative, adjusted by the speed of light squared, equals zero.

• The term \( \frac{\partial^2}{\partial x^2} \) represents how the wave propagates in space.
• The term \( \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \) represents how the wave changes over time.
• The constant \( c \) is the speed of light, highlighting that electromagnetic waves travel at this consistent speed in a vacuum.

The Galilean transformation, which is a straightforward shift in coordinates between two frames, does not preserve the form of the wave equation due to its assumption that time is absolute. This results in discrepancies when considering the propagation of light, as examined in the exercise.

Lorentz Transformation

The Lorentz transformation is key to understanding how physical laws, including wave equations, retain their form across different inertial frames moving at constant velocities relative to each other. Unlike the Galilean transformation, which fails at relativistic speeds, the Lorentz transformation accounts for the constancy of the speed of light.

The transformation is described by the equations:
\[ x' = \gamma (x - vt), \quad t' = \gamma (t - \frac{vx}{c^2}), \]where \( \gamma \) is the Lorentz factor, defined as \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \).

• The Lorentz transformation ensures that the speed of light \( c \) is the same in all reference frames.
• It corrects for time dilation and length contraction, phenomena absent in Galilean physics.
• Using this transformation on the wave equation demonstrates that the form remains unchanged in any inertial frame: \( \frac{\partial^2 E(x', t')}{\partial x'^2} - \frac{1}{c^2} \frac{\partial^2 E(x', t')}{\partial t'^2} = 0 \)

This invariance illustrates a fundamental principle of special relativity, ensuring that the laws of physics, including those describing wave propagation, are the same for observers in different inertial frames.

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that travel through space without the need for a medium. They are described by the wave equation, a crucial component in understanding their behavior.

These waves are part of the electromagnetic spectrum and include:

• Radio waves
• Microwaves
• Infrared
• Visible light
• Ultraviolet
• X-rays
• Gamma rays

All electromagnetic waves propagate at the speed of light \( c \) in a vacuum, which is approximately 299,792 kilometers per second.

• The electric and magnetic fields in these waves are perpendicular to each other and to the direction of propagation.
• The constancy of the speed of electromagnetic waves in a vacuum was a core insight for Einstein's theory of relativity.
• The wave equation reflects this constancy, emphasizing that the physics of electromagnetic waves remains consistent across different reference frames when using the Lorentz transformation.

Understanding electromagnetic waves is fundamental for exploring technologies and phenomena ranging from wifi and radio broadcasting to the behavior of light and radiation in scientific research.