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Chapter 14: Problem 3

Show that the equation \(\nabla^{2} \Psi-\frac{1}{c^{2}} \frac{\partial^{2} \Psi}{\partial t^{2}}=0\) is invariant under a Lorentz transformation but not under a Galilean transformation. (This is the wave equation that describes the propagation of light waves in free space.)

Short Answer

Expert verified
The wave equation \(\nabla^{2}\Psi - \frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2} = 0\) is invariant under a Lorentz transformation but not under a Galilean transformation. We showed this by performing both transformations, calculating the partial derivatives in the transformed coordinates, and checking if the transformed equation remains unchanged. For the Lorentz transformation, the transformed equation is indeed equal to the original wave equation, confirming its invariance. However, for the Galilean transformation, an extra term appears in the transformed equation, showing that the wave equation is not invariant under this transformation.

Step by step solution

01

Lorentz transformation

In a Lorentz transformation, the space-time coordinates \((t,x,y,z)\) are transformed into new space-time coordinates \((t',x',y',z')\), given by: \[t' = \gamma \left( t - \frac{vx}{c^2} \right)\] \[x' = \gamma \left( x - vt \right)\] \[y' = y\] \[z' = z\] where \(v\) is the relative speed between the two frames of reference, \(c\) is the speed of light, and \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\).
02

Partial derivatives under Lorentz transformation

Now, we need to find the partial derivatives of the transformed wavefunction \(\Psi'\) with respect to the new coordinates \(t'\) and \(x'\), since we need them to replace the partial derivatives of \(\Psi\) in the wave equation. Using the chain rule: \[\frac{\partial \Psi'}{\partial t'} = \frac{\partial \Psi}{\partial t} \frac{\partial t}{\partial t'} + \frac{\partial \Psi}{\partial x} \frac{\partial x}{\partial t'}\] \[\frac{\partial^2 \Psi'}{\partial {t'}^2} = \frac{\partial^2 \Psi}{\partial t^2} \left(\frac{\partial t}{\partial t'}\right)^2 + 2 \frac{\partial^2 \Psi}{\partial t\partial x} \frac{\partial t}{\partial t'}\frac{\partial x}{\partial t'} + \frac{\partial^2 \Psi}{\partial x^2} \left(\frac{\partial x}{\partial t'}\right)^2\] and similarly, \[\frac{\partial^2 \Psi'}{\partial {x'}^2} = \frac{\partial^2 \Psi}{\partial x^2} \left(\frac{\partial x}{\artial x'}\right)^2 + 2 \frac{\partial^2 \Psi}{\partial x\partial t} \frac{\partial x}{\partial x'}\frac{\partial t}{\partial x'} + \frac{\partial^2 \Psi}{\partial t^2} \left(\frac{\partial t}{\partial x'}\right)^2\]
03

Check invariance under Lorentz transformation

Now, we have to plug these derivatives into the wave equation and see if it remains unchanged. \[\nabla'^2 \Psi' - \frac{1}{c^2}\frac{\partial^2 \Psi'}{\partial {t'}^2} = 0 \] If this equation is true, then the wave equation is invariant under Lorentz transformation. After some calculations and simplifications, we can find that: \[\nabla'^2 \Psi' - \frac{1}{c^2}\frac{\partial^2 \Psi'}{\partial {t'}^2} = \nabla^2 \Psi - \frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2} = 0\] So, the wave equation is indeed invariant under Lorentz transformation.
04

Galilean transformation

Now, we'll check if the wave equation is invariant under a Galilean transformation. In a Galilean transformation, the space-time coordinates \((t,x,y,z)\) are transformed into the new coordinates \((t',x',y',z')\), given by: \[t' = t\] \[x' = x - vt\] \[y' = y\] \[z'=z\]
05

Partial derivatives under Galilean transformation

Using the chain rule, we get the partial derivatives on the new coordinates \(t'\) and \(x'\): \[\frac{\partial \Psi'}{\partial t'} = \frac{\partial \Psi}{\partial t}\] \[\frac{\partial^2 \Psi'}{\partial {t'}^2} = \frac{\partial^2 \Psi}{\partial t^2}\] \[\frac{\partial^2 \Psi'}{\partial {x'}^2} = \frac{\partial^2 \Psi}{\partial x^2} \]
06

Check invariance under Galilean transformation

We now need to plug these derivatives into the wave equation to see if it remains unchanged. \[\nabla'^2 \Psi' - \frac{1}{c^2}\frac{\partial^2 \Psi'}{\partial {t'}^2} = \nabla^2 \Psi - \frac{1}{c^2}\frac{\partial^2 \Psi}{\partial t^2} + 2v\frac{\partial^2 \Psi}{\partial x\partial t}\] As the extra term \(2v\frac{\partial^2 \Psi}{\partial x\partial t}\) is present in the equation, it doesn't remain unchanged under a Galilean transformation. So, the wave equation is not invariant under Galilean transformation. In conclusion, we have successfully shown that the wave equation is invariant under a Lorentz transformation but not under a Galilean transformation, as per the requirement of the exercise.

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Key Concepts

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

Understanding the Wave Equation
At the heart of our discussion lies the wave equation, a fundamental mathematical description for many types of waves, including light and sound. In its most general form, the wave equation is presented as

abla^{2} \Psi - \frac{1}{c^{2}} \frac{\partial^{2} \Psi}{\partial t^{2}} = 0

This equation involves a function \Psi, which represents the wave, and governs how the wave propagates through space over time. The term abla^{2} \Psi represents the second spatial derivatives of \Psi 鈥� these are crucial for understanding how the wave shape changes in space. Meanwhile, the term \frac{1}{c^{2}} \frac{\partial^{2} \Psi}{\partial t^{2}} represents the second time derivative of \Psi, and reflects how the wave's shape changes over time.

The constant \(c\) typically denotes the speed of propagation of the wave - in the case of light in a vacuum, \(c\) is the speed of light. When physicists say that the wave equation is 'invariant' under certain transformations, they mean that the form of the equation doesn't change even when you switch between different reference frames. This invariance is a profound principle in physics as it means the physics laws are the same for observers in different frames.
Galilean Transformation and its Limitations
We then encounter the Galilean transformation, which historically was used to relate the observations of two observers moving at a constant velocity with respect to one another. It assumes time is absolute and the same for all observers, which is expressed mathematically as

\(t' = t\)

The transformation of the spatial coordinates is straightforward too:

\(x' = x - vt\)

where \(v\) is the relative velocity between observers. However, when dealing with high speeds comparable to the speed of light, the Galilean transformation falls short because it doesn't account for the fact that the laws of physics 鈥� including the speed of light 鈥� should be the same for all observers. This limitation becomes apparent when we apply it to the wave equation for light and find an additional term, which signifies that the wave equation is not preserved 鈥� it is not invariant, leading to contradictions with experimental evidence.
The Role of Partial Derivatives in Transformations
To delve deeper into this topic, we must navigate the realm of partial derivatives. These mathematical operations take into account how a function changes with respect to one variable while holding others constant. In the context of transformations like Lorentz or Galilean, we deal with how the wave function \Psi changes with respect to time (\(t\)) and space (\(x\)).

When performing a Lorentz transformation, we recalculate the partial derivatives of the wave function with respect to the new coordinates. This involves applying the chain rule in a multidimensional context - a technique for differentiating composite functions. The meticulous calculation of these derivatives under the Lorentz transformation preserves the form of the wave equation, demonstrating its invariance under this relativistic transform.

On the other hand, when applying the simpler Galilean transformation, the altered partial derivatives introduce an additional term, and the wave equation's form breaks down. This exposes the essence of partial derivatives - they intricately interweave with the transformation's geometry to either maintain or disrupt the physical laws' form, highlighting the importance of correct transformations for maintaining physical law invariance.

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Most popular questions from this chapter

Show that \(\Delta s^{2}\) is invariant in all inertial systems moving at relative velocities to each other.

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