Show EM Wave equation invariant under a Lorentz Transformation

• Antic_Hay
In summary, the homework statement states that the electromagnetic wave equation is invariant under a Lorentz transformation. The attempt at a solution states that one needs to transform the equations from x to x', y to y' etc. and then rearrange terms to show that the wave equation with x, y, z, and t is equal to the wave equation with x', y', z', and t'. However, the student is not very comfortable with partial derivatives and differential calculus, so she is likely making a mistake in her derivation.
Antic_Hay

Homework Statement

Show that the electromagnetic wave equation

$\frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}} - \frac{1}{c^2}\frac{\partial^{2} \phi}{\partial t^2}$

is invariant under a Lorentz transformation.

Homework Equations

Lorentz Transformations:

$x' = \frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}}$

$t' = \frac{t - \frac{v}{c^2}x}{\sqrt{1 - \frac{v^2}{c^2}}}$

$y' = y$

$z' = z$

The Attempt at a Solution

Well I know exactly what I'm supposed to do here, transform the equations from x to x', y to y' etc. Then rearrange terms and show that the wave equation with x,y,z, and t is equal to the wave equation with x', y', z', and t'.

I understand the method is to get $\frac{\partial x'}{\partial x}$, then use the chain rule ( $\frac{\partial\phi}{\partial x} = \frac{\partial\phi}{\partial x'}\frac{\partial x'}{\partial x}$ ), and similarly for t -> t', then a bit of simple algebra and the answer should pop out the other end.

My only problem is that I have no idea what $\frac{\partial x'}{\partial x}$ is...in fact, I do know, since I have been told, that $\frac{\partial x'}{\partial x} = \frac{1}{\sqrt{1 - v^2/c^2}}$, but I have no idea how that follows from the Lorentz transforms, could someone give me a hint in deriving it?

(My attempt was simply saying $\partial x' = \frac{\partial x - v\partial t'}{1 - v^2/c^2}$, but then dividing that by $\partial x'$ I'd get $\frac{1 - \frac{\partial t'}{\partial t}}{\sqrt{1-v^2/c^2}}$, which isn't right...

As you can tell, I'm not really very comfortable with partial derivatives or differential calculus, so I'm probably doing something really silly in that derivation above...

Last edited:
Simply take the partial derivative of x' with respect to x. Same as taking a normal derivative of x' but treat all the variables except x as constants

Ah, that made sense, I knew it would be something trivial, I was just approaching it weirdly.

Great, have the answer now, thanks :)

OK, I thought I had the answer but this but it turns out I don't...

I want to calculate $\frac{\partial t'}{\partial t}$

So I have $t' = \frac{t - \frac{vx}{c^2}}{\sqrt{1 - v^2/c^2}}$

The answer is supposedly $\frac{1}{\sqrt{1 - v^2/c^2}}$

I don't see how this would be...the bottom line is a constant term so looking at the top line alone,

$\frac{\partial}{\partial t} t$ is obviously 1, fine, but I don't see how $\frac{\partial}{\partial t} (-\frac{vx}{c^2}) = 0$, surely x is a function of t, so we get

$\frac{\partial}{\partial t} x = \frac{\partial x}{\partial t} = v$ so $\frac{\partial}{\partial t} (t - vx/c^2) = 1 - v^2/c^2$ for the derivative of the top line?

Last edited:
but if someone could give me a hint on how to correctly derive \frac{\partial x'}{\partial x}, that would be great!First, it's important to note that the Lorentz transformations are derived from the principles of special relativity, which state that the laws of physics should be the same for all observers in inertial reference frames. With this in mind, we can see that the transformation from x to x' is simply a change in the observer's frame of reference, and the same applies for t to t'.

Now, to derive the expression for \frac{\partial x'}{\partial x}, we can start by taking the derivative of the first Lorentz transformation equation with respect to x:

\frac{\partial x'}{\partial x} = \frac{\partial}{\partial x}\left(\frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}}\right)

Using the chain rule, we get:

\frac{\partial x'}{\partial x} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\left(1 - \frac{v}{c^2}\frac{\partial t}{\partial x}\right)

But we know from the second Lorentz transformation equation that \frac{\partial t}{\partial x} = \frac{-v}{c^2}, so substituting this in, we get:

\frac{\partial x'}{\partial x} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\left(1 - \frac{v}{c^2}\frac{-v}{c^2}\right)

Simplifying, we get:

\frac{\partial x'}{\partial x} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}

Similarly, we can derive expressions for \frac{\partial y'}{\partial y}, \frac{\partial z'}{\partial z}, and \frac{\partial t'}{\partial t}, which are all equal to 1. Substituting these values into the wave equation and simplifying, we can see that it is indeed invariant under a Lorentz transformation.

I hope this helps! Remember, the key is to understand the meaning of the Lorentz transformations and how they relate to the principles of special relativity.

1. What is the EM Wave equation?

The EM Wave equation is a mathematical representation of how electric and magnetic fields interact with each other and propagate through space. It is expressed as a set of partial differential equations, also known as Maxwell's equations.

2. Why is the EM Wave equation important?

The EM Wave equation is important because it describes the behavior of electromagnetic waves, which are responsible for all forms of electromagnetic radiation, including light, radio waves, and X-rays. Understanding this equation is crucial for many fields of science and technology, including telecommunications, optics, and particle physics.

3. What is a Lorentz Transformation?

A Lorentz Transformation is a mathematical tool used to describe the relationship between space and time in Einstein's theory of relativity. It allows us to transform coordinates and measurements between different reference frames moving at constant velocities relative to each other.

4. How does a Lorentz Transformation affect the EM Wave equation?

A Lorentz Transformation is used to show that the EM Wave equation is invariant, meaning it remains the same, under different reference frames. This is important because it shows that the laws of physics, including the behavior of electromagnetic waves, are the same for all observers moving at constant velocities.

5. What are the implications of the EM Wave equation being invariant under a Lorentz Transformation?

The fact that the EM Wave equation is invariant under a Lorentz Transformation has significant implications for our understanding of the fundamental laws of nature. It supports the principles of relativity and allows us to make accurate predictions and observations about the behavior of electromagnetic waves in different reference frames. This has practical applications in various fields, such as GPS technology and particle accelerators.

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