Changes

This is not an easy equation to solve without using a Fourier transform. Therefore, we'll do just that, with G as the transformed function.

This is not an easy equation to solve without using a Fourier transform. Therefore, we'll do just that, with G as the transformed function.

<math>G = \frac{1}{4\pi^2} \int{g(\Delta x,\Delta t)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}d^3x dt}</math>

<math>G(\mathbf{k},\omega) = \frac{1}{4\pi^2} \int{g(\Delta \mathbf{x},\Delta t)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}d^3x dt}</math>

Therefore,

Therefore,

<math>g = \frac{1}{4\pi^2} \int{G(\Delta \mathbf{x},\Delta t)e^{-i\mathbf{k}\cdot\Delta x}e^{i\omega \Delta t}d^3k d\omega}</math>

<math>g(\Delta \mathbf{x},\Delta t) = \frac{1}{4\pi^2} \int{G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta x}e^{i\omega \Delta t}d^3k d\omega}</math>

We will then need to plug this function into the earlier equation. To make this easier, we know that the four-dimensional delta function on the right-hand side can be simplified.

We will then need to plug this function into the earlier equation. To make this easier, we know that the four-dimensional delta function on the right-hand side can be simplified.

Now, we have an equation with integrals on both sides. Since both of these integrals have the same limits and integrands, they must be integrals of equal functions. We can simply drop the integrals.

Now, we have an equation with integrals on both sides. Since both of these integrals have the same limits and integrands, they must be integrals of equal functions. We can simply drop the integrals.

<math> \frac{d^2}{dt^2} \left(\frac{1}{4\pi^2} G(\Delta \mathbf{x},\Delta t)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right)-c^2\frac{d^2}{dx^2}\left(\frac{1}{4\pi^2} G(\Delta \mathbf{x},\Delta t)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right) = \frac{1}{16\pi^4} e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}</math>

<math> \frac{d^2}{dt^2} \left(\frac{1}{4\pi^2} G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right)-c^2\frac{d^2}{dx^2}\left(\frac{1}{4\pi^2} G(\mathbf{k},\omega)e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}\right) = \frac{1}{16\pi^4} e^{-i\mathbf{k}\cdot\Delta \mathbf{x}}e^{i\omega \Delta t}</math>

This is a complicated equation, but it can be solved for G. Once G is calculated, we can apply an inverse Fourier transform and find g; we can then plug this into

This is a complicated equation, but it can be solved for G. Once G is calculated, we can apply an inverse Fourier transform and find g; we can then plug this into