# Wave eqn. Greens func. depends only on rel. distance proof.

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In summary, the Green's function for the wave equation satisfies the differential equation $$\square G(x,x') = \delta^{(4)}(x-x')$$ and must depend only on the relative distance between two events in spacetime due to the symmetry of the problem. This can be proven rigorously by considering the translation invariance of the problem and showing that a shifted Green's function would lead to a contradiction.
center o bass
The Greens function for the wave equation satisfies the differential equation

$$\square G(x,x') = \delta^{(4)}(x-x')$$

where ##\square= \partial_\nu \partial^\nu## is the wave operator and ##x,x'## are four vectors labeling events in spacetime. If we require ##G(x,x') \to 0## as ##x \to \infty## it is often stated that ##G(x,x')## must depend only on the relative distance due to the symmetry of the problem. But I wonder, how can one prove this rigorously?

It follows from the translation invariance of the problem: if the function ##f(x)## satisfies the wave equation ##\partial_{\nu}\partial^{\nu}f(x)=0##, then also the shifted function, ##f(x+c)##, where ##c## is any constant four-vector, satisfies the same equation.

Try forming a "shifted" Green's function ##G(x+c,x'+c)## and show that if it's not equal to ##G(x,x')##, you get a contradiction with the translation invariance.

## 1. What is the wave equation?

The wave equation is a mathematical formula that describes the behavior of waves in a given medium. It is commonly used in physics and engineering to model and analyze various types of wave phenomena, such as sound waves, light waves, and water waves.

## 2. What is the Green's function for the wave equation?

The Green's function for the wave equation is a mathematical solution that helps us understand how waves propagate in a given medium. It is a function of both time and distance, and it describes the response of the medium to a localized disturbance or source.

## 3. Why does the Green's function for the wave equation only depend on relative distance?

This is because the wave equation itself is a translation-invariant differential equation, meaning that it is independent of the origin or reference point. Therefore, the Green's function, which is a solution to the wave equation, must also be independent of the reference point and only depend on the relative distance between the source and the observation point.

## 4. How is the proof for the dependence of the Green's function on relative distance derived?

The proof for the dependence of the Green's function on relative distance is based on the mathematical properties of translation invariance and superposition. By using these properties, we can show that the Green's function is independent of the reference point and only depends on the relative distance between the source and the observation point.

## 5. What implications does the dependence of the Green's function on relative distance have?

This dependence allows us to simplify the mathematical calculations involved in solving the wave equation. It also means that the Green's function can be used to describe the behavior of waves in any medium, regardless of the specific location or reference point.

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