4

From water waves to gravity waves

To understand gravity waves, consider the more easily understood water wave. Look at waves on the surface of a pond on an idyllic summer day. Instead of writing romantic poetry, the nerdy physicist* writes down the equation governing the surface of the water as it varies in time. How to proceed?

The famous French guy, René “I think” Descartes¹ (who used to exist), taught us that two numbers, call them x and y, suffice to locate where we are. At the location specified by (x, y), the surface is described by the height of the water measured from the bottom of the pond. Call the height g (t, x, y), a function that depends on time t and space x and y. See the figure.

Without any breeze whatsoever, the surface of the pond is flat and thus just a constant, say, 1 in some suitable unit:² g (t, x, y) = 1. (As usual in physics, we idealize: the bottom is flat and we are far from the banks in the middle of the pond.)

A wave means that g(t, x, y) is not a constant but varies in time and in space. As mentioned earlier, the physics governing water waves is clear: gravity pulls down the excess water in a crest to fill a nearby trough. Given the underlying physics, we can write down the equation³ governing how fluids behave: it “merely” involves applying Newton’s force laws to fluids.

A picture of a water wave taken at the instant t. The spatial coordinate y points out of the paper and is suppressed.

But writing down an equation is one thing, solving it is another. This equation for fluid flow has not been solved in its full generality to this very day. In fact, a million dollars is yours if you can solve it.4

Easy to see how it might be kind of tough. Let us leave the pond and go down to the beach on a windy day. As wave after wave approaches shore, they surge, curl upon themselves, try to form tunnels beloved by surfers, and break, crashing into a white foam of myriad bubbles. Fluids exhibit a bewildering wealth of behaviors. Well, the same equation that describes waves on an idyllic pond also rules here, so it can’t be easy to tame.

But that equation is easy to tame when we get back from the beach to that pond, where things are nice and easy. The point is that we can now write g (t, x, y) = 1 + h(t, x, y)into that nasty equation and treat h as small compared to 1. Then we are justified in throwing a whole truckload of terms in the resulting mess away. A small number multiplied by an equally small number produces a way smaller number; for example, 0.1 multiplied by 0.1 gives 0.01. Thus, if you encounter a term involving h multiplied by itself (namely h², the square of h) you can chuck that term out the window. Physicists and mathematicians call this the leading approximation.5

Things simplify enormously. We end up with an equation that any decent physics undergraduate can solve.

I tell you all this because the situation here is almost completely analogous to the situation with Einstein gravity. Einstein’s field equation governing the curvature of spacetime is extremely difficult to solve in general, essentially impossible, but it simplifies enormously for gravity waves in the leading approximation. Again, most physics undergrads should be able to solve the equation governing gravity waves.

I detest jargon and avoid it as best as I can, but still it is useful as shorthand to speed up the discussion. The wave on a pond on a calm summer day is said to be in the linear regime. In contrast, the crashing surf at the beach on a stormy day favored by surfers is definitely in the nonlinear regime.

The take-home message: Einstein’s equation is hard to solve in the nonlinear regime, and easy in the linear regime.

The astute reader might have noticed that I have snuck in the word “field,” as in “field of force.” To people not born into physics, the word often sounds mysterious and unfathomable.^* In fact, physicists simply call any function of space and time, such as g (t, x, y) here, a field. More in chapter 6.

In the rest of this book, I tell the story of how Einstein arrived at his field equation for gravity. In the process, I will give you a flavor of what curved spacetime means.

*  A disclaimer: that would not be me.

*  A professional secret: The concept of field is still mysterious and unfathomable, even to physicists contemplating the fundamental puzzles of the universe. See QFT Nut.