Tidal Forces, Gravitational Waves

Let $V(t)$ be the volume of a small ball of test particles in free fall that are initially at rest relative to each other. In the vacuum there is no energy density or pressure, so $\ddot V\vert _{t = 0} = 0$, but the curvature of spacetime can still distort the ball. For example, suppose you drop a small ball of instant coffee when making coffee in the morning. The grains of coffee closer to the earth accelerate towards it a bit more, causing the ball to start stretching in the vertical direction. However, as the grains all accelerate towards the center of the earth, the ball also starts being squashed in the two horizontal directions. Einstein's equation says that if we treat the coffee grains as test particles, these two effects cancel each other when we calculate the second derivative of the ball's volume, leaving us with $\ddot V\vert _{t = 0} = 0$. It is a fun exercise to check this using Newton's theory of gravity!

This stretching/squashing of a ball of falling coffee grains is an example of what people call 'tidal forces'. As the name suggests, another example is the tendency for the ocean to be stretched in one direction and squashed in the other two by the gravitational pull of the moon.

Gravitational waves are another example of how spacetime can be curved even in the vacuum. General relativity predicts that when any heavy object wiggles, it sends out ripples of spacetime curvature which propagate at the speed of light. This is far from obvious starting from our formulation of Einstein's equation! It also predicts that as one of these ripples of curvature passes by, our small ball of initially test particles will be stretched in one transverse direction while being squashed in the other transverse direction. From what we have already said, these effects must precisely cancel when we compute $\ddot V\vert _{t = 0}$.

Hulse and Taylor won the Nobel prize in 1993 for careful observations of a binary neutron star which is slowly spiraling down, just as general relativity predicts it should, as it loses energy by emitting gravitational radiation. Gravitational waves have not been directly observed, but there are a number of projects underway to detect them. For example, the LIGO project will bounce a laser between hanging mirrors in an L-shaped detector, to see how one leg of the detector is stretched while the other is squashed. Both legs are 4 kilometers long, and the detector is designed to be sensitive to a $10^{-18}$-meter change in length of the arms.

© 2006 John Baez and Emory Bunn

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