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# 3.8: The Metric in General Relativity

So far we’ve considered a variety of examples in which the metric is predetermined. This is not the case in general relativity. For example, Einstein published general relativity in 1915, but it was not until 1916 that Schwarzschild found the metric for a spherical, gravitating body such as the sun or the earth.

When masses are present, finding the metric is analogous to finding the electric field made by charges, but the interpretation is more difficult. In the electromagnetic case, the field is found on a preexisting background of space and time. In general relativity, there is no preexisting geometry of spacetime. The metric tells us how to find distances in terms of our coordinates, but the coordinates themselves are completely arbitrary. So what does the metric even mean? This was an issue that caused Einstein great distress and confusion, and at one point, in 1914, it even led him to publish an incorrect, dead-end theory of gravity in which he abandoned coordinate-independence.

With the benefit of hindsight, we can consider these issues in terms of the general description of measurements in relativity given in section 3.4:

1. We can tell whether events and world-lines are incident.
2. We can do measurements in local Lorentz frames.

# The Hole Argument

The main factor that led Einstein to his false start is known as the hole argument. Suppose that we know about the distribution of matter throughout all of spacetime, including a particular region of finite size — the “hole” — which contains no matter. By analogy with other classical field theories, such as electromagnetism, we expect that the metric will be a solution to some kind of differential equation, in which matter acts as the source term. We find a metric $$g(x)$$ that solves the field equations for this set of sources, where $$x$$ is some set of coordinates. Now if the field equations are coordinate-independent, we can introduce a new set of coordinates x', which is identical to $$x$$ outside the hole, but differs from it on the inside. If we reexpress the metric in terms of these new coordinates as $$g'(x')$$, then we are guaranteed that $$g'(x')$$ is also a solution. But furthermore, we can substitute $$x$$ for $$x'$$, and $$g'(x)$$ will still be a solution. For outside the hole there is no difference between the primed and unprimed quantities, and inside the hole there is no mass distribution that has to match the metric’s behavior on a point-by-point basis.

We conclude that in any coordinate-invariant theory, it is impossible to uniquely determine the metric inside such a hole. Einstein initially decided that this was unacceptable, because it showed a lack of determinism; in a classical theory such as general relativity, we ought to be able to predict the evolution of the fields, and it would seem that there is no way to predict the metric inside the hole. He eventually realized that this was an incorrect interpretation. The only type of global observation that general relativity lets us do is measurements of the incidence of world-lines. Relabeling all the points inside the hole doesn’t change any of the incidence relations. For example, if two test particles sent into the region collide at a point x inside the hole, then changing the point’s name to x' doesn’t change the observable fact that they collided.