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A good insight into this GR contribution is to note that the circumference of a circle is not equal to $π$ times its diameter when the space is curved. In this example the circumference is a little less then $2πr$ so after one orbit Mercury arrives back where it started a little early compared to Newtonian expectations.

Is it correct and what definition of radius is it refering to? (Because it certainly isn't the case in Schwarzschild metric) and is there any intuition on how circumference becomes less than $2πr$?

DanielC
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Shaurya Kad
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Given that the relativistic effects are strongest near the massive object, you might turn the question around. How is it that the distance to the center is longer that you would expect from measuring the circumference?

The answer is that mass curves spacetime. That means that distances and time intervals near a massive object are distorted.

This is often illustrated by the infamous rubber sheet analogy. The massive object pushes spacetime "sideways", "stretching" it. The correct part of this is that it does give an idea that radial distance are longer than you might expect. But it immediately gives rise to confusing incorrect ideas.

  • Spacetime is not distorted in a sideways dimension. It is just that distances and times between points are different that for flat spacetime.
  • There are time distortions as well as space distortions. A distant observer will see time running slower near the object.
  • Gravity is not a force that holds a satellite sideways against a solid sheet of spacetime. The orbit is not determined by how the object rolls on this sheet. This is just wrong, and it is unfortunate that the image is so compelling.

For a much improved version of the rubber sheet, see the ScienceClic video A new way to visualize General Relativity.

As for insight on why this occurs, physics just describes how the universe behaves. It does not say why it does so or where the laws come from.

Physics can say that a complicated law is a consequence of simpler laws. But at some point, you have to accept some simplest laws as just the way things are. You can verify them with experiment. Sometimes they make a lot of sense because they agree with every day experience. Sometimes they are very counter intuitive because the every day world does not appear to act that way.

General relativity has a plenty of counter intuitive ideas. The foundational ideas are not too bad, though some of the consequences are very strange.

The weak equivalence principle says that the acceleration of gravity is no different from the apparent acceleration you see when you look at an object being left behind by a rocket far from any mass. The rocket explains the acceleration of the freely floating object by a force pushing it. We call such forces "fictitious" forces. They exist because you ignore your own acceleration. The weak equivalence principle says gravity is a fictitious force. A falling object is freely floating. The ground is pushing you upward, but you are ignoring that.

If you combine the weak equivalence principle with special relativity, curved spacetime follows. This might help, though it starts from conservation of energy and the equivalence principle. Why can't I do this to get infinite energy?

There is also a strong equivalence principle that says not only trajectories are the same for a rocket and on earth, but all laws of physics are the same.

mmesser314
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That depends on your coordinates:

  • In standard coordinates aka reduced circumference coordinates where the local clocks and rulers are stationary with respect to the central mass the noneuclidity is projected into the $g_{\rm r r}$ while the $g_{\rm \theta \theta}$ and $g_{\rm \phi \phi}$ stay euclidean.

  • In raindrop coordinates the $g_{\rm r r}$, $g_{\rm \theta \theta}$ & $g_{\rm \phi \phi}$ are all euclidean since in those coordinates the local clocks and rulers are free falling from infinity so that the kinematic length contraction cancels the gravitational depth expansion. Since the angular metric coefficients were already euclidean and the length contraction is only in direction of motion they stay as they were.

  • In isotropic coordinates the $g_{\rm \bar r \bar r}$, $g_{\rm \theta \theta}$ & $g_{\rm \phi \phi}$ have the same forefactor, and you could also find some coordinates where the radial coordinate equals the proper radius measured with stationary rulers and project the noneuclidity into the angular metric coefficients, but since that would give imaginary results with black holes we normally choose the other way around.

Yukterez
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