# Div G_ab = 0

What physical meaning can be ascribed to the non-divergence of the Einstein tensor? I find it counterintuitive since I associate divergence with field sources (like the electrical field of a proton) and obviously a gravitational field has a source. Is there a parallel with Newton's formulation of gravity that might be instructive?

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## Answers and Replies

What physical meaning can be ascribed to the non-divergence of the Einstein tensor? I find it counterintuitive since I associate divergence with field sources (like the electrical field of a proton) and obviously a gravitational field has a source. Is there a parallel with Newton's formulation of gravity that might be instructive?
Since the Einstein tensor is proportional to the stress-energy-momentum tensor, T it means that energy and momentum is conserved since div T = 0. This holds true even when the cosmological constant is non-zero.

Pete

haushofer
Science Advisor
What physical meaning can be ascribed to the non-divergence of the Einstein tensor? I find it counterintuitive since I associate divergence with field sources (like the electrical field of a proton) and obviously a gravitational field has a source. Is there a parallel with Newton's formulation of gravity that might be instructive?

You can also see them as constraints on the equations of motion. Intuïtively you can understand them as follows: in general relativity one wants to solve for the metric tensor, which is symmetric and thus can have 10 independent entries ( n*(n+1)/2 ). However, one is free to choose the coordinates, and this gives some freedom in your equations ( which can be seen as a gauge-freedom ). The consequences of this can be calculated to be the Bianchi-identities, which give that the Einstein tensor is divergence-free. Note that this is an identity rather than a symmetry; the description of physics doesn't depend on your coordinates.

A parallel with Newton's formulation is a little tricky; but you have to be aware of the fact that the Einstein tensor already contains second order derivatives of the metric, just like Poisson's equation is a second order differential equation of the classical gravitational field ! A sensible parallel would appear to me that due to constraints on the gravitational field the third order divergence of the classical gravitational field would be zero.

So Einstein created the stress-energy tensor first, then made $$G_{ab}$$ to match it?