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## Main Question or Discussion Point

We can ignore the boundary contribution, namely the term that arises when we solve

$$\int d^{4}x \sqrt{-g}g^{ab} \delta R_{ab}$$ when varying the action. According to Wikipedia it's appropriate to do so only when the underlying manifold is both without boundary and compact.

That condition seems reasonable to me, because (as I current understand it) a manifold without boundary is a manifold in which every point has an open neighborhood ##U## homeomorphic to an open subset of ##\mathbb{R}^4##, such that if any point in ##U## has coordinates ##(x,y,z,t)## in ##\mathbb{R}^4##, ##t## is greater than zero.

So the manifold don't having a boundary (or, pheraphs having a boundary at infinity), Stokes' theorem says that the integral vanishes and that integral above vanishes too.

I'd like to know when should we consider closed manifolds in Relativity and when we should not.

What is the physical meaning in terms of the theory of Relativity to have an open/closed manifold?

Also, it seems that the physicists in working on the equations for the action did not realize that we should include the boundary term. Many years later Hawking-Gibbons-York did realize it. Why were the pioneers first considering only closed manifolds, is there a special reason for this choice?

$$\int d^{4}x \sqrt{-g}g^{ab} \delta R_{ab}$$ when varying the action. According to Wikipedia it's appropriate to do so only when the underlying manifold is both without boundary and compact.

That condition seems reasonable to me, because (as I current understand it) a manifold without boundary is a manifold in which every point has an open neighborhood ##U## homeomorphic to an open subset of ##\mathbb{R}^4##, such that if any point in ##U## has coordinates ##(x,y,z,t)## in ##\mathbb{R}^4##, ##t## is greater than zero.

So the manifold don't having a boundary (or, pheraphs having a boundary at infinity), Stokes' theorem says that the integral vanishes and that integral above vanishes too.

I'd like to know when should we consider closed manifolds in Relativity and when we should not.

What is the physical meaning in terms of the theory of Relativity to have an open/closed manifold?

Also, it seems that the physicists in working on the equations for the action did not realize that we should include the boundary term. Many years later Hawking-Gibbons-York did realize it. Why were the pioneers first considering only closed manifolds, is there a special reason for this choice?