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8
votes
1answer
117 views

Evaluating the Einstein-Hilbert action

The Einstein-Hilbert action is given by, $$I = \frac{1}{16\pi G} \int_{M} \mathrm{d}^d x \, \sqrt{-g} \, R \, \, + \, \, \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{d-1}x \, \sqrt{-h} \, K$$ ...
3
votes
2answers
102 views

Variation of Action with time coordinate variations

I was trying to derive equation (65) in the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu23.html This slightly unusual then usual classical mechanics ...
3
votes
1answer
71 views

Field equations in extended EH-GHY action. Is Schwarzschild a solution?

When taking the EH action, $$S_{EH} = \frac{1}{16\pi G}\int_M d^4x \sqrt{-g}R$$ and making a small variation in the metric while ignoring boundary terms, we obtain $$\delta S_{EH} = \frac{1}{16\pi ...
0
votes
1answer
65 views

Why we can set variations for the metric and its derivatives to zero at infinity?

This question is the continuation of the following one. I still don't understand why $(1)$ may be set to zero. This refers to the zero value variations of metric and its derivatives on the infinitely ...
0
votes
0answers
82 views

Induced metric on the boundary of a manifold

The Gibbons-Hawking-York term which supplements the Einstein-Hilbert action is, $$S_{GH} = \frac{1}{8\pi G} \int_{\partial M} d^3 x\sqrt{-h} \, K$$ where $\partial M$ is the boundary of the manifold ...
1
vote
2answers
404 views

How the boundary term in the variation of the action vanishes

In David Tong's QFT lecture notes (Quantum Field Theory: University of Cambridge Part III Mathematical Tripos, Lecture notes 2007, p.8), he states that We can determine the equations of motion by ...
1
vote
1answer
204 views

Einstein action and the second derivatives

I have naive question about Einstein action for field-free case: $$ S = -\frac{1}{16 \pi G}\int \sqrt{-g} d^{4}x g^{\mu \nu}R_{\mu \nu}. $$ It contains the second derivatives of metric. When we want ...
2
votes
1answer
350 views

How do I calculate the induced metric in the Gibbons–Hawking–York boundary term?

This question concerns the expression for the induced metric in the explicit variation of the GHY boundary term. Just how is that expression derived in detail from the definition of the induced metric ...
6
votes
2answers
314 views

Surface terms for field path integrals?

My question relates to something that I´ve seen in many books and appears in all its glory here: Ryder, pg 198 My question is about eq. 6.74. Which I repeat below: $$i \int {\cal D}\phi \frac{\delta ...