2
$\begingroup$

Let me expose my problem, I am trying to perform the explicit variation of the Gibbons-Hawking-York boundary term, $$S_{GH}=\int_{\partial M} d^{n-1}x\sqrt{\left|h\right|}K$$ The problem I have is that in the calculation of $\delta\sqrt{\left|h\right|}$, it seems like I should carry the calculation as if $h$ was the first fundamental form $$h_{\mu\nu}=g_{\mu\nu} - \sigma n_{\mu} n_{\nu} $$ inasmuch as I obtain the good result doing so. First, I use the identity $$ \delta\sqrt{\left|h\right|} = -\frac12 \sqrt{\left|h\right|} h_{\mu\nu} \delta h^{\mu\nu} $$ then I express $\delta h$ in fuction of $\delta g$ $$\delta h^{\mu\nu} = \delta g^{\mu\nu} -\sigma \delta n^\mu n^\nu -\sigma n^\mu\delta n^\nu $$ and using the fact that $h_{\mu\nu} n^\mu = 0$, we obtain

$$\delta\sqrt{\left|h\right|} = -\frac12 \sqrt{\left|h\right|} h_{\mu\nu} \delta g^{\mu\nu}.$$

My problem is that $h$ is not the first fundamental form in the first expression, but the induced metric. The determinant of the first fundamental form is 0 in gaussian normal coordinates, so it seems like I am skipping a step in this derivation, but I just cannot find what (I never had a course in differential geometry, so my understanding of the difference between the first fundamental form and the induced metric is really poor).

$\endgroup$
2
$\begingroup$

I got the answer reading a book from E. Poisson, what I was doing was indeed wrong, you have to start with the induced metric given by $$ h_{ab}= g_{\mu\nu}e^{\mu}_a e^{\nu}_b $$ where $$e^{\mu}_a=\frac{\partial x^{\mu}}{\partial y^a}$$ are the tangent vectors to curves of the hypersurface. Then, you just replace $g$ by $h$ in the usual relation $$\delta\sqrt{\left|h\right|} = -\frac12 \sqrt{\left|h\right|} h_{ab} \delta h^{ab}.$$ Using the Kronecker invariance, one finds

$$\delta\sqrt{\left|h\right|} = \frac12 \sqrt{\left|h\right|} h^{ab} \delta h_{ab}.$$

Then using the fact that $$ \delta h_{ab} =\delta g_{\mu\nu}e^\mu_a e^\nu_b$$ since the tangent vectors are invariant, we finally get $$ \delta\sqrt{\left|h\right|} = \frac12 \sqrt{\left|h\right|} h^{\mu\nu} \delta g_{\mu\nu}$$ where we have used the definition of the PROJECTOR $$h^{\mu\nu} = h^{ab}e^\mu_a e^\nu_b.$$ And this is where the problem was coming, $h^{\mu\nu}$ is NOT the induced metric, but a projector associated to that induced metric.

$\endgroup$
1
$\begingroup$

as far as i know these two expressions are used synonymically.

and your coordinates $h_{\mu \nu}=g_{\mu \nu}-\sigma n_{\mu}n_{\nu}$ arent gaussian coordinates in general.

gaussian are for example flrw metrics like $ds^2=-dt^2+h_{ij}(t)dx^idx^j$ with spacelike $i,j$ where there is no mixed time/space basis term in the second part of the line element, only a scalefactor dependent on $t$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.