Trains

#3 Nambu-Goto action

Given a $d$ -dimensional manifold $\mathcal{M}_{d}$ with a metric $G_{\mu\nu}(X)$ , a string is a one-dimensional object moving inside $\mathcal{M}_{d}$ . The Nambu-Goto action describes the dynamics of these objects. It’s strongly inspired in the relativistic dynamics of a point particle.

A point particle describes a curve when it’s moving through space and time. Such curve (its trajectory) depends on a single parameter, usually the proper time. Likewise, a string describes a two-dimensional surface called world sheet, and depends on two parameters $X^{\mu}(\tau,\sigma)$ . Even though the role of both parameters is arbitrary, one can think of $\tau$ as the proper time, and, for any given $\tau$ , $\sigma$ would label each one of the points that the string is made of.

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#2 – Hilbert – Einstein action – Part 2

Now we compute $\frac{\delta R}{\delta g_{\mu\nu}}$ . First, the Riemann tensor has the following definition:

${R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}{\Gamma^{\rho}}_{\nu\sigma}-\partial_{\nu}{\Gamma^{\rho}}_{\mu\sigma}+ {\Gamma^{\rho}}_{\mu\lambda}{\Gamma^{\lambda}}_{\nu\sigma}-{\Gamma^{\rho}}_{\nu\lambda}{\Gamma^{\lambda}}_{\mu\sigma},$

$\delta{R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\delta{\Gamma^{\rho}}_{\nu\sigma}-\partial_{\nu}\delta{\Gamma^{\rho}}_{\mu\sigma}+\delta{\Gamma^{\rho}}_{\mu\lambda}{\Gamma^{\lambda}}+{\Gamma^{\rho}}_{\mu\lambda}\delta{\Gamma^{\lambda}}_{\nu\sigma}-\delta{\Gamma^{\rho}}_{\nu\lambda}{\Gamma^{\lambda}}_{\mu\sigma}-{\Gamma^{\rho}}_{\nu\lambda}\delta{\Gamma^{\lambda}}_{\mu\sigma}.$

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#2 – Hilbert – Einstein action – Part 1

The Hilbert-Einstein action is:

$S_{HE}=\int\left(\frac{1}{2\kappa}R+\mathcal{L}_{M}\right)\sqrt{-g}\ d^{4}x.$

Einstein equations (the evolution equations of the metrig $\mathbf{g}$ ) can be found if we set $\delta S_{HE}=0$ .

$\delta S=0= \int\left[\frac{1}{2\kappa}\left(\frac{\delta R}{\delta g_{\mu\nu}}\sqrt{-g}+R\frac{\delta\sqrt{-g}}{\delta g_{\mu\nu}}\right)+\frac{\delta (\mathcal{L}_{M}\sqrt{-g})}{\delta g_{\mu\nu}}\right]\delta g_{\mu\nu}\ d^{4}x.$

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#1 – Proof: variation of a determinant

By definition:

${\rm det}A=\frac{1}{n!}\epsilon^{\alpha_{1}\dots\alpha_{n}}\epsilon^{\beta_{1}\beta_{2}\dots\beta_{n}}\ A_{\alpha_{1}\beta_{1}}A_{\alpha_{2}\beta_{2}}\dots A_{\alpha_{n}\beta_{n}}=\epsilon^{\alpha_{1}\dots\alpha_{n}}\ A_{\alpha_{1}1}A_{\alpha_{2}2}\dots A_{\alpha_{n}n}$

$\frac{\partial\ {\rm det}A}{\partial A_{\mu\nu}}=\epsilon^{\alpha_{1}\alpha_{2}\dots\alpha_{n}}\left[\delta^{\mu}_{\alpha_{1}}\delta^{\nu}_{1}A_{\alpha_{2}2}\cdots A_{\alpha_{n}n}+A_{\alpha_{1}1}\delta^{\mu}_{\alpha_{2}}\delta^{\nu}_{2}A_{\alpha_{3}3}\cdots A_{\alpha_{n}n}+\dots+\right.$

$\left.+A_{\alpha_{1}1}A_{\alpha_{2}2}\cdots A_{\alpha_{n-1}n-1}\delta^{\mu}_{\alpha_{n}}\delta^{\nu}_{n}\right].$

Thus:

$\frac{\partial\ {\rm det}A}{\partial A_{\mu\nu}}A_{\mu\nu}=\epsilon^{\alpha_{1}\alpha_{2}\dots\alpha_{n}}\left[\underbrace{A_{\alpha_{1}1}A_{\alpha_{2}2}\cdots A_{\alpha_{n}n}+A_{\alpha_{1}1}A_{\alpha_{2}2}\cdots A_{\alpha_{n}n}+\cdots }_{\rm n\ times}\right]= n\cdot\epsilon^{\alpha_{1}\alpha_{2}\dots\alpha_{n}}\left[A_{\alpha_{1}1}A_{\alpha_{2}2}\cdots A_{\alpha_{n}n}\right]=n\cdot {\rm det}A.$