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## Main Question or Discussion Point

I am reading a book of General Relativity and I am stuck on a demonstration. If I consider the FLRW metric as :

##\text{d}\tau^2=\text{d}t^2-a(t)^2\bigg[\dfrac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^2+\text{sin}^2\theta\text{d}\phi^2)\bigg]##

with ##g_{tt}=1##, ##\quad g_{rr}=\dfrac{a(t)^2}{1-kr^2}## and ##\quad g_{\theta\theta}=\dfrac{g_{\phi\phi}}{\text{sin}^2\theta}=a(t)^2 r^2##

It is said in this book that, despite of the utility of comoving coordinates, the dependance of time in scale factor ##a(t)## can be better understood if we consider a set of coordinates called "Free Fall coordinates" and noted ##(\tilde{x}^\mu, \mu=0,1,2,3)## with a metric locally Lorentzian near to the origin ##\tilde{x}^{\mu}=0## :

##g_{\mu\nu}=\eta_{\mu\nu}+\dfrac{1}{2}g_{\mu\nu,\alpha\beta}(0)\tilde{x}^{\alpha}\tilde{x}^{\beta}+\,.......\quad(eq1)##

with ##\eta_{00}=-\eta_{11}=-\eta_{22}=-\eta_{33}=1\quad\quad## and ##\eta_{\mu\neq\nu}=0##

and ##g_{\mu\nu,\alpha\beta}=\dfrac{\partial^2 g_{\mu\nu}}{\partial \tilde{x}^{\alpha}\partial \tilde{x}^{\beta}}##

Moreover, one takes the expression of classic geodesics :

##\dfrac{\text{d}}{\text{d}\tau}\bigg(g_{\mu\nu}(x)\dfrac{\text{d}x^{\nu}}{\text{d}\tau}\bigg)-\dfrac{1}{2}\dfrac{\partial g_{\lambda\nu}}{\partial x^{\mu}}\dfrac{\text{d}x^{\lambda}}{\text{d}\tau}\dfrac{\text{d}x^{\nu}}{\text{d}\tau}=0\quad\quad\mu=0,1,2,3\quad (eq2)##

The author says that, by applying ##(eq1)## into the relation ##(eq2)##, one gets, at first order, the following relation :

##\dfrac{\text{d}^2 \tilde{x}^{\alpha}}{\text{d}\tau^2} = -\eta^{\alpha\gamma}\bigg[g_{\mu\gamma,\nu\beta}-\dfrac{1}{2}g_{\mu\nu,\gamma\beta}\bigg]\tilde{x}^{\beta}\dfrac{\text{d}\tilde{x}^{\mu}}{\text{d}\tau}\dfrac{\text{d}\tilde{x}^{\nu}}{\text{d}\tau}\quad\quad(eq3)##

I can't manage to obtain the ##eq(3)## from ##eq(1)## and ##eq(2)##, if someone could help me for the details of the demonstration, this would be nice.

Thanks in advance for your help

##\text{d}\tau^2=\text{d}t^2-a(t)^2\bigg[\dfrac{\text{d}r^2}{1-kr^2}+r^2(\text{d}\theta^2+\text{sin}^2\theta\text{d}\phi^2)\bigg]##

with ##g_{tt}=1##, ##\quad g_{rr}=\dfrac{a(t)^2}{1-kr^2}## and ##\quad g_{\theta\theta}=\dfrac{g_{\phi\phi}}{\text{sin}^2\theta}=a(t)^2 r^2##

It is said in this book that, despite of the utility of comoving coordinates, the dependance of time in scale factor ##a(t)## can be better understood if we consider a set of coordinates called "Free Fall coordinates" and noted ##(\tilde{x}^\mu, \mu=0,1,2,3)## with a metric locally Lorentzian near to the origin ##\tilde{x}^{\mu}=0## :

##g_{\mu\nu}=\eta_{\mu\nu}+\dfrac{1}{2}g_{\mu\nu,\alpha\beta}(0)\tilde{x}^{\alpha}\tilde{x}^{\beta}+\,.......\quad(eq1)##

with ##\eta_{00}=-\eta_{11}=-\eta_{22}=-\eta_{33}=1\quad\quad## and ##\eta_{\mu\neq\nu}=0##

and ##g_{\mu\nu,\alpha\beta}=\dfrac{\partial^2 g_{\mu\nu}}{\partial \tilde{x}^{\alpha}\partial \tilde{x}^{\beta}}##

Moreover, one takes the expression of classic geodesics :

##\dfrac{\text{d}}{\text{d}\tau}\bigg(g_{\mu\nu}(x)\dfrac{\text{d}x^{\nu}}{\text{d}\tau}\bigg)-\dfrac{1}{2}\dfrac{\partial g_{\lambda\nu}}{\partial x^{\mu}}\dfrac{\text{d}x^{\lambda}}{\text{d}\tau}\dfrac{\text{d}x^{\nu}}{\text{d}\tau}=0\quad\quad\mu=0,1,2,3\quad (eq2)##

The author says that, by applying ##(eq1)## into the relation ##(eq2)##, one gets, at first order, the following relation :

##\dfrac{\text{d}^2 \tilde{x}^{\alpha}}{\text{d}\tau^2} = -\eta^{\alpha\gamma}\bigg[g_{\mu\gamma,\nu\beta}-\dfrac{1}{2}g_{\mu\nu,\gamma\beta}\bigg]\tilde{x}^{\beta}\dfrac{\text{d}\tilde{x}^{\mu}}{\text{d}\tau}\dfrac{\text{d}\tilde{x}^{\nu}}{\text{d}\tau}\quad\quad(eq3)##

I can't manage to obtain the ##eq(3)## from ##eq(1)## and ##eq(2)##, if someone could help me for the details of the demonstration, this would be nice.

Thanks in advance for your help