Metric for a Collapsing Disk and FLRW I have to obtain a time-dependent metric for a disk which satisfies a simple differential equation but I am stuck with making sure that the physics is correct.

To be explicit, I'll first describe a shrinking disk:
Suppose that we are working in a $4D$ Minkowski spacetime $(\mathcal{M},\eta_{ab})$ where the metric has signature $\text{sign}(\eta)=\text{diag}(-,+,+,+)$ and $\mathcal{M}=\mathbb{R}_t\times\mathbb{R}_{\Sigma}^3$. Consider embedding a compact disk $\Omega_t\hookrightarrow\mathbb{R}^3_{\Sigma}$ which is defined at each time slice by $$\Omega_t=\left\{(x,y)\in\mathbb{R}^2: x(t)^2+y(t)^2\le R(t)^2 \right\}$$ where the radius of the disk solves the differential equation $$\frac{\text{d}R(t)}{\text{d}t} = -\gamma R(t)$$
with the initial and final conditions $R(t_i)=R_i$, $R(t_f)=R_f$, and $R_f<R_i$. The disk starts at some initial radius $(t_i,R_i)$ and shrinks until it reaches the smaller radius $(t_f,R_f)$. The spacetime diagram looks like a cone.
I can solve the differential equation to find that the radius has the explicit form $$R(t) = R_i \exp(-\gamma (t-t_i)) =  R_i\exp\left(\frac{t\,- t_i}{t_f-t_i}\log\left[\frac{R_f}{R_i}\right]\right).$$

I'm now stuck trying to construct a metric for the disk. My initial guess was that it should look like FLRW since it looks like a shrinking universe; $$\text{d}s^2 \overset{?}{=} -\text{d}t^2 + \exp(-2\gamma(t-t_i))\left(\text{d}r^2+ R_i^2\text{d}\theta^2\right)$$
but I am unsure how to derive this metric, assuming that it is in the correct form. Could I get some help with deriving a metric for this disk model? Where is a good starting point?
 A: The metric 
$$ds^2=-dt^2+dx^2+dy^2$$
with:
$$x^2+y^2=R^2\,\quad \Rightarrow \quad y=\sqrt{R(t)^2-x^2}$$
$$dy=-{\frac {x{\it dx}}{\sqrt { \left( R \left( t \right)  \right) ^{2}-{x
}^{2}}}}+{\frac {R \left( t \right)  \left( {\frac {d}{dt}}R \left( t
 \right)  \right) {\it dt}}{\sqrt { \left( R \left( t \right) 
 \right) ^{2}-{x}^{2}}}}
$$
Goto 
$$ds^2= \left( -1+{\frac { \left( R \left( t \right)  \right) ^{2} \left( {
\frac {d}{dt}}R \left( t \right)  \right) ^{2}}{ \left( R \left( t
 \right)  \right) ^{2}-{x}^{2}}} \right) {{\it dt}}^{2}-2\,{\frac {{
\it dx}\,R \left( t \right)  \left( {\frac {d}{dt}}R \left( t \right) 
 \right) x{\it dt}}{ \left( R \left( t \right)  \right) ^{2}-{x}^{2}}}
+{{\it dx}}^{2} \left( 1+{\frac {{x}^{2}}{ \left( R \left( t \right) 
 \right) ^{2}-{x}^{2}}} \right) 
$$
$$g=
 \left[ \begin {array}{cc} -1+{\frac { \left( R \left( t \right)
 \right) ^{2} \left( {\frac {d}{dt}}R \left( t \right)  \right) ^{2}}{
 \left( R \left( t \right)  \right) ^{2}-{x}^{2}}}&-{\frac {R \left( t
 \right)  \left( {\frac {d}{dt}}R \left( t \right)  \right) x}{
 \left( R \left( t \right)  \right) ^{2}-{x}^{2}}}
\\ -{\frac {R \left( t \right)  \left( {\frac {d}{dt
}}R \left( t \right)  \right) x}{ \left( R \left( t \right)  \right) ^
{2}-{x}^{2}}}&1+{\frac {{x}^{2}}{ \left( R \left( t \right)  \right) ^
{2}-{x}^{2}}}\end {array} \right]$$
$$\det(g)={\frac { \left( R \left( t \right)  \right) ^{2} \left( -1+ \left( {
\frac {d}{dt}}R \left( t \right)  \right) ^{2} \right) }{ \left( R
 \left( t \right)  \right) ^{2}-{x}^{2}}}
$$
Edit
Metrix with polar coordinate
with $x=R(t)\,\cos(\phi)\quad, y=R(t)\,\sin(\phi)$ this satisfy the constraint equation $x^2+y^2=R(t)^2$
you get the metric:
$$ds^2=-dt^2+dx^2+dy^2=-dt^2+\left( {\frac {d}{dt}}R \left( t \right)  \right) ^{2}{{\it dt}}^{2}+
 \left( R \left( t \right)  \right) ^{2}{d\phi }^{2}=\left( -1+ \left( {\frac {d}{dt}}R \left( t \right)  \right) ^{2}
 \right) {{\it dt}}^{2}+ \left( R \left( t \right)  \right) ^{2}{d
\phi }^{2}
$$
