Measuring proper distance with the induced metric I'm working on supplementary problem 7.7 in Schutz' General Relativity Solution Manual. It asks to find the proper distance measured by a standard ruler between neighbouring events $A$ and $B$ in terms of the difference in coordinates between events, $dx^i$. The events $A$ and $B$ are simultaneous in some reference frame $x^\mu$.
At first, I thought that the answer was simply the square root of the inner product between $d\vec{x}$ and itself, like in the formula for proper length given in Chapter 6:
$$dl = \sqrt{g_{\alpha\beta} ~ dx^\alpha dx^\beta}$$
However, this was not the case in the solution given. The length in the MCRF (momentarily co-moving reference frame) of $A$ was found, converting the local inertial coordintates $dx^\bar{i}$ to coordinates in the original frame $dx^i$ using the metric $g_{\alpha\beta}$.
\begin{align*}
dl^2&=\eta_{\bar{i}\bar{j}}dx^\bar{i}dx^\bar{j}\\
&=\eta_{\bar{i}\bar{j}}\frac{\partial x^\bar{i}}{\partial x^i}\frac{\partial x^\bar{j}}{\partial x^j}dx^idx^j\\
&=\left(\eta_{\bar{\mu}\bar{\nu}}\frac{\partial x^\bar{\mu}}{\partial x^i}\frac{\partial x^\bar{\nu}}{\partial x^j}-\eta_{\bar{0}\bar{0}}\frac{\partial x^\bar{0}}{\partial x^i}\frac{\partial x^\bar{0}}{\partial x^j}\right)dx^idx^j\\
&=\left(g_{ij}+\frac{\partial x^\bar{0}}{\partial x^i}\frac{\partial x^\bar{0}}{\partial x^j}\right)dx^idx^j\\
&=\left(g_{ij}-\frac{g_{0i}~g_{0j}}{g_{00}}\right)dx^idx^j\\
\end{align*}
The solution identifies this expression in brackets as the induced metric. It is used as the correct proper distance between $A$ and $B$, which is very confusing to me as it differs from the standard ${g_{\alpha\beta} ~ dx^\alpha dx^\beta}$. I can't seem to get them to equate to each other:
\begin{align*}
\left(g_{ij}-\frac{g_{0i}~g_{0j}}{g_{00}}\right)dx^idx^j&=g_{ij}~dx^idx^j-\frac{g_{0i}~g_{0j}}{g_{00}}dx^idx^j\\
&=g_{ij}~dx^idx^j-\left(\frac{g_{0\alpha}~g_{0\beta}}{g_{00}}dx^\alpha dx^\beta -2g_{0i}~dx^i dt-g_{00}~dt^2\right)\\
&=g_{ij}~dx^idx^j+2g_{0i}~dx^i dt+g_{00}~dt^2-\left(\frac{g_{0\alpha}~g_{0\beta}}{g_{00}}dx^\alpha dx^\beta\right)\\
&=g_{\alpha\beta}~dx^\alpha dx^\beta -\left(\frac{g_{0\alpha}~g_{0\beta}}{g_{00}}dx^\alpha d x^\beta\right)\\
&\neq g_{\alpha\beta}~dx^\alpha dx^\beta
\end{align*}
My original guess was that $dl^2 = {g_{\alpha\beta} ~ dx^\alpha dx^\beta}$ is used to calculate lengths along geodesics, while $\left(g_{ij}-\frac{g_{0i}~g_{0j}}{g_{00}}\right)dx^idx^j$ is used along spacelike distances. But  I don't think this is right, because Schutz uses ${g_{\alpha\beta} ~ dx^\alpha dx^\beta}$ for his proper distances multiple times in the text. What's going on?
 A: For these specific events $A$ and $B$, it is actually true
$$\left(g_{ij}-\frac{g_{0i}~g_{0j}}{g_{00}}\right)dx^idx^j = g_{\alpha\beta}~dx^\alpha dx^\beta$$
From the perspective of the original coordinate system however, you do not know how the coordinate increments $dt$ and $dx^i$ between these two events are related. Eventually, you might extract this information from the induced metric.
Let me expand both the LHS and RHS above
$$g_{ij}dx^idx^j-\frac{g_{0i}~g_{0j}}{g_{00}}dx^idx^j = g_{ij}dx^idx^j + 2g_{0i}dtdx^i + g_{00}dt^2$$
meaning that for these two specific events (which are defined as being sufficiently close and simultaneous in a momentarily comoving reference frame) it must be true
$$-\frac{g_{0i}~g_{0j}}{g_{00}}dx^idx^j = 2g_{0i}dtdx^i + g_{00}dt^2$$
So for these two events, increments in coordinate time $dt$ are related to increments $dx^i$ in a specific way. By using the induced metric you can avoid explicitly finding this relationship.
What is used in this exercise is the knowledge that for two sufficiently close events, that are simultaneous in a certain reference frame, we know how to express the proper length.
Proper length is a specific term used to express space-like distances between simultaneous events. Of course, being simultaneous depends on the chosen coordinate system.
Since it is stated in the exercise that the events are sufficiently close, the metric can be approximated as flat, and in the momentarily co-moving coordinate system ($dt', dx^{i\prime}$) where they are simultaneous, there is no time difference between them, i.e $dt'=0$.
The spacetime interval will then give you the distance between them.
\begin{align}\eta'_{\alpha\beta}dx'^\alpha dx'^\beta &= -dt'^2 + dx'^2 + dy'^2 + dz'^2 \\&= dx'^2 + dy'^2 + dz'^2
\end{align}
and you can re-express this as you did above in the original coordinate system, which enables recongising the induced metric.
