Uniqueness constraint(s) on spacetime What additional constraint(s), if any, must be used with the gravitational field equations
$$R_{\mu\nu}=\kappa \left( T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu} \right)$$ 
to uniquely determine the Christoffel symbols or metric?
In analogy to the Helmholtz decomposition of vectors, the EFE only provides information about the divergence of the connection and therefore allows for other tensors which satisfy a modified problem to  to be added to a given solution, but the nonlinear nature makes it unclear if that is true.
Intuitively the Christoffel symbols must be uniquely determined for spacetime to make sense, à la strong cosmic censorship.
(Strictly speaking this has been disproven, but that case both currently goes over my head and my question does not concern such extremes.)
 A: I'm not quite sure whether the following answers your question. If the question is about the initial value problem (Cauchy problem), and ensuring that a unique spacetime follows from a given initial condition, then this is an answer. It is notes by myself based on a treatment I found in Sean Carroll, and I have seen similar things in the book by D'inverno, with references. I wouldn't be surprised if it were also in MTW but I haven't looked. As I understand it, this answers what was called the 'hole problem' which Einstein worried about back in the early days.
We investigate the problem of finding a solution to the equation $G_{ab} = 8 \pi G T_{ab}$
(setting $c=1$), based on some given initial condition. We pick a spacelike hypersurface $\Sigma$ and
let our zeroth coordinate, $t$, have a given value on that surface. We then guess that
it will be necessary to provide the value of $g_{ab}|_\Sigma$ and the time derivative
$\partial_t g_{ab}|_\Sigma$ on the hypersurface in order to specify the 'initial state', since
the field equation is second order in time (spatial derivatives can be extracted from
the given $g_{ab}$ should we need them). However, we have $\nabla_{\mu} G^{\mu b} = 0$,
which means that the time derivatives of $g_{ab}$ cannot be chosen arbitrarily when
specifying the initial conditions; there are constraints. In order to see this,
write $\nabla_{\mu} G^{\mu b} = 0$ in the form
$$
\partial_0 G^{0b} = -\partial_i G^{i\nu} - \Gamma^{\mu}_{\mu\nu} G^{\nu b} - \Gamma^b_{\mu\nu} G^{\mu\nu}.
$$
Observe that there are no third-order time derivatives on the right hand side; it
follows that there are none on the left hand side. Therefore, although the Einstein
tensor involves second-order derivatives of the metric, the specific components
$G^{0b}$ do not involve second-order time derivatives
(a similar conclusion could be obtained about derivatives of $G^{ib}$
with respect to $x^i$, but they don't concern us). It follows that of the
ten independent components of Einstein's equation, the four represented by
$$
G^{0b} = 8 \pi G T^{0b}  
$$
do not give information that can be used to evolve the initial data
$\{ g_{ab}, \, \partial_t g_{ab} \}_\Sigma$. Rather, these components serve as 
constraints
on the initial data. For example, in vacuum the metric tensor has to have
a form such that $G^{0b} = 0$ at $t=0$. Once we have furnished such a metric 
tensor, the field equation will suffice to guarantee that $G^{0b}$ remains
zero thereafter (this is easy to show from the Bianchi identity).
We thus discover that we only have six equations describing the dynamical evolution:
$$
G^{ij} = 8 \pi G T^{ij}.
$$
This is 6 equations in 10 unknowns $g_{ab}(t)$ (recall that $g_{ab}$ is symmetric so has
10 independent components), so there is a 4-fold ambiguity. Considered as a
set of differential equations, one would say that the problem is underdetermined. 
Thus it may seem as if determinism is not respected in GR, but in fact it is. The 4-fold
ambiguity is simply the freedom that must exist, because we are free to choose
coordinates throughout spacetime. The situation is similar to that of solving
electromagnetic problems via the 4-potential. In order to 'pin down' a specific solution
one may adopt a gauge, safe in the knowledge that it will not matter which gauge
is picked. The type of coordinate freedom we meet in GR is itself a type of
gauge freedom, and can be handled by
adopting a gauge such as Lorenz gauge (also called harmonic gauge): 
$$
 \nabla^\lambda \nabla_{\lambda} x^a = 0
$$
This condition represents a differential equation for the metric components
$g^{0b}$; once we have these we can solve the field equations for $g^{ij}$
and thus we have shown that we can expect a unique solution. We have glossed
over the fact that the gauge condition may only be able to be applied over
a finite region; one would then have to divide spacetime up into patches
described by different coordinate systems.
The conclusion is that Einstein's field equation does yield a well-defined initial value problem as long as there exists a suitable Cauchy surface. This final condition is to do with the concept called global hyperbolicity. Basically one requires no closed timelike curves and a global structure of spacetime such that no event can be influenced
by a region of spacetime that was not either specified in the initial conditions
or located in their future light cones. 
If the above was known to you and your question is of a different kind, then perhaps this reply will help you to spell out more clearly what you are trying to find out. 
Note that the solution outlined above, including metric and Christoffel symbols, is uniquely determined only up to gauge invariance (or coordinate freedom; the same thing). 
