Generalizing a flat-spacetime-approach for time dilation to curved spacetimes

I would like to discuss an idea to generalize a flat-spacetime-approach for time dilation to arbitrary curved spacetimes.

Starting Point

Suppose we have – in flat spacetime – one inertial observer with worldline $C_1$ and another non-inertial observer with worldline $C_2$. Using $gamma^{-1} = sqrt{1-v^2} $ we can define the proper time along $C_2$ connecting two events $ E_a, E_b $ as

$$ tau[C_2] = int_{C_2} dtau = int_{T_a}^{T_b} dt , gamma^{-1} $$

We can work in the rest frame of the inertial observer using coordinates

$$ x^mu = (t, x^i) $$

and 4-velocities

$$ u_{(1)}^mu = (1,0,0,0) $$

$$ u_{(2)}^mu = gamma(1,v^i) $$

with

$$ gamma = eta_{munu} , u_{(1)}^{mu} , u_{(2)}^{nu} $$

Generalization

I would like to generalize this to arbitrary curved spacetime, but – before going through the math – please let me know if the general idea is reasonable at all.

We start in the proper frame of observer 1 with

$$ x_{(1)}^mu = (t,0,0,0) $$

We define a spacelike foliation

$$ M = Sigma times mathbb{R} $$

of spacetime $ M $ where time $ t in mathbb{R} $ labels the slices $ Sigma_t $

For every point $ P in C_1 $ with time $ x_{(1)}^0(P) = t $ we have

$$ P = Sigma_t cap C_1 $$

$$ Sigma_t perp u_{(1)}(P)$$

In addition for every point $ Q in C_2 $ with time $ x_{(2)}^0(Q) = t $ we have

$$ Q = Sigma_t cap C_2 $$

I omit the discussion of the induced coordinates $x^i$ for each $ Sigma_t $.

Then for each pair $ P, Q in Sigma_t $ i.e. in the same slice we define a spacelike geodesic $ C_{PQ} in Sigma_t $ (connecting $P$ and $Q$) with tangent vector $ k $ along $ C_{PQ} $

$$ nabla_k k = 0 $$

This defines the parallel transport of $ u_{(2)} $ along $ C_{PQ} $

$$ nabla_k u_{(2)} = 0 $$

At least formally (ref. to Wald) we can invert this parallel transport to define

$$ u_{(2)}(P) = nabla_k^{-1} , u_{(2)}(Q) $$

This allows us to project this transported 4-velocity on $ u_1(P) $

$$ gamma_{Q}(P) = langle u_{(1)}(P), u_{(2)}(P) rangle = langle u_{(1)}(P), nabla_k^{-1} , u_{(2)}(Q) rangle $$

Using $t = tau_1 = t(P) = t(Q)$ one can introduce the integral

$$ tau_{C_1}[C_2] = int_{C_1} dtau_1, gamma^{-1}_{1,2}(tau_1) $$

Questions

1. is this approach reasonable or rather nonsensical?
2. (how) can one show that intended association $ tau_{C_1}[C_2] stackrel{?}{=} tau[C_2] $is justified?
3. (how) can one show that this definition of $tau[C_2]$ is invariant with respect to change of the foliation? (related to diffeomeophism invariance)
4. has this been discussed in detail somewhere? (I remember Wald’s approach to red shift using the inverted transport)