Observers in General Relativity and adapted coordinates

Question

A recent discussion here got me thinking about observers and changes of coordinates in General Relativity, from a more mathematical point of view.

Let us consider a timelike curve $\gamma:I\subset\mathbb R\to M$, where $M$ is a $n$-dimensional pseudo-Riemannian manifold and $\varphi_\alpha: U_\alpha\subset M\to V_\alpha\subset\mathbb R^n$ is a local coordinate system.
By possibly restricting $V_\alpha$ to a suitable subset $V'_\alpha$, it should be possible to find a map $\psi_{\alpha\beta}:V'_\alpha\subset V_\alpha\to V_\beta$ in such a way that \begin{align} (\psi_{\alpha\beta}\circ\varphi_\alpha\circ\gamma)(\lambda)\equiv y^\mu(\lambda)=(y^0(\lambda),\underbrace{0,\ldots,0}_{n-1}). \end{align} The coordinate system defined in this way seems to be "adapted" to the curve $\gamma$, in the sense that an observer sitting on it feels standing still (no velocity on spatial directions). Then I would interpret $\lambda$ as the proper time of the observer.

1. Is this true/correct?

Consider now the following two push forwards \begin{align} (\varphi_\alpha\circ\gamma)_*: &\,T_\lambda I\to T_{\varphi_\alpha\circ\gamma(\lambda)}V_\alpha\\ &\partial/\partial\lambda\to\frac{dx^\mu(\lambda)}{d\lambda}\frac{\partial}{\partial x^\mu}\\ (\psi_{\alpha\beta}\circ\varphi_\alpha\circ\gamma)_*:& \,T_\lambda I\to T_{\psi_{\alpha\beta}\circ\varphi_\alpha\circ\gamma(\lambda)}V_\beta\\ &\partial/\partial\lambda\to\frac{dx^\mu(\lambda)}{d\lambda}\frac{\partial y^a}{\partial x^\mu}\frac{\partial}{\partial y^a}=\frac{dy^a(\lambda)}{d\lambda}\frac{\partial}{\partial y^a}=\frac{dy^0(\lambda)}{d\lambda}\frac{\partial}{\partial y^0} \end{align} In the second case I find that the velocity of the observer standing still on the particle is only in the time direction, as claimed.
However in the first case, coordinates are not adapted to the curve, so the tangent vector to the curve in principle has components along all directions.

2. Is this computation correct?
3. If so, how do we interpret in the first coordinate system the parameter $\lambda$? This cannot be the proper time and I expect this to be unphysical.

Answer 1

Yes, the computation (2) is correct. For (1), indeed, you need $\gamma$ to be a regular curve ($\gamma'\ne 0$ everywhere), then the claim is quite easy. As for (3), I do not see any physical meaning to it.

As an addendum: By shrinking the domains of the charts you can always assume that your local coordinates are such that with respect to these coordinates, your semi-Riemannian metric is a small perturbation of the standard flat Lorentzian metric.

Lastly, O'Neill's book "Semi-Riemannian Geometry", while dated, is still the best mathematically rigorous reference I know for this subject. (There is also a recent book by S.Newman, I did not read it, so cannot comment.)

Answer 2

I would like to post an answer myself, stating my view on the topic, after some time of reflecting about it.
Once the coordinates on the curve $y^\mu(\lambda)$ are chosen as in the question above, the proper time should be the length of the curve in spacetime $$\Delta\tau\equiv\int_\gamma\sqrt{det(\gamma^* g)}\,d\lambda=\int_\gamma\sqrt{g_{\mu\nu}(y(\lambda))\frac{dy^\mu(\lambda)}{d\lambda}\frac{dy^\nu(\lambda)}{d\lambda}}\,d\lambda=\int_\gamma\sqrt{g_{00}(y(\lambda))\frac{dy^0(\lambda)}{d\lambda}\frac{dy^0(\lambda)}{d\lambda}}\,d\lambda,$$ where $\gamma^*g$ is the pullback of the metric $g$ on the curve $\gamma$. There are then two possibilities:
1) Require $y^0(\lambda)=\lambda$ $\implies$ the curve is described precisely by $(\lambda,0,0,0)$, but $d\tau=\sqrt{g_{00}}\, d\lambda$
2) Require $\sqrt{g_{00}\left(\frac{dy^0(\lambda)}{d\lambda}\right)^2}=1,\quad y^0(\lambda)=\int\frac{d\lambda}{\sqrt{g_{00}(y(\lambda))}}$ $\implies$ the curve is described by $(y^0(\lambda),0,0,0)$, but $d\tau=d\lambda$.
Out of the two possibilities, I regard the first one as a more mathematical choice, as it emphasizes the mathematical description of the curve, which however has no physical meaning, whereas the second one identifies, up to constants, the parameter used to describe the curve with the proper time. From a physical standpoint, the second option makes more sense.