Observer as a future-directed, timelike curve on $M$

Question

Let's suppose that $M$ is spacetime manifold. An observer is then defined as a future-directed, timelike curve $\gamma : I\rightarrow M$, together with a frame field $\{e_{\mu}(\tau)\}^4_{\mu = 1}\in T_{\gamma(\tau)}M$ that satisfies $g_{\gamma(\tau)}(e_{\mu}(\tau), e_{\nu}(\tau)) = \eta_{\mu\nu}$.

I have two questions at this point:

• Is it merely a convention to impose that $\displaystyle \frac{d\gamma}{d\lambda}\biggr|_{\lambda = \tau} = e_{0}(\tau)$? Where $e_{0}(\tau)$ is a timelike, unit vector field.

• Is it right to argue that the choice of frame field determines the components of a tensorial quantity at $\gamma(\tau)\in M$, example of which is metric tensor?

Answer 1

The exact definition of an observer can vary, but typically we are trying to model what an actual apparatus would observe.

Some actual apparatus would typically be a massive object and have some specified preferred three directions in space, as well as some scale to define lengths. The classic example would be for instance some device that can send some light signals back and forth onto other objects, the three preferred directions being three axis on that object that define the two angles for those, and the scale is defined by an onboard clock. This allows to define distances by using for instance the Einstein synchronization, where an object's distance is given by the time it takes for a light signal to be reflected as measured by the onboard clock :

\begin{equation} d = \frac{(t_2 - t_1)}{2c} \end{equation}

In such a configuration, the approximation of this machine as an observer would be that it be represented by a single point (like its center of mass), that forms a timelike curve as it moves through spacetime. The three preferred direction form a spacelike 3-frame field on the curve, the onboard clock gives the curve its parametrization. We are simply missing a timelike vector to complete the tetrad and the most natural choice for this is simply the tangent vector of the observer itself, which will in fact give you the appropriate velocity of an object as measured by the apparatus in this configuration, ie if you measure an object being at a distance $d_1$ and then $d_2$ at two different times, its velocity in the local frame of the observer will be the appropriate

\begin{equation} \vec{v} = \frac{e^i(v)}{e^0(v)} \end{equation}

You can define some other class of curves with a completely arbitrary frame over them of course, but then it is not entirely clear what the significance of the timelike basis vector would be in this case in its interpretation as an observer.

Answer 2

To the first question:

The quantity $\gamma '(\lambda)|_{\lambda=r}=e_0(\tau)$, is the tangent vector to the observer's world line. Since the observer in question has a completely time-like world line, the tangent has no choice but to be $e_0(\tau)$ (by this I mean that such a choice is natural and obviously useful), one could indeed choose something else.

To the second:

The components of tensorial quantities are coordinate dependent, which is reflected in the choice of frame or basis of $T_{\gamma(r)}$. The parametrization and choice of frame determines the specifics of the components of tensors.