Coordinate transformation in Schwarzschild Coordinate

Question

In the lab/Schwarzschild frame, the four velocity of an object is $U=(U^t,U^r,0,0)$. What would be the co-ordinate transformation (Lorentz boost) where the radial component of the four velocity will be zero?

Answer 1

This will be a very general comment (and I hope people correct my understanding if something is not right) as I needed to recall some things myself and make sense of it in my head.

The gist of the problem is that we need to understand better what is meant as an observer's measurement in GR and the question of determination of the relative velocity.

Coordinates

To start with, coordinate transformations and Lorentz boosts are different beasts altogether in GR.

Coordinate charts concern the whole spacetime manifold (the freedom of labeling the events): $$\phi_{U}: \mathcal{M} \supset U \rightarrow \mathbb{R}^{4}$$ and coordinate transformations allow to express the geometric (objective!) quantities - think tensor fields $\Gamma(T^{p}_{q}\mathcal{M})$ - in different (yet equivalent) ways.

Once we know that a geometric quantity - the four-velocity of the object - is expressed in the Schwarzschild coordinates as $$\mathbf{v} = v^{t}\partial_{t} + v^{r}\partial_{r}$$ we cannot transform the component away by some coordinate change and somehow still remain in the same coordinate system. We could express the vector field in some different coordinates and ask, whether in these new coordinates some spatial component can be made zero. It no longer has the same interpretation as the old one though.

In short, a coordinate system a priori does not correspond to some family of observers, or even one observer. It does not have, by itself, any meaning. However, on a case by case basis, a construction of a family of observers that somehow relate to the coordinate system can be found (for example Schwarzschild spacetime, Misner, Thorne & Wheeler §23.3, where the static observers $\partial_{t}$ can agree on a common definition of position $(r,\theta,\phi)$ and simultaneity $t$).

The local view

The answer to the question locally is the Lorentz boost.

Lorentz boosts are only working locally in general relativity, on the tangent spaces $T_{p}\mathcal{M}$ at a point, where the metric can be brought to the Minkowskian form. Let the four-velocities of the observer and the object be $\mathbf{u}$ and $\mathbf{v}$.

At a point $p\in \mathcal{M}$ where their wordlines meet, we can meaningfully compare and add or substract their vectors. Their relative three-velocity has the Lorentz factor (Sean Carroll's book) $$ \gamma = - g_{p}(\mathbf{u}_{p},\mathbf{v}_{p})$$ under the metric signature $(-,+,+,+)$.

One then faces the task of finding a vector $\mathbf{u}_{p}$ that would make the $\gamma=-1$. Assume the worldlines meet at a point $p$ which can be described by coordinates $(t_{0},r_{0},\theta_{0},\phi_{0})$.

Since the curves are timelike, they can be normalized to $-1$. We have $\mathbf{v}=v^{t}\partial_{t}+v^{r}\partial_{r}$ and a completely general $\mathbf{u}=v^{\mu}\partial_{\mu}$ The resulting (underdetermined) set of equations to solve is: $$ g_{p}(\mathbf{u}_{p},\mathbf{v}_{p}) = -1 \\ g_{p}(\mathbf{u}_{p},\mathbf{u}_{p}) = -1 \\ g_{p}(\mathbf{v}_{p},\mathbf{v}_{p}) = -1$$ evaluated at said point $p$. Assuming we know the four-velocity $\mathbf{v}$ of the object, from the third equation, we know how its two components must be related. We then have two equations left for 4 undetermined components of $\mathbf{u}$ at point $p$. As you see, there is some freedom in determining the tangent vector at that point worldline. I leave it to you to explore some possibilities.

The global view

For such an observer-dependent measurement to be prolonged in spacetime (i.e. being the opinion of some observer over the course of some of his proper time) I think one needs to consider some manner of prescribing a generalization.

If the observer is distant from the object (which, suppose, emits a lightlike signal that allows for its tracing), we need to consider how one could extend the notion of zero relative velocity in the radial direction to the framework of general relativity.

The notion of of $\frac{\text{d}r}{\text{d}t}$ is meaningless, as coordinates have no physical meaning.

Do we want to maintain some proper distance $$ L = \int_{s_{0}}^{s_{f}}\sqrt{\gamma(X, X)}\text{d}s$$ (calculated on the hypersurfaces along some (chosen in what way?) spacelike curves everywhere tangent to hypersurface) constant between the observer and the object?

That notion would be dependent on the three-metric $\gamma = \iota^{*}g$, ($\iota:\Sigma_{t}\rightarrow M$) induced on a $t=\text{const}$ hypersurface $\Sigma_{t}$ and thus foliation-dependent (3+1 split of the spacetime). See e.g. Landau, Classic Theory of Fields §84. In general, spatial distance is observer-dependent.

The coordinate-time observers $\partial_{t}$ (again, c.f. the relevant setup in MTW §23.3) could agree that some observer and the object maintain proper spatial distance constant, but that is not an objective statement, i.e. obviously not shared by all observers (choose a different slicing of the spacetime - the worldlines will cross the $t'=$const hypersurfaces differently).

Velocity measurement by redshift?

Perhaps one could meaningfully employ the idea of sending photons from the object to the observer and making sure there is no redshift - the wave frequency stays constant - that is, after all, the correct way of proceeding in flat spacetime.

However, in curved spacetime, the redshift will happen even if the observers are at rest. For what follows next, I refer to Straumann's book, §2.9.2.

Assuming the four-velocities of the observer $\mathbf{u}$, the object $\mathbf{v}$ and emitted photons $\mathbf{k}$, we have: $$\frac{\nu_{observed}}{\nu_{emitted}} = \frac{g(\mathbf{k},\mathbf{u})}{g(\mathbf{k},\mathbf{v})} $$ where the numerator and denominator on the RHS are calculated at the appropriate spacetime points of emission and observation.

Now, see that even if both the observer and the emitter were at rest (parallel to $\partial_{t}$) at positions described by $r_{obs}$ and $r_{emit}$, the light signal would undergo (red/blue)shift by the very fact of gravitational field presence: $$\frac{\nu_{observed}}{\nu_{emitted}} = \sqrt{\frac{g_{tt}(r_{emit})}{g_{tt}(r_{obs})} }$$

In the general case, any such redshift measurement would necessarily entail both the gravitational redshift and the Doppler redshift.

I do not know if balancing one with the other is a meaningful notion of generalizing 'no relative velocity in the radial direction'.

Further reading

I realize that my answer might be somewhat unsatisfactory, thus I refer you to some other topics as well. I recommend these very well-written answers on the topic of observers in GR:

https://physics.stackexchange.com/a/322411
https://physics.stackexchange.com/a/307549
How do frames of reference work in general relativity, and are they described by coordinate systems?
https://physics.stackexchange.com/a/384185
https://physics.stackexchange.com/a/400288
Whose coordinates are the Schwarzschild coordinates?
Whose reference frame to use for $d \theta$ near a black hole?
How can Schwarzschild coordinates be measured?