In general relativity, how do we know when the coordinates we compute are physical observables?

Question

(One of) the whole point of general relativity, is that the coordinates we mathematically use are just "labels", that can change and live on a curved surface. But at some point we have to compare the computations to actual empirical evidence using physically observable distances and time. But in my experience this step is rarely made explicit, and at some point in the computation we implicitly transition from "mathematical" coordinates to "physically observable" coordinates.

This is a general question, but I have a specific example in mind: the mercury perihelion. For every derivation I saw (for example this one but I also quickly read through the original Einstein one ) they use spherical coordinates and the Schwarzschild metric. The coordinates are $r$, $\phi$ and $\theta$. But those are just labels, they a priori don’t have a relation to the radius and angles we observe from earth. However it seems the final $\phi$ difference computed is indeed what we observe. In other words at no point it is justified why the label $\phi$ and the actually observed angle we observe from Earth correspond.

Answer 1

In other words at no point it is justified why the label $\phi$ and the actually observed angle we observe from earth correspond.

The reason they align is because we choose for them to align. We could very well use $(t,x,y,z)$ or $(t,r,\theta,\phi)$ or $(t,r,\theta,z)$ or whatever other coordinates we choose. You can relate all these simple coordinates via simple transformations that give you $\phi$-changes from $x$-changes or vice versa; the coordinates aren’t actually special or physically-meaningful, they’re just a way to refer to different points in spacetime.

In fact, when you start dealing with differential geometry, you sometimes don’t even need to refer to a coordinate system at all. Coordinate-freedom is an idea discussed at length in Gravitation, so I would recommend reading that if you have a chance and are interested.

In some cases, however, it is astronomically-easier to use a particular set of coordinates over others. In particular, spherically-symmetric systems deserve spherical coordinates $(t,r,\theta,\phi)$, cylindrically-symmetric systems deserve cylindrical coordinates $(t,r,\theta,z)$, etc. but that doesn’t mean that any set of coordinates is more physically meaningful than any other. They’re all just different ways to assign vectors to points on the manifold. In fact you can even use $\theta$ and $\phi$ in nonstandard ways including starting from rectangular coordinates and saying that $\theta=x+10\ln(x^2+y^2)$, and you would just have a new coordinate system. In most cases we explicitly choose to have the coordinates correspond in an intuitive way to the points on the manifold, like in spherical coordinates.

Answer 2

In the spherically symmetric Schwarzschild spacetime geometry the theoretical calculation of the perihelion precession is done in terms of the Schwarzschild/Droste coordinates of a distant, stationary observer, not an observer on Earth.

Since Earth's orbit is significantly larger than Mercury's and velocities in the Solar System are $\ll c$, the time interval between perihelia measured from Earth is approximately, but not exactly, what would be measured/inferred by the distant "Schwarzschild observer". There is a difference in potential between an Earth-bound observer and a Schwarzschild observer and a relative velocity difference between Mercury, Earth and the Sun that leads to time dilation. The correction to this time interval would follow the usual process for correcting a proper time interval to $\Delta t$ using the Schwarzschild metric. $$ \Delta \tau = \Delta t\left[ c^2\left(1 - \frac{r_s}{r}\right) - \left(1 - \frac{r_s}{r}\right)^{-1} \left(\frac{dr}{dt}\right)^2 - r^2 \left(\frac{d\theta}{dt}\right)^2 - r^2 \sin^2 \theta \left( \frac{d\phi}{dt}\right)^2 \right]^{1/2} $$

The measured rate of perihelion advance, $\Delta \phi/\Delta \tau$ can then be corrected and compared to theory.

In principle, there could be some ambiguity about the $r, \theta, \phi, t$ coordinates of the event corresponding to the perihelion, but only if observing that event close to the limb of the Sun. In practice, the advance is estimated by summing the advance over many orbits so the affects of small positional perturbations and Shapiro delay due to the light signals curving in the Schwarzschild metric on their way to Earth are much diminished.

In terms of a general procedure, you have to decide on a set of coordinates that are going to be used to describe the interval between two events. You then have to use the metric to work out how to transform between these coordinates and a set of local inertial coordinates (or vice versa) in which the measurements are actually made.

Answer 3

The coordinates and the physical times and distances as measured with clocks and rulers relate via the metric tensor, so the distance between $\rm x_1$ and $\rm x_2$ (assuming the other 3 coordinates stay constant) is not necessarily $\rm |x_1-x_2|$ but $\rm \int_{x_1}^{x_2} \sqrt{\pm g_{xx}} \ dx$.

In Schwarzschild/Droste coordinates (no crossterms) the clocks and rulers are stationary with respect to the center of mass, in Gullstrand/Painlevé (with a $\rm g_{t r} $ crossterm reflecting the negative escape velocity) they are radially free falling from infinity, and so on.

If the path is along more than one direction you have to integrate them all, that's the purpose of the line element $\rm ds^2= \ ...$, so you do the integral instead of the usual substraction.

In your example regarding the perihelion precession the $\rm g_{\theta \theta}=r^2$, and the $\rm g_{\phi \phi}$ is also euclidean. Almost all coordinates in that regard have proper circumference where the spatial distortion is projected into the radial coordinate (except in isotropic or exotic coordinates), so the angular precession can be read off directly.

Answer 4

The coordinates we use mathematically are just "labels" that can change and apply to curved surfaces.

There is an important point that I feel other answers have not emphasized enough:

At each point in a general spacetime, we can define a local inertial frame, formally using a set of orthonomal basis vectors called a tetrad. We can then describe physics at each point using local coordinates, called Riemann normal coordinates. In these coordinates, the metric tensor has the form of the Minkowski metric at that point, and the Christoffel symbols vanish at that point.

Therefore you, assumed as an inertial observer, can approximate^ your immediate surroundings or local laboratory as a small patch of Minkowski spacetime described by these "usual" coordinates. This does not make these coordinates any more or less physical: they are still just mathematical constructs; an arbitrary choice for convenience. By this I mean we pick coordinates to label a physical position, usually based on what is easy to measure.

As ProfRob's answer said, you then need to, in general, transform between your own local set of coordinates to the global coordinates describing the global metric - for example, coordinates you might pick for the Schwarzchild metric.

The process of transforming between your local coordinates and the global coordinates of the metric is discussed in more detail in this related question and this one.

^ : In the weak field limit of gravity, which holds pretty much as long as you're not falling into a black hole.

Answer 5

The textbook, Semi-Riemannian Geometry by O'Neill says:

No concept expressed in coordinates is physically significant unless it can be shown to be independent of the choice of spacetime coordinates.

This is why in the covariant definition of a tensor we have to state a transformation law of how a tensor transforms when we change the coordinate system. Then we can say that, pace the above observation, that tensors are physical.

However, tensors are complicated beasts and the basic idea of covariance is hidden within the tensor definition. We can do the same for position. It should be obvious from experience that position is a physical observable. There is a covariant description of position that refers to the coordinate system and the coordinates of the position under question. We have to also include how these coordinates change as we change the coordinate system.

Making this formal yields the concept of a manifold. A good exposition of this is in the text above or in Lee's Topological Manifolds. You asked:

In General Relativity, how do we know when the coordinates we compute are physical coordinates?

For your specific question, we assume for every coordinate system $u$ on spacetime we have a specific point $p^u$ picked out. Now for any two coordinate systems we also have a transition function $t^v_u$ which maps one coordinate system to another. Then we require that $p^v = t^v_u.p^u$. Then if this is satisfied there is a point $p$ in spacetime whose coordinates in the coordinate system $u$ is $p^u$. This is your physical point $p$.

There is a technical proviso in the preceding in that we should show that each coordinate system should be cts. This means we need a topology on each chart on spacetime. And that these topologies match up on overlaps.