GR Multibody Problem: Comparing Experimental Measurements

[Igorr]

I have a generic question on solution procedure. Suppose I consider a system of several point-like bodies interacting only via gravitation. I formulate PDE+ODE system, containing EFE and geodesic motion of the bodies. Since EFE do not define metric uniquely, I need to impose a particular coordinate condition on metric, say synchronous coordinates. I set initial data for positions and velocities of the bodies and solve the system numerically, e.g. with post-Newtonian expansion or as PDE on the grid. As a result I have the metric and geodesic lines.

It's clear that the answer is given upto general coordinate transformations. Solving it in another gauge (say harmonic coordinates), I will have a different answer. In the sense that it is the same Lorentz manifold but in different parametrization. In other way, I take a solution in synchronous coordinates, apply arbitrary coordinate transformations to it, and have a general answer.

My question is about comparison of this result with experimental measurements. Suppose, the motion of the bodies is measured by observation from a distant point. E.g. by measuring directions with a telescope and distances with a radar, or by other means. Suppose the motion of the bodies is reconstructed from these data and recorded as vector functions x_i(t). Suppose also that this reconstruction was not based on GR, space-time has been considered as flat, i.e. the light was assumed to propagate at a constant speed along straight lines.

Now I want to superimpose the answer from GR computations and measure a difference from the experiment, due to GR effects.

What to do with general coordinate transformations present in the answer?

One idea is to reconstruct the light rays used in telescope and radar measurements, now as light-like geodesic lines in the metric I have computed. In this way I can predict, what the observer will see, assuming the bodies move along their GR solutions, and compare this with the measured data. The picture, which the observer will see, will not depend on general coordinate transformations in other points and will not depend on the gauge conditions used in the computations. However, it would be better to do comparison not on the level of raw measurements, but on the level of reconstructed vector functions x_i(t).

Is there any standard procedure for comparing the results of GR computations of multibody system with the measurements? Some preferred gauge to solve the eqs? Some preferred coordinate system to transform a solution?

 

[pervect]

Taking a broad overview and using an analogy, a space-time metric is a map of space-time rather similar to the way an ordinary map would be a map of the surface of the Earth. The space-time map of General Relativity gives proper intervals, so I'd suggest testing the "goodness" of the map by measuring the proper intervals between agreed-on landmarks, and comparing the measurements to the theoretical predictions.

Proper intervals could be proper distances or proper times, but you can generally reduce proper distance measruements to proper time measurements via radar techniques.

This is somewhat based on a paper by Misner, "Precis of General Relativity", http://arxiv.org/abs/gr-qc/9508043, which is devoted to explaining the principles lying behind an idealized GPS system. While this may not be an exact fit for answering your quesiton. It's the closest thing that comes to mind at the moment. I've probably put my own "spin" on the paper, too, but it's the source of the inspiration to the answer I gave.

As far as coordinates go, you can use whatever coordinates you like. Coordinates are a useful tool for expressing a map and identifying a landmark, but they're not fundamental. So the existence of different coordinate systems that are related by diffeomorphisms doesn't really matter - choose any coordinates you like, the point is that their predictions about intervals are the same, and ultimately, measuring the Lorentz interval is the observer-independent expression of what you can measure experimentally.

 

[PAllen]

One thing that divorces choice of coordinates for computational convenience 'where the action is' from computing observables (which are coordinate invariant, but can be difficult to compute in some coordinates) for distant observers, is that for non-cosmological situations you assume asymptotic flatness, and any reasonable coordinates you choose approach standard Minkowski coordinates 'at infinity'. You are assuming that e.g. for analyzing a binary pulsar from Earth it is a good enough approximation to assume that you have an isolated distant observer with the binary pulsar the only thing in the 'universe'. Then, no matter what coordinates you find convenient to analyze 'the action', far a way you can interpret coordinates as in SR, an avoid computing everything rigorously as an invariant. If you want to ask questions about what happens to a rocket where the dynamics are extreme, you have no choice but to compute invariants. For the distant observer, you can finesse this.

For cosmology (where asymptotic flatness is substantively false), you typically pick coordinates that make analysis much easier (in particular, coordinates that display the maximum symmetry of the model - e.g. homogeneity and isotropy). Then, you must explicitly compute observations as invariants. The direct observations are typically redshift / luminosity relationships, distribution of objects (e.g. scale of deviations from isotropy- homogeneity on smaller scales), etc. Things like distances and recession velocities are just (useful) computed quantities that are NOT invariant and cannot be measured in any plausible way.

 

[Igorr]

OK, this is a good example, a binary pulsar, whose trajectories are studied by an observer from a large distance. The purpose is to compare GR solution with the observed trajectory and also with Newtonian approximation to the same problem.

GR solution is defined upto arbitrary diffeomorphisms. There are diffeomorphisms identical at large distance and non-identical in the region of binary pulsar. These diffemorphisms can completely change the shapes of trajectories of the binary pulsar. Which GR trajectories should be taken for comparison with the observation and Newtonian approximation?

The same question formulated in other way. The motion of Mercury around the Sun in Newtonian approximation is an ellipse, in GR in the first order approximation is precessing ellipse. Furthermore, GR solution is an arbitrary coordinate deformation of precessing ellipse. Why for comparison with experiment and Newtonian approximation precessing ellipse is taken, but not any other solution from the family of gauge-equivalent solutions?

 

[PeterDonis]

These diffemorphisms can completely change the shapes of trajectories of the binary pulsar.

They don't change the actual shapes; they only change how the same shapes are represented in the chosen coordinates. But anything you actually compare with experiment is an invariant, something that is the same in all coordinates. You just have to figure out how to extract the appropriate invariants from the coordinates you have chosen. (A big part of this is picking coordinates that make it easy to extract the invariants you're interested in; see below.)

Consider an analogy: I can use many different kinds of flat maps to represent the surface of the Earth. For example, a Mercator projection or a stereographic projection. A given piece of the Earth's surface, for example Greenland, will look very different on these two flat maps. But if I want to compute the area of Greenland in order to compare with measurements I've made, I can do it with both maps; I just have to adjust appropriately for the distortion in each map. For example, Greenland looks much larger in a Mercator projection than it does in a stereographic projection centered on the North Pole; but when I adjust for the distortion of the Mercator projection (by using the appropriate metric), I can still compute the correct area.

We do the same thing in GR when comparing theory with experiment; we use whatever coordinates make the computation easiest, but the computation still ends up giving invariants, which are what we actually compare with experiment. See below.

Why for comparison with experiment and Newtonian approximation precessing ellipse is taken, but not any other solution from the family of gauge-equivalent solutions?

The precessing ellipse is not what we actually compare with observations, because we don't observe Mercury moving in a precessing ellipse. We observe it moving across Earth's sky in a particular pattern. In the two models (GR and Newtonian approximation), the precessing ellipse is a calculational convenience, to make it easier to compute where Mercury will appear in Earth's sky at particular times. But the position of Mercury in Earth's sky as a function of time by Earth clocks is the actual observable, and that is an invariant.

If you wanted, you could pick coordinates in which Mercury's spatial orbit looked more complicated than a precessing ellipse. For example, you could use the Ptolemaic system in which the Earth is at the center of coordinates, and in which Mercury's path, like the paths of all the other planets and the Sun, is modeled as epicycles moving on circles. By adding more and more epicycles, you could in principle extract predictions of where Mercury would appear in Earth's sky as a function of time that were within experimental error. But the calculations would be a lot more complicated. We use the "precessing ellipse" model because it's simpler, not because it's "preferred" in any physical sense.

 

[pervect]

OK, this is a good example, a binary pulsar, whose trajectories are studied by an observer from a large distance. The purpose is to compare GR solution with the observed trajectory and also with Newtonian approximation to the same problem.

GR solution is defined upto arbitrary diffeomorphisms. There are diffeomorphisms identical at large distance and non-identical in the region of binary pulsar. These diffemorphisms can completely change the shapes of trajectories of the binary pulsar. Which GR trajectories should be taken for comparison with the observation and Newtonian approximation?

The same question formulated in other way. The motion of Mercury around the Sun in Newtonian approximation is an ellipse, in GR in the first order approximation is precessing ellipse. Furthermore, GR solution is an arbitrary coordinate deformation of precessing ellipse. Why for comparison with experiment and Newtonian approximation precessing ellipse is taken, but not any other solution from the family of gauge-equivalent solutions?

What you'd probably mesure for the inspiral is something like the orbital period. The mesurement isn't going to be affected by the diffeomorphisms. Perhaps what you're really after when asking about the "shape" is not measurements, but what coordinates to use? There are a couple of standard coordinate choices that can help you to develop intuition into the physics, for instance fermi-normal coordinates. But I don't want to get into that unless it's really your question. At the moment you're asking about measurements, and the answer as far as measurements go is that they are unchanged by the diffeomorphisms.

 

[Igorr]

Please tell me am I right:

In Newtonian mechnics as well as in special relativity one can perform observations of a body motion from a distant point, then reconstruct the motion in the form of worldline x(t). In GR one can reconstruct a worldline only upto diffeomorphisms of surrounding space-time. So one can never superimpose Newtonian orbits and GR orbits and perform direct comparisions between them. One can measure only invariants (like angles between light geodesics coming to the observer) and must do any comparisons between these invariants only.

There are two difficulties then:

(1) The measurements raw data can be unavailable, only the worldlines reconstructed in SR. So one needs at first to recompute the measurements data back from the worldlines.

(2) One always needs to compute light geodesics to the observer, which can be more difficult than to compute the motion of studied bodies. E.g. some of the bodies can be non-relativistic and their motion can be computed analytically, while the light rays over curved space-time can require numerical integration.

Could these problems be resolved by a choice of coordinate system? A system that fixes the arbitrariness of diffeomorphisms and better reflects what a distant observer will see. So I'm interested to know more about fermi-normal coordinates if they help comparing GR solutions with SR solutions or with SR-based reconstruction of experimental data.

The other possibility is that computation of light geodesics (2) for particular problems and particular coordinate systems can bring just a negligible correction between GR and SR results. E.g. for slowly moving bodies one can neglect light delays and differences in visible positions between Newton and SR descriptions. It sounds plausible that in the same approximation one can also neglect a curvature of light rays and its influence to visible positions. Theoretically there are still so curved coordinate systems which can require computations (2), but for normally used coordinate systems like synchronous and harmonic the effects introduced by computation (2) can be negligible. I guess that under these circumstances the differences between the trajectories in synchronous and harmonic coords will be also negligible. Then one can use any of these coordinate systems for computations and compare the trajectories to SR directly. It only remains to control that light geodesics do not deviate too much from SR and do not distort the real observables.

Concerning the period of motion as an invariant. To be on safe side, one needs to define it as a period of motion visible by a distant observer, in own time of this observer. This will generally require to reconstruct light geodesics (2) or show that these corrections are neglectable. Comparing to an alternative definition, a period in own time of a moving body, it is not directly measurable by a distant observer. Also, periodical worldline is well defined in SR, but in GR arbitrary diffeomorphisms can destroy periodicity. Period of return to the same point of space is not defined in GR, since "the same point of space in different time" is not defined invariantly under diffeomorphisms.

 

[PeterDonis]

In Newtonian mechnics as well as in special relativity one can perform observations of a body motion from a distant point, then reconstruct the motion in the form of worldline x(t).

No. You can't reconstruct the worldline from observations alone. You have to also make a choice of coordinates. This is just as true in Newtonian mechanics as it is in GR.

In GR one can reconstruct a worldline only upto diffeomorphisms of surrounding space-time.

No. Choosing coordinates removes the "up to diffeomorphisms". And if you don't choose coordinates, you can't reconstruct the worldline.

The measurements raw data can be unavailable,

Um, what? If the "measurements raw data" is unavailable, how can you do anything at all?

some of the bodies can be non-relativistic and their motion can be computed analytically, while the light rays over curved space-time can require numerical integration.

I don't understand. We can often find closed-form analytical expressions for the paths of light rays, and conversely, just because an object is "non-relativistic" does not mean we will always have a closed-form expression for its motion.

Could these problems be resolved by a choice of coordinate system? A system that fixes the arbitrariness of diffeomorphisms and better reflects what a distant observer will see.

As I and others have pointed out, this is backwards. The actual observations--where in the sky a particular object appears at a particular time by Earth clocks--are the starting point. Choosing coordinates in order to describe those observations in terms of a worldline x(t) instead of in terms of positions on the sky as a function of Earth clock time comes after that; it is a convenience for calculation and visualization. You don't need to choose coordinates to describe the actual observations; they are independent of any choice of coordinates.

Concerning the period of motion as an invariant. To be on safe side, one needs to define it as a period of motion visible by a distant observer, in own time of this observer.

Yes. This is what will be invariant.

This will generally require to reconstruct light geodesics (2)

No, it doesn't. It just requires making the observations. Once again, the observations come first. Trying to express them in terms of particular coordinates, or translate them into the motion of the source in particular coordinates, comes after that, and is only a convenience; it is not necessary to make the observations or to know that they are invariant.

a period in own time of a moving body, it is not directly measurable by a distant observer.

True; but it is measurable by an observer on the body itself. For example, we can measure the period of Earth's motion around the Sun, in this sense, here on Earth.

 

[pervect]

Please tell me am I right:

In Newtonian mechanics as well as in special relativity one can perform observations of a body motion from a distant point, then reconstruct the motion in the form of worldline x(t). In GR one can reconstruct a worldline only upto diffeomorphisms of surrounding space-time. So one can never superimpose Newtonian orbits and GR orbits and perform direct comparisions between them. One can measure only invariants (like angles between light geodesics coming to the observer) and must do any comparisons between these invariants only.

I'll translate this into my tensor-based viewpoint, and you can tell me if you agree, disagree, or don't follow what I'm saying at all. (Or perhaps there's some posssibility I missed). In Newtonian mechanics, as well as special relativity, the geometry of space-time is fixed. So you don't have to worry about how the geometry of space-time affects your measurements, because you assume it's flat.

In GR, the geometry of space-time can vary and impact your measurements, so observations that would fix the trajectory of an object in a Newtonian/SR context won't tell the whole story in a GR context. You want to not only determine the path of the object in a fixed geometry, you need to determine the path of an object and the nature of the geometry.

To give a simplifed non-SR example. Suppose you have a 2d spatial manifold in Newtonian space (with the corresponding Newtonian notion of absolute time). And you measure the angle and distance of an object as a function of time.

When you know a priori that the spatial geometry is a flat plane, these measurements determines the course of the object. Whether or not the measurements tell you "the course of the object" in the case where you don't know the underlying geometry is a matter of semantics. But certainly, if you don't know whether the 2-d manifold is a plane or the surface of a sphere or some other curved manifold, then having observations of the distance and angle do not give you a complete picture. They don't tell the whole story. One needs in general some way to reconstruct the geometry as well as the trajectory.

It's not entirely clear how to determine the geometry with measurements made in the manner you advocate, i.e. by a single central observer. It's perhaps possible to determine the geometry by performing these same measurements on a set (or congruence) of such objects - the matter would require some thought. Certainly one can reconstruct the geometry if one has a congruence of objects that can also measure their distances to the other nearby objects in the congruence. At least I'm reasonably certain - I'm actually considering the special case where the congruence of objects are following geodesics (so it's a geodesic congruence), and one uses the measurements of the distances between objects in the congruence to calculate the geodesic deviation, which then tells you what the Riemann curvature tensor is, which answers the questions about the geometry.

But in my mind the issue isn't determining "the course of the object", it's determining both the course of the object and the geometry it's moving through.

 

[PeterDonis]

In Newtonian mechanics, as well as special relativity, the geometry of space-time is fixed.

This is not quite correct. In Newtonian mechanics, the geometry of space is fixed. There is no such thing as "the geometry of spacetime" in Newtonian mechanics, because there is no geometric invariant involving events that happen at different times.

In GR, the geometry of space-time can vary and impact your measurements, so observations that would fix the trajectory of an object in a Newtonian/SR context won't tell the whole story in a GR context.

I see what you're saying, but I'm not sure I would put it this way. Thinking of the geometry of spacetime as "varying and impacting your measurements" IMO puts things backwards. Your measurements are the fixed data; the question is how many measurements you need to fix the geometry of spacetime as well as the trajectory of the object within that geometry. Or, to put it another way that doesn't require picking coordinates, how many measurements you need to accurately predict future measurements. In the GR case, you would need more measurements because there are more degrees of freedom (the geometry as well as the trajectory) than in the Newtonian case (where the geometry is fixed).

It's not entirely clear how to determine the geometry with measurements made in the manner you advocate, i.e. by a single central observer.

Bear in mind that the observer can send light signals and reflect them off of other objects, so that the times and directions of both emission and reception are available as data, instead of just the time and direction of reception.

 

[Igorr]

I agree, SR space-time is flat and the shapes of worldlines are understood modulo Poincare transform, while in GR the worldlines and metric are modulo diffeomorphisms; that's why it's more difficult to interpret them.

I just try to adopt a practical point of view, common in celestial mechanics. It's normal there to draw trajectories and to consider corrections to trajectories e.g. from influence of one planet to the other. I would expect that GR in the first order will act here also as a small correction to the trajectory. It would not require complete change of language so one can talk only about raw observables, like the path of the planet's image on the focal plane of the telescope.

For simplicity, let me consider the following situation. One computes trajectory of a planet in GR using synchronous coordinates. Then one computes trajectory using harmonic coordinates. I suppose they will not coincide, since they differ by diffeomeorphisms and nobody proved yet that variations vanish or are small. One draws the both trajectories together with an ellipse from Newtonian mechanics and is puzzled what compare with what. Then GR specialist comes and recommends to evaluate an observable, angle of light geodesic coming to a distant observer, in own time of this observer, or in common language: angle between the planet and the Sun as visible by a telescope. After a month of computations the curves are ready and one sees that the curves for synchronous and harmonic coordinates coincide exactly and slightly differ from Newtonian curve. GR specialist says this is the difference between GR and Newtonian mechanics. On the questions can one avoid computation of light geodesics and can one translate this difference to the difference of trajectories, GR specialist answers NOPE and NOPE.

I just want to learn a common practice, is it a bit similar to this?

Concerning to a specific coordinate system, I can imagine one which consists completely of GR invariants and is close to common astronomical coordinates. E.g. time is own time of a distant observer, space origin is position of the observer, the space is described by spherical coordinates, where direction defines a tangent to light geodesics coming to the observer and radius displays a distance measured by a radar or other means common in astronomy. I would be happy if you say that such system already exists and save me from inventing my own.

I wish everybody Merry Christmas and Happy Holidays!

 

[PeterDonis]

I just try to adopt a practical point of view, common in celestial mechanics. It's normal there to draw trajectories and to consider corrections to trajectories e.g. from influence of one planet to the other.

But we don't observe trajectories. We observe positions on the sky vs. Earth clock time. So if you want to talk about comparison with experiment, which is what you said you wanted to talk about, you need to understand how the trajectories, or whatever it is you are using in your model, are computed from the experimental data. In order to do that, you have to talk about the data as well as the trajectories.

I would expect that GR in the first order will act here also as a small correction to the trajectory. It would not require complete change of language so one can talk only about raw observables, like the path of the planet's image on the focal plane of the telescope.

Nobody is saying you can only talk about the raw observables. I am saying that you can't only talk about trajectories, because we don't observe them, and you are asking about how we compare our models with what we actually observe.

One computes trajectory of a planet in GR using synchronous coordinates. Then one computes trajectory using harmonic coordinates. I suppose they will not coincide, since they differ by diffeomeorphisms and nobody proved yet that variations vanish or are small. One draws the both trajectories together with an ellipse from Newtonian mechanics and is puzzled what compare with what.

This is not what you do to compare models with observations. You don't compare the GR model with the Newtonian model. You compare the Newtonian model with observations, and you compare the GR model with observations. If it turns out that the GR model matches the observations better, then you adopt the GR model in preference to the Newtonian model. (For example, the GR model matches the observed motions of the planets better.) To compare the models with observations, you have to compute the predicted observations within each model, which means you have to figure out how to compute the appropriate invariants in each model. Computing the invariants gets rid of the "up to diffeomorphism" ambiguity in the models; models using different coordinates will have different "shapes" for planetary orbits, but will also have different formulas for computing where on Earth's sky at a given Earth clock time a planet will appear; so the computed predictions will be the same in both cases. And for comparing models with observations, it's only the computed predictions that matter, not the "shape" of the orbits in the model.

The comparison you are talking about here, where you look at the "shape" of planetary orbits in different coordinates and wonder which one is the "right" shape, is model to model, not model to observations; but in your OP you said you wanted to talk about comparing model with observations. Which is it?

If you are going to compare model with model, obviously you have to adjust the GR model to use coordinates that are compatible with the coordinates you are using in the Newtonian model. (Note that there is more than one possible choice of coordinates in the Newtonian model as well.) For example, when we say that planetary orbits are perfect ellipses in the Newtonian model, whereas they show perihelion precession in the GR model, we are implicitly adopting what are called "barycentric" coordinates (coordinates in which the barycenter of the solar system is at rest, and which are not rotating with respect to the distant stars) in both models. But we're not doing that to compare the models with observations; we're doing it to compare the models with each other by choosing a common coordinate system for both.

 

[pervect]

Maybe this will help - or maybe not.

The issue of a solution being unique only up to a diffeormorphism also occurs in special relativity. For example, people don't usually get overly concerned if they see a solution to a problem in polar coordinates (r,theta), or cartesian coordinates (x,y), they just pick whichever one makes the problem easiest to solve. I don't think there's much real angst about the solution being a "different solution" depending on the coordinate choice, it's considered to be the "same solution" in different coordinates.

The transform from polar coordinates to cartesian coordinates is one example of a diffeomorphism. So there's nothing new about having "the same" solution appear in different coordinates.

Without the mathematical techniques of GR, only a relatively small number of familiar coordinate systems are used, so people tend to choose from a very limited set, and develop some intuition about the physical significance of their coordinates.

Now rather than saying x=r cos theta and y=r sin theta as one would to convert a solution from polar coordinates to cartesian coordinates, one can express the solution (assumed for the time being to still be on a flat plane) as p= f(r,theta) and q=g(r,theta), where p and q are arbitrary coordinates and f and g are arbitrary functions. It's nothing really new physically, the math takes some getting used to, especlaly if one uses tensors, though for an introduction one can avoid tensors and just use algebra. (I believe "Exploring black Holes" is one of an increasing number of undergraduate treatments of GR that uses this kind of approach, though unfortunately I don't own it. The first few chapters are available online on Taylor's website though.)

To have some physical meaning to the solution, though, one need to know the metric. This would just be dx^2 + dy^2 for Cartesian coordinatesm, or dr^2 + r^2 d theta^2 for polar coordinates. Loosely speaking, the metric just gives the distance between two nearby points as a function of the differential displacements, i.e. dx and dy, or dr and d theta. It turns out the metric is all one needs to give physical meaning to the solution. In fact, specifying the metric is a good way of specifying coordinates. See for instance Misner's "Precis of General Relativity" which I linked to before. I won't track down the link if nobody's going to read it, but if you wnt to read it and can't find it on arxiv with a search from the title and author, ask and I'll post the url.

On a flat surface, one can always make a familiar coordinate choice to get a familiar metric. On a general curved surface, this isn't possible, though some particular simple curved surfaces (such as the surface of a sphere) have a simple metric that become familiar with through experience.

I've often thought that it'd be useful to do as a warm-up to GR some problems with spherical geometry. I still think this is feasible and a good idea, but it seems there aren't many (or perhaps any) textbooks or papers that go through the details, so at the moment it's mainly just a suggestion for self-study.

Anyway, I hope that helps. It's gotten a bit long, so let me recap the important points. Diffeomorphisms aren't unique to GR, they occur commonly as a result of choosing what coordinates to use, for example polar or cartesian coordinates. I believe a lot of the confusion lies in how people think about coordinates - especially in how they think about generalized coordinates. It seems like writing about this doesn't help as much as it could, it's just something people have to come to terms with, the notion that coordinates are just labels, and that there are an infinite number of ways of labelling the points that doesn't change the underyling physics, because the different labels reflect the same physical system.

 

[Ibix]

Anyway, I hope that helps. It's gotten a bit long, so let me recap the important points. Diffeomorphisms aren't unique to GR, they occur commonly as a result of choosing what coordinates to use, for example polar or cartesian coordinates. I believe a lot of the confusion lies in how people think about coordinates - especially in how they think about generalized coordinates. It seems like writing about this doesn't help as much as it could, it's just something people have to come to terms with, the notion that coordinates are just labels, and that there are an infinite number of ways of labelling the points that doesn't change the underyling physics, because the different labels reflect the same physical system.

Very useful, at least to me - thank you. The point about "coming to terms with" feels very familiar, and vaguely like "coming to terms with" the fact that you can't just say "simultaneous".