# Experimental determination of the metric tensor

Does anyone know a reference with a discussion on the experimental determination of the metric tensor of spacetime?
I only know the one in "The theory of relativity" by Møller, pages 237-240.
https://archive.org/details/theoryofrelativi029229mbp

pervect
Staff Emeritus
Short answer - I'm not aware of any such reference, but I'd be interested in what you found. I don't have time to see if I can find Moeller online at the moment, I'll check into it when I can.

Longer answer. From Misner's "Precis of general relativity", abstract at http://arxiv.org/abs/gr-qc/9508043

There is no other source of information about the coordinates apart from the expression for the metric. It is also not possible to define the coordinate system unambiguously in any way that does not require a unique expression for the metric.

That aside, it seems to me that in order to experimentally measure the metric, you need to first specify coordinates. Misner suggests that you specify coordinates do this via specifying a metric (he might or might not go to far by claiming that specifying coordinates requires a metric. This strikes me occasonally as seeming a bit too strong, but I haven't really made up my mind). It does seems to me you'd need some other approach to specifying coordinates other than defining a metric if you wish to measure the metric.

It might be worth looking at a few specific examples, GPS (discussed in Misner's paper), and the inernatioanal celestial reference frame / system, a system that uses a large number of extra-galactic radio sources to serve as a GPS-like way of taking "fixes" on positions.

In both cases, the metric isn't something that is measured - it's a tool that's used to turn measurements we actually take (time of arrival of GPS signals, angular position of extragalactic radio signals) and get a set of coordinate values of events (location on the Earth's surface, celestial location in the ICRS). So these particular examples don't show the metric as something that is measured, they show it as a tool you use to interpret other primitive measurements into a larger theoretical framework.

The book by Møller is available for free at the link I posted in the OP. The discussion concerns itself with the theoretical possibility of determining the metric by measurements in spacetime, it doesnt propose any method to perform such measurements, nor does it argue that it would be a practicable task. I just found the discussion interesting and wanted to see it exposed by other authors.

bcrowell
Staff Emeritus
Gold Member
How about thinking of it this way. Suppose that in Newtonian gravity, someone asks you to measure the gravitational potential $\Phi$ (defined by its relation $g=\nabla\Phi$ to the gravitational field g). Potential energies always include an arbitrary additive constant, so your reply to this person is going to be that it's meaningless to talk about measuring the potential at one point, but it is possible to measure its spatial variation (its first derivative).

The metric in relativity is essentially the relativistic version of the gravitational potential, so things play out fairly similarly, although with a twist. You certainly can't measure the metric at one point; the most you can do is to *choose* the metric at one point. Choosing coordinates is one way of choosing the metric. For example, I can always choose an inertial (free-falling) reference frame, construct Minkowski coordinates at a point, and say that the metric there is of the usual form diag(1,-1,-1,-1). The first derivatives of the metric are essentially the gravitational field, but I've also made that equal zero by choosing an inertial frame. Therefore the most I can do is measure the second derivative of the metric, which is how we get measures of curvature.

A concrete example would be detection of gravitational waves. These devices don't measure the metric, they measure oscillations of the metric.

How about thinking of it this way. Suppose that in Newtonian gravity, someone asks you to measure the gravitational potential $\Phi$ (defined by its relation $g=\nabla\Phi$ to the gravitational field g). Potential energies always include an arbitrary additive constant, so your reply to this person is going to be that it's meaningless to talk about measuring the potential at one point, but it is possible to measure its spatial variation (its first derivative).

The metric in relativity is essentially the relativistic version of the gravitational potential, so things play out fairly similarly, although with a twist. You certainly can't measure the metric at one point; the most you can do is to *choose* the metric at one point. Choosing coordinates is one way of choosing the metric. For example, I can always choose an inertial (free-falling) reference frame, construct Minkowski coordinates at a point, and say that the metric there is of the usual form diag(1,-1,-1,-1). The first derivatives of the metric are essentially the gravitational field, but I've also made that equal zero by choosing an inertial frame. Therefore the most I can do is measure the second derivative of the metric, which is how we get measures of curvature.

A concrete example would be detection of gravitational waves. These devices don't measure the metric, they measure oscillations of the metric.

I guess you're talking about gauge freedom?
I understand your point, but the discussion in the book assumes one has set up coordinates beforehand, so that measurements can in fact be put into correspondence with the components of the metric at one point. Now of course measuring the metric at one point wont tell you much since as you point out you can always choose it to be diag(1,-1,-1,-1). The discussion in the book is about performing measurements at every event in spacetime, much in the way one would theoretically determine the metric of a plane by measuring infinitesimal distances in three directions at every point.

pervect
Staff Emeritus
Lets see if this helps any. Lets consider, for the moment, that you are in the flat space-time of SR. You hopefully already know, that there is an invariant interval, called the Lorentz interval between two nearby events P and P' in space-time, with events being points whose position is specified both in space and time. If P has the coordinates t,x,y,z and P' has the coordinates ##t' = t+\delta t, x' = x+\delta x, y' = y+\delta y, z'=z + \delta z## you can write this invariant interval as ##L^2 = \left( \delta x \right)^2 + \left( \delta y \right)^2 + \left( \delta z \right)^2 - \left( \delta t \right)^2##, which is a quadratic form, and that this interval will be the same between any two observers.

Since you are interested in experiment, lets see how you might find this invariant interval experimentally, without necessarily knowing what the coordinates are - so we'll attempt to talk about what the above means without using coordinates. Basically any experiment you can do is going to take some time to perform the measurement, so you'll have to work around this issue. If you consider the above formula you realize that what you need to do is measure the distance between P and P', square it, find the time difference between P and P', square that, and subtract the squared time from the squared distance. Moeller suggests that you consider a frame in which both P and P' are at rest to perform these measurements. It seems to me that that isn't strictly necessary, though it's convenient if you want to specify a particular way to measure it, in general the rules of SR allow you to use any frame to measure the Lorentz interval - its independent of the observer.

I won't go into the details of how you measure distance, except to say abstractly you can do it with radar, though this process, when you do it experimentally, takes time which you have to account for. Measuring the time difference is similar, it's important to note that you need to use Einstein's midpoint simultaneity convention or some convention that's equivalent to this to determine the notion of simultaneity in whatever frame you choose (Moeller suggest the rest frame) to perform the measurement in. It would get tedious for me to specify in great detail how you go about measuring distances and time differences, hopefully the idea is familiar enough that going into this sort of tedious detail would be unnecessary (as well as boring). The only mildly tricky part about this is to be sure to use Einstein's simultaneity convention to determine time differences.

So hopefully this is sufficient to give you some idea of how you might measure the Lorentz interval between P and P' experimentally in special relativity, without necessarily having a coordinate system and using the coordinate based definition earlier, by measuring the distance between P and P , squaring it, subtracting the square of the time difference, and taking the square root.

What happens when we go to GR? Experimentally, if P and P' are close, nothing is different. The reason nothing is different is that the effects of curvature are second order - it's similar to the way that the Earth's surface is curved, but you can use flat maps for your local neighborhood without any appreciable errors. So as long as you make sure P and P' are close, you just use the same techniques as you did before.

Of course, to be able to talk about measuring the metric at all, you need some operational way of assigning coordinates to events, a procedure that allows you to measure at what coordinates an event occurs at.

It isn't particularly obvious how you'd do this, but for the sake of answering the question, lets just say you have some way of doing it. Given that, to find the metric, you take a bunch of nearby events, and measure the Lorentz interval between them. You then find a curve-fit that expresses the invariant Lorentz interval between two nearby events as a function (which will turn out to be a quadratic form) of the coordinate differences. Why is this function a quadratic form? Well, we already know it was a quadratic form in SR, and because we are considering only small displacements, we expect only linear transformations, and the linear transformation applied to a quadratic form yields another quadratic form.

To spell this out in more detail, if you have generalized coordinates p,q,r,s for some point P, and coordinates p', q', r', s' with ##p' = p + \delta p, q' = q+\delta q, r' = r + \delta r, s' = s + \delta s##, then you have some function of ##\delta p, \delta q, \delta r, \delta s## that gives you the Lorentz interval between P and P'.

The coefficients of , the quadratic form that fits or generates the function f are then just the metric coefficients you were trying to measure.

But it appears that this situation is a bit artificial. We don't actually ever have a very large array of events whereby we measure the distances and time differences between them to "measure the metric". Instead, this is a theoretical framework, and we instead look at things we can measure, like the Shaipiro time delays, light bending, the precession of orbits, that we predict from our theoretical framework, (which includes some prediction of the metric coefficients), and what we do experiemntally is to make sure that our theoeretical framework gives the correct predictions for the simpler things we do measure.

Also note that the values of the metric coefficients are really a reflection of the coordinate choices, the part that we assumed we had some way of doing, without specifying exactly what that way was.