Measuring Metric Tensor: Apriori Specification or Geometric Object?

[Dale]

Is there any standard way for measuring the metric tensor. Does it require the a priori specification of a coordinate system relative to which we measure the components of the metric, or is there a way to measure the underlying geometric object?

 

[Chris Hillman]

Suggest two options: Riemann normal chart or Coll canonical chart

Hi Dale, good question, in fact I have long planned a BRS concerning a closely related question, "how can you measure the curvature tensor of our spacetime?" There are several papers in the gtr literature (try journals like Gen. Rel. Gravitation) on this, and some textbooks such as de Felice and Clarke, Relativity on Curved Manifolds, discuss it too.

Anyway, this is one of those questions which arise just frequently enough that I really should have set down my thoughts in permanent form, and just infrequently enough that each time I find I have forgotten what I said previously, so you'll have to bear with me if I stumble around a bit.

The first thing which springs to mind is the legend of how Gauss was hired to map Hanover and Denmark, and experimented by trying to measure precisely the distances between three mountaintops, hoping to verify by a new method the average radius of the Earth, by applying his then new formula relating the curvature of a sphere to the area and sides of a geodesic triangle drawn on the sphere.

The zeroth thing which sprang to mind, however (meaning that it was already on my mind), is the Riemann normal chart, which gives both an expression for the line element near some point P, valid to second order in small distances from P, in which the second order correction term is the Gaussian curvature at P! The expression is (for a two-dimensional Riemannian manifold, in a Cartesian style normal chart)
$$ds^2 = dx^2 +dy^2 - \frac{K}{3} \, \left( -y \, dx + x \, dy \right)^2 = \| d\vec{x} \|^2 - \frac{K}{3} \| \vec{x} \wedge d\vec{x} \|^2$$
where K is the Gaussian curvature at P (equals the Riemann tensor frame field component $R_{1212}$, for any orthonormal frame at P, but the coordinate basis component $R_{xyxy}$ will usually differ from the frame field component.). This formula says that the curvature at P provides a second order correction to simply using the euclidean formula for small distances measured near P. This correction term is negative (positive) according to whether K is positive (negative).

(If you consider a small disk in S^2 or H^2 the sign of the correction term should make sense!)

See the figure below and notice that the units make sense since K has units of reciprocal area and the magnitude of $\vec{x} \wedge d\vec{x}$ is the area of the indicated triangle. As usual, to find the length between two points on the curve, measured along the curve, you are supposed to take the square root and integrate along our curve using an appropriate parameter. If the curve happens to be a geodesic, we can try to argue that the resulting distance along the curve can be regarded as "the distance", but this runs into all kinds of problems.

From the form of the line element as I wrote it, you can see that it depends (up to second order in the "distance" $\| d\vec{x} \|$ from P) only on the Gaussian curvature at P, so this chart is reasonably "canonical"--- as long as one doesn't move very far from P or demand/require third order accuracy! Also, if we think of P as a "center of attraction" we recognize the triangle as a Kepler triangle, and its area as the area appearing in Kepler's law. (Not that I suggest pushing this too hard, since here P plays the role of an arbitrarily chosen point and our curve is "nothing special".)

For those of you who use Maxima, here is a Ctensor file you can run in batch mode under wxmaxima:

/* 
E^2 manifold; Riemann normal chart
Chart covers
	entire plane			if K < 0 or K = 0
	disk 	x^2+y^2 < K/3 		if K > 0

Gaussian curvature is
	R_(1212) = K*(1-2/9*K*(x^2+y^2))/(1-1/3*K*(x^2+y^2))^2
		 ~ K*(1 + 2/3*K*(x^2+y^2)) + HOT
		 = K at origin
Note that the Christoffel symbols vanish at the origin.

For this model surface, line element is exactly
	ds^2 = (dx^2 + dy^2) - K/3 (-y dx + x dy)^2
	     = || dX ||^2 - K/3 || X /\ dX ||^2 + O(||X||^3)
For any Riemannian two-manifold, near any point O with Gaussian curvature K, 
in a Riemann normal chart about O this expression holds, 
valid to 2nd order in || P ||, the euclidean distance from O.

Reason: curvature at point O determines 2nd partials of metric at O
NOTE: implies all E^2 manifolds "isotropic to 2nd order"

Area form
	omega = sqrt(1-K/3*(x^2+y^2)) dx /\ dy

*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 2;
/* list the coordinates */
ct_coords: [x,y];
/* define variables */
/* define background metric */
lfg: ident(2);
/* define the coframe */
fri: zeromatrix(2,2);
fri[1,1]:  x/sqrt(x^2+y^2);
fri[1,2]:  y/sqrt(x^2+y^2);
fri[2,1]: -y/sqrt(x^2+y^2)*sqrt(1-1/3*K*(x^2+y^2));
fri[2,2]:  x/sqrt(x^2+y^2)*sqrt(1-1/3*K*(x^2+y^2));
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
ratsimp(lriem[1,2,1,2]);
ratsimp(taylor(%,x,0,3));
ratsimp(taylor(%,y,0,3));

This confirms that up to first order in x,y the Gaussian curvature is K throughout the small neighborhood of P where the chart is valid.

There are Riemann normal charts for higher dimensional Riemannian (and Lorentzian) manifolds, and the expression for the line element (up to second order in "distance" from P) is a bit more complicated, but it still involves "correction terms" of the form "Riemann component times area of a Kepler triangle". Discovering this required some brilliant "high school algebra" manipulations on the part of Riemann.

One reason why Riemann normal charts are important is that they shed much light on the question: "what information is needed to specify a Riemannian manifold, up to coordinate transformations?" For a Riemannian two-manifold, the theory of conformal mapping (a theory in which Riemann again played a key role!) shows that we can always find an isothermal chart in which the line element has the form
$$ds^2 = \exp(f(x,y)) \, \left( dx^2+dy^2)$$
so that only one scalar function is needed. We could have guessed that from the Riemann normal chart for a two-manifold, in which the curvature at P determines the metric (approximately) near P.

In higher dimensions, an ingenenious parameter counting argument due to Riemann (and later used by Hilbert who apparently taught it to Einstein) shows that

• for a Riemannian two-manifold you need the second partials of
  • one function of two variables,
• for a Riemannian three-manifold you need the second partials of
  • three functions of three variables
  • three functions of two variables
• for a Riemannian four-manifold you need the second partials of
  • six functions of four variables
  • eight functions of three variables
  • six functions of two variables

The Riemann normal chart for a two-manifold shows that in a Taylor expansion near some point P, the first "free" second partials are second order partials, which is more or less why the curvature tensor and metric tensor contain the same information in this case.

For dimension greater than two, in general no isothermal chart exists, but there are (many, many) charts of a certain "canonical" form which exhibit where some of these functions are used. The "leading terms" in Riemann's count suggests charts in which the metric tensor has the form

• Riemannian two-manifolds: one function of two variables
  $$\left[ \begin{array}{cc} \exp(f) & 0 \\ 0 & \exp(f) \end{array} \right]$$
• Riemannian three-manifolds: three functions of three variables
  $$\left[ \begin{array}{cc|c} \exp(f) & 0 & a \\ 0 & \exp(f) & b \\ \hline a & b & 1 \end{array} \right]$$
• Riemannian four-manifolds: six functions of four variables
  $$\left[ \begin{array}{cc|cc} \exp(f) & 0 & p & q \\ 0 & \exp(f) & a & b \\ \hline p & a & \exp(g) & 0 \\ q & b & 0 & \exp(g) \end{array} \right]$$

Einstein himself applied Riemann's technique to "count" the variety of Lorentzian manifolds. The "leading terms" in Einstein's count suggests charts in which the metric tensor has the form

• Lorentzian two-manifolds: one function of two variables (double null chart!)
  $$\left[ \begin{array}{cc} 0 & \exp(f) \\ \exp(f) & 0 \end{array} \right]$$
• Lorentzian three-manifolds: three functions of three variables
  $$\left[ \begin{array}{cc|c} 0 & \exp(f) & a \\ \exp(f) & 0 & b \\ \hline a & b & 1 \end{array} \right]$$
• Lorentzian four-manifolds: six functions of four variables
  $$\left[ \begin{array}{cc|cc} 0 & \exp(f) & p & q \\ \exp(f) & 0 & a & b \\ \hline p & a & \exp(g) & 0 \\ q & b & 0 & \exp(g) \end{array} \right]$$

In these charts, the special case where all the functions vanish identically should look familiar!

As you probably noticed, the Riemann tensor of a Lorentzian or Riemannian four-manifold has twenty algebraically independent components; the counting argument suggests that we only need specify some of these on certain hyperslices or slices of hyperslices, and from this data we can reconstruct the metric tensor from the curvature tensor. A scheme of Rainer Sachs shows how to do this, using NP formalism.

So from this perspective, a good answer to your question might be to suggest a new question, "how can we measure the curvature tensor at the point P?" Following Riemann's approach, we should seek a scheme which simultaneously measures curvature components at some point and also determines the metric near that point (up to second order in the "distance" from the point).

For two manifolds, one method of measuring curvature is suggested by Diquet's formula relating the area deficit of a small disk drawn on a Riemannian two-manifold (not neccessarily a sphere) "centered" at P to the Gaussian curvature at P.

For measuring the curvature tensor in a spacetime, several schemes have been suggested, but AFAIK none of them deserve to be called "canonical".

The second thing which springs to mind is recent work by B. Coll and his students on providing a foundation for third generation satellite beaconing navigation systems, ones which will permit mapping of events in time and space to cm accuracy in the solar system, not just near the surface of the Earth. This is quite fascinating and deserves a BRS in itself. The basic idea is to use time signals sent regularly from a set of beacons to define a Coll canonical chart. For a two-dimensional locally flat Lorentzian manifold, in the simple case of two inertial (but not comoving) beacons, the line element can be written in the Coll chart as
$$ds^2 = -k_1 \, k_2 \, du^1 \, du^2$$
where [itex]k_1, \, k_2[/tex] are the frequency shift factors for signals from the two beacons, as received by an inertial observer (not comoving with either beacon, in general) whom we can call "the static observer". We can boost the frame with a boost parameter depending on u^1, u^2, and then we see that (if our observer is between the two beacons) we get the same metric: not only does the product k_1 \, k_2 not depend on which inertial observer we use, it doesn't even depend on whether we use an inertial observer at all!

For those of you who use Maxima, here is a Ctensor file you can run as a batch file under wxmaxima:

/* 
E11 manifold; Coll canonical normal chart for two inertial beacons
	u^1 = time signal from 1st beacon
	k_1 = frequency shift of 1st time signal msrd by inertial observer
	u^2 = time signal from 2nd beacon
	k_2 = frequency shift of 2nd time signal msrd by inertial observer	
	v = velocity of obsvr relative inertial obsvr comoving with both
		becaons (a function of u^1, u^2)

Form of metric is entirely independent of v

*/
load(ctensor);
cframe_flag: true;
ratchristof: true;
ctrgsimp: true;
/* define the dimension */
dim: 2;
/* list the coordinates */
ct_coords: [u1,u2];
/* define variables */
constant(k1,k2);
depends(v,[u1,u2]);
/* define background metric */
lfg: ident(2);
lfg[1,1]: -1;
/* define the coframe */
fri: zeromatrix(2,2);
fri[1,1]:  -sqrt(1-v)/sqrt(1+v)*k1/2;
fri[1,2]:  -sqrt(1+v)/sqrt(1-v)*k2/2;
fri[2,1]:  -sqrt(1-v)/sqrt(1+v)*k1/2;
fri[2,2]:   sqrt(1+v)/sqrt(1-v)*k2/2;
/* setup the spacetime definition */
cmetric();
/* display matrix whose rows give coframe covectors */
fri;
/* compute a matrix whose rows give frame vectors */
fr;
/* metric tensor g_(ab) */
lg;
/* compute g^(ab) */
ug: invert(lg);
christof(false);
/* Compute fully covariant Riemann components R_(mijk) = riem[i,k,j,m] */
lriemann(true);
cgeodesic(true);

Here, u^1 has the intuitive meaning that an observer an an event with coordinate u^1 = 123 has just received a time signal reading "I am beacon 1 and my time is 123 seconds", and this signal has been frequency shifted by the amount $k_1$ (static observer) or [itex]k_1 \, \frac{\sqrt{1+v}}{\sqrt{1-v}}[/tex] (arbitrary observer to right of first beacon). Combining the observed values at event p of [itex]u^1, \, u^2, \, k_1, \, k_2[/tex], our arbitrarily moving observer at p knows both the Coll chart label of p and the line element near p. So in the special case of

• two dimensional flat spacetime
• two inertial (but not neccessarily comoving) beacons with reliable clocks

this spacetime charting scheme is practical and simple. But there is a hint of trouble ahead: we need to assume our arbitrarily moving observer has a world line in the region between the world lines of the two beacons! (Only then does the Doppler shift due to the boost we introduced above cancel when take the product of the observed frequency shifts of the two time signals.)

More generally, one can consider an arbitrarily moving pair of beacons. For curved manifolds things are more complicated. And for higher dimensions things are still more complicated--- at least if you seek exact results.

Curiously, what has been lacking to date is a suitable approximation scheme, which is clearly what one really wants. For one thing, we might want to use more beacons than the dimension of spacetime, as a check.

But in any case, the point is that, following Coll's approach, we should try to work out a scheme in which we simultaneously construct a chart and its metric tensor.

For me, one of the fascinating aspects of this work is that it hearkens back to the roots of my Ph.D. thesis, even the roots of mathematics itself. That is, the theory of Penrose tilings depends heavily upon simple continued fractions, whose theory was worked out by Theaetetus, a brilliant Athenian teenager who had been hired by his native city state to concoct an improved calendrical system. Theaetetus came up with a simple general method for finding the "most efficient" rational approximations to any real number. That is, we want to find an approximation which uses small integer denominators but which is also very accurate. Then one can use what Gauss later called "mod n" to work out a calendar with appropriate periodicities, such that the best date for the harvest won't "migrate" quickly wrt the calendrical dates, which was a serious problem with earlier systems. This achievement was the start of Greek mathematics; and also the start of the theory of algorithms (the idea upon which Theaetetus founded the theory of simple continued fractions is also the idea behind the "euclidean" algorithm). Had Theaetetus lived longer, Greek mathematics might well not have become "stuck" on using purely "euclidean" geometrical methods.

Anyway, one of the huge problems bedeviling both calendars and global mapping of the Earth is due to topology: the international date line continues to confuse modern historians writing about say the Gulf of Tonkin incident, who in principle need to figure out, with each document they examine, whether the times mentioned refer to Washington D.C. time and date, Gulf of Tonkin time and date, or (as is often the case) reflect confusion on the part of intelligence analysts and White House personnel about such time/space charting issues.

At one point I even started to draft an expository paper with the provacative title "Is Bartolome Coll the new Copernicus?", which I think really does capture the fundamental nature of this problem and its remarkable place in intellectual history. The point of the title is that the Coll chart (and related schemes previously suggested by several other researchers) offers a completely novel approach which promises to completely evade topological problems, and which also fully dethrones accidental features of one small solar system from timekeeping and location charting. At the expense of introducing accidental features due to which region--- defined by the world lines of the beacons--- our observer lies in. But if our beacons are distant pulsars, and we are only trying to navigate around our dinky little solar system, this is not likely to become problematic!

However, such a solution would only become practical when some new Riemann concocts a suitable approximation scheme. I haven't sufficient optimism to think about it, but maybe one of you will

Further reading:
E. T. Bell, Men of Mathematics, Dover reprint, chapter 14.

K. F. Gauss, General Investigations of Curved Surfaces, Dover reprint of Princeton translation, 1902, originally given as a talk in Latin (not German!) in 1827 at the scientific society of Goettingen, but not published for another 25+ years.

Michael Spivak, Comprehensive Introduction to Differential Geometry, 2nd edition, Publish or Perish Press, 1979, Vol. II, chapters 3 &4. (Chapter 3 includes a proof of Diquet's formula and chapter 4 includes a translation of B. F. Riemann, "On the Hypotheses which Lie at the Foundations of Geometry", 1854, the famous lecture which so impressed Gauss, in which Riemann introduced the Riemann normal chart and Riemannian geometry.)

arxiv.org/abs/gr-qc/0606044

Relativistic Positioning Systems: The Emission Coordinates
Bartolomé Coll and José MarÍa Pozo

arxiv.org/abs/0905.3798

Gravimetry, Relativity, and the Global Navigation Satellite Systems
Albert Tarantola, Ludek Klimes, Jose Maria Pozo, and Bartolome Coll

I know, I haven't tried to answer the question--- let me know whether either Riemann normal chart or Coll canonical chart sound like what you are looking for, and I'll try to say more.

Figures (left to right):

• sketch of length along a curve, up to second order in distance from P, according to the Riemann normal chart centered at P
• sketch of construction of Coll canonical chart and spacetime charting, for a Lorentzian two-manifold

 

[PAllen]

I've mentioned it before, but my favorite approach to this is the derivation of 5 point curvature detector in J. L. Synge 1960 book on GR. He derives that if you measure super accurate distances (10 distances in all, required, if memory serves) among 5 generally arbitrary points, there is a relation between them which must hold if the curvature tensor is zero. If it doesn't hold, you have information about the curvature tensor. If you do this for points in different configurations, you can build up more information about the curvature tensor.

Unfortunately, near earth, the precision required is way beyond feasible. I recall needing distance measurements accurate to 1 part in 10^29.