Evidence of curvature in presence of mass can be seen is what are called "Einstein rings". For this phenomenon to occur, observer and two different galaxies are required to fall on a straight line; and also the observer has to be situated on one of the end points. In the case where space-time is flat, we can see only one galaxy and the second one is hidden behind it. However due to the presence of curvature, hidden galaxy becomes visible in the form of a ring. Radius of the ring can be calculated using general relativity.

The ring effect was first mentioned in academic literature by Orest Chwolson in 1924. -WIKIPEDIA. https://en.wikipedia.org/wiki/Einstein_ring

Thanks for your answer. I know we can observe curvature via grabitational lensing. I would like to hear from a general relativity theorist, though, about the specific approach I mentioned: measuring the ratio of area to radius of a sphere surrounding a gravitating mass.

Yes the answer of Biswajoy Brahmachariis very good and the question of Steve Mcgrew "measuring the ratio of area to radius of a sphere surrounding a gravitating mass" is good also. But we have a big problem : what does   "the ratio of area to radius of a sphere" mean? Nothing in GR. But using Painlevé-Gullstrand for of Schwarzshild form of metric, may be we can do it.

The only usefull method is to measure the period of a satellite of the mass. But we need another equation.

Michel, could you please explain why "the ratio of area to radius of a sphere" has no meaning in GR?

In the situation I offered, there is no event horizon and no singularity at r=0.

Excuse-me Steve, it is not the ratio which as no physical meaning, it is the radius because the radial distance has no physical meaning.

Michel, all the papers I have read that conclude there is no physical meaning to radial distance seem to begin with the assumption of a black hole with an event horizon and a singularity at the center.  In the example I offered, there might be an extremely strong gravitational field, but there would be no event horizon or singularity because the gravitational field would weaken steadily to zero at the center inside the spherical mass.

For a weak gravitational field it the same mathematical problem than for a strong one, but the error is very low and so no physical problem.

The main problem is about "R" : How you compute the "long distance R away from the center" ?

@ Steve

In DNAg (see attached) there is a straight forward mathematical answer.

R' =R/(1+ 3GM/Rc^2).

So that give's you the radius reduction, and clearly  for the (surface) area that term needs to be squared.

This actually gives more accurate results than GTR - and can be applied to black holes.

Ignore the inevitable protests from the fixated general relativists. If you try it and you will see the mathematics actually works better than GTR - and is a lot easier.

https://www.researchgate.net/publication/269689554_The_formulation_of_Dynamic_Newtonian_advanced_gravity_DNAg

Article The formulation of Dynamic Newtonian advanced gravity, DNAg

The proper distance d(r) up to the the radial coordinate r and the proper surface area A(r) of the surface at fixed r around a spherically symmetric mass (like a small spherical star) are obtained using the Schwarzschild metric. While d(r) is given by the integral of  1/sqrt(1-2GM/c^2 r) over the limits  0 to r, the area A(r) is simply 4 pi r^2. Physically the relation between d(r) and A(r) can be measured by estimating how the bolometric flux of the radiation from the star falls off with the physical distance d(r) given a fixed bolometric luminosity of the star. 

Schwartschild metric measures the curvature you requires. It only depends o mass and distance. Mercury perihelium advance represent the deficit between area and distance. So far so good.

Steve, have a look at how radius is defined in the Schwarzschild Metric (link). Your big problem will be to find a practical method of actually measuring the radius and area without making assumptions about how they are related.

https://en.wikipedia.org/wiki/Schwarzschild_metric#The_Schwarzschild_metric

Suppose there is a spherical mass that is *almost* a black hole.

Steve, the real object which best fits your description is a neutron star.

Since neutron stars (and their pulsars sub-group) are well-studied, both observationally and in theoretical modelling, the definitive answer should be straightforward to find. Papers about 3d relativistic modelling of pulsar magnetospheres would be a good place to look.

Measuring the area of a sphere should be fairly straightforward if there is a distance standard - such as the wavelength of one particular emission line of hydrogen.  Measuring the radius when the center is occupied by a dense mass or a black hole is not so easy.  However, it should not be difficult to compare the areas of two concentric spheres whose (unknown) radii differ by a measured distance.  Curvature might then be determined from dA/dR rather than A/R.

SM: Measuring the area of a sphere should be fairly straightforward

• How would you measure the area of the surface of a sphere centred on the Sun?
• How would you make sure you area measurement was on the surface without being able to measure the radius?
• What accuracy could you achieve?

What you propose is theoretically valid by completely impractical.

The easier way to measure the curvature is to find the sum of the internal angles of a triangle. Imagine three laser beams crossing and making a triangle with the Sun at the centre, for example from near Earth to each of the STEREO probes and a third beam between them. If each beam is bent as it passes the Sun, the sum of those bends would  add to 180 degrees to give the total. However, we already know how much a single beam bends passing the Sun, that was measured by Eddington in the 1920's and has been repeated many times since, for example by Hipparcos. Those existing measurements will give you your answer.

Guess I should confess: what I'm reaching for is a different perspective on the inverse square law for the electric field around a massive charged particle, and how it might be modified very near the particle.  A simple explanation for the 1/R^2 dependence of the E field around a charge is that the integral of the radial component of the E field over the area of a sphere equals the charge inside the sphere -- and the area of the sphere varies as R^2.  IF the area of the sphere appears depend on R differently from 1/R^2 due to the presence of mass at the center of the sphere, then the apparent charge inside the sphere will appear different at different distances from the center.

Steve,

If I understand your question correctly I think the following is the problem you would be facing and which many before me has alluded to:

In general relativity one solves the Einstein equations for the metric tensor AND the energy momentum tensor simultaneously. What this means is that one cannot define an isolated system as a star (for instance) consisting of some form of matter and calculate the warping of space a distance away from the centre --- because the space upon which one would define the radius is also an unknown of the theory. There is no background space upon which the star "sits" and where radial distance would be meaningful. In electromagnetism one solves Maxwells equations in an already well defined Euclidean back ground space, and hence the radial coordinate makes sense. It is the same in Newtonian mechanic. But unfortunately not in Einsteins equations. This has to do with the non-linearity of the equations. Was that helpful?

Could measuring angles between points surrounding the mass, rather than measuring distances, provide a way to determine curvature?