I don't understand why we are able to see and measure curvature /
warping of space at all.

The Earth's surface is curved and this can be observed via the vast number of pictures of the Earth from space that now exist.

However, the surface curvature can also be "seen" via measurements on the surface itself.

For example, if one were start at the North Pole and travel in a "straight line" (a great circle) to the equator, then move east along the equator for a quarter of the circumference, and then move North (always along a great circle), one would eventually reach the starting point at the North Pole.

But look, one would have formed a "triangle" with the interior angles adding up to 270 degrees! This is one way that *intrinsic* curvature is measured.

Simply put, intrinsic curvature is mathematically characterized by the Riemann Curvature Tensor and *observed* via geodesic deviation.

Of course Coulomb's law has to be adapted! And it is therefore fortunate that there exist manifestly *covariant* formulations of electromagnetism that do not care how spacetime is curved. However, we should first briefly observe that Coulomb's law is not one of the fundamental laws of electromagnetism, though it has played a great role in its inception:

Coulomb's law is just the solution of Maxwell's equations for a point charge and no current in flat Minkowski space. Maxwell's equations can jointly be generalized to arbitrary spacetimes:

The electric field strength is a 2-form $F = \frac{1}{2}F_{\mu\nu}\mathrm{d}x^\mu \wedge \mathrm{d}x^\nu$ on spacetime, and *electric current* is a 3-form $J = \frac{1}{6}J_{\mu\nu\rho}\mathrm{d}x^\mu \wedge \mathrm{d}x^\nu \wedge \mathrm{d}x^\rho$, as the Hodge dual of the usual vector current. Maxwell's equations now simply read

$$ \mathrm{d}F = 0 \; \text{and} \; \mathrm{d}\star F = J$$

where, since the Hodge star depends on the metric, the curvature of spacetime indeed influences the form of our laws.

It has to be noted that, on arbitrary spacetimes, the notion of having "separate laws" for electric on magnetic fields doesn't really make sense anymore, since they mix in (almost) arbitrary ways, depending on the metric. You can still get the electric and magnetic fields as components $F^{0i}$ and $F^{ij}$ of the field strength, but you won't be writing any nice, frame-independent laws for them. Maxwell's equations do not dissolve nicely into "Gauss' law", "Faraday's law" or such things in a general setting.

## Best Answer

While "warping of space" is a perfectly good way to think about it, to be precise we talk about the "curvature of spacetime". General relativity is the standard modern theory of gravity, and it describes this curvature with something called the Riemann tensor (although there are numerous other good ways to talk about curvature). A tensor is — to put it very roughly — a

collectionof numbers, and you can't boil it down to just one number. But each of the numbers in this Riemann tensor has units of $1/\mathrm{distance}^2$. Basically, the Riemann tensor measures the rate of change of the rate of change (yes, both) of space's shape as you move along different directions — so each of those rates of change brings in one factor of $1/\mathrm{distance}$, which is how you get a total of $1/\mathrm{distance}^2$.There's no known maximum limit to how big the curvature can be. In our current theory, at least, there are places where it can become infinite. These are called gravitational singularities. For example, we believe that singularities are found inside of black holes. Of course, we also believe that the theory of general relativity is incomplete, and needs to be combined with quantum theory somehow. When this is done, most physicists expect that singularities will disappear and be replaced by some crazy quantum phenomenon. As just a vague ballpark guess, most physicists would expect that craziness to kick in once the curvature gets anywhere close to $1/\ell_{\mathrm{P}}^2$, where $\ell_{\mathrm{P}}$ is the Planck length — which might be something like a limit to the curvature.

In certain special cases, you can measure warping in different ways. Relatively recently, one important measurement of warping has been the detection of gravitational waves. Because they are very weak, we don't have to deal with a lot of the headaches that you normally run into in general relativity, which means that we can get away with measuring the warpage with something called the strain. The strain is the ratio between how much the gravitational wave changes the length of an object it passes and the normal length of that object. The units of strain, therefore, are just $\mathrm{distance}/\mathrm{distance}$ — which is to say that it's dimensionless.