In 3D Euclidean space, the distance (squared) between two points is the sum of the squares of coordinate separations – i.e., the Pythagorean theorem. So the distance s between (x_1,y_1,z_1) and (x_2,y_2,z_2) is given by s = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}. Hereafter, we can talk just about s^2 and not s to avoid the clumsy square root sign. We can also zoom in to an infinitesimal displacement so that we can talk about how the space behaves locally as a point. For this we can use differential symbols like “dx” to mean “an infinitesimal step in the x direction”, which is the calculus/limiting case of a more chunky step \Delta x.
Putting all that together, the “metric” – how you measure in the space – in boring 3D Euclidean space is
ds^2 = dx^2 + dy^2 +dz^2 .
Keep in mind that s is related to the concept of distance between points (and ds the infinitesimal version that encapsulates the local behavior of the geometry at hand).
In (3+1)D special relativity, distances in just the space portion aren’t fundamental. We are instead concerned with distances in spacetime. In special relativity, distances in spacetime are related to the metric
ds^2 = -(c\,dt)^2+dx^2 + dy^2 +dz^2 .
The negative sign on the time term is worth noting, but it’s actually not that interesting a feature for the purposes of this question. Also, you may find in your own research two different sign conventions, where the expression on the right is equal to -(ds)^2. This, too, is uninteresting for the purposes of this question, but it can be confusing.
Both of the metrics above represent “flat” – not curved – spaces, since the contribution of each coordinate step is independent of the coordinate values themselves and of the other step sizes. A step dy always contributes the same no matter what location (t,x,y,z) you are at and no matter how big a step you are taking in the other coordinate directions.
In general relativity, the presence of energy, momentum, and flows of those modify the spacetime metric. The Einstein field equations are the differential equations that relate the “sources” to the metric, although the metric itself is somewhat buried in the tidy form of those equations. (It’s in there, though!)
For spherical sources of gravity, it will be useful to switch from Cartesian coordinates to spherical coordinates. So, we’ll measure distances not in terms of steps in (t,x,y,z) but rather in (t,r,\theta,\phi). Here, r is the radial distance from the center of our source object (star or planet or whatever); \theta is the polar angle (like latitude, but with 0 at the north pole); and \phi is the azimuthal angle (like longitude).
The flat metric in these coordinates looks messier than in Cartesian coordinates:
ds^2 = -(c\,dt)^2+dr^2 + r^2d\theta^2 + r^2(\sin\theta)^2 d\phi^2
but it’s still just as flat as the previous metric. It’s just that a step in (say) \theta is worth more distance when you are at higher r, and a step in \phi is worth less distance when you are near the poles. The various extra bits in the metric are just these boring aspects of spherical coordinates in a flat space. So, this metric is our starting point now.
We can drop a spherically symmetric, non-rotating mass M at the center of our coordinate system. This leads to the metric
ds^2 = -\left(1-\frac{2GM}{rc^2}\right)(c\,dt)^2+\left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 + r^2d\theta^2 + r^2(\sin\theta)^2 d\phi^2 ,
where G is Newton’s gravitational constant. Now the distance contribution from steps in the t and r directions are modified in an r-dependent way relative to the flat case. But the angular pieces are untouched and there is no “cross” dependence anywhere, meaning the contribution of a step in r (say) doesn’t depend on how much you are stepping in \theta (say).
Finally, we introduce a spinning source, and you get the so-call Kerr metric. It is not worth reproducing the very complicated expression here – see the wiki link – but two key features include:
- The coefficients now all also depend on the angular momentum present and the \theta coordinate.
- There is now cross-dependence in the coordinate intervals!
For that last point, there is now a term that looks like
... +\, (\mathrm{coefficients})\, dt \,d\phi .
This means distances in the (+t,+\phi) direction will work differently than distances in the (+t,-\phi) direction, and how much this happens also depends on where you are in r and \theta (as embedded in the coefficients).
This connects directly to the previously noted feature that travel around the spinning object looks different one way versus the other. More generally, all these coordinate-value and step-value couplings lead to observable deviations in behavior from the non-spinning case. Lense-Thirring precession is another specific consequence, as mentioned above, wherein a rotating object like a gyroscope will exhibit anomalous precession when near a rotating massive object.
