How is spacetime locally Lorentzian?

Question

Following up on the questions raised here:

If you take all the matter and energy out of a significantly large volume of spacetime, what you'll be left with is a small chunk of spacetime that - in the assumptions of $\lambda CDM$ - will be Lorentzian. By Lorentzian, we mean that two particles with no mass and at rest with respect to each other and separated by a distance, $\zeta$, will stay separated by $\zeta$ for all time.

I can't reconcile this with the equations of GR. GR has this λ constant which exists on all scales. Whatever λ is - and we really don't know - this property is intrinsic to all spacetime. That is, it can't be removed. The net effect of this intrinsic property is that space grows exponentially with time.

Now put two test particles with no mass in this completely empty, 'local' space. Won't they accelerate away from each other? Reduce the scale and repeat the experiment. The acceleration is the same. How is this spacetime, even on the smallest scale, Lorentzian if test particles act like it's curved?

Answer 1

The point is that "locally flat", "locally Riemannian", "locally Lorentzian", do not mean that all geometric properties tend to become the corresponding ones in flat space when restricting to a sufficiently small neighborhood of a point. Roughly speaking, this is only valid for those properties that depend on the metric coefficients up to the first derivatives computed at the given point. Mathematically speaking this fact corresponds to the existence of normal neighbourhoods of points.

As soon as a property depends on higher derivatives it cannot be cancelled by restricting in an arbitrarily small neighborhood of a point. The typical example is the relative acceleration of "infinitesimally" close geodesics. Notice that it is computed with respect to the affine parameter so that it exists also on a sphere where that parameter has the dimension of space and not time. This acceleration depends on the Riemann tensor (made of second derivatives of the metric).

Finally, physically speaking, the curvature is the relative acceleration of free falling bodies in GR: the Riemann curvature tensor at an event can be completely measured by measuring that relative acceleration of congruences of timelike geodesic at that event.

Answer 2

The surface of a sphere is locally Euclidean.

Consider two points on the equator of the Earth maybe a meter apart. Start them on northward trajectories. For a very short time, they will appear to be at constant distance. But by the time they reach 5°N latitude, you will notice that the distance between them is smaller by about half a millimeter. But, crucially, you had to travel some 500 km to see this effect.

This is the difference between local and global properties. The idea of calculus, if you like, is that a curve is locally a straight line. Calculus fails with noisy structures, and you have to invent something called stochastic calculus to patch it, because noise when you zoom in does not look like a straight line. But calculus succeeds on a circle because if you zoom in at a point on the circle, it looks more and more like a straight line.

Differential geometry is just making it clear what sort of calculus you're doing, in the sense of asking, if I zoom in more and more on spacetime, what exactly does it look like? What is my “manifold” equivalent for a “straight line” here? For the sphere, it is a two-dimensional euclidean space, if I look very very closely at the points on the sphere I can pretend that the sphere is made up of a patchwork of these little 2D euclidean spaces, caveat being that I would have to go to an infinitely small mesh to truly cover the sphere with them.

I hope that clarifies a little bit. Just like in the circle we build a curve out of little straight lines, and in the sphere we build curves out of little flat planes, the idea of general relativity is that we can build any big space-time out of little hyperbolically-flat spacetimes.