Question: Looking at a Penrose diagram for Minkowski space, you would think that you could draw a null geodesic running from $i^0$, along $\mathscr{I}^+$, and ending up on $i^+$. In fact there would be many such geodesics, due to the rotational symmetry that is hidden on the two-dimensional diagram. Similarly we could draw null geodesics running along $\mathscr{I}^-$.

These seem like nonphysical artifacts that we don't want around. It would be weird if there were some dynamics going on "off stage" like this that we could never detect. We don't have a metric on the boundary, so it seems like there is no real definition of a geodesic. Also, we can't apply the Einstein field equations at the boundary, since they are written as differential equations that only make sense if you have an open neighborhood surrounding any given point.

On the other hand, we *do* have null geodesics that *intersect* null infinity. So it seems a little delicate to set things up mathematically so that null geodesics can get to null infinity but can't have any motion *within* null infinity.

Is there some way that it's encoded into the formalism that we don't have such things going on? Is it in one of the complicated technical conditions written into the definition of asymptotic flatness? How do we make sure that it *is* allowed for geodesics to intersect null infinity, but not allowed for them to dwell there?

related: [How to make sense of $\mathcal{I}^-$ as a Cauchy surface rigorously?](https://physics.stackexchange.com/q/455877/)

Answer: 
This feels like something that is a definitional thing to me that is however you make it. Infinitesimally in from conformal null infinity, the spacetime is infinitesimally close to Minkowski space, and the normal to your outbound null vector is definitely a null 3-surface, so it makes sense to me to say that conformal infinity is a null 3-surface too. but yeah, it's not a point on the underlying manifold, so it's a bit dishonest to extend the metric there.
It sounds like you're thinking about using null infinity as a Cauchy surface, and worried about the null geodesics within infinity representing a particle propagating "through" the initial condition?

If so, here's an answer.

Are there null geodesics running through $\mathscr{J}^-$? Yes. 

(The only reason there wouldn't be is if you disallow it for technical reasons, which is reasonable, but not really the point.)

Is that a problem? No. The issue I think you're worried about has nothing to do with $\mathscr{J}$ being at infinity, and everything to do with it being null. And it has a solution.

Long story short: For a massless KG equation, or something else with null-propagating solutions, a null Cauchy surface is something called a [characteristic surface](https://en.wikipedia.org/wiki/Method_of_characteristics) of the problem. Instead of providing the usual $\phi, \dot{\phi}$ initial data on the null surface, you only need to provide $\phi$. Because the initial data is specified differently on a surface like this, there's not an issue with modes propagating along the boundary surface.

If you want me to elaborate on that leave a comment, I do have an example in mind. (Or work it out yourself... the massless 2d KG equation has a general solution $f(t-x)+g(t+x)$. What happens if you specify initial data on a pair of null surfaces?)