# Why must the spacetime we inhabit be a geodesically complete manifold?

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Pencilvester

## Main Question or Discussion Point

Can someone tell me how we know that our physical universe is geodesically complete? In response to a question I had about why we assign any meaning to the other side of a black hole’s event horizon (or its interior), I got an answer prompting me to look into the concept of geodesic completeness. I found a little bit about it in a book titled “Semi-Riemannian Geometry with Applications to Relativity” by O’Neill. I found the definition of a geodesically complete manifold and a few examples of complete and incomplete manifolds (the Schwarzschild half-plane being of the incomplete variety). So I now understand what geodesic completeness means, and I understand that considering our physical spacetime to end at the event horizon of a black hole implies the manifold we live on is geodesically incomplete, but I still don’t understand why I should have a problem with that. Is there some physical evidence that tells us that we should?
Related question: If we consider a black hole’s mass to be concentrated beneath the event horizon (at a singularity or otherwise), and gravitational effects propagate through space with speed c, and anything with a speed less than or equal to c cannot cross an event horizon from below, how does a black hole affect any other mass gravitationally?

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phyzguy
Can someone tell me how we know that our physical universe is geodesically complete? In response to a question I had about why we assign any meaning to the other side of a black hole’s event horizon (or its interior), I got an answer prompting me to look into the concept of geodesic completeness. I found a little bit about it in a book titled “Semi-Riemannian Geometry with Applications to Relativity” by O’Neill. I found the definition of a geodesically complete manifold and a few examples of complete and incomplete manifolds (the Schwarzschild half-plane being of the incomplete variety). So I now understand what geodesic completeness means, and I understand that considering our physical spacetime to end at the event horizon of a black hole implies the manifold we live on is geodesically incomplete, but I still don’t understand why I should have a problem with that. Is there some physical evidence that tells us that we should?
I don't think we have any evidence one way or the other. I think it's possible to take the view that the mass inside a black hole has "left our universe". In that sense, what's on the other side of the event horizon could be viewed as irrelevant. Others may have a different take.

Related question: If we consider a black hole’s mass to be concentrated beneath the event horizon (at a singularity or otherwise), and gravitational effects propagate through space with speed c, and anything with a speed less than or equal to c cannot cross an event horizon from below, how does a black hole affect any other mass gravitationally?
It's only changes in the gravitational field that propagate at the speed of light. So any changes to the mass distribution that occur after the mass falls in can have no impact outside the event horizon. But the boundary conditions at the event horizon still hold and constrain the geometry outside.

PeterDonis
Mentor
If we consider a black hole’s mass to be concentrated beneath the event horizon (at a singularity or otherwise), and gravitational effects propagate through space with speed c, and anything with a speed less than or equal to c cannot cross an event horizon from below, how does a black hole affect any other mass gravitationally?
See here:

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_gravity.html

We have also had previous PF threads on this, though it's been a while.

Dale
Mentor
considering our physical spacetime to end at the event horizon of a black hole implies the manifold we live on is geodesically incomplete, but I still don’t understand why I should have a problem with that. Is there some physical evidence that tells us that we should?
It would mean that spacetime would just end for no locally discernible reason. We simply have never seen any data which would suggest that spacetime just ends somewhere or sometime for no reason. Why would spacetime just stop somewhere with no physical cause there?

Pencilvester
It would mean that spacetime would just end for no locally discernible reason. We simply have never seen any data which would suggest that spacetime just ends somewhere or sometime for no reason. Why would spacetime just stop somewhere with no physical cause there?
Makes sense. But how do we reconcile the two facts that in-falling objects take an infinite amount of coordinate time to cross the horizon and a black hole has a finite lifespan? Okay, a thought just occurred to me: worldlines of particles with constant acceleration that are evenly spaced at a fixed proper distances along with their surfaces of simultaneity (or lines if we just look at t and r) can represent a valid coordinate system for spacetime near a black hole, with the lines of simultaneity approaching the event horizon, yes? While an accelerated observer never will reach a time at which his line of simultaneity shows a freely falling particle “crossing the horizon,” there will come a time for this observer when the light he shines at the particle never gets reflected back to him, making communication between him and the particle utterly impossible, so in a very real sense, the particle has crossed the event horizon at that point, yes? So it’s seeming to me like coordinate time, when viewed from large enough separations in distance and/or time, is what I shouldn’t assign any meaning to, which, yes, I realize is like a core principle of Einstein’s relativity that I’ve been completely ignoring. Am I getting anything wrong?

Dale
Mentor
But how do we reconcile the two facts that in-falling objects take an infinite amount of coordinate time to cross the horizon and a black hole has a finite lifespan?
What is there to reconcile? The Schwarzschild coordinates are bad at the EH, just like longitude is bad at the poles. Don't use those coordinates at locations where they are bad.

martinbn
The standard cosmological model is geodesically incomplete. The big bang singularity is such incompleteness.

Pencilvester
worldlines of particles with constant acceleration that are evenly spaced at a fixed proper distances along with their surfaces of simultaneity (or lines if we just look at t and r) can represent a valid coordinate system for spacetime near a black hole, with the lines of simultaneity approaching the event horizon, yes?
Related question about this that I’ve had ever since I was playing around with special relativity: a rocket ship with constant acceleration really will experience an effective event horizon behind it (of course very, very far behind it if the acceleration is tolerable for humans), right? Is there a name for this kind of horizon?

Dale
Mentor
The standard cosmological model is geodesically incomplete. The big bang singularity is such incompleteness.
But geodesics that end at curvature singularities are reasonable. Meaning, there is a local reason why the geodesics end there, namely the infinite local curvature

Dale
Mentor
Related question about this that I’ve had ever since I was playing around with special relativity: a rocket ship with constant acceleration really will experience an effective event horizon behind it (of course very, very far behind it if the acceleration is tolerable for humans), right? Is there a name for this kind of horizon?
Yes, this is called the Rindler horizon.

Pencilvester
martinbn
But geodesics that end at curvature singularities are reasonable. Meaning, there is a local reason why the geodesics end there, namely the infinite local curvature
Fair enough.

PeterDonis
Mentor
how do we reconcile the two facts that in-falling objects take an infinite amount of coordinate time to cross the horizon and a black hole has a finite lifespan?
If the black hole has a finite lifespan (because it evaporates), then it no longer takes an infinite amount of coordinate time (in coordinates similar to Schwarzschild coordinates) for an object to cross the horizon.

robphy
Homework Helper
Gold Member
So I now understand what geodesic completeness means, and I understand that considering our physical spacetime to end at the event horizon of a black hole implies the manifold we live on is geodesically incomplete,
Our physical spacetime would not end at the event horizon.
An infalling observer can cross the event horizon and continue to live.... until reaching the singularity within.
Although we would no longer receive messages from that observer who has crossed over,
we can still influence that observer [in his future] and have other observers join him.

Our physical spacetime would not end at the event horizon.
An infalling observer can cross the event horizon and continue to live.... until reaching the singularity within.
Although we would no longer receive messages from that observer who has crossed over,
we can still influence that observer [in his future]
and have other observers join him.
Great point! I think there is also the entropy and stuff all adds up so imo that means it's within and part of the universe physically. I used to think that inside black holes is physically moot, but you changed my perspective.

Do any of the other "fields" have something equivalent to a blackhole?