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

So they are great for describing an observer far away from the hole (who can't really observe anything happening at the horizon anyway, since it takes an infinite amount of time for anything to happen), but they are not good for describing what happens at the horizon itself (where an observer is in freefall).A black hole's lifespan is infinite, it just keeps on existing for as long as the universe exists. But if you mean the time it takes for an observer to see something happen as the black hole forms, that's a different question. And the answer depends on the observer's position and motion. If they are far away (but not infinitely
  • #1
Pencilvester
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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  • #2
Pencilvester said:
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.
 
  • #3
Pencilvester said:
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.
 
  • #4
Pencilvester said:
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?
 
  • #5
Dale said:
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?
 
  • #6
Pencilvester said:
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.
 
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  • #7
The standard cosmological model is geodesically incomplete. The big bang singularity is such incompleteness.
 
  • #8
Pencilvester said:
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?
 
  • #9
martinbn said:
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
 
  • #10
Pencilvester said:
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.
 
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  • #11
Dale said:
Yes, this is called the Rindler horizon.

Thanks!
 
  • #12
Dale said:
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.
 
  • #13
Pencilvester said:
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.
 
  • #14
Pencilvester said:
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.
 
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  • #15
robphy said:
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 black hole?
 

1. What is a geodesically complete manifold?

A geodesically complete manifold is a mathematical concept used in the field of general relativity to describe the spacetime we inhabit. It is a type of manifold (a mathematical space that looks like Euclidean space from a small enough scale) where all geodesics (the shortest paths between two points) can be extended infinitely without reaching a boundary.

2. Why is it important for the spacetime we inhabit to be a geodesically complete manifold?

This is important because it allows us to describe the behavior of objects in our universe using the principles of general relativity. If the spacetime we inhabit were not geodesically complete, we would not be able to accurately predict the motion and interactions of objects in our universe.

3. How do we know that the spacetime we inhabit is a geodesically complete manifold?

This is a question that is still being studied and debated by scientists. However, one way we can determine this is by analyzing the behavior of objects and particles in our universe and seeing if they follow the principles of general relativity. So far, all evidence points to the fact that the spacetime we inhabit is indeed a geodesically complete manifold.

4. What are the implications of a non-geodesically complete manifold for our understanding of the universe?

If it were discovered that the spacetime we inhabit is not a geodesically complete manifold, it would have significant implications for our understanding of the universe. It could mean that our current theories and models of the universe, such as general relativity, would need to be revised or replaced with new ones that can account for a non-geodesically complete spacetime.

5. How does a geodesically complete manifold relate to the concept of curvature in spacetime?

A geodesically complete manifold is closely related to the concept of curvature in spacetime. In a geodesically complete manifold, the curvature of spacetime is constant and uniform. This means that the laws of physics and the behavior of objects are consistent throughout the entire manifold. In contrast, a non-geodesically complete manifold would have varying levels of curvature and could lead to inconsistencies in the laws of physics and the behavior of objects in different regions.

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