On the nature of the infinite fall toward the EH

[PeterDonis]

I'm on your side here, but I think that there is an unresolved issue (at least for me) as to how you know that a solution to GR equations is "complete".

You know it's complete when you can't analytically extend the manifold any further. One way of testing this, as you note later in your post, is to test whether the patch you are looking at is geodesically complete.

Suppose we take the "patch" consisting of the region of the Schwarzschild geometry

• r > 2GM/c²
• -∞ < t < +∞

and we declare that that's our universe. There is nothing else. There is no interior. What is wrong with that?

The manifold described by this patch is geodesically incomplete, which means it can be analytically extended.

If there is a geodesic leading to a boundary, and nothing singular happens at that boundary, then we need to describe what happens on the other side of the boundary, to "complete" the geodesic. But is that just an aesthetic consideration, or is there some reason we must have geodesic completeness?

Because not having it would mean that objects moving on geodesic worldlines would just "disappear" at a finite value of their proper time, without any physical reason. This violates energy-momentum conservation: where does the energy and momentum carried by the object go?

One could also give a similar argument using spacetime itself: if a spacetime were not geodesically complete, then the law that the covariant divergence of the stress-energy tensor must be zero would be violated at the boundary at which geodesics were incomplete. (In the case of Schwarzschild spacetime, the SET is identically zero because the spacetime is vacuum, but that does not prevent one from computing its covariant divergence.)

 

[PAllen]

Ohh Stevendaryl, you just hit on one of my biggest philosophical reasons for having a distaste for black holes. I believe reality should be describable mathematically using a single coordinate system (no patches, no infinities, etc). THAT would be beautiful to me.

Stevendaryl responded to one aspect of this - that it is a silly requirement that would deny 2-spheres from being legitimate objects. However, in the case of the BH, it is even sillier in that there are many coordinate systems that cover the interior and exterior in one coordinate patch with no infinities except at the central singularity: Kruskal, Eddington-Finkelstein, Gullestrand-Panlieve, Lemaitre.

 

[stevendaryl]

Because not having it would mean that objects moving on geodesic worldlines would just "disappear" at a finite value of their proper time, without any physical reason. This violates energy-momentum conservation: where does the energy and momentum carried by the object go?

That's aesthetically unpleasant, but it's not really a problem. You can amend it to say that the differential form of the law of conservation of energy-momentum applies only in the interior.

One could also give a similar argument using spacetime itself: if a spacetime were not geodesically complete, then the law that the covariant divergence of the stress-energy tensor must be zero would be violated at the boundary at which geodesics were incomplete. (In the case of Schwarzschild spacetime, the SET is identically zero because the spacetime is vacuum, but that does not prevent one from computing its covariant divergence.)

Well, isn't that a little circular? The law is not a first-principle, but is PROVABLE using the assumption of geodesic completeness. If you don't assume geodesic completeness, then that law isn't provable. But it's still provable in the interior.

 

[rjbeery]

Stevendaryl responded to one aspect of this - that it is a silly requirement that would deny 2-spheres from being legitimate objects. However, in the case of the BH, it is even sillier in that there are many coordinate systems that cover the interior and exterior in one coordinate patch with no infinities except at the central singularity: Kruskal, Eddington-Finkelstein, Gullestrand-Panlieve, Lemaitre.

I can deny that 2-spheres are anything but idealized mathematical models, or I can account for them in three dimensions.

 

[PeterDonis]

That's aesthetically unpleasant, but it's not really a problem. You can amend it to say that the differential form of the law of conservation of energy-momentum applies only in the interior.

You can't do that without also modifying the rest of the theory; the law of energy-momentum conservation is not an independent assumption. See below.

The law is not a first-principle, but is PROVABLE using the assumption of geodesic completeness. If you don't assume geodesic completeness, then that law isn't provable.

Huh? The law is a mathematical identity, the Bianchi identity, that is satisfied by the Einstein tensor; therefore, by the Einstein Field Equation, it is also satisfied by the stress-energy tensor. There is no assumption of geodesic completeness that I'm aware of that is required to prove the Bianchi identity or to derive the EFE.

Unless you mean that one could simply declare by fiat that we don't allow derivatives to be defined at all on the boundary (since the Bianchi identity involves derivatives of the metric). But I'm not sure you can even get away with that without violating other continuity requirements on the manifold; in other words, you'd have to declare by fiat that the manifold structure of spacetime is not applicable at the boundary. I would have to think about that some more.

 

[Dale]

he would indeed conclude that it reached c at the EH. Does that differ substantially from him declaring her to be on a null worldline?

Yes, it differs substantially. Coordinate velocities are frame variant quantities and can easily exceed c. A null tangent vector is frame invariant and is only possible for massless particles.

I can easily come up with a coordinate system where my coordinate velocity sitting here typing this response is c, but there is no coordinate system where my worldline is null. [EDIT: and why settle for c, I can make a coordinate system where my v>>c, woohoo FTL travel solved!]

What coordinate velocity would Bob assign to Alice as she crossed the EH?

That depends entirely on the coordinate chart selected. However, note that you could not select Schwarzschild coordinates for this since they don't cover the EH. The closest you could do in Schwarzschild coordinates is the limit of Alice's velocity as she approached the EH.

 

[stevendaryl]

You can't do that without also modifying the rest of the theory; the law of energy-momentum conservation is not an independent assumption. See below.

Huh? The law is a mathematical identity, the Bianchi identity, that is satisfied by the Einstein tensor; therefore, by the Einstein Field Equation, it is also satisfied by the stress-energy tensor. There is no assumption of geodesic completeness that I'm aware of that is required to prove the Bianchi identity or to derive the EFE.

I feel that what you're saying is circular. Yes, you can prove that the differential form of the conservation of energy-momentum holds, but it doesn't imply anything about geodesics continuing.

We have a "patch" P, with a boundary B. We propose the (quite weird, I admit) rule that any geodesic that intersects B ceases to exist on the "far" side of B. How can that rule possibly violate a tensor identity?

Unless you mean that one could simply declare by fiat that we don't allow derivatives to be defined at all on the boundary (since the Bianchi identity involves derivatives of the metric). But I'm not sure you can even get away with that without violating other continuity requirements on the manifold; in other words, you'd have to declare by fiat that the manifold structure of spacetime is not applicable at the boundary. I would have to think about that some more.

Right, it would be a different kind of manifold. Derivatives are only defined in the interior.

 

[Dale]

I believe reality should be describable mathematically using a single coordinate system (no patches, no infinities, etc).

My kids believe in Santa Claus.

Do you have any evidence supporting your belief? If so, which coordinate system is the "one"?

 

[PeterDonis]

The closest you could do in Schwarzschild coordinates is the limit of Alice's velocity as she approached the EH.

And just for extra fun, the limit as r -> 2m of Alice's coordinate velocity, dr/dt, is *zero*, not c, in Schwarzschild coordinates. Alice's coordinate velocity goes to c at r = 2m in *Painleve* coordinates. None of which changes anything physically, but we might as well get all the coordinate velocities out on the table for what it's worth.

 

[stevendaryl]

I can deny that 2-spheres are anything but idealized mathematical models, or I can account for them in three dimensions.

Right, you can declare that the only geometry is Euclidean geometry, but there is no reason to do that.

 

[Dale]

Right, it would be a different kind of manifold. Derivatives are only defined in the interior.

Manifolds have open boundaries, so I think that you can take derivatives all the way to the edge. That said, I don't know anything about geodesic completeness.

 

[Dale]

For example, the surface of a sphere cannot be described by a single patch. But there is nothing weird about the surface of a sphere.

I can deny that 2-spheres are anything but idealized mathematical models, or I can account for them in three dimensions.

Stevendaryl's statement is true for any manifold which is topologically the same as a sphere, which could very well be true for the universe as a whole. Also, the embedding space works for a 2-sphere but doesn't help in GR since we don't know of any 5th or higher dimensions in which to account for spacetime manifolds.

 

[PAllen]

Forget the the preferred frame, it's just confusing things. What coordinate velocity would Bob assign to Alice as she crossed the EH?

Whatever he wants. Pick any value, and there is a coordinate system the produces that value. Remember, there is no specific coordinate system Bob must use. You can adopt, as a reasonable rule, that Bob should use a coordinate system that matches his local inertial frame near near each event on his world line. But since there is no such thing as a global inertial frame, that still leaves great freedom for how coordinates are assigned further and further from Bob's world line. It is, indeed, easy to construct a coordinate system that approaches local inertial coordinates near Bob's world line and assigns any coordinate velocity you want to Alice at horizon crossing. As with any such coordinate question, yours has no physical meaning.

A physical question would be e.g. what redshift does Bob see for Alice as Alice approaches the horizon. And the coordinate independent answer is obviously redshift factor approaches infinite.

 

[stevendaryl]

Manifolds have open boundaries, so I think that you can take derivatives all the way to the edge. That said, I don't know anything about geodesic completeness.

That's a good point. That's a counter-argument to PeterDonis' claim that the EFE implies geodesic completeness. If the manifold is an open set, then the EFE would be satisfied at every point in the manifold, whether or not there is geodesic completeness. Similarly, the Bianchi identities would be satisfied at every point. So I don't think that anything would be violated by simply declaring that nothing exists outside the manifold.

 

[martinbn]

About geodesic completeness, the theorems of Hawking and Penrose show that "quite often" there are geodesicly incomplete unextendable manifolds.

 

[stevendaryl]

About geodesic completeness, the theorems of Hawking and Penrose show that "quite often" there are geodesicly incomplete unextendable manifolds.

Isn't the usual Schwarzschild geometry geodesically incomplete? You can't extend geodesics beyond the singularity, can you?

Or does "geodesic completeness" only require that any geodesic that does not pass through a singularity must be complete?

 

[PeterDonis]

Or does "geodesic completeness" only require that any geodesic that does not pass through a singularity must be complete?

This is the definition of "geodesic completeness" that I was using, yes. The technically correct mathematical definition would call a geodesic that ends on the singularity incomplete (because the proper time to the singularity is finite), but physically that isn't interpreted the same way as the geodesic incompleteness of exterior Schwarzschild coordinates at the horizon. At the curvature singularity at r = 0, geometric invariants are infinite (more precisely, they increase without bound as r -> 0). That isn't true at r = 2m. The physical "requirement" of geodesic completeness only applies at boundaries where the geometric invariants are finite.

 

[PAllen]

That's a good point. That's a counter-argument to PeterDonis' claim that the EFE implies geodesic completeness. If the manifold is an open set, then the EFE would be satisfied at every point in the manifold, whether or not there is geodesic completeness. Similarly, the Bianchi identities would be satisfied at every point. So I don't think that anything would be violated by simply declaring that nothing exists outside the manifold.

I think the best way to approach this is as follows:

1) As you are presumably aware, you can derive inclusive coordinate systems directly from the EFE and see that e.g. exterior SC is simply a subset of one of these.

2) So to rule out 'regions you don't like' you must modify GR. One variant I proposed, that, I think, fully expresses the desired boundary condition (including the requirement that it is open) is:

-----
Consider what this modification might look like, classically, and assuming we want to keep the coordinate independent nature of the equations of GR.1) We must add a couple of new axioms the theory: Universes containing naked singularities are prohibited (as a corollary, closed universes are prohibited because event horizons cannot technically be defined for them; the required new law I give next cannot be stated for a closed universe). Eternal WH-BH are prohibited. (Much stronger than 'we think not physically plausible').

2) We supplement the EFE with a new universal boundary law: The universe is bounded such that the world line of any particle or fluid element always has null paths extending from it to null infinity.

-------

Aesthetically, why should we add this 'universal boundary law' to the EFE?

Physically, I think the sharpest problem is shown by a collapsing shell of matter. One may posit a shell that would one light year as it reached EH. Inside, we have a 1 ly region following exactly the laws of SR (exactly flat spacetime). Now add a solar system in this region (small local deviations from pure SR). On an 'earth' in this solar system, Alice has dropped a ball. All normal laws of physics in the region say this ball hits the floor at time t1. However, it happens that the Universal Boundary Law kicks in and says the the ball will approach but never reach the half way point in its fall (because if it reached the half way point, the shell would have reached a radius such that no null path from from the ball can reach null infinity; thus the ball would be assigned infinite SC type time coordinate on approach to the half way point).

Geodesic completeness is equivalent to excluding such physical absurdities (as well as ruling out my proposed 'Universal Boundary Law'). Any situation where geodesics end for no local reason (e.g. singularity) are equivalent to dropped ball stopping for no reason.

 

[PeterDonis]

We have a "patch" P, with a boundary B. We propose the (quite weird, I admit) rule that any geodesic that intersects B ceases to exist on the "far" side of B. How can that rule possibly violate a tensor identity?

It does if the manifold includes B, because derivatives can't be defined at B if the manifold "ceases to exist" on the far side of B. But it doesn't if, as DaleSpam pointed out, the manifold is an open set*, because then B would not be included in the manifold. In that case, no geodesic would intersect B; it would approach B closer and closer, without ever reaching it, because B itself is not in the manifold. This is true for both of the cases we have discussed in Schwarzschild spacetime: the exterior Schwarzschild coordinate patch does not include r = 2m (it is an open set with r -> 2m), and patches like the Painleve patch that include the horizon and the black hole interior do not include the singularity (they are open sets with r -> 0). In that case, you are correct that it is not an actual contradiction to suppose that B and the region beyond B "don't exist".

All this is mathematical, though, and doesn't address the question of whether a proposed manifold that is geodesically incomplete is physically reasonable.

* - I believe DaleSpam is right that the technical definition of "manifold" (at least the one that is used in GR) requires manifolds to be open sets, but I haven't looked it up to confirm.

 

[rjbeery]

Yes, it differs substantially. Coordinate velocities are frame variant quantities and can easily exceed c. A null tangent vector is frame invariant and is only possible for massless particles.

I can easily come up with a coordinate system where my coordinate velocity sitting here typing this response is c, but there is no coordinate system where my worldline is null. [EDIT: and why settle for c, I can make a coordinate system where my v>>c, woohoo FTL travel solved!]

That's interesting. How would you do so without involving rotating frames (or black holes )?

That depends entirely on the coordinate chart selected. However, note that you could not select Schwarzschild coordinates for this since they don't cover the EH. The closest you could do in Schwarzschild coordinates is the limit of Alice's velocity as she approached the EH.

I object to this. You're saying SC coordinates don't cover the EH but they do at the limit, and what you end up with is a coordinate velocity of the infalling object = c. This is equivalent to saying that the escape velocity at that point is (just above) c, hence impossible.

 

[PAllen]

I object to this. You're saying SC coordinates don't cover the EH but they do at the limit, and what you end up with is a coordinate velocity of the infalling object = c. This is equivalent to saying that the escape velocity at that point is (just above) c, hence impossible.

NO, as Peter pointed out, the limiting SC coordinate velocity of a radial infaller is zero not c, on approach to the EH. Showing just how meaningless it is to talk about coordinate velocity as a physical thing.

 

[Dale]

That's interesting. How would you do so without involving rotating frames (or black holes )?

I don't know why you would exclude rotating frames. But anyway, e.g. start with the usual Minkowski coordinates (t,x,y,z) for the rest frame of an object in flat spacetime. Then use the following coordinate transformation:
T=t
X=x+1000000ct
Y=y
Z=z

I object to this. You're saying SC coordinates don't cover the EH but they do at the limit, and what you end up with is a coordinate velocity of the infalling object = c. This is equivalent to saying that the escape velocity at that point is (just above) c, hence impossible.

This is related to the lecture notes I posted earlier, please read through them. Note how coordinate charts and manifolds are defined on open sets, meaning that they do not include the boundary. The EH is a boundary for SC coordinates, so they coordinates do not include the EH. There are some solid mathematical reasons for this, please read the notes.

Also, as PeterDonis pointed out earlier the coordinate velocity in SC coordinates goes to 0, not c.

 

[PeterDonis]

* - I believe DaleSpam is right that the technical definition of "manifold" (at least the one that is used in GR) requires manifolds to be open sets, but I haven't looked it up to confirm.

This is related to the lecture notes I posted earlier, please read through them. Note how coordinate charts and manifolds are defined on open sets, meaning that they do not include the boundary.

Easier to look it up there then to dig out my copy of MTW. Yes, I see that in Chapter 2 Carroll goes into detail about manifolds being defined on open sets.

 

[PAllen]

Something that someone has mention is completeness of geodesics. If there is a geodesic leading to a boundary, and nothing singular happens at that boundary, then we need to describe what happens on the other side of the boundary, to "complete" the geodesic. But is that just an aesthetic consideration, or is there some reason we must have geodesic completeness?

And, to relate this to my physical motivation for geodesic completeness, imagine the geodesics representing free fall of a lab towards a supermassive BH horizon. In the lab, a student is observing and timing a spring oscillator. They see that the next peak should occur at t1 (last one at t0). Instead, the oscillator approaches, but never reaches the half way point of its oscillation, for no local physical region. Any time you have incomplete geodesics for no local reason, you can set up a scenario of this type.

 

[martinbn]

* - I believe DaleSpam is right that the technical definition of "manifold" (at least the one that is used in GR) requires manifolds to be open sets, but I haven't looked it up to confirm.

Any topological space is open and closed, part of the definition of topology.

 

[rjbeery]

NO, as Peter pointed out, the limiting SC coordinate velocity of a radial infaller is zero not c, on approach to the EH. Showing just how meaningless it is to talk about coordinate velocity as a physical thing.

Of course I meant the velocity of the infaller is zero. They "freeze", and never appear to cross the EH. Locally, the escape velocity is c, remotely the coordinate velocity is zero and the escape velocity is anything > 0.

Philosophically, it's curious though: if we say that coordinate velocity is not a physical thing then can we even say that any sort of velocity is a physical thing? Velocity seems to "mean something" and have an a physicality to it, yes?

 

[PAllen]

Of course I meant the velocity of the infaller is zero. They "freeze", and never appear to cross the EH. Locally, the escape velocity is c, remotely the coordinate velocity is zero and the escape velocity is anything > 0.

Philosophically, it's curious though: if we say that coordinate velocity is not a physical thing then can we even say that any sort of velocity is a physical thing? Velocity seems to "mean something" and have an a physicality to it, yes?

Only velocity comparisons are meaningful in relativity. In GR, only local velocity comparisons have unambiguous definition. To compare velocities of distant world lines, you have to bring one tangent close to the other to compare - that is done via parallel transport. In flat spacetime, parallel transport is path independent, so there is a unique definition [given a choice of 'now', which is a whole other matter]. In GR, the result is path dependent so there is just no preferred way to compare velocities of distant objects.

Also, note, that despite the ambiguity, you can pick any of an infinite number of paths along which to transport the 4-velocity of an infaller at the EH, or up to the singularity, to some distant static world line. If you do this, no matter what choice you make, the comparison both for horizon crossing and up to the singularity, will always be < c.

 

[PeterDonis]

if we say that coordinate velocity is not a physical thing then can we even say that any sort of velocity is a physical thing? Velocity seems to "mean something" and have an a physicality to it, yes?

The 4-velocity of a timelike worldline can be defined in an invariant way (it's the worldline's tangent vector), so it qualifies as a "physical thing". But that's a 4-vector, not a 3-vector, so it doesn't directly tell you anything about "velocity" in the ordinary sense. (One can also define a null tangent 4-vector to the worldline of a light ray; that is not usually called a 4-velocity because "4-velocity" normally means a unit vector, i.e., one with length 1, or c in conventional units, not zero.)

The relative velocity of two worldlines, at least one of which is timelike, at a particular event where they cross can also be defined in an invariant way, by taking the inner product of their two tangent vectors. If both worldlines are timelike, this will always give a result less than c. If one is timelike and one is null, this will always give a result equal to c. This happens, for example, at the event where a timelike object crosses the horizon: the inner product of the worldline's timelike tangent vector with the null tangent vector to the horizon gives c. (This is often misinterpreted as saying that the object "moves at c" when it crosses the horizon; in fact it's the *horizon* that is "moving at c".) So relative velocity in this sense also qualifies as a "physical thing".

These are the two senses of "velocity" that have direct physical meanings. Note that neither of them corresponds to coordinate velocity. Note also that both of them are local: a worldline's tangent vector has to be evaluated at a particular event, and the inner product of two worldlines' tangent vectors has to be evaluated at the event where they cross.

 

[harrylin]

It's no more of a paradox than the twin "paradox". In fact, it's more or less an extreme version of said paradox - A thinks it takes an infinite amount of time for something to happen, B thinks its' finite.

If you mean the SR twin paradox: once more, that is very different as the (t, t') sets are finite and agree with each other. It's different however with Einstein's GR twin paradox which is much more interesting and relevant as background for this topic. It would distract too much from this topic to discuss it here, but I encourage you to study it.

Similar "paradoxes" occur outside relativity, Zeno's paradox is very similar, and the answer is very similar as well. Basically one can map a finite interval of the real numbers (say 0-1) to an infinite interval (0-infinity) with a 1:1 mapping. Thus having an infinite expanse of coordinate time means nothing. Having an infinite amount of proper time does have physical significance, but the proper time here is fnite.

I always considered Zeno's paradox as a joke - it may have been serious for philosophers, but not for physicists IMHO.

 

[PeterDonis]

Einstein's GR twin paradox

What are you referring to here? If you just mean the part of the Usenet Physics FAQ entry on the twin paradox that talks about the equivalence principle, that's not a different paradox, it's the same twin paradox analyzed from another viewpoint. If, OTOH, you mean something else, can you give a reference?