Minkowski metric beyond the event horizon

[disregardthat]

My question is regarding how spacetime looks like beyond the event horizon of a black hole, in particular how distances behave. In the Minkowski diagram of a black hole, all paths leads to the singularity. But what is the magnitude of the distances involved here? Let's say a neutron star is slowly accumulating mass, and eventually the Schwarzschild radius overtakes the radius of the star, causing it to collapse into a black hole. Now all forces are overtaken by the curvature of spacetime, and all matter converges towards the singularity. But what distance (in the Minkowski metric) does matter on the boundary of the star have to travel to get there? Does it ever arrive?

I'm sort of imagining an endless well (looking like the graph of y = -1/x^2) down which matter is traveling, slowly getting closer to the singularity, but still infinitely far away. What does the math say about this?

 

[haushofer]

Why do you mention the minkowski metric while talking about black holes?

You need to use a coordinate patch in which you can calculate distances from the event horizon to the singularity.

 

[jbriggs444]

But what distance (in the Minkowski metric) does matter on the boundary of the star have to travel to get there?

Infalling material traverses a time-like path that ends at the singularity in a finite proper time. The time required can vary depending on the exact trajectory but is on the order of the Schwarzschild radius divided by the speed of light. For a solar mass black hole, it does not take long.

 

[PeterDonis]

In the Minkowski diagram of a black hole

Do you mean a Kruskal diagram?

https://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

If so, you should be aware that this is not a Minkowski diagram, because the metric of spacetime containing a black hole is not the Minkowski metric.

 

[Orodruin]

Do you mean a Kruskal diagram?

Or even a Penrose diagram?

 

[disregardthat]

By minkowski diagram I meant a diagram depicting the schwarzchild coordinates (wasn't aware of the proper name for it). And I did mean the schwarzchild metric.

@jbriggs444 How do we interpret proper time in this context? Isnt the interior of the black hole disconnected from all worldlines of our universe? I'm interested in the frame of reference for a particle on the boundary of the star, does its journey end in finite time? And what distance does it cover (in its own frame of reference).

 

[jbriggs444]

By minkowski diagram I meant a diagram depicting the schwarzchild coordinates (wasn't aware of the proper name for it). And I did mean the schwarzchild metric.

@jbriggs444 How do we interpret proper time in this context? Isnt the interior of the black hole disconnected from all worldlines of our universe? I'm interested in the frame of reference for a particle on the boundary of the star, does its journey end in finite time?

"Proper time" is the accumulated elapsed time in the rest frame of the infalling particle.

 

[disregardthat]

Ok, and the proper distance?

 

[jbriggs444]

Ok, and the proper distance?

Zero, of course. In its rest frame it is, by definition, at rest. Unmoving.

In addition, it can be freely falling the whole way -- proper acceleration of zero.

 

[disregardthat]

That makes sense. Is there a sensible way (in general relativity) in which a particle in free fall (inside a black hole) traverses space, covering distance in some sense?

Analogous to the distance traversed by an object in free fall towards the earth, in the frame of reference of an object stationary with respect to the surface.

 

[jbriggs444]

That makes sense. Is there a sensible way (in general relativity) in which a particle in free fall (inside a black hole) traverses space, covering distance in some sense?

Sure. Set up a coordinate system and count how much distance is traversed in that coordinate system. But such a distance will be no more meaningful than the zero result you already have.

Edit: @PeterDonis seems to be making a stronger answer in the negative. He knows this stuff better than I.

Analogous to the distance traversed by an object in free fall towards the earth, in the frame of reference of an object stationary with respect to the surface.

One can pick a coordinate system in which the Earth's surface is stationary and obtain a distance measurement. I know of no such physically meaningful stationary reference in the case of a black hole.

The singularity is not a good reference point. The event horizon is not stationary in any locally inertial frame.

 

[PeterDonis]

Analogous to the distance traversed by an object in free fall towards the earth, in the frame of reference of an object stationary with respect to the surface.

As @jbriggs444 implies, there is no such frame in the interior of a black hole.

 

[disregardthat]

@jbriggs444 How about in these terms: Suppose A and B are two initially stationary particles on a line from the center of the star to the boundary, where A is on the boundary, and where A measures an initial Schwarzschild distance d to B. How does the formation of a black hole affect this distance (in the frame of reference of A)? Does it stay constant, or does it quickly approach 0 (or [itex]\infty[/itex])?

EDIT: I'm talking about the time interval [itex][t_0,t)[/itex], where t is the proper time at which A has converged at the singularity.

 

[PeterDonis]

Suppose A and B are two initially stationary particles on a line from the center of the star to the boundary, where A is on the boundary, and where A measures an initial Schwarzschild distance d to B. How does the formation of a black hole affect this distance (in the frame of reference of A)?

During the formation of the black hole, the particle at B falls into the particle at A; so the distance between them goes to zero for the humdrum reason that they fall together.

If you are thinking that particle B can somehow stay at the "boundary" while the black hole forms, that's impossible. Nothing can "hover" at the boundary (event horizon) of a black hole. The horizon is an outgoing lightlike surface: radially outgoing light just manages to stay at the same radial coordinate. No ordinary particle can move at the speed of light, so all ordinary particles fall into the singularity.

I'm talking about the time interval ##[t_0,t)##, where t is the proper time at which A has converged at the singularity.

That doesn't change the answer I gave above.

 

[PeterDonis]

in the frame of reference of A

I'll comment on this separately: there is no "frame of reference of A" that has the properties you are assuming (such as having a "space" that doesn't change with "time"). However, you can choose coordinates in which the process of the star collapsing to the black hole is homogeneous, i.e., in these coordinates the density of the object remains uniform throughout as it collapses (we're assuming that the star started out with uniform density, which is not a very practical assumption, but it's a useful idealization). These coordinates are the ones I was implicitly using in my previous post. The first investigation of this type of model (which is also highly idealized as being perfectly spherically symmetric) was done by Oppenheimer and Snyder in 1939.

 

[disregardthat]

I'll comment on this separately: there is no "frame of reference of A" that has the properties you are assuming (such as having a "space" that doesn't change with "time").

I'm not sure what you mean by the "properties I'm assuming". Do you agree that it makes sense to speak of the frame of reference of A, and that in the Schwarzschild metric, it can meaningfully assign a distance to particle B, even after the formation of a black hole?

During the formation of the black hole, the particle at B falls into the particle at A; so the distance between them goes to zero for the humdrum reason that they fall together.

If you are thinking that particle B can somehow stay at the "boundary" while the black hole forms, that's impossible. Nothing can "hover" at the boundary (event horizon) of a black hole. The horizon is an outgoing lightlike surface: radially outgoing light just manages to stay at the same radial coordinate. No ordinary particle can move at the speed of light, so all ordinary particles fall into the singularity.

This is not what I had in mind. I am (naively) attempting to understand how the singularity deforms spacetime by considering a pair of particles falling into the singularity. In my mind it is conceivable that the immense curvature affects Schwarzschild distance in slightly unintuitive ways.

Again, in analogy with a non-black hole gravitational body (and perhaps this is the question I really should have asked): Let's say A and B are two particles on a line from the center of a massive object. A and B are initially at rest with respect to the center S of this object, and with a separation of [itex]d = d(A,B)[/itex]. Let's say that the initial distance [itex]r_0 = d(A,S)[/itex]. is large. Now, in the time interval [itex][t_0,t_1)[/itex], where [itex]t_1[/itex] is the time at which B comes into contact with the surface, how does the distance between A and B in the Schwarzschild metric evolve?

My motivation is from the perspective of special relativity:

In the frame of reference of S, there is at all times a slightly higher gravitational force applied to B than to A, and thus the acceleration of B is higher than the acceleration of A towards S. So [itex]d_t(A,B)[/itex] gets bigger as [itex]t \to t_1[/itex]. In the frame of reference from A however, the corresponding distance [itex]d'_{t}(A,B)[/itex] is contracted inverse proportionally to the lorentz factor at each instant t. This distance [itex]d'_t(A,B)[/itex] can either be equal, higher or lower than the distance [itex]d_t(A,B)[/itex] measured by S at any time. I have been unable to compute this though. The result would depend on the the time t, the initial distance d between A and B, and the initial distance [itex]r_0[/itex] to S, and the mass M of S. The question becomes whether some configurations of [itex]d,r_0[/itex] and [itex]M[/itex] can flip the sign of [itex]d_t(A,B)-d'_t(A,B)[/itex] at some time [itex]t_0 < t < t_1[/itex].

EDIT: Of course, I don't expect that such computations are approximations to the GR solution. Intuitively I expect the Schwarzschild distance to remain constant, since both A and B are in free fall, and approximately at rest with each other at time t_0 for r_0 large.

 

[Ibix]

I don't think your line of reasoning works inside a black hole. As I understand it there is no obvious way to describe what you mean by "now", so asking "how far apart are A and B now?" doesn't have a well-defined answer.

It's worth noting that the interval between the event horizon and the singularity is time-like, not space-like. In other words, there isn't a distance to the singularity, there's a time.

Peter's point about the frame of reference is that spacetime near a gravitating mass only looks like SR over small volume. The definition of a small volume is one where you do not notice any differential acceleration, because that's an effect of curved spacetime. If you notice it, an SR frame isn't a useful approximation.

 

[disregardthat]

I don't think your line of reasoning works inside a black hole. As I understand it there is no obvious way to describe what you mean by "now", so asking "how far apart are A and B now?" doesn't have a well-defined answer.

Yes, I get that, and I only brought up the special relativity example as an analogy and motivation. In this context (even non-black hole gravitational bodies), I suspect that SR won't give a meaningful approximation. On the other hand, while simultaneity poses a problem, I do expect that the set of distances from A to B as time passes in the reference frame of A to be well-defined. At least it should be possible to determine whether the Schwarzschild distance d(A,B) grows, shrinks or remains constant as A and B are in free fall.

 

[Ibix]

On the other hand, while simultaneity poses a problem, I do expect that the set of distances from A to B as time passes in the reference frame of A is well-defined.

No. Because you can't define an SR reference frame covering A and B.

What you are trying to ask is: "how far away is B when A's clock reads 0? What about when it reads 1? 2? 3?" Etcetera. But inside the event horizon there is no unambiguous way to define what point on B's worldline is "at the same time as" a given event on A's worldline. So, while you can certainly calculate the interval between any event on A's worldline and any event on B's worldline, no pair is "at the same time" in any meaningful sense. So the answer is pretty much "the length of a piece of string". And you can choose which piece of string you use - there is none picked out by the physics.

 

[disregardthat]

No. Because you can't define an SR reference frame covering A and B.

I'm not attempting to understand black holes from an SR perspective.

What you are trying to ask is: "how far away is B when A's clock reads 0? What about when it reads 1? 2? 3?" Etcetera. But inside the event horizon there is no unambiguous way to define what point on B's worldline is "at the same time as" a given event on A's worldline. So, while you can certainly calculate the interval between any event on A's worldline and any event on B's worldline, no pair is "at the same time" in any meaningful sense. So the answer is pretty much "the length of a piece of string". And you can choose which piece of string you use - there is none picked out by the physics.

Assuming a non-black hole gravitational body: Let me pose the question as follows: A can meaningfully measure a set of distances to B in its own frame of reference, by regularly emitting and receiving photons being reflected from B, and then measure the time it took. As A and B are falling towards S, does this kind of "clock" tick faster or slower for A as time progresses (maybe slower initially, and then faster, depending on the mass of S)? I can't see any ambiguity in this setup caused by issues with simultaneity. I'm interested in the same question extended to the situation of a black hole, but this is really a two-part question now.

 

[Ibix]

Assuming a non-black hole gravitational body: Let me pose the question as follows: A can meaningfully measure a set of distances to B in its own frame of reference, by regularly emitting and receiving photons being reflected from B, and then measure the time it took. As A and B are falling towards S, does this kind of "clock" tick faster or slower for A as time progresses? I can't see any ambiguity in this setup caused by issues with simultaneity.

I haven't done the maths, but I expect the repeats would be longer and longer, as they are in the Newtonian regime. The question is: why? Is it because the distance between the ships is increasing? Or is it because the spacetime geometry is changing between the ships as they fall in?

Outside the event horizon you can answer the question by finding some observers for whom local spacetime geometry is not changing. These are hovering observers, and we can use their notion of "at the same time" to define a notion of "space" and separate out the time change due to distance-in-space.

Inside the event horizon, however, there can be no hovering observers. In fact, there are no observers for whom spacetime is not changing around them. So there is no one to tell you "x% of the time delay is due to distance, y% due to time dilation/spacetime curvature" (formally: there are no time-like Killing vector fields inside the event horizon). You are free to attribute all of the change in pulse return time to the differing spacetime geometry between A and B, or all of it to changing distance, or any mix thereof.

That's my understanding at least. If Peter or pervect tells you different, believe them...

 

[disregardthat]

I haven't done the maths, but I expect the repeats would be longer and longer, as they are in the Newtonian regime. The question is: why? Is it because the distance between the ships is increasing? Or is it because the spacetime geometry is changing between the ships as they fall in?

What do you mean by a Newtonian regime? I am not sure one can approximate the answer to that with Newtonian mechanics. Especially if the mass of S is large.

Outside the event horizon you can answer the question by finding some observers for whom local spacetime geometry is not changing. These are hovering observers, and we can use their notion of "at the same time" to define a notion of "space" and separate out the time change due to distance-in-space.

The question as presently stated does not require a third frame of reference, I believe. It refers exclusively to a system of two particles and a gravitational body in the frame of reference of A.

 

[PeterDonis]

Do you agree that it makes sense to speak of the frame of reference of A, and that in the Schwarzschild metric, it can meaningfully assign a distance to particle B, even after the formation of a black hole?

No. This is a case where you really need to look at the math; your intuitions are leading you astray.

My motivation is from the perspective of special relativity:

I'm confused; you also said this:

I'm not attempting to understand black holes from an SR perspective.

The latter is the correct answer, since you can't understand black holes from an SR perspective. But that means your previous attempt at doing that won't work.

In the frame of reference of S, there is at all times a slightly higher gravitational force applied to B than to A

This won't work, because there isn't a well-defined "frame of reference of S", and there isn't a well-defined "gravitational force", at or inside the horizon of a black hole.

you can't define an SR reference frame covering A and B.

It's worse than that. You can't even define a "reference frame" at all that covers A and B. The best you can do is find a global coordinate chart that covers both their worldlines. But you won't be able to use that chart to construct a "reference frame" that allows you to define "distances" the way @disregardthat is trying to do.

Further comments in a follow-up post.

 

[disregardthat]

@PeterDonis Like I said, it was my motivation for my question, and did not involve black holes at all (S is not a black hole singularity in that example). Since I do not know general relativity, I merely attempted to look at what my (limited) knowledge of special relativity could tell me in such a situation, which was not much. I am eager to see your comment on the situation where there is not reference to any reference frame other than A (and if it even then can not be meaningfully be applied to the situation of a black hole, then I would like to know the answer when S is simply a non-black hole gravitational body).

 

[Ibix]

What do you mean by a Newtonian regime? I am not sure one can approximate the answer to that with Newtonian mechanics. Especially if the mass of S is large.

Far away from a mass and if A and B are traveling slowly then general relativity looks very like Newtonian gravity. Remember that the GR correction for Mercury's orbital precession is only 43 seconds of arc per century. You can simply disregard all of the complexity and use Newton to work out what happens if you drop two objects into the Sun on any timescale less than a decade. In this case, normal intuition applies and the distance between A and B grows and the pulse return time grows. All of the complexity we've been discussing applies, strictly speaking, but the error from ignoring it is tiny. You have a slight question mark about how light behaves, but as long as the velocities of A and B relative to the Sun are low, any plausible answer doesn't actually make a qualitative difference to the outcome.

Of course, if you have a denser or more massive body, you can no longer disregard the worries about changing curvature. But I suspect that you will find that the pulse return time continues to grow.

The question as presently stated does not require a third frame of reference, I believe. It refers exclusively to a system of two particles and a gravitational body in the frame of reference of A.

It depends what you want to know. Does the pulse return time grow? You can answer that one in a coordinate independent way, yes. As I said I expect it continues to grow, although I have not confirmed that mathematically.

Where you need to introduce coordinates (not a reference frame! That's strictly an SR concept, as Peter points out) is if you want to ask "is the pulse return time changing because the distance between A and B is changing?" And you can't really answer that from A's point of view because the spacetime geometry around him is changing so the concept of "distance as defined by A" isn't definable. So to answer this question - which was where you started, I believe - you do need to introduce a coordinate system that is static. And you can do that outside the event horizon by pegging the coordinates to hovering observers. But there cannot be hovering observers inside the event horizon.

 

[PeterDonis]

(S is not a black hole singularity in that example).

Then I'm confused about what scenario you are asking about. I understood you to be asking about a star that collapses to a black hole.

 

[PeterDonis]

That's my understanding at least.

It's mine as well. This post is a good summary of the issues involved.

 

[PeterDonis]

A can meaningfully measure a set of distances to B in its own frame of reference, by regularly emitting and receiving photons being reflected from B, and then measure the time it took.

This works fine as a definition of "distance", but it requires that it be possible for light to make repeated round trips between A and B. If one of the two is inside the horizon of a black hole, and the other is outside, this is not possible.

 

[Ibix]

This works fine as a definition of "distance", but it requires that it be possible for light to make repeated round trips between A and B. If one of the two is inside the horizon of a black hole, and the other is outside, this is not possible.

As long as the second one free-falls in after the first one it'll work, I think. Although the fact that it just fails completely if the second one aborts its fall after the first has crossed the horizon is a rather dramatic illustration of the point.

But how do you mean that it works fine as a measure of distance? Are you just naively multiplying the return time by c/2, as per Dolby and Gull's radar time methodology? Isn't that pretty much a reasonable-but-completely-arbitrary decision?

 

[PAllen]

This works fine as a definition of "distance", but it requires that it be possible for light to make repeated round trips between A and B. If one of the two is inside the horizon of a black hole, and the other is outside, this is not possible.

I thought A and B were posed as free falling. Then they can continue exchanging signals until A ‘dies’. If they started out close enough, these radar coordinates will be near Minkowski until both are well inside the horizon. Signal intervals between them will grow, and a geometric argument can be given that this growth is primarily a distance growth as long as they remain reasonably close. This argument is that if they had only radial separation when both were just outside the horizon, then when both are inside, their separation along the extra spacelike killing direction (besides the two angular killing directions which hold constant for them) will be increasing (ultimately without bound). Since this distance is measured along a killing symmetry, I think it is reasonable to attach some physical significance to it. This is also the direction in which anybody will be stretched, while being compressed in the angular directions.

 

[disregardthat]

Far away from a mass and if A and B are traveling slowly then general relativity looks very like Newtonian gravity. Remember that the GR correction for Mercury's orbital precession is only 43 seconds of arc per century. You can simply disregard all of the complexity and use Newton to work out what happens if you drop two objects into the Sun on any timescale less than a decade. In this case, normal intuition applies and the distance between A and B grows and the pulse return time grows. All of the complexity we've been discussing applies, strictly speaking, but the error from ignoring it is tiny. You have a slight question mark about how light behaves, but as long as the velocities of A and B relative to the Sun are low, any plausible answer doesn't actually make a qualitative difference to the outcome.

The velocities are low, but so is the velocity differential. The difference in acceleration for two objects A and B with a tiny separation d is small, and I don't quite see why it should be sufficiently larger than any resulting relativistic effects.

Where you need to introduce coordinates (not a reference frame! That's strictly an SR concept, as Peter points out) is if you want to ask "is the pulse return time changing because the distance between A and B is changing?" And you can't really answer that from A's point of view because the spacetime geometry around him is changing so the concept of "distance as defined by A" isn't definable. So to answer this question - which was where you started, I believe - you do need to introduce a coordinate system that is static. And you can do that outside the event horizon by pegging the coordinates to hovering observers. But there cannot be hovering observers inside the event horizon.

I see, so I gather that "frame of reference" in GR is basically a concept which is only meaningful for a point with approximately flat spacetime. Is there no meaningful way to speak of a "frame of reference" for a particle A in curved space?

Then I'm confused about what scenario you are asking about. I understood you to be asking about a star that collapses to a black hole.

That second question was part of my discussion with (EDIT: you!). I explicitly pointed out a non-black hole point mass for that scenario, and I feel like there should be no cause for confusion in my post on this particular issue.

This works fine as a definition of "distance", but it requires that it be possible for light to make repeated round trips between A and B. If one of the two is inside the horizon of a black hole, and the other is outside, this is not possible.

Will this definition of distance work if both A and B are inside the event horizon, and how will it behave? I'm also interested in the situation where A and B are falling towards a non-black hole.

 

[PeterDonis]

how do you mean that it works fine as a measure of distance?

I mean you can define "distance" this way and it will give you consistent results. The word "distance" is an ordinary language word and does not have a single precise meaning; the round-trip light travel time method is one way of giving it one, but of course not the only way (and the different possible ways are not all consistent with each other).

I thought A and B were posed as free falling.

I'm no longer sure exactly what scenario the OP is asking about. In the case that A and B are both free-falling, yes, they can continue to exchange light signals until one hits the singularity, provided they are not separated too far initially (if their initial separation is large enough, A will hit the singularity before even one round-trip light signal).

This argument is that if they had only radial separation when both were just outside the horizon, then when both are inside, their separation along the extra spacelike killing direction (besides the two angular killing directions which hold constant for them) will be increasing (ultimately without bound).

I'm not sure about the "ultimately without bound". It seems to me that their separation along the other spacelike Killing direction should approach a finite limit. More precisely, if for any radial coordinate outside the horizon, the events at which the two objects cross that radial coordinate are separated by a finite Killing time (since the fourth Killing field is timelike outside the horizon), then the limit of their separation along the fourth Killing field inside the horizon, as both approach the singularity, will be finite (although of course it can be made arbitrarily large by allowing the time separation at which the objects cross a particular radial coordinate outside the horizon to be arbitrarily large).

 

[PAllen]

I'm not sure about the "ultimately without bound". It seems to me that their separation along the other spacelike Killing direction should approach a finite limit. More precisely, if for any radial coordinate outside the horizon, the events at which the two objects cross that radial coordinate are separated by a finite Killing time (since the fourth Killing field is timelike outside the horizon), then the limit of their separation along the fourth Killing field inside the horizon, as both approach the singularity, will be finite (although of course it can be made arbitrarily large by allowing the time separation at which the objects cross a particular radial coordinate outside the horizon to be arbitrarily large).

I think you’re right about this. The stretch would still be finite when one hits the singularity.