# I Minkowski metric beyond the event horizon

1. Nov 1, 2017

### 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?

2. Nov 1, 2017

### 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.

3. Nov 1, 2017

### jbriggs444

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 Schwarzchild radius divided by the speed of light. For a solar mass black hole, it does not take long.

4. Nov 1, 2017

5. Nov 1, 2017

### Orodruin

Staff Emeritus
Or even a Penrose diagram?

6. Nov 1, 2017

### 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).

7. Nov 1, 2017

### jbriggs444

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

8. Nov 1, 2017

### disregardthat

Ok, and the proper distance?

9. Nov 1, 2017

### jbriggs444

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.

10. Nov 1, 2017

### 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.

11. Nov 1, 2017

### jbriggs444

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.

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.

12. Nov 1, 2017

### Staff: Mentor

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

13. Nov 1, 2017

### 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 $\infty$)?

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

14. Nov 1, 2017

### Staff: Mentor

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.

That doesn't change the answer I gave above.

15. Nov 1, 2017

### Staff: Mentor

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.

16. Nov 2, 2017

### disregardthat

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 Schwarschild metric, it can meaningfully assign a distance to particle B, even after the formation of a black hole?

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 Schwarzchild 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 $d = d(A,B)$. Let's say that the initial distance $r_0 = d(A,S)$. is large. Now, in the time interval $[t_0,t_1)$, where $t_1$ is the time at which B comes into contact with the surface, how does the distance between A and B in the Schwarzchild 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 $d_t(A,B)$ gets bigger as $t \to t_1$. In the frame of reference from A however, the corresponding distance $d'_{t}(A,B)$ is contracted inverse proportionally to the lorentz factor at each instant t. This distance $d'_t(A,B)$ can either be equal, higher or lower than the distance $d_t(A,B)$ 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 $r_0$ to S, and the mass M of S. The question becomes whether some configurations of $d,r_0$ and $M$ can flip the sign of $d_t(A,B)-d'_t(A,B)$ at some time $t_0 < t < t_1$.

EDIT: Of course, I don't expect that such computations are approximations to the GR solution. Intuitively I expect the Schwarzchild 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.

Last edited: Nov 2, 2017
17. Nov 2, 2017

### 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.

18. Nov 2, 2017

### disregardthat

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 Schwarzchild distance d(A,B) grows, shrinks or remains constant as A and B are in free fall.

19. Nov 2, 2017

### Ibix

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.

20. Nov 2, 2017

### disregardthat

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

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.