B How do Black Holes Grow? A Far-Away Observer's Perspective

[Ibix]

Is the coordinate speed of light close to the black hole less than c?

For a radial null path the Schwarzschild metric tells us that $0=\left(1-\frac{R_S}r\right)dt^2-\left(1-\frac{R_S}r\right)^{-1}dr^2$, which means that in these coordinates the coordinate speed of light is $\frac{dr}{dt}=\pm\left(1-\frac{R_S}r\right)$ (minus sign for inward-going rays). This tends to zero as $r$ approaches $R_S$.

 

[Rene Dekker]

Not only is the coordinate speed of light less than c, it is also anisotropic in Schwarzschild coordinates.

Yes, I see it myself now. It is $(1 - \frac{r_s}{r}) c$ in the radial direction, correct?
I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.

 

[Ibix]

It's possible to write the Schwarzschild metric in isotropic coordinates (wiki lists them in the Alternative Coordinates section). It's just not always convenient, and they don't cross the horizon either.

 

[DrGreg]

I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.

In Kruskal–Szekeres coordinates the radial speed of light is $c$ everywhere, i.e. inside, outside and on the event horizon (though not the tangential speed).

 

[PeroK]

Yes, I see it myself now. It is $(1 - \frac{r_s}{r}) c$ in the radial direction, correct?
I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.

You are still fundamentally confused by the relationship between a geometry, which is the same geometry in all coordinate systems; and coordinates.

A black hole is a black hole regardless of your coordinate system.

Your other mistake is to elevate coordinate dependent things to the statis of universal laws that must apply in all coordinate systems.

 

[Dale]

Yes, I see it myself now. It is $(1 - \frac{r_s}{r}) c$ in the radial direction, correct?

Yes, that is correct.

I would have thought that it would be more practical to always use a coordinate system where the coordinate speed of light is c everywhere. But I guess in such a system it is not possible to describe things like a black hole.

There are fundamental limitations to the types of spacetimes for which such a coordinate system is possible. And even in well-behaved spacetimes often such coordinates cannot cover the entire spacetime.

 

[Rene Dekker]

You are still fundamentally confused by the relationship between a geometry, which is the same geometry in all coordinate systems; and coordinates.

What I am definitely confused by, is the habit of changing coordinate system if they don't suit your needs. In my mind, coordinates are related to measurements, and you can only (and must) change a coordinate system, if it does not correspond to the measurements you are making.

I can fully understand that different coordinate systems can describe the same reality in different ways, like cartesian and polar coordinates. But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.

Like in this discussion, when one says: "there is no event on the event horizon that corresponds with 'now' for the remote observer". And then somebody else says: "But we can just choose another coordinate system, and then there will be a 'now' on the event horizon". That is something I don't yet grasp. The event horizon is either part of reality or it isn't. If it is part of reality, there should be events that happen at the same time as events on Earth. I cannot understand that you can simply whip that into or out of existence by changing coordinate systems.
And yes, I understand that simultaneity is a much more fluid concept in GR than it is in Newtonian physics and Special Relativity. But my mind cannot follow why you can just change that by choosing another coordinate system, without making any physical changes in reality, or taking another observer's viewpoint.

But those are just my personal struggles with the theory, and I love to learn more about it. It was a great discussion, and I learned a lot. I thank everybody for the insightful comments.

 

[PeroK]

I can fully understand that different coordinate systems can describe the same reality in different ways, like cartesian and polar coordinates. But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.

Yes, but this is modern physics (relativity) and we have not separate time and space, but a single spacetime continuum! Spacetime distances are invariant. But, unlike classical physics, spatial distances and time intervals are not invariant. They are coordinate dependent.

This is why it's important to learn the basics before trying to study black holes!

 

[Ibix]

Like in this discussion, when one says: "there is no event on the event horizon that corresponds with 'now' for the remote observer". And then somebody else says: "But we can just choose another coordinate system, and then there will be a 'now' on the event horizon". That is something I don't yet grasp.

The point is that "now, over there" doesn't have any possible assumption-free definition. You need to decide what you mean by "now, over there", which you can do in several different ways. And once you've decided what you mean you can set up a measurement process that reflects that definition - or simply define a mathematical transform between convenient physical measurements and convenient coordinates.

The teaching of SR based on inertial frames obscures this somewhat, because people come away with the notion that there's "my frame's time" and "your frame's time" and there's a Right Way to define these things. There isn't. Draw a Minkowski diagram and draw an arbitrary curve on it, with the only restriction being that the gradient must be everywhere strictly less than one. That's a valid definition of "now" for any event on that line, albeit a pointlessly complicated one. Note that this process allows you to draw the straight line simultaneity plane of a moving Einstein frame and claim it for your own. That's fine - nastier maths, but fine.

GR, with its curved spacetime, gives you a reason to use more complicated definitions than Einstein's simple "bounce a light pulse off it guys" approach. And sometimes you absolutely have to do it because "bounce a light pulse off it" can't work to define "now" at an event horizon because the pulse never comes back.

 

[PeterDonis]

In my mind, coordinates are related to measurements

And that is something you will have to unlearn in GR, because it is simply not possible to always find coordinate charts in a curved spacetime that work the way your intuitions say they should work.

The event horizon is either part of reality or it isn't.

It is.

If it is part of reality, there should be events that happen at the same time as events on Earth.

This is the part you will need to unlearn. I understand that your intuitions are telling you this should be true. That means you need to retrain your intuitions.

One avenue towards retraining is to substitute the phrase "spacelike separated" for "at the same time". The advantage of this is that "spacelike separated" is an invariant--whether or not two events are spacelike separated from each other does not depend on your choice of coordinates. So if we rephrase your statement here to "there are events on the horizon that are spacelike separated from events on Earth", that statement is true without qualification.

I cannot understand that you can simply whip that into or out of existence by changing coordinate systems.

That's not what you're doing. The horizon is there regardless of what coordinates you choose; it's part of reality. But "simultaneity" is not part of reality. It's a human convention.

And yes, I understand that simultaneity is a much more fluid concept in GR than it is in Newtonian physics

Yes.

and Special Relativity.

No. SR has the same "fluidity" about simultaneity that GR does. Unfortunately many treatments of SR do not properly emphasize this fact.

my mind cannot follow why you can just change that by choosing another coordinate system, without making any physical changes in reality, or taking another observer's viewpoint.

Because simultaneity is not a "real thing"; it's a human convention. Changing coordinates just means changing the convention.

 

[Nugatory]

If it is part of reality, there should be events that happen at the same time as events on Earth. I cannot understand that you can simply whip that into or out of existence by changing coordinate systems.

The counterintuitive thing here is buried in that phrase "at the same time". "At the same time" is just an informal way of saying "has the same time coordinate", so its meaning depends on the way that we assign time coordinates to events.

This is true even in a perfectly boring flat spacetime, no gravity, no curvature: Google for "relativity of simultaneity" to see how it works.

Or for a quick handwaving explanation: Say a bomb at rest relative to you explodes one light-second away from you, and the light from the explosion reaches your eyes when your wristwatch reads 12:00. It is absolutely natural (to the point that it would be perverse to suggest otherwise) to say that the explosion happened at the same time that your wristwatch read 11:59:59 and then the light took one second to get you, right? That's basically how we assign time coordinates to events not right under our nose. Well, someone moving relative to you, looking at your wristwatch through a telescope and using the same "time I see it, minus light travel time" logic will find that events "explosion" and "wristwatch read 11:59:59" did not happen at the same time.

 

[Dale]

What I am definitely confused by, is the habit of changing coordinate system if they don't suit your needs. In my mind, coordinates are related to measurements

You have already been taught that this is incorrect. Coordinates are basically just arbitrary labels, and they do not have any intrinsic relationship to measurements.

and you can only (and must) change a coordinate system, if it does not correspond to the measurements you are making.

You can adopt any smooth and invertible mapping as your coordinate system at any time for any reason or no reason. Usually we pick a coordinate system for convenience. Usually convenience is determined by how well it simplifies the math.

But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.

Indeed, the measurements are invariant and any coordinate chart that covers a region will agree on the outcome of any measurements in that region. The problem is that the Schwarzschild coordinate chart does not cover the event horizon. So it cannot be used to make any statements about the horizon.

The event horizon is either part of reality or it isn't.

It is part of reality, but coordinates are not. So it is not a reflection on reality if a given coordinate chart does not cover the event horizon.

If it is part of reality, there should be events that happen at the same time as events on Earth

There have been studies that show that the single most difficult concept in learning relativity is understanding that the concept of “at the same time” is a human concept. It is not part of reality. The idea of “now” is part of the coordinates (simultaneity) so it is not a part of nature. This is a hard thing to accept, but it is well founded both experimentally and theoretically.

I don't know that we can make this easier, but we can teach correct principles and sympathize with your struggle in learning this difficult concept

 

[Vanadium 50]

What I am definitely confused by, is the habit of changing coordinate system if they don't suit your need

When you visit a new city do you use their addresses or what they would be in your home town? (Let's see...12 North Avenue would be 1,005,872 South Street and....)

 

[Demystifier]

Sure. You can use the simultaneity convention of any of the following:

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

https://en.m.wikipedia.org/wiki/Lemaître_coordinates

https://en.m.wikipedia.org/wiki/Eddington–Finkelstein_coordinates

https://en.m.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

Or any other simultaneity convention you like. Your simultaneity convention is arbitrary.

I see your point. However, I would like to point out that these simultaneity conventions cannot be interpreted as synchronization conventions. A synchronization convention is an experimental procedure, while these simultaneity conventions cannot be used to experimentally synchronize two clocks, one outside and the other inside the black hole. It does not imply that these simultaneity conventions are wrong, but to avoid confusion (including my own) it is important to keep in mind the difference between simultaneity conventions and synchronization conventions.

 

[Ibix]

I see your point. However, I would like to point out that these simultaneity conventions cannot be interpreted as synchronization conventions. A synchronization convention is an experimental procedure, while these simultaneity conventions cannot be used to experimentally synchronize two clocks, one outside and the other inside the black hole. It does not imply that these simultaneity conventions are wrong, but to avoid confusion (including my own) it is important to keep in mind the difference between simultaneity conventions and synchronization conventions.

They're distinct from synchronization conventions available to permanently external observers, yes. Free-falling observers (see Gullstrand-Painleve coordinates) may well synchronise clocks using these conventions, since they cross the horizon themselves. They can't complete the synchronisation process while the clocks are on opposite sides of the horizon, but they can cross the horizon before, during, or after it.

 

[vanhees71]

What I am definitely confused by, is the habit of changing coordinate system if they don't suit your needs. In my mind, coordinates are related to measurements, and you can only (and must) change a coordinate system, if it does not correspond to the measurements you are making.

Coordinates are not related to measurements. They are just arbitrary parameters mapping the spacetime manifold locally to open sets of $\mathbb{R}^4$. From a mathematical point of view the "general covariance" of relativity, i.e., the local diffeomorphism invariance, is a gauge symmetry and in this sense coordinates are gauge-dependent mathematical entities and thus cannot be directly observables.

To cover a spacetime completely you usually need more than one such map ("coordinate chart"), i.e., an atlas in the sense of differentiable manifolds. As with an atlas consisting of maps of the Earth there are many possible atlasses of a spacetime, which itself however is independent of the choice of a specific atlas.

I can fully understand that different coordinate systems can describe the same reality in different ways, like cartesian and polar coordinates. But the measurements (distances, proper time, etc) that you calculate with those coordinates should be the same. The should reflect reality.

Exactly! I.e., only (local) gauge-invariant quantities (tensors) directly describe observables. It's as in electrodynamics: The four-potential is not directly an observable, because it's gauge dependent. You can calculate observables from it, i.e., the electromagnetic field, which is a tensor, which is independent of the choice of gauge.

 

[Dale]

I see your point. However, I would like to point out that these simultaneity conventions cannot be interpreted as synchronization conventions. A synchronization convention is an experimental procedure

Oh, interesting. I have never seen this distinction before. It is a reasonable distinction, but it is novel to me.

these simultaneity conventions cannot be used to experimentally synchronize two clocks, one outside and the other inside the black hole

I would need to see a proof that this is true. Of course, you cannot use Einstein's synchronization procedure to do so, but to say that there exists no possible experimental synchronization procedure seems like quite a stretch.

I would think that it suffices to have one master clock broadcast a time signal and other clocks adjust their clocks accordingly by solving the null geodesic equation in the given coordinates. I.e. Einstein synchronization is based on knowing that the speed of light is c in inertial coordinates while in other coordinates it is not necessarily c but is nevertheless known. But I don't have a proof of the generality of such a procedure, nor a counter-proof.

 

[Demystifier]

I would think that it suffices to have one master clock broadcast a time signal and other clocks adjust their clocks accordingly by solving the null geodesic equation in the given coordinates.

It seems that you are right. Originally I thought that synchronization should involve an exchange of information between two clocks, but there is actually no need for that, one way information should be enough. If the master clock is outside the black hole, it should work. More precisely, it would work for physical black hole created in a gravitational collapse, but it would not work for a maximal Kruskal extension containing both the black and the white hole.