Challenging the notion of "crossing" a black hole's event horizon

[euquila]

TL;DR Summary

I'd like to explore a perspective that challenges our usual way of talking about objects "falling into" black holes. While we know we can never directly observe anything crossing an event horizon from the outside, I want to question whether anything actually "crosses" the horizon at all - even from its own reference frame.

The conventional picture​

We often describe two perspectives of infalling matter:

1. Outside observers see the object asymptotically approach but never cross the horizon, with signals becoming increasingly redshifted
2. The infalling object supposedly experiences a finite proper time to cross the horizon

The challenge​

Here's the key insight that makes me question this picture: From the perspective of an infalling object, as it approaches the horizon, it would observe the rest of the universe's time accelerating dramatically due to gravitational time dilation. The closer it gets to the horizon, the faster the external universe would appear to evolve.

This suggests a symmetry to the horizon's unreachability:

• External observers see infalling objects approach but never cross
• Infalling objects see the universe's evolution speed up without bound as they approach

In both cases, reaching the horizon seems to require completing an infinite process. Just as external observers would need to wait an infinite time to see something cross, perhaps the infalling object would "see" the entire future evolution of the universe play out before it could reach the horizon.

The question this raises​

Should we think of event horizons as truly asymptotic boundaries that nothing can actually cross?

The case of photons​

This becomes particularly interesting when considering photons directed straight toward a black hole. Since photons don't experience proper time and always travel at c, the time dilation argument might not apply in the same way. However, incoming photons would experience increasingly extreme blue-shifting as they approach the horizon. Could this lead to some kind of limiting behavior that still prevents actual horizon crossing?

I'm interested in hearing the community's thoughts on this perspective. Am I missing something fundamental, or could this be a more accurate way to think about black hole horizons?

Note: This is a conceptual discussion that might require proper mathematical treatment to fully explore. I'm particularly interested in whether there are any published papers that have explored similar ideas.

 

[PeterDonis]

Am I missing something fundamental

Yes.

This is a conceptual discussion that might require proper mathematical treatment to fully explore.

A proper mathematical treatment shows that objects do cross the horizon and your idea is fundamentally wrong.

See this Insights article:

https://www.physicsforums.com/insights/black-holes-really-exist/

Your idea is the first misconception about black hole formation that is discussed in the article.

 

[PeterDonis]

The infalling object supposedly experiences a finite proper time to cross the horizon

Not "supposedly". This has been known as a fact since the classic 1939 paper by Oppenheimer & Snyder which showed the calculation.

 

[renormalize]

Here's the key insight that makes me question this picture: From the perspective of an infalling object, as it approaches the horizon, it would observe the rest of the universe's time accelerating dramatically due to gravitational time dilation. The closer it gets to the horizon, the faster the external universe would appear to evolve.

But suppose your infalling observer is contained inside a windowless vessel and only observes a stopwatch that he holds in his hand. With no external clues, how would he notice the approach-to and passage-of the horizon?

 

[PeterDonis]

incoming photons would experience increasingly extreme blue-shifting as they approach the horizon

Not relative to an observer far away. Relative to that observer they would experience increasing redshift.

Relative to an observer "hovering" at a constant altitude just above the hole's horizon, incoming light is blueshifted, but only as long as it is above the observer; once it falls past the observer, it starts becoming redshifted.

Could this lead to some kind of limiting behavior that still prevents actual horizon crossing?

No.

 

[PeterDonis]

perhaps the infalling object would "see" the entire future evolution of the universe play out before it could reach the horizon

No, it wouldn't. The math is clear on this.

Should we think of event horizons as truly asymptotic boundaries that nothing can actually cross?

No.

 

[Dale]

From the perspective of an infalling object, as it approaches the horizon, it would observe the rest of the universe's time accelerating dramatically due to gravitational time dilation.

The math of GR is pretty clear on these results. (Clear, not easy)

What you are describing here is not the perspective of an in falling object. What you are describing is the perspective of a series of hovering observers each closer to the horizon than the last.

But an infalling observer is not hovering. They are moving with respect to the hovering observers, and moving faster with respect to each subsequent hovering observer they pass than the previous one.

The closer it gets to the horizon, the faster the external universe would appear to evolve
…
perhaps the infalling object would "see" the entire future evolution of the universe play out before it could reach the horizon

The net result is that the infalling observer does not experience an infinite blueshift. The shift is finite. They cross the horizon in a finite proper time, and having received a finite amount of signals from the outside.

 

[Ibix]

Since others have explained the various flaws in the OP's reasoning, it's maybe worth pointing out that there's a closed-form solution to the equations in this case. The process may be above B-level but the result can be studied with a scientific calculator.

From Carroll's lecture notes equations 7.47 and 7.48 for a timelike ($\epsilon=1$, $\lambda=\tau$) radial ($L=0$) path,$$\frac{dr}{d\tau}=-\sqrt{E^2-\left(1-\frac{r_S}{r}\right)}$$where $r_S=2GM$ is the Schwarzschild radius (remember Carroll uses $c=1$) and we have chosen the negative root for an infalling body. We want the infall to start at a finite radius $r=r_0$, so requiring $dr/dt=0$ at this radius gives us $$\frac{dr}{d\tau}=-\sqrt{\frac{r_S}{r_0}+\frac{r_S}{r}}$$Plug this into Wolfram Alpha or your favourite ODE solver to get$$\tau=\sqrt{\frac{r_0^3}{r_S}}\arctan \left(\frac{\sqrt{r_{0}-r}}{\sqrt{r}}\right)+\sqrt{\frac{r_0}{r_S}r(r_0-r)}$$There is an additive constant of integration, which I have set to zero to have the proper time $\tau=0$ when $r=r_0$ - i.e., we choose to start the infaller's clock as they start to fall. This is a messy expression, but clearly finite at $r=r_S$. Thus the object falls into the hole in finite proper time.

Because we chose $c=1$, if we work in light seconds the time comes out in seconds. Conveniently, $r_S$ for the Sun is almost exactly $10^{-5}\mathrm{ls}$. Dropping from an arbitrarily chosen $r_0=1\mathrm{ls}$ takes a little under 497s. You can play around with other numbers if you want.

Side note: you might be a little suspicious of this result because the work was done in Schwarzschild coordinates, which aren't valid at the horizon. You could take the limit $r\rightarrow r_S$ if you wanted, or repeat the work in some other coordinate system that is valid at the horizon, such as Kruskal-Szekeres coordinates. You will obtain the same result.

 

[euquila]

Thank you all for the thoughtful responses. I'd like to clarify my argument and address the points raised:

Regarding the article's "first misconception" about infinite coordinate time - my argument is different. I'm not concerned with coordinate time versus proper time, nor am I questioning the mathematical calculations showing finite proper time to reach the horizon. The Oppenheimer & Snyder calculations and the closed-form solution presented are mathematically sound.

Instead, I'm raising a physical paradox about what the infalling observer must actually experience during that finite proper time journey. As the observer falls inward, they would observe the external universe's evolution accelerating dramatically due to gravitational time dilation. To complete the journey to the horizon (even in finite proper time), they would need to observe the entire future evolution of the universe.

Regarding the windowless vessel thought experiment - I don't believe this resolves the paradox because the vessel itself (including its atoms, the electromagnetic forces holding it together, and even the stopwatch's mechanisms) is part of the universal causal structure. There's no way to isolate from the time dilation effect - it's fundamental to the physics of the situation.

This reminds me of the Quantum Zeno Effect, where mathematical calculations show a quantum system should transition in finite time, but continuous observation prevents the transition from occurring. The mathematics is correct, but the physical reality of observation prevents the predicted evolution. Similarly, while the math shows finite proper time to the horizon, the physical reality of experiencing the universe's accelerated evolution might prevent the journey's completion.

About the photon blue/redshift discussion - I was specifically considering a single inbound photon and its increasing energy as it falls deeper into the gravitational well, not what various observers would measure. This raises questions about whether there might be physical limits to this energy increase that prevent horizon crossing.

However, this argument admittedly treads on shaky conceptual ground. Speaking of a photon's "own perspective" is problematic since photons travel at c and don't experience proper time. Different observers would measure different energies for the same photon, making it unclear whether we can meaningfully talk about the photon's "own" energy increase independent of observers. While the blue-shifting argument is intriguing, it may not be as robust as the time dilation argument for massive objects.

My core question remains: Even if we can calculate a finite proper time mathematically, how can that journey physically complete if it requires observing an infinite future evolution during that finite time? This seems different from the coordinate time issues addressed in the article's first misconception.

A finite proper time calculation alone doesn't resolve how a journey can physically complete if it requires observing an infinite future evolution during that finite time. We cannot know with certainty whether this creates an insurmountable physical barrier to horizon crossing, even if mathematical solutions allow it. The distinction between mathematical possibility and physical realizability may be important to understanding the true nature of black holes and their horizons.

While this argument remains in the realm of thought experiment and may not lead to testable predictions, it suggests we should be cautious about definitive statements regarding horizon crossing. Much of the literature discusses matter crossing event horizons based on mathematical solutions, but there may be a gap between these solutions and physical reality. This isn't to claim the mathematics is wrong, but rather to suggest that the physical realizability of horizon crossing might be more subtle than often presented.

 

[Dale]

Even if we can calculate a finite proper time mathematically, how can that journey physically complete if it requires observing an infinite future evolution during that finite time?

As I stated in my answer, an infalling observer does not observe an infinite future evolution. They observe a finite evolution of the universe in a finite proper time:

What you are describing here is not the perspective of an in falling object. … an infalling observer is not hovering. … the infalling observer does not experience an infinite blueshift. The shift is finite. They cross the horizon in a finite proper time, and having received a finite amount of signals from the outside.

So

I'm raising a physical paradox about what the infalling observer must actually experience during that finite proper time

This is incorrect. You are making a mistake in your analysis. The paradox is not part of GR.

The mathematics is correct

The description of the paradox is inconsistent with the mathematics of the theory.

 

[renormalize]

Regarding the windowless vessel thought experiment - I don't believe this resolves the paradox because the vessel itself (including its atoms, the electromagnetic forces holding it together, and even the stopwatch's mechanisms) is part of the universal causal structure. There's no way to isolate from the time dilation effect - it's fundamental to the physics of the situation.

That's just word salad. If the infalling observer can only see his watch ticking one-second per second and he carries an accelerometer that always shows zero g's, how can he detect the approach and passage of the event horizon? (Be specific!) And how is his situation different from an astronaut in freefall while orbiting the earth, which is moving 370 km/sec relative to the cosmic background radiation? Can said astronaut perceive and measure any effect of time dilation from moving at that velocity? From the point of view of the observers, each of which is cutoff from external observations, how are the two situations measurably distinct?

 

[euquila]

I see - I may have confused two different scenarios. A sequence of hovering observers stationed at different distances from the horizon would see an increasingly accelerated universe evolution as they get closer. But an infalling observer is different because they're moving relative to these hovering observers, and with an increasing velocity between each step / hovering observer. The relative motion of the infalling observer means that it would only experience a finite blueshift and finite universe evolution and would cross the horizon in finite proper time, receiving only a finite number of signals from the outside before crossing the horizon.

 

[euquila]

That's just word salad. If the infalling observer can only see his watch ticking one-second per second and he carries an accelerometer that always shows zero g's, how can he detect the approach and passage of the event horizon? (Be specific!) And how is his situation different from an astronaut in freefall while orbiting the earth, which is moving 370 km/sec relative to the cosmic background radiation? Can said astronaut perceive and measure any effect of time dilation from moving at that velocity? From the point of view of the observers, each of which is cutoff from external observations, how are the two situations measurably distinct?

Yes, you make an excellent point. There would be no way to locally detect the approach nor passage through the horizon. I was fixated on this idea that the universe would "play out" completely before crossing. Now I see I was incorrectly applying observations to infalling observers that would only apply to hovering observers.

 

[Ibix]

Instead, I'm raising a physical paradox about what the infalling observer must actually experience during that finite proper time journey.

The paradox is in your analysis, I'm afraid. To separate out gravitational time dilation from the more general proper time versus coordinate time calculation you must assume that the observer is at constant altitude. You have an infaller, yet you are using an approach that assumes a constant altitude. It's no wonder you arrive at a contradiction when you start with one.

Fundamentally, time dilation is the ratio of the arc length along a worldline between two planes of constant coordinate time to the difference in coordinate time between the two planes. That ratio depends on the "angle" (actually the rapidity) between the worldline of interest and the worldline of an observer not moving in your coordinates, so observers in different states of motion have different time dilation even if they are at the same place. "Gravitational time dilation" is what you get when the rapidity is zero, and it goes to infinity at the event horizon because the worldline of a hovering observer is null there so you get a division by zero. But the infaller is not following that null worldline, so doesn't care about the failure of your maths for some other family of worldlines. Their proper time remains finite.

TL;DR: time dilation is not a property of where you are in space, but rather a property of where you are and how you are moving. Your failure to consider the latter point leads you into the contradiction you are encountering.

 

[euquila]

Yes, I see the infalling observer's worldline is different from the null worldlines fo the hovering observers near the horizon. Time dilation depends on both the position AND the motion and the hovering observer would need to move at c just to stay in place. But an infalling observer never needs to accelerate and their wordline doesn't become null at the horizon.

 

[Nugatory]

As the observer falls inward, they would observe the external universe's evolution accelerating dramatically due to gravitational time dilation. To complete the journey to the horizon (even in finite proper time), they would need to observe the entire future evolution of the universe.

That is not right. It’s perhaps easiest to see this with a Kruskal diagram: draw the timelike worldline of the infalling observer, then the past light cone of the event where that worldline intersects the horizon. That light cone does not include the entire future of the universe.

 

[Dale]

A sequence of hovering observers stationed at different distances from the horizon would see an increasingly accelerated universe evolution as they get closer. But an infalling observer is different because they're moving relative to these hovering observers, and with an increasing velocity between each step / hovering observer.

Yes, exactly

 

[timmdeeg]

Here's the key insight that makes me question this picture: From the perspective of an infalling object, as it approaches the horizon, it would observe the rest of the universe's time accelerating dramatically due to gravitational time dilation.

This is true for an stationary observer just outside the event horizon. For an infalling observer the Doppler shift has to be taken into account in addition. He would see the far outside redshifted with z=1 while crossing the event horizon.

P.S. see post #25

 

[Dale]

He would see the far outside redshifted with z=1 while crossing the event horizon.

Hmm, is z=1 true in general or only for an observer falling from infinity?

 

[timmdeeg]

Hmm, is z=1 true in general or only for an observer falling from infinity?

Because of the lightlike surface of the event horizon I think that doesn't matter. The observer at r_S would measure c relative to a photon stationary there.
I have done the math in 2016 and can show it if of interest, only handwritten however. But if I remember correctly z=1 is noted somewhere in the web.

 

[Ibix]

Hmm, is z=1 true in general or only for an observer falling from infinity?

Consider two observers in relative motion who pass each other at the horizon. I can't immediately see how they could receive the same frequency, primarily because if the light's wave vector is $k^a$ one observer would measure frequency $k^au_a$ and the other $k^av_a$ and I don't see why those would be the same unless $u^a=v^a$.

 

[Dale]

Consider two observers in relative motion who pass each other at the horizon. I can't immediately see how they could receive the same frequency, primarily because if the light's wave vector is $k^a$ one observer would measure frequency $k^au_a$ and the other $k^av_a$ and I don't see why those would be the same unless $u^a=v^a$.

That is my line of thinking too.

 

[pervect]

Thank you all for the thoughtful responses. I'd like to clarify my argument and address the points raised:

Regarding the article's "first misconception" about infinite coordinate time - my argument is different. I'm not concerned with coordinate time versus proper time, nor am I questioning the mathematical calculations showing finite proper time to reach the horizon. The Oppenheimer & Snyder calculations and the closed-form solution presented are mathematically sound.

Instead, I'm raising a physical paradox about what the infalling observer must actually experience during that finite proper time journey. As the observer falls inward, they would observe the external universe's evolution accelerating dramatically due to gravitational time dilation. To complete the journey to the horizon (even in finite proper time), they would need to observe the entire future evolution of the universe.

I'll draw a useful distinction here. What an infalling observer "actually" experiences are observations, such as radio signals or photographs through telescopes.

Putting those observations together in a mental model is not something I would call an "experience". It's one step beyond, I would call it a a perception (though philosphy isn't my strongest subject).

If you were to argue that the Schwarzschild coordinates are not conducive to forming a mental model or picture of what's happening, I'd agree, and I might even venture the idea that Kruskal coordinates are better and less likely to lead to incorrect conclusions (such as the one you drew about objects not passing the horizon). But there is no physical paradox here - there are just some interpretations

If you were to say that you had issues with your mental model of what happens when a clock falls through the event horizon, I'd certainly agree. If you were interested in a better model, I might be able to help. But if you say that the failure of your mental model has some physical significance, I'll disagree unless you can show me some actual physical measurment by a physical instrument and simply say that in my opinion the problem is in your mental model, not the physics. I'd also suggest learning more physics, as other have, though I can see where that suggestion, might seem dismissive and unpleasant. I don't want to be insulting or undiplomatic, but I also don't want to be so careful of peoople's feelings that I can't get my point across.

 

[PeterDonis]

Hmm, is z=1 true in general or only for an observer falling from infinity?

Only for an observer falling in from rest at infinity.

Because of the lightlike surface of the event horizon I think that doesn't matter.

Yes, it does.

The observer at r_S would measure c relative to a photon stationary there.

Which tells you nothing useful, because any timelike observer will measure that, so you can't use it to tell whether those observers have anything else in common, or not.

I have done the math in 2016 and can show it if of interest

You will need to if you really think your claim is correct. But I strongly suggest that you check your work first.

Consider two observers in relative motion who pass each other at the horizon. I can't immediately see how they could receive the same frequency, primarily because if the light's wave vector is $k^a$ one observer would measure frequency $k^au_a$ and the other $k^av_a$ and I don't see why those would be the same unless $u^a=v^a$.

Exactly. You can even do the computation explicitly in Painleve coordinates.

 

[timmdeeg]

You will need to if you really think your claim is correct. But I strongly suggest that you check your work first.

I couldn't find it, so I have to withdraw this claim.

I'm sorry for having caused confusion.

 

[Dale]

This is true for an stationary observer just outside the event horizon. For an infalling observer the Doppler shift has to be taken into account in addition.

This part of your statement is still correct, so you were not far off. It is just that the Doppler shift only gives z=1 for observers falling "from infinity". So let's make sure to not "over-correct"! You were conceptually correct.