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Black hole mass can't be visualized as a property of the black hole interior

I still don't get why this [public] NASA video with sound that I posted to YouTube has acquired 1.8 million views – and is therefore my most viewed video although a few others with a higher added value of mine came close – but lots of people simply seem to be terrified when they are told that a black hole sounds like a vacuum cleaner that will devour them LOL. Most of the 2,200+ comments seem to be compatible with this explanation.

Jitter – and Erwin – have brought a confusing point about black holes that I have actually encountered many times in the past.

Jitter: When a large percentage of the mass of the interior of a BH is turned into gravitons/waves during a collission, then wont future LIGO detectors potentially be able to view the interior?

[...] Thanks Lubos. I still don't understand how scientists can say "nothing can escape a BH" yet scientists claim that LIGO observed lots of mass escaping from two BHs. Sorry but it just does not make sense to me. Can anyone help me understand this? At least as much as I understand the need for Eigenstates to explain attraction. ;-)

Erwin: Hi Luboš, I think Jitter asks because of "...the merger of two black holes with masses of 35 times and 30 times the mass of the Sun (in the source frame), resulting in a post-merger black hole of 62 solar masses..."

[...] I think it boils down to the question: what do we mean by "the mass of a BH", "is" this mass inside the event horizon?
The dots represent my (so far mostly unsuccessful) efforts to convey the basic point that no observations done in the asymptotic region i.e. outside the black hole(s) – and the LIGO experiment is a top example of that – simply cannot see inside the black hole. The black hole interior is defined as the region where it's impossible to look, even in principle, and black hole itself is defined as an object that creates such a (non-vanishing) inaccessible region of the spacetime.

I didn't give up so I tried to post another answer to Erwin's last question. Is the mass a "stuff" that may be counted by counting something in the black hole interior?

The answer is No.

Dear Erwin - and Jitter,

you might try to imagine that the black hole mass \(M\) has the value it has because of "something inside". But in general relativity, you can't ever get an accurate method to calculate the precise mass by "integrating some quantity in the interior" of the black hole – or a neutron star, for that matter, or any other celestial object.

Instead, the mass of the black hole – but also the mass of the neutron star, a planet, or anything else - may only be measured accurately when you're outside, and it may be defined e.g. as the ADM mass. The ADM mass is defined as the numerator \(M\) in \(M/r\) which determines the deviation of the \(g_{00}\) component of the metric tensor \(g_{\mu\nu}\) at infinity. At (\(r\) goes to) infinity,\[

g_{00} = 1 - \frac{2 GM}{r} + \text{small corrections}.

\] So the value of the mass is given by how much the object - black hole, neutron star, or anything else – curves the spacetime outside the object. And it becomes easiest to isolate the mass \(M\) by looking how much the object curves the *very distant* spacetime of large \(r\) i.e. \(r\gg 2GM\).

In particular, the black hole mass can't be calculated as any integral of mass/energy density such as \(T_{00}\). In fact, the energy density \(T_{00}\) is exactly zero – the whole stress-energy tensor is zero in the whole black hole spacetime, the black hole spacetime is Ricci-flat so it solves the *vacuum* Einstein's equations (without a source) \(R_{\mu\nu}=0\).

And if you wanted to say that the whole mass \(M\) is obtained from the singularity, you won't be able to get a good calculation because the integral over the singularity would be singular. Moreover, the space and time are really interchanged inside the black hole (the signs of the components \(g_{rr}\) and \(g_{tt}\) get inverted for \(r \lt 2GM\)) so the exercise is in no way equivalent to a simple 3D volume integral of \(M\delta(x)\delta(y)\delta(z)\). The Schwarzschild singularity, to pick the "simplest" black hole, is a moment in time, not a place in space. It is the final moment of life for the infalling observers. In a locally (conformally) Minkowski patch near the singularity with some causally Minkowskian coordinates \(t,x,y,z\) and \(r=|(x,y,z)|\), the Schwarzschild singularity looks like a \(t=t_f\) hypersurface, not as \(r=0\).

So again, I hoped that my previous comment made it clear, and everyone should have known it before that comment, anyway – but it wasn't the case. So I hope you will try to read this comment and understand it – Erwin and Jitter. And maybe others. The value of the black hole mass or a celestial object's mass has nothing to do with the properties of the interior. Instead, it is given by how much the object curves the exterior spacetime. The exterior spacetime is continuously connected to the interior. But that doesn't mean that the interior is the "cause" of the curvature in the exterior. In the black hole case, this interpretation of the cause-and-effect is strictly prohibited because the external spacetime does not belong to the future light cone of the interior, so it cannot be affected by the interior. So the events and mass densities in the black hole interior, and especially the singularity, cannot be the cause of any measurement done outside, e.g. the measurement of the black hole mass or neutron star mass done by LIGO. LIGO only sees external properties of the objects – what has existed around those objects since the beginning.

The Penrose causal diagram for a regular 4D Schwarzschild (neutral) black hole. Well, an evaporating one – that only adds the tooth at the top. Time generally goes up. Light moves to the Northwest and Northeast, massive objects have to move closer to the North (vertically up) than either West or East. Superluminal motion in directions closer to the East or West is prohibited. Each point describes a sphere with the extra coordinates \(\theta,\phi\) that are suppressed because the black hole spacetime is spherically symmetric. However, the components of the metric tensor \(g_{\tau\tau}\) and \(g_{\rho\rho}\) where \((\rho,\tau)\) are the two Cartesian coordinates on the picture are rescaled by a general scalar function. Near the right end of the tooth-like spacelike singularity, this scalar factor is huge, so most of the history of the black hole – and its volume – is concentrated over there. The yellow line is the word line of the surface of a collapsing star. That point, like any doomed observer, is guaranteed to hit the singularity where the curvature goes to infinity. The orange life is the world line of an observer who managed to stay outside. The red triangle is the black hole interior. It is separated from the light brown exterior by the green event horizon. Nothing from the black hole interior, the red triangle, can ever influence the "infinite future" region where e.g. the LIGO experiments are located.

The causal relationship is exactly the opposite one. The black hole singularity is a consequence (not the cause) of the too strong gravitational field that has existed around the object (e.g. a star that has gravitationally collapsed). The black hole singularity is a consequence – it belongs to the causal future of the collapsing star – and the collapsing star already had the \(1-2GM/r\) component g_{00} of the metric tensor to start with. \(GM/r\) was too comparable to one – \(M\) was too large or r was too small – and that's why the star collapsed and led to the birth of the event horizon as well as the singularity. The singularity is a consequence, not the cause, of the mass.

Is it getting clear now? The whole interior is absolutely irrelevant and invisible for the experiments such as LIGO that are done outside the black hole. You can't see inside any black hole, not even with LIGO. I really mean it. The unobservability of the black hole interior is the #1 thing you should know about the black hole – it is really the definition of the black hole. And you seem to misunderstand it. It seems that you're working hard to overlook and fool yourself and manipulate yourself into some misunderstanding of this very straightforward and simple fact I am trying to convey for the 4th time now. ;-)

In Newtonian physics, you may think that the gravitational field is just some "consequence" – a shadow of the object, a derived thing, perhaps just a virtual one that we only imagine to exist, and only the object with its intrinsic mass is "real". But that's not a possible story in general relativity. In general relativity, the gravitational fields must be considered at least as real as the mass sources – and for the LIGO measurements, they're more real or more relevant. LIGO sees pure gravitational waves that are just a curved spacetime and they may exist even if \(T_{\mu\nu}=0\) everywhere.

The gravitational fields – the spacetime with some geometry encoded in \(g_{\mu\nu} (x,y,z,t)\) – exists on top of the matter with nonzero \(T_{00}\). In the black hole case, all the \(T_{\mu\nu}\) ultimately goes to zero almost everywhere (I say "almost" because the claim may be debated near the singularity). It's only the metric tensor that remembers the interesting information about the two merging black holes and the final black hole that they combine into. And these gravitational fields encoded in \(g_{\mu\nu}(x,y,z,t)\) have the property that before the merger, they look like\[

g_{00} = 1 - \frac{2GM_1}{r_1} - \frac{2GM_2}{r_2}

\] i.e. the superposition of two gravitational potentials from two black holes with masses \(M_1,M_2\) – where \(r_1,r_2\) are distances from their centers. And at later times after the merger is mostly completed and the unified new object no longer radiates much, \(g_{00}\) may be approximated as \[

1 - \frac{2GM}{r}

\] for the unified \(M\) and a new \(r\) – the distance from the center of the new larger black hole – where \(M\) is slightly smaller than \(M_1+M_2\) because some mass-energy was radiated away in the form of gravitational waves.

But the merging black holes are described by pure geometry – a curved spacetime. There is no "matter" that sources anything that matters for the LIGO observation. The matter from the 2 stars has collapsed a long time before the two black holes came close to start the merger. The two smaller initial black holes have been stabilized for millions of years before they came close to each other and started to consider a wedding (merger). For these millions of years, there were no traces of the stellar matter.

The neutron stars have a nonzero stress-energy tensor \(T_{\mu\nu}\), unlike the black holes. But this material source is almost entirely irrelevant for the larger first part of the whole gravitational wave from the merger – it only matters in the final stages when the two neutron stars touch and merge. In the first part, the gravitational waves from the two orbiting neutron stars, only the external gravitational fields – the geometry similar to the black hole case - matters. The gravitational field curved like \(1-2GM/r\) far from the massive objects was "always there", long before the existence of a black hole was decided. The \(1-2GM/r\) field was there even when the mass was carried by a "gas cloud" that wasn't a star yet. There was some value of \(M\) – defined from the numerator of \(1-2GM/r\) in \(g_{00}\) – and it could only decrease when the object emitted some mass – ejections of massive particles but mostly electromagnetic and gravitational waves.

There were some relevant values of \(M_1,M_2,M\) shortly before and after the merger. But they were just snapshots of the gravitational field around the two initial black hole or the single final one. The gravitational wave seen by LIGO is a consequence of the spacetime geometry involving the two merging black holes. But the events near the singularity are a consequence of the interactions of the objects with masses \(M_1,M_2\) or the heavier stellar masses \(M_{s1},M_{s2}\), too.

What is wrong to say is the statement the LIGO observation is a consequence of some matter or events inside the black hole. It is simply not. There's no well-defined "overall mass-energy" distributed in the black hole interior, and even if there were one, it just couldn't affect the black hole exterior such as the LIGO experiment.

By the way, I started this blog post with the NASA video that sounds like a vacuum cleaner. That sound was obtained in NASA by rescaling some (visible by telescopes) X-ray spectrum coming from that black hole to us into audible frequencies. Hundreds of the 2,200 comments posted under the video also asked how it was possible that there's sound coming from a black hole although nothing can escape a black hole – which is almost the same question as Jitter's and Erwin's question that I was answering above.

(On YouTube, at some moment, I gave up efforts to answer those questions because the reservoir of the new people who ask the same thing seems unlimited, and neither my previous answers nor the FAQ added to my video seemed to help at all.)

Well, what's prohibited is a signal coming from the black hole interior. But the sound we hear is the signal coming from the vicinity (i.e. a part of the exterior), not interior, of the black hole – it encodes the X-rays from the dinner as the black hole devours the matter (mostly some gas) around. So there's really no contradiction with causality. All the sound is coming from the dinner that gets accelerated before it crosses the event horizon – before it gets into the interior. We can't see or hear what the gas is doing after it gets to the black hole interior.

I won't proofread this blog post because I've proofread a similar Disqus comment and the resulting \(\LaTeX\)ed result seems OK after 10 seconds. Please let me know if you find mistakes.

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