# Where is the mass of a particle located?

• I
• HansH
In summary: So, in summary, the mass of a black hole is determined by two integral quantities: the ADM mass, which is the mass you would get if you put test bodies in orbit about the object and measured their orbital parameters, and the Komar mass, which corresponds to the intuitive idea of adding up the contributions of all the little pieces of matter inside the object to get a total mass.
HansH
TL;DR Summary
atoms have mass. But where is this mass exactly located. is it a fixed point or is it a statistical distribution so in fact spread out over a certain volume?
in https://www.physicsforums.com/threa...ue-to-evaporation.1051054/page-2#post-6869765 there was a discussion about where the mass inside a black hole is located. according to general relativity it is located in an infinite small point, which causes spacetime to be infinitely bend and the theory to beak down. But could it be that the mass of a particle is not located in a single point but in fact spreads out over a certain volume? That could then probably prevent the singularity. In order to answer that question we should probably start at more 'normal' situations of atoms and subatomic particles.
and then probably answer this question for special situations such as a neutron star. Could we then also say something about mass inside black holes?

Matter is composed of fields and those fields have field density and momentum density. The momentum and energy densities can be combined to obtain mass density.

Of course, quantum mechanically this is all done through operators on the wavefunction, so it is probabilistic.

Could this mean anything usefull to conclude about the centre of a black hole. if you have a mass density then the mass inside a volume of zero is zero. If all mass is in one single point the mass in that point is the full mass. This could mean the difference between a curvature of spacetime of infinite versus a curvature of zero. so a singularity or not.

Do you know of any good quantum mechanical descriptions of BHs?

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Dale and Hyperfine
HansH said:
Could this mean anything usefull to conclude about the centre of a black hole.
Not yet. We don’t yet have a quantum theory of gravity. Maybe sometime in the future we will, but not now.

No as this is still not clear. But as long as it is not known then what is a better assumption: a single point where all mass is located or a statistically determined volume? (although we probbly don't know how to determine this volume or the mass distribution)

As we know that nature works with fields and wave functions I would expect that direction makes more sense than a single point. That it would be a single point is also not proven I suppose. General relativity can deal with both I assume, so it does not make GR different.

so basic question for me would be: what makes us having a preference for 1 single point with infinite mass density to the other solution of a distributed mass over a volume?

HansH said:
according to general relativity it is located in an infinite small point
No, this is not correct. The short answer is that the "mass" of a black hole is a global geometric property of the spacetime. It cannot be localized to a particular point or region in the spacetime. For a longer answer, see below.

HansH said:
In order to answer that question we should probably start at more 'normal' situations of atoms and subatomic particles.
and then probably answer this question for special situations such as a neutron star.
These things all differ from a black hole in one crucial respect: they have matter in them. A black hole is vacuum everywhere. That makes a difference when one is trying to figure out what the "mass" of the object means.

That said, there is a method of determining "mass" that will work equally well for both kinds of objects, provided we consider them in isolation. In isolation, we can model the object using an asymptotically flat spacetime, i.e., a spacetime whose metric becomes the flat Minkowski metric as the distance from the object goes to infinity. Any asymptotically flat spacetime has an integral quantity defined on it called the "ADM mass", which turns out to be equal to the "mass" of the object the way you would ordinarily think of it (for example, it is the mass you would get if you put test bodies in orbit about the object and measured their orbital parameters). This works for both ordinary objects like stars and planets and black holes. For black holes, it gives you the ##M## that appears in the metric.

For objects which are stationary, i.e., which can be modeled as a finite spatial distribution of matter surrounded by vacuum, such that the distribution does not change with time, there is another integral quantity called the "Komar mass", which corresponds to the intuitive idea of adding up the contributions of all the little pieces of matter inside the object to get a total mass. For objects like stars and planets, considered in isolation, this method gives the same answer as the ADM mass, above. However, this method cannot be used for black holes, because considered as isolated objects in themselves, while they are stationary, they contain no matter; and if we extend our model to include the matter that collapsed to form the hole, that matter is not stationary (because it collapsed), so it has no well-defined Komar mass. So the ADM mass is the only mass integral quantity that works for black holes.

phinds
HansH said:
what makes us having a preference for 1 single point with infinite mass density to the other solution of a distributed mass over a volume?
For black holes, neither of these are true. See my post #7.

HansH said:
what is a better assumption: a single point where all mass is located or a statistically determined volume?
You are misunderstanding here. First, a black hole with a singularity at ##r=0## is not an assumption, it is a derived result. Second, the singularity is not a point, it is more like a moment in time, so the concept of volume is nonsensical anyway. Third, the singularity itself isn’t part of the manifold so saying that it has mass or contains matter is problematic.

Really, there is nothing sensible that can be said about this topic until we get a quantum theory of gravity. Then we can derive the corresponding result in that theory and see if it differs from the GR result or not.

bhobba
PeterDonis said:
The short answer is that the "mass" of a black hole is a global geometric property of the spacetime. It cannot be localized to a particular point or region in the spacetime. For a longer answer, see below.
using the following part of your input:

"Komar mass", which corresponds to the intuitive idea of adding up the contributions of all the little pieces of matter inside the object to get a total mass. while they are stationary. they contain no matter; and if we extend our model to include the matter that collapsed to form the hole, that matter is not stationary (because it collapsed), so it has no well-defined Komar mass. So the ADM mass is the only mass integral quantity that works for black holes.

However, this method cannot be used for black holes, because considered as isolated objects in themselves, while they are stationary, they contain no matter; and if we extend our model to include the matter that collapsed to form the hole, that matter is not stationary (because it collapsed), so it has no well-defined Komar mass.

For me the logical reasoning and your input above together with the behavior of quantum mechanics that mass is distributed would then lead to the following:

assuming mass to be able to to be contracted without limits we get the situation you describe with no "Komar mass" in the spacetime region and an ADM mass seen from outside the black hole.

But assuming mass that cannot contract without limits (when it has a statistical determined volume) and while the large star that forms the black hole star collapses, I expect the curvature would reach a certain maximum (being determined by the "Komar mass" inside a sphere with certain radius) and then reduces to 0 in the center where the radius is 0 and a given mass density . (similar to stars, white dwarf or neutron star) so as the curvature does not go to infinity, the mass stops contracting in the region where the curvature reduces from a maximum to 0, and a stable mass distribution (although still extreme density) remains. And then a stable situation would still give a valid "Komar mass".

so based on that I conclude we could have 2 ways to describe the internal structure of the black hole, depending om the assumption that mass can contract infinite or not.

so then again I come back to the question what makes us so sure that mass can contract infinite under gravity.

Dale said:
a black hole with a singularity at ##r=0## is not an assumption, it is a derived result.
but any derived result is based on assumptions (or proven results if you like) I assume the singularity as derived result comes from the input that mass can be contracted infinitely?

HansH said:
but any derived result is based on assumptions (or proven results if you like) I assume the singularity as derived result comes from the input that mass can be contracted infinitely?
No it does not. It comes from solving the Einstein field equations for a spherically symmetric distribution of mass.

Nugatory said:
for a spherically symmetric distribution of mass.
so a distribution of mass. But how far can this mass be contracted?

HansH said:
depending on the assumption that mass can contract infinite or not.
so a distribution of mass. But how far can this mass be contracted?
This is nonsense, because contraction means a reduction of volume, and volume is not a property that mass has. Talking about “mass contracting” is like talking about “energy contracting” - just a concatenation of words without meaning.

We can sensibly talk about the amount of mass within a given volume (although there are some pitfalls here) in some circumstances but the center of a black hole is not one them. The problem is that without a quantum theory of gravity we cannot even define the volume around the singularity.

dextercioby and Dale
HansH said:
so then again I come back to the question what makes us so sure that mass can contract infinite under gravity
We are not at all sure of that.

HansH said:
but any derived result is based on assumptions (or proven results if you like) I assume the singularity as derived result comes from the input that mass can be contracted infinitely?
No. The singularity comes from the following assumptions:

1) the Einstein field equations
2) spherical symmetry
3) vacuum spacetime
4) geodesic completeness

No properties of matter are involved. There is, in fact, no matter anywhere in the spacetime (assumption 3), and hence no assumptions about any equation of state (the behavior of matter).

dextercioby and PeterDonis
Dale said:
We are not at all sure of that. No. The singularity comes from the following assumptions:

1) the Einstein field equations
2) spherical symmetry
3) vacuum spacetime
4) geodesic completeness

No properties of matter are involved. There is, in fact, no matter anywhere in the spacetime (assumption 3), and hence no assumptions about any equation of state.
I Think your reasoning is starting from the situation where the matter is not in the spacetime anymore. When matter is in spacetime beween the moment that a star starts collapsing and the final state where this leads to, it curves spacetime if I am richt?

so if during the collapsing process a situation occurs where the mass distribution causes no further curvature, then this is the stable final state.

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Nugatory said:
volume is not a property that mass has.
if mass is statistically spread out due to quantum effects it is? then the mass of any particle has a minimum effective volume?
Nugatory said:
The problem is that without a quantum theory of gravity we cannot even define the volume around the singularity.
or a quantum theory about how mass of particles is spread out over a volume. Then with GR it can be calculated how spacetime curves and how the mass is finally spread in a stable final situation?

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HansH said:
using the following part of your input

HansH said:
assuming mass to be able to to be contracted without limits
That's not an assumption in GR, it's a derived result given a particular set of conditions.

HansH said:
assuming mass that cannot contract without limits
You can't just assume this. You have to show what kind of "mass" (i.e., what equation of state of matter) will cause the contraction to stop. There are solutions that have this property; for example, do a PF forum search on "Bardeen black hole". They're just not solutions that, at our present state of knowledge, we have any evidence actually occur when massive objects collapse under their own gravity. One of the key objectives of quantum gravity research is to see under what conditions quantum fields can produce such effects.

HansH said:
I conclude we could have 2 ways to describe the internal structure of the black hole, depending om the assumption that mass can contract infinite or not
No, you have two possible kinds of objects--a traditional black hole, with an event horizon and a singularity; or an object like a Bardeen "black hole", which is not really a black hole at all because it has no event horizon and no singularity, but from the outside it can look like a traditional black hole for very long time scales (comparable to Hawking evaporation time scales).

HansH said:
I Think your reasoning is starting from the situation where the matter is not in the spacetime anymore.
No, it's not. The spacetime includes a region that contains matter and a region that does not; but the matter region is never "not in the spacetime anymore". That would make no sense, since the spacetime contains an entire history; it's not something that can change or that regions can be "added to" or "removed from". The regions are just there in the 4-dimensional geometry.

https://www.physicsforums.com/insig...-model-of-gravitational-collapse-an-overview/

They describe the model of gravitational collapse that Oppenheimer and Snyder first discovered in 1939, and which has been the subject of much research ever since.

bhobba
HansH said:
atoms have mass. But where is this mass exactly located.
HansH said:
where the mass inside a black hole is located
This thread is in the QM forum, which would indicate that we should be discussing the first question. But most of our actual discussion has been about the second question, which really should be discussed in the relativity forum since we have no theory of quantum gravity on which to base a discussion.

@HansH, which of the two questions above do you want to discuss?

Dale
HansH said:
I Think your reasoning is starting from the situation where the matter is not in the spacetime anymore.
Yes, that is the standard Schwarzschild spacetime.

HansH said:
When matter is in spacetime beween the moment that a star starts collapsing and the final state where this leads to, it curves spacetime if I am richt?
What spacetime are you talking about? Perhaps the Oppenheimer Schneider spacetime?

That does involve an equation of state that models matter. It is not a particularly realistic equation of state, even classically. And it certainly isn’t a quantum mechanical equation of state.

In any case, we are back to the point that we simply do not have a quantum theory of gravity. There simply isn’t anything else to say. Once we have such a theory then we can derive the answer. Until then any answer is purely guessing.

HansH said:
a quantum theory about how mass of particles is spread out over a volume. Then with GR it can be calculated how spacetime curves and how the mass is finally spread in a stable final situation?
When you find a professional scientific reference that shows such a calculation we can discuss it. Just PM me with a link and I will be glad to reopen the thread.

Until then, this thread is closed. There is simply nothing that we can discuss on the topic without speculation.

bhobba

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