Comments about "Penrose-Hawking singularity theorems" in Wikipedia

This document contains comments about the article https://en.wikipedia.org/wiki/Penrose-Hawking_singularity_theorems in Wikipedia

• The text in italics is copied from that url
  
• Immediate followed by some comments
  

In the last paragraph I explain my own opinion.

Contents

• 1. Singularity
• 2. Interpretation and significance
• 3. Elements of the theorems
• 4. Nature of a singularity
• 4.1 Assumptions of the theorems
• 4.2 Tools employed
• 4.3 Versions
• 4.4 Modified gravity
• 5. See also

Reflection

• Reflection 1 -
• Reflection 2 -
• Reflection 3 -

Introduction

The article starts with the following sentence.

1. Singularity

A singularity in solutions of the Einstein field equations is one of three things:

• Spacelike singularities: The singularity lies in the future or past of all events within a certain region. The Big Bang singularity and the typical singularity inside a non-rotating, uncharged Schwarzschild black hole are spacelike.
• Timelike singularities: These are singularities that can be avoided by an observer because they are not necessarily in the future of all events. An observer might be able to move around a timelike singularity. These are less common in known solutions of the Einstein field equations.
• Null singularities: These singularities occur on light-like or null surfaces. An example might be found in certain types of black hole interiors, such as the Cauchy horizon of a charged (Reissner–Nordström) or rotating (Kerr) black hole.

It is important to inform the reader how it is decided, based on observations, which is a specific black hole.

A singularity can be either strong or weak:

• Weak singularities: A weak singularity is one where the tidal forces (which are responsible for the spaghettification in black holes) are not necessarily infinite. An observer falling into a weak singularity might not be torn apart before reaching the singularity, although the laws of physics would still break down there. The Cauchy horizon inside a charged or rotating black hole might be an example of a weak singularity.
• Strong singularities: A strong singularity is one where tidal forces become infinite. In a strong singularity, any object would be destroyed by infinite tidal forces as it approaches the singularity. The singularity at the center of a Schwarzschild black hole is an example of a strong singularity.

The center of a black hole cann't be called a singularity
Of course you could, but than it becomes important to define which BH's are a singularity and which are not.

2. Interpretation and significance

In general relativity, a singularity is a place that objects or light rays can reach in a finite time where the curvature becomes infinite, or spacetime stops being a manifold. Singularities can be found in all the black-hole spacetimes, the Schwarzschild metric, the Reissner–Nordström metric, the Kerr metric and the Kerr–Newman metric, and in all cosmological solutions that do not have a scalar field energy or a cosmological constant.

It is important what a singularity physical means.

One cannot predict what might come "out" of a big-bang singularity in our past, or what happens to an observer that falls "in" to a black-hole singularity in the future, so they require a modification of physical law. Before Penrose, it was conceivable that singularities only form in contrived situations. For example, in the collapse of a star to form a black hole, if the star is spinning and thus possesses some angular momentum, maybe the centrifugal force partly counteracts gravity and keeps a singularity from forming. The singularity theorems prove that this cannot happen, and that a singularity will always form once an event horizon forms.

IMO the only (?) way to prove a physical effect (event) is when it be predicted or observed in an experiment.

An interesting "philosophical" feature of general relativity is revealed by the singularity theorems. Because general relativity predicts the inevitable occurrence of singularities, the theory is not complete without a specification for what happens to matter that hits the singularity.

How can a theory predicts a singularity (an infinity ?), when it is not know what a singularity physical means?

One can extend general relativity to a unified field theory, such as the Einstein–Maxwell–Dirac system, where no such singularities occur.

They should be prevent in the first case, because they are not physical.

3. Elements of the theorems

Penrose concluded that whenever there is a sphere where all the outgoing (and ingoing) light rays are initially converging, the boundary of the future of that region will end after a finite extension, because all the null geodesics will converge. This is significant, because the outgoing light rays for any sphere inside the horizon of a black hole solution are all converging, so the boundary of the future of this region is either compact or comes from nowhere. The future of the interior either ends after a finite extension, or has a boundary that is eventually generated by new light rays that cannot be traced back to the original sphere.

Very complex physical description.

4. Nature of a singularity

4.1 Assumptions of the theorems

4.2 Tools employed

This is relevant for singularities thanks to the following argument:

1. Suppose we have a spacetime that is globally hyperbolic, and two points p and q that can be connected by a timelike or null curve. Then there exists a geodesic of maximal length connecting 'p' and 'q'. Call this geodesic 'gamma'
2. The geodesic 'gamma' can be varied to a longer curve if another geodesic from 'p' intersects 'gamma' at another point, called a conjugate point.
3. From the focusing theorem, we know that all geodesics from 'p' have conjugate points at finite values of the affine parameter. In particular, this is true for the geodesic of maximal length. But this is a contradiction – one can therefore conclude that the spacetime is geodesically incomplete.

What is the true purpose of this discussion?

4.3 Versions

4.4 Modified gravity

In modified gravity, the Einstein field equations do not hold and so these singularities do not necessarily arise.

You first need a clear definition of what exactly is a singularity. If a singularity exists, it much be more than mathematical concept.

5. See also

Following is a list with "Comments in Wikipedia" about related subjects

• Relativity of Simultaneity Commments
• One-way speed of light Commments

Reflection 1

Reflection 2

Reflection 3

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