# B Examples of theoretical proofs overturned by evidence?

#### Cerenkov

Summary
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1970.0021
The Singularities of Gravitational Collapse and Cosmology
Hello.

The Singularities of Gravitational Collapse and Cosmology

The above paper by Hawking and Penrose is presented in the form of a mathematical proof. To my knowledge the predictions it makes about the initial cosmological singularity have not been tested by experiment, so there is no data or body that evidence that might rule it out. However, I was wondering if the members of PF can cite any examples of other theoretical proofs in cosmology and/or astrophysics that have been ruled out by evidence?

Cerenkov.

Homework Helper

#### Klystron

Gold Member
After thermodynamic theory was formulated but before radioactivity was discovered, many scientists attempted to calculate the age of planet Earth and infer the age of the Solar system based on temperature and geological data. Given the available temperature data including solar and geothermal studies, the best calculations led to an estimated age of tens to hundreds of thousand years.

Once radioactive elements were discovered and with improved understanding of planetary structure, the old theories and age estimates were superseded by models with data indicating an age in thousands of millions years. Within my lifetime theories of plate tectonics and other processes developed from improved data provide a more accurate estimated age of the Earth confirmed by Lunar and meteorite samples.

• Nik_2213

#### PeterDonis

Mentor
To my knowledge the predictions it makes about the initial cosmological singularity have not been tested by experiment, so there is no data or body that evidence that might rule it out.
It's a mathematical theorem; if the premises are true, the conclusion must be true.

What can be investigated by data is whether the premises are true. And we already have at least two examples of conditions which have evidence to support them, which violate the key premise, that energy conditions are satisfied. Dark energy (aka cosmological constant) and inflation (more precisely, the "inflaton" field that caused inflation if inflation happened, which current evidence indicates is probable) both violate energy conditions, so universes that contain those are not subject to the singularity theorem.

#### Cerenkov

My thanks to Scott, Klystron and PeterDonis for their replies.

To PeterDonis...

I didn't realize that inflation and dark energy violated the energy conditions of the H - P singularity theorems. That's v. interesting. If I recall, Hawking and Penrose stipulated three kinds of energy condition; the Strong, the Weak and the Generic energy conditions. Are all three violated?

Also, since Hawking and Penrose must have been aware of this, can you please point me to any of their papers or articles where they respond to the possibility of such violations? Since both inflation and dark energy have been around for decades I'm guessing that they would have addressed this issue. Thank you.

Lastly, I'd like to put a counterfactual scenario and a question to you, please Peter. (Hoping that this doesn't fall foul of the forum guidelines about speculation.) If the 2014 Bicep2 findings had shown the B-mode polarization, that would have (as far as I understand it) been the 'smoking gun' for inflation. Which would have therefore indicated that an "inflaton" field had existed in the early universe. Which would therefore have been a violation of the H - P energy condition/s. Therefore overturning their singularity theories.

So, is that line of reasoning correct?

Thank you.

Cerenkov.

#### PeterDonis

Mentor
If I recall, Hawking and Penrose stipulated three kinds of energy condition; the Strong, the Weak and the Generic energy conditions. Are all three violated?
There are actually four: null, weak, dominant, and strong:

The one that is most relevant for cosmology is the strong energy condition, since it is the one that applies to the case of a timelike geodesic congruence such as the congruence of "comoving" observers in an FRW spacetime. A cosmological constant or inflaton field can be considered to be a perfect fluid with $p = - \rho$, which, if you look at the "perfect fluids" section of the Wikipedia article, violates the strong energy condition, but not the others. So in an FRW spacetime with a cosmological constant or inflaton field, the singularity theorems do not hold since one of the premises is not satisfied.

since Hawking and Penrose must have been aware of this, can you please point me to any of their papers or articles where they respond to the possibility of such violations?
I'm not sure what you mean. Any mathematical theorem has premises, and does not hold true if any of its premises are violated. This is a commonplace fact. Why would you expect them to explicitly point out this commonplace fact?

If you mean, did they discuss what was required for the premises to be true, in particular the energy conditions, I believe that is discussed in Hawking & Ellis.

Therefore overturning their singularity theories.
You're misunderstanding the point of the Hawking-Penrose theorems. They were not "theories" in the sense of claiming that our actual universe must satisfy them. They were simply mathematical theorems that established certain conditions under which spacetimes in GR would be geodesically incomplete. Hawking & Penrose never claimed that our actual universe had to satisfy those conditions. They simply derived an interesting consequence that would be true if the conditions were satisfied.

#### Cerenkov

There are actually four: null, weak, dominant, and strong:

The one that is most relevant for cosmology is the strong energy condition, since it is the one that applies to the case of a timelike geodesic congruence such as the congruence of "comoving" observers in an FRW spacetime. A cosmological constant or inflaton field can be considered to be a perfect fluid with $p = - \rho$, which, if you look at the "perfect fluids" section of the Wikipedia article, violates the strong energy condition, but not the others. So in an FRW spacetime with a cosmological constant or inflaton field, the singularity theorems do not hold since one of the premises is not satisfied.

Thank you Peter. Four energy conditions.

I'm not sure what you mean. Any mathematical theorem has premises, and does not hold true if any of its premises are violated. This is a commonplace fact. Why would you expect them to explicitly point out this commonplace fact?

If you mean, did they discuss what was required for the premises to be true, in particular the energy conditions, I believe that is discussed in Hawking & Ellis.

You're misunderstanding the point of the Hawking-Penrose theorems. They were not "theories" in the sense of claiming that our actual universe must satisfy them. They were simply mathematical theorems that established certain conditions under which spacetimes in GR would be geodesically incomplete. Hawking & Penrose never claimed that our actual universe had to satisfy those conditions. They simply derived an interesting consequence that would be true if the conditions were satisfied
.
I see. That's an important distinction, which a layperson like myself can easily overlook.
Hawking and Penrose formulated their singularity theories purely as mathematical exercises - never looking for them to be validated or ruled out by future experiments and data?

So, since neither Hawking nor Penrose actually claimed that their singularity theories are actually describing our universe, then it's quite wrong for anyone else, scientist or layperson, to claim that they are?

And, by the same measure, it would be equally wrong for anyone to take what they've written about their singularity theorems as actually describing the physical reality of either black hole singularities or the initial (white hole) singularity?

Thanks again.

Cerenkov.

#### PeterDonis

Mentor
Hawking and Penrose formulated their singularity theories purely as mathematical exercises - never looking for them to be validated or ruled out by future experiments and data?
Obviously they undertook those particular mathematical exercises because they thought they were of interest for physics, in particular General Relativity.

since neither Hawking nor Penrose actually claimed that their singularity theories are actually describing our universe, then it's quite wrong for anyone else, scientist or layperson, to claim that they are?
Not if they can demonstrate that our actual universe satisfies the premises. But showing that our actual universe does not satisfy the premises does not "disprove" the mathematical theorem. It just shows that the premises are not satisfied in our actual universe.

by the same measure, it would be equally wrong for anyone to take what they've written about their singularity theorems as actually describing the physical reality of either black hole singularities
It depends on whether actual black holes satisfy the premises.

or the initial (white hole) singularity?
The initial singularity of the universe is not a white hole singularity.

It's also worth noting that an unstated premise of any mathematical theorem in classical GR, as far as actual physics is concerned, is that classical GR is the correct physical theory for the regime in question. Most physicists believe that GR breaks down when spacetime curvatures become large enough (roughly speaking, on the order of the Planck scale), and if that is in fact the case, the Hawking-Penrose singularity theorems would not apply to our actual universe, since those theorems assume classical GR.

#### Cerenkov

Obviously they undertook those particular mathematical exercises because they thought they were of interest for physics, in particular General Relativity.

Thank you.

Not if they can demonstrate that our actual universe satisfies the premises. But showing that our actual universe does not satisfy the premises does not "disprove" the mathematical theorem. It just shows that the premises are not satisfied in our actual universe.

And nobody has yet demonstrated that our actual universe satisfies the premises underpinning the H - P singularity theorems?

It depends on whether actual black holes satisfy the premises.

And are we yet able to say if they do?

The initial singularity of the universe is not a white hole singularity.

It's also worth noting that an unstated premise of any mathematical theorem in classical GR, as far as actual physics is concerned, is that classical GR is the correct physical theory for the regime in question. Most physicists believe that GR breaks down when spacetime curvatures become large enough (roughly speaking, on the order of the Planck scale), and if that is in fact the case, the Hawking-Penrose singularity theorems would not apply to our actual universe, since those theorems assume classical GR.
I see. To my naive eyes it seems that the H - P singularity theorems are saying that the spacetime curvatures become infinite. Yet the Planck scale, although incredibly small, is not infinitely small. So these theorems appear to me (in my naivety) to be making a prediction about something smaller than the scale at which they are expected to break down. How can this be? The Planck scale isn't infinitely small.

As you can tell Peter, I'm confused. Could you help me out please?

Thanks,

Cerenkov.

#### PeterDonis

Mentor
To my naive eyes it seems that the H - P singularity theorems are saying that the spacetime curvatures become infinite.
More precisely, that curvature increases without bound as a finite affine parameter is approached along a family of geodesics. Which, if the geodesics are timelike, means that an observer following one would encounter curvature increasing without bound in a finite time by his clock.

the Planck scale, although incredibly small, is not infinitely small
That's right, it's a finite scale.

So these theorems appear to me (in my naivety) to be making a prediction about something smaller than the scale at which they are expected to break down.
"Expected" if you don't believe that GR continues to be valid at arbitrary scales. But one of the key reasons why physicists don't believe that is precisely that the singularity theorems show that, if their premises are true, GR implies that some observers will encounter curvature increasing without bound in a finite time by their clock. Which does not seem physically reasonable; hence physicists expect that GR will not continue to be valid at arbitrary scales.

In short, you have things backwards. It's not that Hawking and Penrose set out to prove a theorem about conditions that nobody thought were physically reasonable. It's that Hawking and Penrose proved a theorem that, when its implications were considered, made a lot of physicists think, hey, that's not physically reasonable. Which in turn led them to think that the theory, classical GR, on which the theorem was based might not be valid at all scales.

#### Cerenkov

More precisely, that curvature increases without bound as a finite affine parameter is approached along a family of geodesics. Which, if the geodesics are timelike, means that an observer following one would encounter curvature increasing without bound in a finite time by his clock.

Thank you.

That's right, it's a finite scale.

Thank you.

"Expected" if you don't believe that GR continues to be valid at arbitrary scales. But one of the key reasons why physicists don't believe that is precisely that the singularity theorems show that, if their premises are true, GR implies that some observers will encounter curvature increasing without bound in a finite time by their clock. Which does not seem physically reasonable; hence physicists expect that GR will not continue to be valid at arbitrary scales.

Thank you.

In short, you have things backwards. It's not that Hawking and Penrose set out to prove a theorem about conditions that nobody thought were physically reasonable. It's that Hawking and Penrose proved a theorem that, when its implications were considered, made a lot of physicists think, hey, that's not physically reasonable. Which in turn led them to think that the theory, classical GR, on which the theorem was based might not be valid at all scales.
Thank you Peter.

You've given me a great deal to think about.

Cerenkov.

#### pinball1970

Gold Member
However, I was wondering if the members of PF can cite any examples of other theoretical proofs in cosmology and/or astrophysics that have been ruled out by evidence?

Cerenkov.

#### Cerenkov

Thanks for this pinball1970.

Yes, I understand that the Steady State theory was ruled out by evidence.

Cerenkov.

#### Cerenkov

Hello again Peter.

Having slept on it, I think I'm beginning to see what you've been trying to explain to me. I now reckon that I phrased my initial question (examples of theoretical proofs overturned by evidence) on the back of some kind of misunderstanding. Was I comparing apples and oranges? A theoretical proof is a different kind of animal from an empirically-based theory. Y/N?

Overnight I Googled the 'Scientific Method' and found this diagram. I've tried to fit what I know about Hawking and Penrose into this and come up with this revised version. I think I'm on pretty solid ground for the first three steps, but in the light what you explained yesterday, I'm still struggling to understand where their singularity theorems go in the above process. If anywhere. ???

Thanks,

Cerenkov.

#### George Jones

Staff Emeritus
Gold Member
@Cerenkov , I like what Robert Geroch, a top mathematical physicist, wrote about theories in one of his books:

Robert Geroch said:
It seems to me that "theories of physics" have, in the main, gotten a terrible press. The view has somehow come to be rampant that such theories are precise, highly logical, ultimately "proved". In my opinion, at least, this is simply not the case - not the case for general relativity and not the case for any other theory in physics. First, theories, in my view, consist of an enormous number of ideas, arguments, hunches, vague feelings, value judgements, and so on, all arranged in a maze. These various ingredients are connected in a complicated way. It is this entire body of material that is "the theory". One's mental picture of the theory is this nebulous mass taken as a whole. In presenting the theory, however, one can hardly attempt to present a "nebulous mass taken as a whole". One is thus forced to rearrange it so that it is linear, consisting of one point after another, each connected in some more or less direct way with its predecessor. What is supposed to happen is that one who learns the theory, presented in this linear way, then proceeds to form his own "nebulous mass taken as a whole". The points are all rearranged, numerous new connections between these points are introduced, hunches and vague feelings come into play, and so on. In one's own approach to the theory, one normally makes no attempt to isolate a few of these points to be called "postulates". One makes no attempt to derive the rest of the theory from postulates. (What, indeed, could it mean to "derive" something about the physical world?) One makes no attempt to "prove" the theory, or any part of it. (I don't even know what a "proof" could mean in this context. I wouldn't recognize a "proof" of a physical theory if I saw one.)

I like also Nobel-laureate Steven Weinberg's characterization of the scientfific method as practiced by top reseach scientists:

Steven Weinberg said:
... scientific research is more honestly reported as a tangle of deduction, induction, and guesswork

#### pinball1970

Gold Member
Thanks for this pinball1970.

Yes, I understand that the Steady State theory was ruled out by evidence.

Cerenkov.
Ok noted. Neil Turok wrote a pop Science book and from memory inflation was not a requirement in his model of the early universe. Not certain of this but I think he claimed gravitational waves would not be detected. There has been confirmation of GW now so does that disprove his theory?

#### PeterDonis

Mentor
A theoretical proof is a different kind of animal from an empirically-based theory. Y/N?
Yes. A proof is a proof--if the premises are true, the conclusion must be true as a matter of logic. You can't test a proof by experiment; you can only look at the proof's logic and decide whether you agree that it's valid. Everybody agrees that the logic of the proof of the Hawking-Penrose singularity theorem is valid.

An empirically-based theory is a model that makes predictions, which are then tested by comparing them with experiment. The model might be based on deductive logical reasoning, like the proof of a mathematical theorem, but it also might not. It might make its predictions using approximations, heuristics, guesses, whatever. It doesn't matter. All that matters is how well the model's predictions match the experimental data. Of course the ideal case is where you have a model that makes precise quantitative predictions based on precise logical reasoning from simple premises, which then match the data to a very high precision. But most scientific models fall short of the ideal case, often by a lot.

General Relativity is an empirically-based theory. The Hawking-Penrose singularity theorem is a proof that happens to illustrate an interesting property of a certain class of solutions in General Relativity (the class that satisfies all of the premises of the theorem). You can't test the theorem by experiment, since it's just a mathematical theorem. But you can certainly test GR by experiment.

#### PeterDonis

Mentor
Neil Turok wrote a pop Science book
Pop science books are not valid references for PF discussion.

#### Cerenkov

To George Jones and pinball1970...

Many thanks for your interesting input. However, if I may draw your attention to the central point of my query in this thread - please click on the link I've given above to read Hawking and Penrose's paper, 'The Singularities of Gravitational Collapse and Cosmology.' My point being that this paper does include a proof. As you will see, the paper is broken down into the following sections.

A preamble.
1. Introduction.
2. Definitions and Lemmas.
3. The Theorem.
Corollary.
Proof of the Theorem.
References.
Appendix.

Ooops! Just noticed that PeterDonis has replied, so I'm going to finish this post off here.

Thanks again,

Cerenkov.

#### Cerenkov

Yes. A proof is a proof--if the premises are true, the conclusion must be true as a matter of logic. You can't test a proof by experiment; you can only look at the proof's logic and decide whether you agree that it's valid. Everybody agrees that the logic of the proof of the Hawking-Penrose singularity theorem is valid.

An empirically-based theory is a model that makes predictions, which are then tested by comparing them with experiment. The model might be based on deductive logical reasoning, like the proof of a mathematical theorem, but it also might not. It might make its predictions using approximations, heuristics, guesses, whatever. It doesn't matter. All that matters is how well the model's predictions match the experimental data. Of course the ideal case is where you have a model that makes precise quantitative predictions based on precise logical reasoning from simple premises, which then match the data to a very high precision. But most scientific models fall short of the ideal case, often by a lot.

General Relativity is an empirically-based theory. The Hawking-Penrose singularity theorem is a proof that happens to illustrate an interesting property of a certain class of solutions in General Relativity (the class that satisfies all of the premises of the theorem). You can't test the theorem by experiment, since it's just a mathematical theorem. But you can certainly test GR by experiment.

Hello again Peter and thanks for your response. It's helping me clarify my thinking.

Ok, so on the back of what you've just written, would it be fair to say that the H - P singularity theorems cannot be tested themselves, but they stand or fall on the testing of GR? Or putting it another way, so long as GR stands, then the H - P singularity theorems are considered to be valid?

Cerenkov.

#### Buzz Bloom

Gold Member
General Relativity is an empirically-based theory. The Hawking-Penrose singularity theorem is a proof that happens to illustrate an interesting property of a certain class of solutions in General Relativity (the class that satisfies all of the premises of the theorem). You can't test the theorem by experiment, since it's just a mathematical theorem. But you can certainly test GR by experiment.
Hi Peter:

I have made an effort to undestand the quote above by looking at two references below.
Singularity section​
During inflation, the universe violates the dominant energy condition, in which the energy is larger than the pressure.​
The dominant energy condition stipulates that, in addition to the weak energy condition holding true, for every future-pointing causal vector field (either timelike or null) T_Arrow , the vector field − Tab Yb must be a future-pointing causal vector. That is, mass–energy can never be observed to be flowing faster than light.​

Do the texts I quote from the two references properly interpret what this math theorem says about the physics?

Regards,
Buzz

#### PeterDonis

Mentor
would it be fair to say that the H - P singularity theorems cannot be tested themselves, but they stand or fall on the testing of GR?
Why do you say "stand or fall"? Mathematical theorems are mathematical theorems. The H-P singularity theorems are still true mathematical theorems even if it turns out that there is nowhere in our actual universe in which their premises are satisfied.

More generally, why do you care about "standing or falling"? What problem are you trying to solve, and why does it matter for that problem whether the singularity theorems "stand or fall"? Math is math and physics is physics; math is a tool used in physics, but the tool and the user are not the same.

#### Cerenkov

A couple more questions, if I may please, Peter.

If I were to make a direct comparison between the H - P singularity theorems and Inflationary theory, Ekpyrotic theory and Steady State theory, that would be incorrect, wouldn't it? The first isn't an empirical theory, while the others are. That would be like comparing oranges and apples, right?

Secondly, if I were to claim that the first must be true, because it is proven, then I'd be making an error. A mathematical proof (even though it relates to the origin of the universe) cannot be taken as legitimately describing the universe in the same way that an empirical does. Is that so?

Thanks,

Cerenkov.

#### Cerenkov

Ok, we've cross posted, Peter. Please give me a little while to respond. Thank you.

#### PeterDonis

Mentor
During inflation, the universe violates the dominant energy condition
I don't think this is correct. Per my post #6, the strong energy condition is the one that is violated by a cosmological constant or a scalar inflaton field.