Is such a flaw conceivable in GR?

[AbstractDreamer]

Just now, exchemist said:

I think you may be confusing abstract mathematics with physics. Calculus is used all the time in QM. The uncertainty principle arises from pairs of non-commuting operators for certain observable properties. This concept is quite compatible with calculus. In fact some of the operators concerned contain things like derivatives. 

Pairs of non-commuting operators...
Like position and momentum
Like pressure and volume
Like space and time?

 

[exchemist]

6 minutes ago, AbstractDreamer said:

Pairs of non-commuting operators...
Like position and momentum
Like pressure and volume
Like space and time?

 

Don’t be an idiot.

[studiot]

Just now, AbstractDreamer said:

Shall we waste more time arguing over irrelevances? 

Instead of shouting everybody else down how about you try listening?

In particular I didn't make this an argument.

Just now, AbstractDreamer said:

 I don't accept the argument that because its a limitation, we shouldn't discuss it as a flaw.

That is most definitely not what I said.

 

Just now, AbstractDreamer said:

Conjugate or not depends on what the axes represent in observables.   You cannot make a blanket statement at x and y are not conjugate before you apply units to them.  Well you can, but you'd be wrong.  What if x was position and y was momentum?   Would you then agree x and y are conjugate?

I will ignore the first sentence as you clearly do not know what conugate means in the context of QM.

But I do agree that you can labe the axes any way you like.

If you lable them as you describe in the underlined sentence, it becomes impossible to plot any points at all using those axes.
That is a direct result of the uncertainty principle.

 

On the other hand if you prefer to hold a civilized discussion then I am quite happy to expand on my explanations further.

[AbstractDreamer]

1 minute ago, studiot said:

Instead of shouting everybody else down how about you try listening?

In particular I didn't make this an argument.

That is most definitely not what I said.

 

I will ignore the first sentence as you clearly do not know what conugate means in the context of QM.

But I do agree that you can labe the axes any way you like.

If you lable them as you describe in the underlined sentence, it becomes impossible to plot any points at all using those axes.
That is a direct result of the uncertainty principle.

 

On the other hand if you prefer to hold a civilized discussion then I am quite happy to expand on my explanations further.

Honestly, I'm just reciprocating the attitude shown to me.  I'm not shouting, and its not at everybody.  I'm reciprocating the attitude that SwansonT showed me, when saying to me that I need to speak the language of physics or nobody would understand me, with the demeaning implication I couldn't speak the language.  When in actual fact, what I said was perfectly understandable.

Lets talk about Calculus, conjugate variables, Space and Time, and General Relativity.

 

[studiot]

Just now, AbstractDreamer said:

Lets talk about Calculus, conjugate variables, Space and Time, and General Relativity.

Go on ?

 

And what about your misreading of my previous posting ?

[AbstractDreamer]

1 minute ago, studiot said:

Go on ?

 

And what about your misreading of my previous posting ?

I dont know what else im expected to say.  I've already made my position very very clear.

GR is a model of Space and Time founded upon Calculus.   Space and Time are conjugate variables.   Calculus with conjugate variables break the Uncertainty Principle.  Therefore GR breaks the Uncertainty Principle.

Call it a flaw, limitation, incompleteness, whatever you like.  Refute me, instead of arguing over irrelevances (not you) or calling me an idiot (also not you).
 

[studiot]

Just now, AbstractDreamer said:

  I've already made my position very very clear.

Yes and that position is that you are unwilling to correctly read what others tell you or even to ask them what they mean by a statement.

You seem to prefer immediate outright denial.

 

If you are saying that I did claim that GR is flawed pleased post the quote .

 

Otherwise reread what I actually said about about GR and apologise.

[swansont]

1 hour ago, AbstractDreamer said:

But if calculus is used on conjugate variables, then it contradicts Uncertainty Principle.

No, it actually defines the uncertainty principle. The fact that calculus uses infinitesimals is irrelevant; there’s nothing that says you can’t measure any particular variable to arbitrary precision. The HUP only restricts you from simultaneously measuring one other, specific variable to arbitrary precision at the same time. 

37 minutes ago, AbstractDreamer said:

Calculus with conjugate variables break the Uncertainty Principle.  Therefore GR breaks the Uncertainty Principle.

The HUP is a ramification of QM and GR is a classical theory. You could just as easily claim that classical mechanics is flawed, and it would be just as much bollocks. We know classical theories break down at certain scales. It defines the limit of their applicability. We don’t have any theories that are universal in what they apply to. We don’t use QM to solve for an object being dropped off of a building, or calculating the trajectory of an orbit. You use the model that’s most useful.

[studiot]

What I actually said was

1 hour ago, studiot said:

GR is just a model.

Complaining that GR is flawed is a bit like trying to put a screw in with a hammer and complaining that the hammer is flawed.

 

When you offered this were you confusing the mathematic process of taking a limit with limitations such as the limitation on square roots "There are no real square roots of negative numbers"  ?

 

Just now, AbstractDreamer said:

Call it a flaw, limitation, incompleteness, whatever you like. 

 

1 hour ago, studiot said:

Mathematically we have learned to handle the 'impossible' division by zero.

For instance the point scalar density has a definite and measurable value which coincides with the scalar limit of Mass / volume as the volume approaches 0.

 

[AbstractDreamer]

42 minutes ago, studiot said:

If you are saying that I did claim that GR is flawed pleased post the quote .

Otherwise reread what I actually said about about GR and apologise.

It is as irrelevant if GR is flawed as it is if GR is limited.   I am sorry if you want to talk about what you claimed or not.   You did not claim GR is flawed I agree.  

My first point was about GR's premise on a differentiable manifold.  I've made my case. OP asked about flaws, where it breaks down. Everyone here doesn't think its a flaw, because they prefer to call it a defining limitations.  Semantics.   Limitations are the boundaries where it breaks down.  So OP was really asking what are the limitations and why are they the way they are.

And differentiable manifolds is one such limitation - my case - as are singularities

20 minutes ago, swansont said:

The HUP only restricts you from simultaneously measuring one other, specific variable to arbitrary precision at the same time. 

Including simultaneously measuring Space and one other specific variable say, Time? to arbitrary precision at the same time?
 

26 minutes ago, swansont said:

It defines the limit of their applicability. 

Well there you go.  it defines the limit of their applicability.  The boundary where it breaks down.  What the OP intended to mean when he said "flaw".    

[studiot]

Just now, AbstractDreamer said:

Including simultaneously measuring Space and one other specific variable say, Time? to arbitrary precision at the same time?

This is where you should go back and read up on the HUP.

I also recommend you read more carefully exactly what swansont said.  (Hint "specific")

Conjugate variables for the purpose of the HUP have dimensions ML²T^-1  when multiplied together

So

 

Momentum times Position

MLT^-1   x  L  =   ML²T^-1 

Energy times Time

ML²T^-2   x  T  =   ML²T^-1

But you also said

Just now, AbstractDreamer said:

Pairs of non-commuting operators...
Like position and momentum
Like pressure and volume
Like space and time?

 

So

Pressure times Volume

ML^-1T^-2   x  L³  =   ML²T^-2

and

Space times Time

L³  x  T   =   L³T

 

Neither of which satisfy the HUP conditions of conjugate variables.

Edited Sunday at 11:34 PM by studiot
Add missing exponent to energy dimension

[swansont]

14 minutes ago, AbstractDreamer said:

Including simultaneously measuring Space and one other specific variable say, Time? to arbitrary precision at the same time?

“space” isn’t a variable, position is. Its conjugate variable is momentum. Time is paired with energy. You can measure time and position to arbitrary precision at the same time. You’d be limited by instrumentation, not the HUP.

18 minutes ago, AbstractDreamer said:

Well there you go.  it defines the limit of their applicability.  The boundary where it breaks down.  What the OP intended to mean when he said "flaw".    

I wasn’t addressing the OP, I was addressing your claim about calculus somehow being incompatible with the HUP

1 hour ago, AbstractDreamer said:

I'm reciprocating the attitude that SwansonT showed me, when saying to me that I need to speak the language of physics or nobody would understand me, with the demeaning implication I couldn't speak the language.  When in actual fact, what I said was perfectly understandable.

I never said you couldn’t, I pointed out that you weren’t. If you feel that having an error pointed out demeaning I’m sorry, but being corrected is the price of admission to discussions like this. 

You don’t get to decide what's understandable by others.

[AbstractDreamer]

Ok and  gravitation potential and mass density are paired?

So GR metric tensor for spacetime curvature, energy, momentum and stress that produce those differential equations.  None of those differential equations involve conjugate variables?
 

[KJW]

2 hours ago, AbstractDreamer said:

Space and Time are conjugate variables.

No, they are not.

 

 

Edited Monday at 12:41 AM by KJW

[KJW]

For arbitrary function [math]\psi[/math], and [math]x^\alpha[/math] and [math]\dfrac{\partial}{\partial x^\beta}[/math] as operators:

[math]\dfrac{\partial}{\partial x^\beta} \Big(x^\alpha \psi\Big) - x^\alpha \dfrac{\partial}{\partial x^\beta} \psi = \delta^\alpha_\beta\ \psi[/math]

Thus:

[math]x^\alpha[/math] and [math]\dfrac{\partial}{\partial x^\beta}[/math] are conjugate variables (operators) for [math]\alpha = \beta[/math].

Note that:

[math]p_\gamma = -i \hbar \dfrac{\partial}{\partial x^\gamma}[/math]

So that:

[math][x^\alpha,p_\beta] = x^\alpha p_\beta - p_\beta x^\alpha = i \hbar\ \delta^\alpha_\beta[/math]

 

Edited Monday at 01:47 AM by KJW

[Markus Hanke]

8 hours ago, AbstractDreamer said:

Pairs of non-commuting operators...
Like position and momentum
Like pressure and volume
Like space and time?

I don’t understand this comment - in GR, these things aren’t operators (or even observables in the QM sense), so it isn’t clear to me what you even mean by “non-commuting” in this context. We just have a differentiable manifold with four locally linearly independent basis vectors, plus a connection and a metric; there’s nothing in this basic structure really that is meaningfully relatable via Fourier transforms. 

Even in quantum mechanics, specifically for your last example, the correct pairing would be time and energy. Time and position do commute, with the caveat that treating time as an operator comes with its own complications in QM.

I think it’s also important to remember that GR is from the ground up designed to be a purely classical theory, and classicality precisely implies that there are no non-commuting observables.

Edited Monday at 06:10 AM by Markus Hanke
PS. Cross-posted with KJW, whose comment I didn’t see for some reason.

[studiot]

Just now, Markus Hanke said:

I don’t understand this comment - in GR, these things aren’t operators (or even observables in the QM sense), so it isn’t clear to me what you even mean by “non-commuting” in this context. We just have a differentiable manifold with four locally linearly independent basis vectors, plus a connection and a metric; there’s nothing in this basic structure really that is meaningfully relatable via Fourier transforms. 

Even in quantum mechanics, specifically for your last example, the correct pairing would be time and energy. Time and position do commute, with the caveat that treating time as an operator comes with its own complications in QM.

I think it’s also important to remember that GR is from the ground up designed to be a purely classical theory, and classicality precisely implies that there are no non-commuting observables.

Good insights getting straight to the heart of the matter. +1

 

4 hours ago, AbstractDreamer said:

Ok and  gravitation potential and mass density are paired?

Work it out for yourself.

Gravitational potential  times Mass Density

ML²T^-2  x   ML^-3  =  ?

[studiot]

1 hour ago, studiot said:

11 hours ago, AbstractDreamer said:

Ok and  gravitation potential and mass density are paired?

Work it out for yourself.

Gravitational potential  times Mass Density

ML²T^-2  x   ML^-3  =  ?

 

Actually I must own up to a definition error here

I gave the dimensions for gravitational potential energy, not gravitational potential (which is potentential energy per kg) so the multiplication is

 

Gravitational potential energy times Mass Density

ML²T^-2  x   ML^-3  =  ?

Gravitational potential  times Mass Density

L²T^-2  x   ML^-3  =  ?

 

neither of which work out to

 

ML²T^-1 

Edited Monday at 05:02 PM by studiot

[Markus Hanke]

22 hours ago, studiot said:

Good insights getting straight to the heart of the matter. +1

Just for the record though - there are things in GR where the concept of commutativity is meaningful and useful. An obvious example would be the directional covariant derivative, which doesn’t commute - this is precisely how Riemann curvature is defined in the context of GR.

[Ammaniya]

Einstein's equations remain valid. --- Is there an error in Einstein's theories? Could curving space and time be impossible? Some argue that it is equivalent to striving to empty something. Nothing can move at the speed of light. Excessive curvature in space requires a lot of energy. This could imply that black holes do not arise easily. Perhaps Einstein's math was imperfect. However, science is constantly verifying these assumptions. It's hard to say for sure right now. New ideas are constantly being investigated!

[Genady]

2 hours ago, Markus Hanke said:

Just for the record though - there are things in GR where the concept of commutativity is meaningful and useful. An obvious example would be the directional covariant derivative, which doesn’t commute - this is precisely how Riemann curvature is defined in the context of GR.

One does not need to deal with GR or QM to have meaningful and useful non-commuting operations. They are everywhere. For example, rotations in Euclidean 3D space do not commute.

[studiot]

Just now, Genady said:

One does not need to deal with GR or QM to have meaningful and useful non-commuting operations. They are everywhere. For example, rotations in Euclidean 3D space do not commute.

That is true for finite rotations.

However the calculus and for instance KJW's presentation is based on the fact that infinitesimals in the limit rotations do commute and differential geometry works at all.

And GR is based on diff geom.

That is why partials form linear combinations.

Also the discussion seems to be being dragged further and further away from the OP, which was about what happens in that limit.

And there is a fundamental issue between GR and particle physics that no one has thus far brought up.

 

[Genady]

39 minutes ago, studiot said:

there is a fundamental issue between GR and particle physics that no one has thus far brought up.

It is an old and well-known problem. What is in it to be discussed here?

[studiot]

Just now, Genady said:

It is an old and well-known problem. What is in it to be discussed here?

Obviously not well enough known to be noted here.

 

The issue is a practical one, that both GR and QM share since both assume infinite divisibility of space.

The issue is that the smallest particles we have identified are about 15 orders of magnitude greater in size than the planck length and 20 orders of magnitude greater than we have successfully been able to probe. ( note I am measuring size by L units, not M units  ie diameter not mass).

If you want to probe the mathematics of the region between this sizes I recommend this book which take you from Brirkhoff and Von Neuman (1936) through Segal (1947) to Kakutani(1948) and Gleason (1953) and Bogachev (1998) for mathematical models of what happens with Borel sets in (possibly infinite Hilbert spaces (manifolds).

The question of the meaning and existance of A*B and A + B and A-B and commutators is examined in great detail leading to Segals axiomatic statement of QM.

It is how ever admitted that (axiom VII) the justification is 'that it works'.

As my last reference indicated work has proceed since Mackay's 1963 original.

 

 

[Khanzhoren]

On 1/7/2025 at 5:54 PM, Halc said:

On 1/7/2025 at 5:54 PM, Halc said:

A solid is definitely classical. But what I should have said is that size doesn't apply to fundamental things like an electron.  All such things are quantum, but technically a horse is a quantum thing as well, so not all quantum things are without meaningful size.

 

As it is known, many classical properties (defined, known, and measured within the framework of classical physics) of solids or other similar systems are truly understandable only by using quantum physics through condensed matter physics. The properties that manifest "classically", including the concept of size, are consequences of underlying quantum phenomena. We can therefore possibly have a quantum definition of size that (rigorously) differs from the elementary classical definition , but which reduces to the equivalent concept in classical physics under certain conditions: this is what is done when defining the size of an atom or a molecule for example.

Even for something considered as fundamental as an electron, quantum field theory states that it is subject to vacuum fluctuations, these properties are intrinsically linked to its interactions with virtual particles and the associated creation-annihilation processes according to the energy scales considered (...) So even the description of the properties of a single particle considered as fundamental need a many particles (quantum) theory . Does the concept of size really make no sense in this context? Perhaps we could eventually introduce a quantum concept of  "size" for an electron which could be an extension of the classical concept and reduces to the latter in certain conditions? Maybe this size depends on the energy scale? The question is probably still open ?

Now what about things like the "gravitational singularities" that are discussed in this thread? It is known that a black hole, for example, results from the gravitational collapse of a system formed by a large number of particles (it is a composite system ... like a molecule or a solid). Couldn't we talk about its "size" in relation to a quantum description taking into account the localizations and movements of its constituents (governed by uncertainty relations, among other things)?

Edited 13 hours ago by Khanzhoren