**Ask a monkey a physics question thread**

[MadMojoMonkey]

To understand gravity is to understand that it's a purely geometric theory. Wrapping your head around the distortion isn't easy, and the distinction between space and time isn't clear for different observers, so it's still an absolute beast and when I say "understand" I mean at the most basic level. But that's still more than the other forces.

I'm of the opinion that the other forces are also geometric in nature and can be described as straight-line constant motion though insanely curved spatial dimensions. This is why I like string theory sorry hypothesis, because it brings into importance these other dimensions. But I'm not bound by rigorous science so maybe that's why I'm happy to indulge in it.

No major notes from me, here.

You're in good company with wanting string hypotheses to be a good description of reality. Some very clever people who've done amazingly smart things agree with you.

And I, to an extent, want that, too. I'm just not pretending that the failed attempts we've assembled thus far as the whole of humanity are "good." They are only "good" to the extent that they are forced to be good. Once you step outside the forcing, they're all bad. I.e. they can only explain what they are specifically manufactured to explain. They make no "extra" predictions that bear out to experimentation. This is not the hallmark of good science. It doesn't mean that any of the string hypotheses are wrong, but it means they're not useful.

The hallmark of good science is like... look at GR. Einstein wanted to explain 1 thing, but then his explanation made a host of wild predictions. When we look to confirm or refute those predictions, we find them readily (not easily) confirmed. This is what I'm holding my breath for with string hypotheses. They can explain known things, but also predict unknown things... and those unknown things can be experimented on and confirmed or refuted. I need a prediction that is not made by preexisting physics that is confirmed experimentally before I put any string hypotheses on any pedestals.

[OngBonga]

I guess the best proof for string theory would be gravity not perfectly obeying the inverse square law at small scales, and not necessarily as tiny as Planck scales, that's a prediction that no other theory, as best I'm aware, could readily explain. And it seems to me it must be a prediction, because to precisely obey the inverse square law implies exactly three spatial dimensions at all scales. Whether we can measure any deviation is another matter.

[MadMojoMonkey]

There's certainly a lack of deep understanding of gravity on the smallest scales.

I don't follow your statement here:
"it seems to me it must be a prediction, because to precisely obey the inverse square law implies exactly three spatial dimensions at all scales."

Why would that matter?

[OngBonga]

"it seems to me it must be a prediction, because to precisely obey the inverse square law implies exactly three spatial dimensions at all scales."

Inverse square is a very 3D function. If the universe were 2D then it would be intensity = d/r, in 3D it's d/r^2 (inverse square), if 4D it would be d/r^3 (inverse cube). To precisely obey inverse square is mathematical confirmation that gravity only propagates through exactly three dimensions. If that relationship weakens at small scales, if we begin to drift very slightly above a power of 2 as we approach the Planck scale, that's compelling evidence that there exist very small dimensions that gravity can propagate through.

[OngBonga]

It matters because string theory proposes that there exists these small dimensions and that gravity exists within these dimensions. So gravity cannot obey inverse square law precisely if this is true. You could perhaps argue that these dimensions exist but gravity doesn't propagate through them, but then it's not a quantum theory of gravity.

[OngBonga]

If gravity does propagate through small dimensions without drifting from an inverse square law, then that is hugely problematic when it comes to conservation of energy.

[OngBonga]

Any such dimensions cannot be infinite like the three spatial dimensions we're familiar with, otherwise we'd have noticed because gravity would be orders of magnitude weaker, especially if there's 11 infinite dimensions. These other dimensions must be closed and very very small for us to have not noticed yet, since gravity appears to precisely obey inverse square law. It's a question of measurement.

I would hope it's not the only prediction string theory makes because otherwise if it's wrong we'll be waiting a long time to confirm it's wrong.

[MadMojoMonkey]

Hmm... Can a 2D surface contain an n-dimensional volume? Or does it have to be an n-1 dimensional "surface" that bounds an n-dimensional "volume"?

You're probably right, there. Can't contain a 3d surface with 1D lines. Not unless you make the lines infinitely closely spaced such that they form a 2D surface, but that's still a 2D surface.

[OngBonga]

Hmm... Can a 2D surface contain an n-dimensional volume?

I have no idea, but it seems to me string theory proposes this to be the case. Well, it proposes a 3D volume can contain a 4th dimensional extravolume (I just made that word up). If gravity propagates through a fourth dimension, and that dimension is infinite, then gravity would obey the inverse cube law. Clearly that isn't the case. It is either precisely inverse square (and therefore 3 dimensions) or it is very close to inverse square (three infinite dimensions and one or more closed dimensions). If it's somehow lower than inverse square (<3D) that would make absolutely no sense and conservation of energy is already being violated.

If I were to wildly speculate like a stoner, I'd be inclined to believe that any extra dimensions are complex in nature. I can't wrap my head around complex numbers and how they relate to the real world, but I also have ultimate faith in the absolute truth of mathematics. Discovering i was a seismic moment in maths, it solved so many equations that many assumed were unsolvable.

The reason I think complex numbers are at play here is because the complex plane is very circular in nature. If you iterate multiplications of i you go round in a mathematical circle.

i*i = -1
-1*i = -i
-i * i = 1
1*i = i

I think Schrödinger himself was shocked that i emerged in the wave function. Or was it Heisenberg? One of them two. It was shocking for them to see what they believed was a mathematical curiosity emerging in the physical universe. It's basically proof that i is very much a real number.

[MadMojoMonkey]

The imaginary number i is useful, but no more real than any other so-called "real" number.
Which is to say... it's more a philosophical question than one of any matter or consequence.
Numbers are useful, whether or not they're "real."

There are other objects that square to -1, in other mathematical spaces aside from the complex numbers.

In a sense, i can be seen as a rotation operator of the number line. Which is useful when working with numbers.
There are similar objects in other math disciplines which represent rotations around axes.
There are other objects still which represent rotations in planes, which are mathematical duals of the axes objects in 3-space.

[OngBonga]

The number 1 is very real, assuming the universe is quantised. And any number where the equation x = 1x is true is also very real, since it's basically an extension of the number 1. By "real" I mean exists as a self consistent non-human concept in the universe, not the mathematical concept of "real numbers". The number 1 is not useful, it's fundamental. Without 1, there is only 0. That means a universe that doesn't exist.

There's nothing arbitrary or even philosophical about quantum energy levels.

[OngBonga]

I mean I guess 0.5 is a human concept if you want to argue the universe only does integers, but 0.5 is defined relative to 1, as is every other number such as pi, phi, i and xi (dunno if the last one is actually a number).

Infinity might or might not be real. That can't be defined as a function of 1. 1*infinity is indeed infinity, but infinity/infinity isn't 1 unless they are identical infinities. This is mathematical philosophy.

There are other objects that square to -1, in other mathematical spaces aside from the complex numbers.

I'm gonna have to dig into this.

[MadMojoMonkey]

The number 1 is very real, assuming the universe is quantised. And any number where the equation x = 1x is true is also very real, since it's basically an extension of the number 1. By "real" I mean exists as a self consistent non-human concept in the universe, not the mathematical concept of "real numbers". The number 1 is not useful, it's fundamental. Without 1, there is only 0. That means a universe that doesn't exist.

All of which is a philosophical argument.

How can you prove that any number is a non-sapient concept?
Many brilliant mathematicians have tried, but there's always a leap of faith at the start.

Something like, axiom 1) there is an empty set
WTF?!?

OK... so moving on.. oh hey! it's all of math!

But it always starts with some nonsensical statement that we just assume to be true. Axioms are exactly that... things we assume to be true w/o any motivation or proof.

[MadMojoMonkey]

I'm gonna have to dig into this.

Geometric algebra is a good start.
You don't need to get too deep into it before you see that 2-component multivectors (xy, yz, zx) represent 90 degree rotations in those planes. These are the duals (compliments) to rotations about an axis in 3D. And allow us to talk about rotations in 2D spaces w/o invoking a 3rd dimension for an axis of rotation and without invoking "imaginary" numbers.

The dual is like... in 3D rotation about an axis is equivalent to rotations in a plane. Indeed, when we teach 2D rotations, we impose an axis of rotation that is perpendicular to that plane of rotation. This is confusing to many students. It only works because in 3D an oriented plane contains the same number of degrees of freedom as a vector, and so we can link these together in a 1:1 mapping and use vectors.

This allows us to teach calculus w/o invoking multivectors, but there is the cost of ambiguity around why a 2D rotation necessitates a 3D axis... and the answer is that it technically doesn't, but we either have to teach you the entire framework of multivectors or stick with just regular vectors and set it normal to the plane.

All of this breaks down in 4D and higher dimensions. A rotation in a plane no longer is a rotation about an axis, it's a rotation about another plane. That's hard to visualize, but you can see why it's that way.



As a working example, let's look at what is (xy)^2, where xy is a multivector representing an oriented direction of 90 degree rotation in the xy plane. We'll need to impose the framework of how to multiply 2 vectors xy, which geometric algebra defines thus

xy = x [dot product] y + x [wedge product] y

The wedge product in 3D is the dual of the cross product. I.e. it's an oriented area, an area with a defined positive rotation direction. This is equivalent to an oriented area perpendicular to the cross product, with the same magnitude. I.e. it contains the same information as the cross product, but it is actually the dual of the cross product.

In this example, x and y are of unit length and are perpendicular to each other.
xx = x dot x + x wedge x = 1 + 0 = 1
yy = 1
xy = x dot y + x wedge y = 0 + xy
This object is it's own thing, a positive rotation in the xy-plane.

SINCE these objects are of unit length and perpendicular to each other, we have an additional property.
xy = -yx

So here we go
(xy)^2 = (xy)(xy)
(xy)(xy) = xyxy
I forgot to mention that multivectors are associative - i.e. x(xy) = (xx)y

xyxy = x(yx)y
x(yx)y = x(-xy)y
x(-xy)y = -(xx)(yy)
-(xx)(yy) = -(1)(1)
-(1)(1) = -1

Therefore (xy)^2 = -1
And the same holds for (yz)^2 and (zx)^2

[OngBonga]

How can you prove that any number is a non-sapient concept?

By proving that photons come in quanta? Humans didn't invent mathematics, we discovered it. Photons, and indeed all energy, comes in integer values. So the universe created the number line from 1 to however many energy quanta there are in the entire universe. That mathematical framework exists without life observing it and describing it. We might have expanded on it and there's an argument that fractions, zero, negative numbers and complex numbers are sapient, but the number 1, and multiples of it, are not.

That wall of math is a bit much for me to take in, sorry. I wish I could follow it but I can't.

[OngBonga]

Take pi as an example. This is a number that is directly related to the number 1 in that pi is the circumference of a circle with a diameter of 1. Since pi is famously irrational, there is no way of perfectly visualising this ratio. Thus, in a quantised universe, pi is an impossible number. Perfect circles don't exist. This is philosophy.

But 1 exists, and so does 2. No philosophy required, no mental gymnastics. Just quantised energy, actual physical matter.

[MadMojoMonkey]

That wall of math is a bit much for me to take in, sorry. I wish I could follow it but I can't.

The simple way to put it is that any object or operation that represents a 90 degree rotation, MUST square to -1.
Because applying it twice is a 180 degree rotation in *some* plane (about some axis if in 3D).


Something familiar:
Multiplying a real or complex number by i rotates it 90 degrees (about the origin) in the complex plane, so the fact that it squares to -1 must be true. If you rotate a point in the complex plane by 180 degrees, that is equivalent to multiplying each of its components by -1.
e.g.
(1+i)*i = i-1
(i-1)*i = -1-i

multiplying by i twice is the same as multiplying each of the components by -1.


Something new?:
This is true for anything A which rotates any other thing B by 90 degrees. Applying A twice (A^2) results in a 180 degree rotation in some plane, which is equivalent of multiplying the 2 components of object B which it shares in common with the plane by -1.

An example of that:
Given some point (x,y,z), if I rotate that point "in the xy plane" by 180 degrees, that would be (-x,-y,z), just what we'd expect if we said we were rotating about the z-axis by 180 degrees.

The utility of talking about rotations in a plane, rather than about an axis, is that rotation in planes extends to higher dimensions than 3D, whereas rotation about an axis does not.

[MadMojoMonkey]

Take pi as an example. This is a number that is directly related to the number 1 in that pi is the circumference of a circle with a diameter of 1. Since pi is famously irrational, there is no way of perfectly visualising this ratio. Thus, in a quantised universe, pi is an impossible number. Perfect circles don't exist. This is philosophy.

But 1 exists, and so does 2. No philosophy required, no mental gymnastics. Just quantised energy, actual physical matter.

I guess I get where you're coming from, but it still doesn't seem like a proof to me.

If an empty set exists, then I can imagine an empty set within that empty set, and I have created nothing, nor invoked a universe or atoms or energy levels. All that requires is that I can imagine partitioned nothingness. If partitioned nothingness isn't an absolutely absurd idea, then IDK what is.

Nonetheless, that's all we need. Anything which can springboard us into a constructing a sequence. Nesting empty sets within empty sets and labeling their "nested depth" is adequate. All we need is a notion of sequence. What we call each step in the sequence becomes the number line, whether it's one, two, three, or eins, tzwei, drei, or whatever.

From that you get all of math. Not exactly, but you get addition, subtraction, multiplication, division, exponentiation, complex numbers, rationals, irrationals, etc. You don't get *all* of math, but you get a nice huge chunk of it.


But still, using numbers is a sapient way of understanding things. What's to say it isn't some defect in humanity that keeps us from seeing that everything truly is 1, as many spiritual leaders have assured us..?

[OngBonga]

But still, using numbers is a sapient way of understanding things.

I'm not sure if you're using the word "sapient" here to refer to general intelligence of life, or specifically humans. If it's the latter, it's just wrong. Any animal that knows how many offspring it has is using numbers. They didn't invent a number system so they knew if any of their kids were missing, they use the one that already existed, the same one we use. We just have a much deeper understanding of the implications of that system than the animal does.

I strongly disagree that numbers are a purely intellectual concept. You might as well just go right ahead and say electrons are a human concept too.

[MadMojoMonkey]

I don't need to count to miss my brothers, lol.

[OngBonga]

I don't need to count to miss my brothers, lol.

You need to have a basic ability to count to know how many brothers you miss! And no matter what you call the numbers 1 and 2, it means exactly the same and they relate to each other exactly the same, to you, a German, and a duck counting its ducklings. It also means the same to the universe when "counting" energy quanta.

The idea it's intelligence is basically like saying that you need to consciously count in order for the maths to be real, to be aware of the numbers and their relation to one another. I disagree with that. If this were true, then different cultures would have different number systems. We might have different ways of describing the mathematical framework, but it's the same framework. So it's a discovery, not an invention. It existed before we found it.

[MadMojoMonkey]

I don't think I do need a sense of numbers to know this. The number of brothers I miss is irrelevant, and not even a part of my emotional attachment. I only know that I miss them. I can miss multiple things w/o any sense of how many things I miss.

The argument that humans behave in human ways doesn't lend to any reality, IMO.
The notion that Earth creatures do Earth creature things isn't any sound argument, and neither is a notion that all creatures in the universe do the same things as far as numbers are concerned, IMO.

Trying to make an argument that something intangible and unmeasurable is real is no easy task.


Show me 1. Point to 1. Help me identify and define 1 in practical terms of observations.
Certainly, you can show me 1 of something, and 1 of something else, ad nauseum, but you can't show me 1.
You can only keep waving your hands around about what you mean until I concede that I understand you.

You can just as easily describe to me a unicorn, and tell me all the properties of unicorns, and show me pictures of unicorns and all sorts of things describing unicorns. None of that makes unicorns real.

Utility is not proof of reality.

[OngBonga]

Show me 1. Point to 1. Help me identify and define 1 in practical terms of observations.

I already did, quantum mechanics, and of all people to respect that, I'd expect you to be that person.

I've shown it and pointed to it. I can define it as "the minimum possible energy value of a photon".

I'm not showing you one of something. I'm showing you that nature created what can be called the number 1, and the number 2. We created the bits in between.

Utility is not proof of reality.

What is proof of reality? What does "real" mean to you if it's not "something that exists in the universe"? Do you need to be able to, in theory, physically touch something for it to be real? That would mean gravity isn't real. Or do we just need to be able to observe it? We surely "observe" numbers.

Why is gravity more real than the number 1?

[OngBonga]

I mean, nature didn't create the number 1, nature created integers and we named them. But we're not having a discussion about semantics here as best I'm aware. It's not about what we call it, it's about the concept of a number 1 existing without human observation.

[OngBonga]

This is a totally different discussion when we talk about fractions and negative numbers. These are human extensions of the number line that are very useful and certainly consistent but they don't exist in the universe in any way that distinguishes it from the number 1. We humans can say well 0.5 is half on 1, so if 1 exists, so does 0.5, but this is philosophy. Nature doesn't do 0.5, there's no half a photon. What we call 0.5 could just as easily be redefined to 1 and so long as we redefine every other number in proportion to it, the mathematical framework still works just fine.

If the universe isn't quantised, then it's also a different conversation. But as best as we know, it is quantised, which means a set of integers emerges without a human having to understand it.

[MadMojoMonkey]

You have it right, as far as this isn't semantics, it doesn't matter what we name a number, all that is is a title on a position in a sequence.

I'm not compelled by a human model of the universe being proof of some fundamental property of the universe. We know our models are flawed, and we're not sure exactly how or where.

And I don't think I care. Back to what I said at the start, it doesn't matter whether or not numbers are real, numbers are useful.
But asserting that utility as a proof takes it a step too far, IMO.


FWIW, if sequences are not nonsensical things to talk about, then numbers aren't either.
If there is any sense in the statement, "I am not you," then numbers exist.

It's just that we have no proof of those statements being sensible aside from the fact that we like them.

[MadMojoMonkey]

I'd like to correct a misconception you've said repeatedly in this conversation.

Not all energy levels are quantized. There is no minimum energy of a photon. A photon at some hypothetical minimum would have to gradually lose energy as it travels through expanding spacetime (according to our current models), and thus it will decrease.

It is not at all clear what this apparent violation of conservation of energy is resolved by. It's a glaring flaw in that law which is otherwise very reliable. Photons lose energy as they travel through expanding spacetime, and it is not at all clear where that energy goes.

Kinetic energy isn't quantized, either. An object's speed is not universal. It can only be defined relative to another object. It can take on literally any value based on what it is in relation to.

Distances are not quantized. Directions are not quantized. We'd see crystal artifacts when looking out at space if it were, and we don't.

In short, the only things we can reliably say are almost certainly quantized are certain properties in *bound* systems. An electron in an atom can have only certain values of its kinetic energy, but an electron not bound to an atom can have any energy.

[OngBonga]

Photons lose energy as they travel through expanding spacetime, and it is not at all clear where that energy goes.

This is curious. Surely it's not "losing" energy, it's just being spread out?

Kinetic energy isn't quantized, either. An object's speed is not universal. It can only be defined relative to another object. It can take on literally any value based on what it is in relation to.

Kinetic energy is only half the story. The sum of kinetic energy and potential energy is universal. Kinetic energy and potential energy are the same thing viewed from different frames of reference. We might disagree on how much kinetic energy an object has, and we might disagree how much potential energy it has, but we won't disagree on the sum of both.

As for if it's quantised, well let's move on to this...

Distances are not quantized.

Distance loses meaning at sub-Planck scales. It might be the case that the Planck length IS the minimum possible distance. This seems like an open question in physics, so I'm not sure how you can say with certainty that distance isn't quantised.

We'd see crystal artifacts when looking out at space if it were, and we don't.

We would? Why?

[OngBonga]

A photon at some hypothetical minimum would have to gradually lose energy as it travels through expanding spacetime (according to our current models), and thus it will decrease.

More on this. As I understand it, the photon is being "stretched" (for lack of a better word) by expanding spacetime. So as its frequency decreases, its wavelength increases proportionally.

So no energy is lost. Is this not the accepted model? It's like stretching an elastic band. Its mass remains constant, but its density does not.

[OngBonga]

Slightly off on a tangent but the mass of a stretched elastic band is probably very slightly more than the mass of a slack band. By stretching it you're adding energy to it, and energy and mass are equivalent. But obviously spacetime isn't literally stretching a photon in the same way, there's nobody adding energy to it. There's no reason to think a "stretched" photon experiences tension.

[MadMojoMonkey]

This is curious. Surely it's not "losing" energy, it's just being spread out?

To where? It's one of the only things I know of that isn't rigorously following the conservation of energy by some known mechanisms.

Kinetic energy is only half the story. The sum of kinetic energy and potential energy is universal. Kinetic energy and potential energy are the same thing viewed from different frames of reference. We might disagree on how much kinetic energy an object has, and we might disagree how much potential energy it has, but we won't disagree on the sum of both.

As for if it's quantised, well let's move on to this...

The point stands that kinetic energy isn't quantized. Kinetic and Potential Energy are properties of systems, not individual objects.
What you've argued is that since the total energy is constant and kinetic energy is not quantized, then at least 1 other energy is not quantized as well.

Distance loses meaning at sub-Planck scales. It might be the case that the Planck length IS the minimum possible distance. This seems like an open question in physics, so I'm not sure how you can say with certainty that distance isn't quantised.

Because we do not see crystal artifacts when we look into space across any time/distance scales.

We would? Why?

Because the reason we see artifacts when looking through crystals is exactly the fact that the geometry of the crystal's atomic structure quantizes the spacial degrees of freedom. It implies geometric structures like planes and holes. When a wave bounces off a crystal, it has a probabilistic chance to bounce off of any of the implied planes in the crystal lattice. Those planes are not all parallel. Imagine a square grid. Lines at 0 or 90 degrees are implied lines, but also lines of integer fraction slopes are implied lines. Same in 3D.

This is a purely geometric property of waves. If they cannot propagate through a continuous medium, they will diffract or reflect. Any geometric lack of homogeneity imparts a change to the way waves propagate through that medium.

So what we'd see looking at the sky would be weird spikes and dots and diffraction based on the quantized geometry of the space through which the waves propagate. That would tell us the nature of the quantized space. Much like we use gravitational lensing to tell us about parts of space that we cannot directly observe.

E.g. the bullet cluster. We see gravitational lensing though we do not see the source of the mass. That geometric lack of homogeneity is something we can detect.

[MadMojoMonkey]

More on this. As I understand it, the photon is being "stretched" (for lack of a better word) by expanding spacetime. So as its frequency decreases, its wavelength increases proportionally.

So no energy is lost. Is this not the accepted model? It's like stretching an elastic band. Its mass remains constant, but its density does not.

Decreasing frequency is the same thing as increasing wavelength for any wave that moves at constant speed.
Since all photons move at c, this applies to all photons.

For photons
[wavelentgh] = c / [frequency]
or
[wavelength] * [frequency] = c
with c being appropriately the speed of light, which is a constant

The energy of a photon is
E_ph = h*[frequency] = h*c/[wavelength]

where h is Plank's constant (not the reduced constant)

Photons traveling through expanding spacetime have their wavelength increased (in the denominator), which, since they move at constant speed, means their frequency is reduced. All of which means the photons lose energy.


But here's the fucky part.
In a photon's non-inertial reference frame (frames moving at c are non-inertial), the universe is infinitely thin in its direction of travel and no time passes between when it is emitted and absorbed. Furthermore, the position of a photon is ill defined in any reference frame. The momentum of the photon is linked to its frequency as well, and since the momentum of the photon is fully defined, the position is fully undefined. Because of position momentum uncertainty.

So the sense of exactly what does it mean to say a photon is moving through expanding spacetime is not trivial.

[OngBonga]

To where?

Just going to bed but I'll quickly answer this for now.

It's better to imagine energy in a volume. A photon has x energy in y volume. As y increases, that is as spacetime expands, x decreases, that is the energy of the photon decreases. But the total energy in the volume remains constant. The photon is "bigger", it's been "stretched", so the energy is spread out over a larger volume.

I don't think that's literally what's happening, because photons don't have size in any meaningful way, but it's what it looks like if you're not moving at c. Space isn't expanding from the FoR of the photon because it doesn't experience space, so from its FoR, nothing happened, it didn't lose energy, it didn't get stretched. It's an illusion that we sub-c entities observe. So basically I'm arguing space isn't expanding, that it's just a relative illusion.

So probably bollocks.

Good night!

[MadMojoMonkey]

If space isn't expanding, then why does it look like it's expanding?
Not only expanding, but accelerating in its expansion?

[OngBonga]

If space isn't expanding, then why does it look like it's expanding?
Not only expanding, but accelerating in its expansion?

It's not if you're a photon. To expand not only requires space, but also time, neither of which a photon experiences.

If someone is falling from a plane, as you watch the speed of their fall increase, you will say they are accelerating. Einstein would disagree and say the observer is the one accelerating. So we know acceleration is an illusion, it's relative.

idk why space looks like it's expanding, but why is our FoR more important than the photon's? Both frames of reference are equally valid.

[MadMojoMonkey]

It's not if you're a photon. To expand not only requires space, but also time, neither of which a photon experiences.

If someone is falling from a plane, as you watch the speed of their fall increase, you will say they are accelerating. Einstein would disagree and say the observer is the one accelerating. So we know acceleration is an illusion, it's relative.

idk why space looks like it's expanding, but why is our FoR more important than the photon's? Both frames of reference are equally valid.

Yeah. It's all a mind fuck with photons.

Just 'cause something is a property of relativity, doesn't mean it's an illusion..? Just that it's open to interpretation.
Einstein would say it's not invariant. Einstein did a lot of work on what are the things which are invariant, which all observers agree upon. Understanding invariants helps us to better understand things which do vary.

Like you said, all frames are equally valid. Whether you see something as unchanging or changing may or may not be a local phenomenon, but we assume there is an objective universe which perspective doesn't alter, and thus all perspectives are valid.

The problem with photons is that there are obvious contradictions, which you pointed out. A photon does not experience space or time, and yet, we observe that photons lose energy when traveling through spacetime. Hence the conundrum at the heart of this discussion.

We seem to be seeing conservation of energy not doing what it otherwise always does, and to an object that it seems like it shouldn't even be able to touch.

Something is missing from our understanding.

[OngBonga]

Just 'cause something is a property of relativity, doesn't mean it's an illusion..? Just that it's open to interpretation.

Einstein used better language than I did! He understood that he needed to properly define things, not least because everyone hangs off his every word. I'm just a dude with a spliff.

By "illusion" I simply mean things aren't always as they appear. Gravitational lensing would certainly qualify as an illusion, even though it's very real. Gravity itself is an illusion, we observe acceleration when it's actually constant speed in straight lines in curved space, but gravity is still real. I don't mean "illusion" to describe something we observe that doesn't really exist. Just that our intuitive interpretation of something is inaccurate.

Something is missing from our understanding.

If light is "losing energy", then is this not evidence of extradimensions? We know how much space is expanding, right? So we can calculate what we should expect the energy of a photon to be after is has passed through a known amount of space. It's not going to be inverse square because the expansion is, mathematically, a new dimension, but if we know the value of this expansion then we can factor this in. If there's still something missing, then that for me looks good for string theory. It's going to be really difficult for us to detect gravity in these extra dimensions, but light? Much easier because we can actually see it, measure it with astonishing accuracy. If we couldn't measure it that accurately we wouldn't even be talking about an apparent violation of conservation of energy, because it would be assumed to be measurement issues.

I think string hypothesis might have an answer to your problem here buddy.

[OngBonga]

It could also be that space isn't expanding at precisely the rate we think it is.

Light isn't "losing energy", we're simply failing to account for that energy.

[MadMojoMonkey]

Einstein used better language than I did! He understood that he needed to properly define things, not least because everyone hangs off his every word.

Einstein was interested in invariants - those measurable properties upon which all observers agree. Though they may need to do some math to convert their direct measurement into the invariant one in order to compare.

I think string hypothesis might have an answer to your problem here buddy.

Maybe.
If so, it's pretty daunting that after many decades of attempting to find a working string theory, none has yet been found.

Something is missing, whether strings or otherwise. Extra dimensions may be a property of spacetime, but not strings. String hypotheses have to shoehorn in extra dimensions to not break everything. That's not evidence of a good model. That's just brainstorming for solutions. If those solutions yield a contradiction with existing data, they gotta go. If they predict something different than the current model, then that difference needs to be tested on.

If where the prediction is inconsistent is not even theoretically measurable, then maybe those descriptions are equivalent descriptions, and it's just a human failing to see the 1-ness of them. Famously Schroedinger and Heisenberg said the other was wrong, until someone finally showed that you could derive either one from the other, and therefore, those are the same statements wrapped in different trench coats.

I've never said there will not be a working string theory. I've only said that there currently is not one, despite many decades of valiant effort by some of the smartest and most dedicated people able to work on it.

[ash190]

Imagine you're in a vacuum, and you have a perfectly elastic ball
you drop it from a height 'h.', after hitting the ground, it bounces back up to a fraction of its original height, say 'b.'

calculate the COR for this ball

[OngBonga]

Not enough information.

[OngBonga]

Questions -

Is the ground perfectly elastic?

Is the ball rotating? Is the ground rotating?

What are the mass ratios between ball, ground and observer? Are there any other masses in this universe?

Are all objects in this universe in thermal equilibrium?

[MadMojoMonkey]

Imagine you're in a vacuum, and you have a perfectly elastic ball
you drop it from a height 'h.', after hitting the ground, it bounces back up to a fraction of its original height, say 'b.'

calculate the COR for this ball

For everyone else, CoR is Coefficient of Restitution.


without googling, it's b/h, right?


post googling, ah... no it's not energy ratio, it's speed ratio before and after bounce
Energy is kinetic there, so the ratio would be of the square root of the max height
so
COR = SQRT(b/h)

[MadMojoMonkey]

Questions -

Is the ground perfectly elastic?

Is the ball rotating? Is the ground rotating?

What are the mass ratios between ball, ground and observer? Are there any other masses in this universe?

Are all objects in this universe in thermal equilibrium?

The "normal" assumption in this case is that the ground's mass is much much greater than the ball's mass, and that it is perfectly rigid.
Neither is really true, but when they invoked gravity (well, not necessarily gravity, but some acceleration) in their question, w/o further information, we the solver are free to insert the simplest interpretation of what they meant.

Since no rotation was stipulated, again, we are free to assume whatever is simplest to solve/answer the question. Non-rotating bodies simplify things, so we'll assume that.

The masses are a red herring, all that matters is the local g field is "uniform," so again, we'll assume a "very large" planet that is "very massive." Effectively, we treat the ground as having infinite mass and infinite rigidity. This is simplest. It cannot be accelerated or deformed.

We are free to assume that the floor and the ball are in thermal equilibrium, as that is simplest. I don't think the rest of the universe matters, except insofar that we will, of course, assume that the floor/ball system is isolated from external interactions like heat transfer into either object from the outside universe.

[OngBonga]

I thought it was b/h, but that's because I assumed the scale from 0 to 1 was linear. But still, assuming ideal conditions with perfect elasticity and no other influences, the answer is still a COR of 1, I believe. The ball bounces to precisely to the same location from where it was dropped. Fun fact - the ground also falls slightly towards the ball, and then bounces back the same distance it fell, which gives it a COR of 1 too.

The masses are a red herring.

I don't believe so. We have three masses, because "you" are dropping the ball. Even if we have perfect elasticity, an absurd concept in its own right, we have a third gravity influence altering the velocity of both the ball and ground.

Effectively, we treat the ground as having infinite mass...

Infinite mass? The COR of a ball "bouncing" off an infinite mass is surely precisely zero.

I don't think the rest of the universe matters...

See above. It does matter because other objects in the universe mean non-uniform gravitational influences. But if we assume a perfectly isolated system, with a ghost dropping the ball onto a much much larger ball (not actually important, balls can be same size), made of the same perfectly elastic material with perfectly uniform density, and everything else being perfect so there is only ball ground and bounce, then it's COR=1 surely.

[OngBonga]

I think the concept of "ground" is open to interpretation.

You're imaging a flat barrier while I'm imagining a planet-like object. So I'm visualising a mutual gravitational interaction between the two masses, while you're basically imagining a ball bouncing off the immovable and unpenetrateable edge of the universe.

[OngBonga]

Thing is though, we're to assume that the laws of physics apply, it's just we're making some seriously absurd assumptions regarding isolated systems and the ideal conditions.

If the laws of physics apply, then the ball falls because of gravity, and gravity happens because of mass. We must assume, for the laws of physics to not be violated, that the ball has a non-zero finite mass and so does the ground, and that the ball falls because of gravity, in a predictable way. This is why the ball falls. If the ground has infinite mass, well this seriously complicates gravity and probably a fuck ton of other physics concepts, such that the idea of a ball bouncing off such a surface is ludicrous. The energy released when the ball (this is a pretty loose term in such a gravity field because it's not going to be a sphere) hits the ground will probably destroy the universe, certainly in a ball ground and observer universe.

[MadMojoMonkey]

I thought it was b/h, but that's because I assumed the scale from 0 to 1 was linear.

I mean... it is linear in speed, it's just nonlinear in energy.

Max height of an object in projectile motion is a measurement of it's total mechanical energy (simplified to Potential + Kinetic, here). It is at rest in a potential energy field at some distance h from what we've chosen to call 0. At height 0, we model that 100% of its potential energy is turned into kinetic energy. That part of the model is linear in energy.

At the elastic collision, we have a speed that is before and after. The COR is linear in this speed ratio, with 1 being a perfectly elastic collision, and 0 being a perfectly inelastic collision.

Then we're back into the projectile motion regime and it's linear in energy again, with total energy constant.

But still, assuming ideal conditions with perfect elasticity and no other influences, the answer is still a COR of 1, I believe. The ball bounces to precisely to the same location from where it was dropped. Fun fact - the ground also falls slightly towards the ball, and then bounces back the same distance it fell, which gives it a COR of 1 too.

Rereading the original question, you got it.
Saying the ball is perfectly elastic is one thing, but the ground itself wasn't said, and the ball-ground interaction is what's under scrutiny.
But if we assume the ground is perfectly rigid, then the ball being perfectly elastic seals the deal. COR = 1.

It's kinda a trick question because perfect elasticity means COR = 1, and knowing how to calculate COR when b != h is a red herring.

I don't believe so. We have three masses, because "you" are dropping the ball.

By assuming the ground's mass is "much much greater" than the ball, we also assume it's much much greater than our own mass, as well.
In effect, we're assuming the ground's mass is infinite; it is un-accelerate-able, and a suitable place to attach a Newtonian inertial reference frame.

Which, as you say, is an absurd approximation.
Or is it?
The Earth's mass is dozens of orders of magnitude higher than my mass, which is assumed to be within a couple orders of magnitude of the ball.

If we're assuming the reason the ball accelerates is due to gravity, then the assumption that we're on a roughly Earth-sized planet seems apt. Under those conditions, the fault in our approximation will show up only dozens of decimal places deep. So we're technically wrong, but not measurably wrong. So the wrongness is quantified as "acceptable."

Even if we have perfect elasticity, an absurd concept in its own right, we have a third gravity influence altering the velocity of both the ball and ground.

In that sense, all knowledge is absurd.

Perfect elasticity isn't entirely absurd, no more than any other model (guess / approximation) of reality.

Think of 2 magnets' like poles repelling each other.
Mostly, what is deformed that is the origin of the force?
The magnetic field.
The (electro)magnetic field is perfectly elastic. Any energy put into deforming it is given back in full.

Infinite mass? The COR of a ball "bouncing" off an infinite mass is surely precisely zero.

Precisely 0 is perfectly inelastic, when the ball sticks to the wall.
The "after" speed is 0, and that goes in the numerator of the COR fraction.

See above. It does matter because other objects in the universe mean non-uniform gravitational influences. But if we assume a perfectly isolated system, with a ghost dropping the ball onto a much much larger ball (not actually important, balls can be same size), made of the same perfectly elastic material with perfectly uniform density, and everything else being perfect so there is only ball ground and bounce, then it's COR=1 surely.

Provided the given statement that the balls a perfectly elastic, yes.

If the origin of the force that causes the acceleration during the bounce is the deformation of macroscopic objects (like balls), then the COR will always be less than 1. The deformation will cause some internal heating, which is energy lost from the mechanical interaction.

[MadMojoMonkey]

Thing is though, we're to assume that the laws of physics apply, it's just we're making some seriously absurd assumptions regarding isolated systems and the ideal conditions.

If the laws of physics apply, then the ball falls because of gravity, and gravity happens because of mass. We must assume, for the laws of physics to not be violated, that the ball has a non-zero finite mass and so does the ground, and that the ball falls because of gravity, in a predictable way. This is why the ball falls. If the ground has infinite mass, well this seriously complicates gravity and probably a fuck ton of other physics concepts, such that the idea of a ball bouncing off such a surface is ludicrous. The energy released when the ball (this is a pretty loose term in such a gravity field because it's not going to be a sphere) hits the ground will probably destroy the universe, certainly in a ball ground and observer universe.

These are valid points.

If the ground's gravitational mass is infinite, the ball's energy upon contact is infinite, and it will have long since dissociated from being anything you could reasonably call a "ball." Unless you mean a spaghettified ball of plasma, I guess.

All we're really assuming is that the ground's *inertial* mass is infinite.
IRL, there's no fundamental reason the inertial mass, the exertional gravitational mass, and the responsive gravitational mass must all be the same thing. At least, there's nothing yet in the human model which predicts these 3 masses should be the same.
They just happen to be the same, as far as humans can measure.


All we really need to assume is some constant acceleration and a reasonably rigid bouncing surface which is perpendicular to the acceleration.

[OngBonga]

Rather than assuming an infinite mass, it's better to just measure h as a fixed point from the ground, which means h moves if the ground moves. That essentially fixes the ground in the sense that it makes it mathematically immovable, while not complicating gravity and energy!

The (electro)magnetic field is perfectly elastic. Any energy put into deforming it is given back in full.

This is a really good point.

the deformation will cause some internal heating, which is energy lost from the mechanical interaction.

At least heat loss will be reduced to the bare minimum blackbody radiation, seeing as we're in a vacuum and we're assuming the ground is in thermal equilibrium with the ball!

[OngBonga]

The distinction between a ground that moves and doesn't move is actually critically important when we consider the COR of the ground in this scenario. If the ground is some immovable perfectly rigid object, it has a COR of 0. If it moves just a single Planck length due to gravitational interactions, and then back again when it bounces off the ball, it has a COR of 1.

I prefer the latter solution because an immovable object is one ideal assumption more than necessary. It's not even possible in a perfectly ideal universe, it's just a made up concept that has no meaning in physics.

[MadMojoMonkey]

At least heat loss will be reduced to the bare minimum blackbody radiation, seeing as we're in a vacuum and we're assuming the ground is in thermal equilibrium with the ball!

My point is that the internal energy of the ball has gone up, and that energy came from somewhere.

If anything that is not the mechanical energy goes up, then the mechanical energy must go down to compensate, such that total energy is conserved.

[MadMojoMonkey]

The distinction between a ground that moves and doesn't move is actually critically important when we consider the COR of the ground in this scenario. If the ground is some immovable perfectly rigid object, it has a COR of 0. If it moves just a single Planck length due to gravitational interactions, and then back again when it bounces off the ball, it has a COR of 1.

I prefer the latter solution because an immovable object is one ideal assumption more than necessary. It's not even possible in a perfectly ideal universe, it's just a made up concept that has no meaning in physics.

I need to google if COR is a single-particle property, or a 2-body interaction property.
I feel like it's the latter, but can't commit.


Easiest to just ignore the complications and treat the ground as "unaccelerateable" which is also nonsense, but it gets us there.

E.g. if I'm to perform this experiment in my office, then the mass of the ball is a couple tenths of a kg, whereas the mass of the Earth is 6(10)^24 kg. My estimation that the motion of the Earth is 0, and the motion of the ball is all that matters is technically wrong, but I'd be unable to measure that wrongness using any reasonable measurement device.

Newton's 3rd law says that the force the Earth exerts on the ball is equal in magnitude and opposite in direction to the force the ball exerts on the Earth. Equal and opposite forces forming a 3rd law pair.

F = ma
so
F/m = a

The same F applying to 2 drastically different masses will yield accelerations differing by as many orders of magnitude.

If the ball bounces 1 m, the Earth's bounce in the opposite direction adds another ~10^-25 m to the bounce.
That'd take 10 billion of those to get to the diameter of a single proton (~10^-15 m).

So, in practice, we're technically wrong to not include the acceleration of the Earth, but just as practically, we'd be unable to measure that difference, even if we accounted for it with our math.

The upshot is that we can make our lives a lot easier by ignoring it in our math and still being right enough.

[MadMojoMonkey]

"The COR is a property of a pair of objects in a collision, not a single object. If a given object collides with two different objects, each collision would have its own COR."

From the wikipedia page.

So yeah. "the COR of the ball" is on the face of it an absurd statement. I can only interpret it to mean all the interactions between pairs of particles that make up the ball are perfectly elastic... which is true even in the case of thermal heating, so ... still absurd.

I think a better way to say what they meant would be to say something about the ball's internal energy being constant throughout the bounces. That "internal energy" covers all sorts of temperature, pressure and volume changes.

[OngBonga]

I feel like we got quite a lot of mileage out of that question posed by a random internet user. Cheers Ash.


It is interesting though. If you have two objects of vastly different mass "bounce" off each other such that the mathematical motion of the larger body is less than the Planck length, has the larger body moved at all? How would it be possible for such a transfer of energy to happen? Maybe in this case the ground does behave like a perfectly rigid surface.

[MadMojoMonkey]

I wonder if there's anything I can possibly say that will make you believe that the only thing the Plank length tells us is that our model stops working in the limits of the very very tiny. It makes no predictions.

That's the only significance of the Plank Length... whatever it actually is. It's just a trick of math that gives a length from certain measured values. Interpretation of that value may or may not be meaningful in physics.

In this case, there is some meaning in the sense that below some very tiny length scales, our Standard Model breaks down and stops making any sensible predictions.

That doesn't mean anything at all about the physics below those scales. It's just a statement about human ignorance or the limitations of models.

[MadMojoMonkey]

Maybe in this case the ground does behave like a perfectly rigid surface.

Perfect rigidity is problematic in the same ways, it just takes a bit of Einstein to see it.
You can google Born Rigidity (named after physicist Max Born) to see why it's problematic, but it's pretty heady to understand.
IDK, though... your knowledge of Einstein's Relativity is way above the average Joe, so maybe you can dig it.

If an object is perfectly rigid, then it cannot experience length contraction, but then it cannot accelerate at all.
This takes more than a passing glance to prove, and an old model is to imagine 2 identical rockets connected by a thread. The 2 rockets are exactly the same and accelerate in exactly the same direction such that the tension on the string is constant. For all intents and purposes, they are 1 rigid object, just a funny shape.

The thread is needlessly thin to show that it simply must change length or break... and if it breaks, then that's a problem... 'cause it means that a perfectly rigid body must change shape in order to not break... which is a contradiction with "perfectly rigid."



Ergo, it cannot be real. It would be the hypothetical immovable object.

[OngBonga]

I wonder if there's anything I can possibly say that will make you believe that the only thing the Plank length tells us is that our model stops working in the limits of the very very tiny. It makes no predictions.

I wonder when it will dawn on you that the Planck length is more than just a length!

https://www.youtube.com/watch?v=bjVfL8uNkUk&t=2s

Maybe Arvin can explain better than I can. Skip to 6 minutes if you don't want to be patronised.

He is at pains to point out that we simply do not know what happens at sub-Planck scales. However, it's not a question of measurement accuracy, it's a fundamental limit at which our models stop working. Our 4-dimensional model of the universe doesn't work at sub-Planck scales. That means length as we understand has absolutely no meaning. It's not a math trick, no more than c or g are math tricks. It's a fundamental constant of nature.

If an object is perfectly rigid, then it cannot experience length contraction

I need to think this one through!

[OngBonga]

c tells us the speed of causality... basically how fast we move through time.

g tells us how much a given mass will bend spacetime.

h-bar tells us something too, we just haven't really figured out what.

[MadMojoMonkey]

He is at pains to point out that we simply do not know what happens at sub-Planck scales. However, it's not a question of measurement accuracy, it's a fundamental limit at which our models stop working. Our 4-dimensional model of the universe doesn't work at sub-Planck scales. That means length as we understand has absolutely no meaning.

That doesn't mean anything at all about the physics below those scales. It's just a statement about human ignorance or the limitations of models.

It's not a math trick, no more than c or g are math tricks. It's a fundamental constant of nature.

I need to think this one through!

c and G are not math tricks. They're constants you have to go out and measure. There's no predictive model of what those values *should* be, there's only the busywork of actually doing the science and measuring them.

Google the Buckingham Pi method.
Google other "Planck" units. Like Planck energy, etc. All found by the same method as the Planck Length, but whether or not they tell us anything about the universe is not guaranteed.

[MadMojoMonkey]

c tells us the speed of causality... basically how fast we move through time.

g tells us how much a given mass will bend spacetime.

h-bar tells us something too, we just haven't really figured out what.

Why use h-bar and not h?

The difference is only a factor of 2 pi, but which one you choose to use in the calculation of the Planck Length will obviously change the value you calculate.

So which one gives the "real" Planck Length? Why is that the one?

[OngBonga]

So which one gives the "real" Planck Length? Why is that the one?

I have no idea, but maybe you can translate this into something I can grasp...

The whole discussion begs the question: If ℏ
is so convenient, why do we have h
around?


As usual, "historical reasons".


Planck originally invented h
as a proportionality constant. The problem he was solving was blackbody radiation, for which the experimental data came from spectroscopy people. And spectroscopy people used ν
(for frequency, for that or wavelengths were what they measured). So the data was tabulated in frequency. So, when he formulated his postulate, he used E=nhν
for his quantization.


In modern theory, we prefer working with ω
rather than ν
, because it is annoying to write sin(2πνt)
rather that sin(ωt)
. With angular frequencies, the quantization postulate becomes:


E=nh2πω


Now life sucks. So we invented the shorthand:


E=nℏω

[OngBonga]

Google other "Planck" units. Like Planck energy, etc. All found by the same method as the Planck Length, but whether or not they tell us anything about the universe is not guaranteed.

Well I'm not arguing this, and it's clear the Planck mass is not a fundamental lower bound of any sort because it's massive (on these scales). Probably Planck volume is fundamental, and Planck time, but we're still talking spacetime units here. So this constant appears to be telling us something about spacetime.

[OngBonga]

Some other reading that's bending my brain...

(somewhat cropped)

Nevertheless, the importance of Planck's constant went far beyond its initial use. It is the fundamental constant associated with quantum theory. Any theory that does not contain Planck's constant is not a quantum theory. Furthermore, if you take Planck's constant to zero, you can convert a quantum theory to a classical theory. This gives us some insight into the physical nature of Planck's constant.

Let's first consider angular momentum. Planck's constant must pop up in the treatment of angular momentum given that its units are those associated with angular momentum. In fact, it is the quantum unit of angular momentum. Any change in angular momentum occurs in integer units of Planck's constant. Thus angular momentum is quantised. But there is more to quantum theory than just angular momentum. This gives some insight into Planck's constant, but is not sufficiently general.

Planck's constant is often simplified to representing a quantum of energy. This is not strictly correct. Instead, it provides the energy of a quantum object with a specific frequency. It cannot be a quantum of energy because the units are incorrect. Furthermore, this is not just applicable to photons, which are quanta of the electromagnetic field.

And then there's this which is really something for you and not me because it's fucking French to me...

Phase space is a convenient way of expressing dynamics of systems. A phase space corresponds to considering position and momentum as orthogonal coordinate axes. Then the dynamics of a mechanical system traces out some sort of pattern in this phase space. For example, a simple pendulum will trace out an ellipse in phase space. For a classical system, the state at any time can be represented by a point in this phase space. However, for a quantum system, this is not the case. Planck's constant represents a volume in phase space. Therefore it cannot be a point! In fact, this corresponds precisely to the Heisenberg uncertainty principle. A property of phase space volumes is that they are invariant if energy is conserved. That means if you squeeze one axis, the other expands to compensate to retain the volume. Thus, the quantum of phase space volume represented by Planck's constant can be squeezed and stretched, but retains its volume. In short, the introduction of Planck's constant directly gives the Heisenberg uncertainty principle. Furthermore, this explains why the classical limit corresponds to taking Planck's constant to zero, as this allows the phase space volume to disappear back to a point, where both position and momentum are precisely defined.

I won't go into the other interpretations of Planck's constant related to the Action. One often reads that Planck's constant represents a quantum of the Action. However, it is not so clear what that actually means. It certainly doesn't mean that systems evolve in quantised steps of the action. The evolution of systems is smooth. Rather, one must consider the path integral formulation of quantum theory as introduced by Feynman. This quantum of the action represents some sense of the extent of a classical path among the sum over all paths. So in a sense it is a measure of the breadth of the Action. I personally do not find this too enlightening, so won't pursue this further.


For me, the biggest insight comes through the phase space analysis. The interpretation of Planck's constant as a fundamental quantum of phase space volume directly gives us the Heisenberg uncertainty principle.

Overall, Planck's constant serves a number of roles, which is why it is difficult to pin down its physical meaning. It is a measure of the fundamental quantum of angular momentum; a measure of a quantum of phase space volume; and a measure of a quantum of the Action. Moreover, and perhaps most importantly, it provides a conversion between classical and quantum Fourier theory, which is the essence of quantum theory. That's really doing a lot of heavy lifting for a single constant.

[MadMojoMonkey]

I have no idea, but maybe you can translate this into something I can grasp...

Well, they got part of it wrong, but I'm willing to let it slide for their otherwise assuming a tone I rather enjoyed.

They said E = n h 2 pi w
Should be E = n h w/(2 pi)

The relationship between frequency (cycles per second), v, and angular frequency (radians per second), w, is
w = 2 pi v
so
v = w/(2 pi)


Planck's equation for the energy of a photon of a given frequency, v, is
E = hv

For n photons, it's
E = nhv


Substituting w for v brings along that divisor of 2 pi, and the energy of n photons with given angular frequency is
E = n h w/(2 pi)

But we can shift the denominator of 2 pi to anywhere in that equation, since it's 3 multiplied terms over a denominator.
So
E = n h [w/(2 pi)]
becomes
E = n [h/(2 pi)] w

which we now simplify by giving this new constant divided by pi a new name, h_bar
And thus
E = n ℏ w


In some equations, h is more convenient, in other equations, ℏ is better.

[MadMojoMonkey]

The phase space thing isn't so hard to understand. Just a fancy name for a familiar thing.

Replace the x and y labels on your axis with any 2 other things. Tada. You made a phase space!

Here, we're replacing x with x ... uhh... it's fine. Don't worry about it.
And we're replacing y with p. P for momentum, obv. sheesh.

So instead of an xy graph, we have an xp graph. Phase space!

Now, the uncertainty relation they're talking about without saying it is that old, familiar
delta_x * delta_p >= ℏ

In our graph, the delta_x is a range of values in the x-axis, and likewise, the delta_p is a range of values in the p-axis.
Drawing these out, we see that they make a rectangle area on our phase space.

The thing you posted consistently misnamed this area as a volume.

Since the uncertainty relation is greater than a non-0 value, that means we can't shrink the rectangle down to a point, which would allow us to know the position and momentum to arbitrary precision at the same time.

If it were >= 0, then that's just the classical world where it *can* go to 0. I.e. the area can always be bigger than the minimum.

We can always know very little about either position or momentum. But as we try to know both better and better, we find that at a certain point... reducing the uncertainty in one direction (x or p) always increases it in the other direction. Such that there is a minimum area that rectangle can have.

[OngBonga]

It's not easy finding something reliable on this topic. Those posts are cropped from a discussion asking what the physical significance of the Planck constant is.

Is he being accurate when he claims that angular momentum is quantised? And is he being accurate when he claims that a Planck constant of zero turns a quantum theory into a classical theory? Because if those two claims are indeed accurate, then that's pretty solid evidence that we're talking about a constant of nature, not a simple length.

I think it's easy to get caught up in the "length" aspect of it, in the same way people talk of the speed of light where actually it's not about light, it's about causality. The speed of light emerges from the speed of causality. Likewise, the Planck length emerges from quantum mechanics.

If I were to guess which Planck length is "correct", I'd say h-bar because it's smaller, and if we can have a "smaller" length than the smallest length, there's a problem with the entire argument right there. If h is right, then h-bar is meaningless, but the math holds up under theoretical scrutiny as best I'm aware, otherwise they wouldn't use h-bar because the theories would break at these "smaller than h" scales.

But I'm speculating hard here. I don't know the significance of the Planck units. It just seems apparent that there is a major significance there.

[MadMojoMonkey]

It's not easy finding something reliable on this topic. Those posts are cropped from a discussion asking what the physical significance of the Planck constant is.

I feel like in the latter parts of this post, you've conflated the Planck constant and the Planck length as the same thing.

I agree that there's a lot of misinformation about the significance of the Planck Length. Also, there's a lot of misinformation about the nature of quantum mechanics out there. Sifting through the nonsense to find a single, consistent model isn't easy if the internet is your primary source.

Is he being accurate when he claims that angular momentum is quantised?

100% yes

The quantization of angular momentum is a rather big deal, as it is part of the explanation for why electrons refuse to jump to certain energy levels in atoms. In order to jump, they must absorb or emit a photon, which has angular momentum. This means a electron in a bound system cannot jump to an energy level unless the difference in angular momentum between its start and end differ by exactly the same angular momentum as the photon emitted or absorbed.

All photons have the same angular momentum, and this does not change as they travel through expanding spacetime.

and it's amazing that the law of conservation of angular momentum remains perfectly un-violated in all known, verifiable experiments and observations.

And is he being accurate when he claims that a Planck constant of zero turns a quantum theory into a classical theory?

In this one specific example, yes. But more broadly, no.

If h = 0, then photons have no energy. That's not a classical theory.

Because if those two claims are indeed accurate, then that's pretty solid evidence that we're talking about a constant of nature, not a simple length.

Here, I'm starting to question if you have the Planck constant and the Planck length mixed up.

Whether the larger or smaller Planck Length is considered fundamental, it's a constant of nature either way.
Both are constants of nature, regardless of where they fit into our model.

I think it's easy to get caught up in the "length" aspect of it, in the same way people talk of the speed of light where actually it's not about light, it's about causality. The speed of light emerges from the speed of causality. Likewise, the Planck length emerges from quantum mechanics.

It emerges from a combination of QM and GR. Newton's big-G is in the calculation of the Planck Length, which was absorbed by Einstein's Relativity.

There are many places where QM and GR work well together, but not all. Some of the places they don't work together are so problematic as to motivate many many brilliant physicists to explore strings.

It's not clear if the Planck Length is an appropriate combination of QM and GR, except in that it puts a lower bound on length scales at which our model can no longer produce predictions of the outcomes of experiments. I.e. it don't work there.

If I were to guess which Planck length is "correct", I'd say h-bar because it's smaller, and if we can have a "smaller" length than the smallest length, there's a problem with the entire argument right there. If h is right, then h-bar is meaningless, but the math holds up under theoretical scrutiny as best I'm aware, otherwise they wouldn't use h-bar because the theories would break at these "smaller than h" scales.

But I'm speculating hard here. I don't know the significance of the Planck units. It just seems apparent that there is a major significance there.

Both h and h_bar are meaningful. Which is easiest to use depends on the exact equation you're solving.

Kinda like pi and tau are both equally good, and which is easier to work with depends on what equation you're solving. With pi and tau, what's in question is whether the radius or the diameter of a circle is a better descriptor of a circle. But it's a moot point because both radius and diameter encode the exact same information about the circle.

Pi is used for historical reasons. Ancient people could measure the diameter of a circle and they could measure the circumference. The easy measurability of these made pi the obvious choice for them. circumference / diameter = pi.

Nowadays, mathematicians prefer to describe circles by their radius. Circumference / radius = tau

Which you use is irrelevant. They encode the exact same information about the circle.


Whether or not there is meaning in any Planck units depends on how humans can interpret them. It's not guaranteed that there is meaning in any of them, since they are produced by asking the question, "Can I combine these numbers with gobbledygook units in such a way that I get a number with only 1 unit?" The Buckingham Pi method says, "Yes, you can do that as long as you have enough gobbledygook."

The test of whether those units are meaningful is whether we can reproduce that number using the axioms of the model.

[OngBonga]

I feel like in the latter parts of this post, you've conflated the Planck constant and the Planck length as the same thing.

I can see how I'm giving this impression but it's because I'm lazy with my language. The Planck constant is a constant of nature, the Planck length is a unit that emerges from it. But what seems clear is that something happens at this exact scale (either h or h-bar, idk) in the same way something happens at the event horizon of a black hole... physics "changes" in the sense we do not have a working model to describe and predict behaviour, it's a barrier when the laws of physics as we know them are no longer valid.

Now if we look at it from that pov, do you think the event horizon is just a mathematical trick? Or is this region of gravitational curvature really that curved such that moving away from the gravity source requires a velocity exceeding c? All our observations suggest the latter. We can talk about what someone sees if from their FoR if they drift across the event horizon, but the tidal forces felt by anything larger than a singularity makes this concept utterly absurd in its own right.

The math appears to be telling us that something important happens at the Planck scale, there's an abrupt change from classical to quantum.

100% yes

If angular momentum is quantised then surely so too is spacetime. How can an object rotate perfectly smoothly through spacetime if the energy value changes in integers? That motion from one minimum quanta to the next is rather like the second hand on a clock ticking, only the space between the ticks doesn't exist. If that space does exist, if the ticker moves through smooth space at a consistent rate, then angular momentum is smooth, and therefore not quantum. That's my interpretation anyway. The motion from one quanta to the next is instant, if it isn't there's an intermediary period between the quanta, that is, fractions of integers, ie not quantum.

Here, I'm starting to question if you have the Planck constant and the Planck length mixed up.

Is spacetime is quantised, there's a minimum distance and time. These are constants, right?

Whether the larger or smaller Planck Length is considered fundamental, it's a constant of nature either way.

Yes but only one can be "fundamental". This is kinda like saying that both one and two electrons are constant so there's nothing special about one electron.

My assumption is that spacetime is quantised and the implications of this is that there is a minimum distance, both in space and in time. Given our models break down at the Planck length (h or h-bnar, idk), then the Planck length seems the leading candidate for the length of space quanta, and the Planck second the length of the time quanta.

[OngBonga]

Probably there's nothing special about the Planck length, nor the Planck second, rather what's special is the Planck lengthsecond. That's what in quantised. Spacetime.

If this is true, the universe is a 4D ticking clock, and motion is instant between one instant in spacetime to the next, taking 1 Planck second to move 1 Planck length (how can it be slower?), which incidentally is the speed of light (so how can it be faster?), because the Planck second is defined as the time it takes light to move 1 Planck length. So if we're essentially moving from one quanta of spacetime to the next at the speed of light, that's instant. We don't move at the speed of light, but spacetime does. So in this context the Planck lengthsecond is the quanta of energy "dragging" through spacetime in discrete values, which translates into our less-than-c motion through space.

[OngBonga]

It's sunday morning spliff time, can you tell?

[OngBonga]

Our motion through spacetime is always c. Our motion through space is not. So if we set c to 1, our motion through time is the reciprocal of our motion through space. If we move slower than c in space, we move faster than c in time, such that the product of the two is equal to 1 (c). I think this complicates matters.

Basically we move forwards in space and backwards in time. That is my Sunday morning hypothesis.

[MadMojoMonkey]

I think I see what you're asking. Help me if I've misinterpreted you.

The radius of the event horizon can be calculated. In that sense, it's a trick of math. But what separates it from being a "true" trick of math is where the thought process starts. To calculate, e.g. the Swarzchild radius, he (Swartzchild) started with Einstein's GR equations and asked the question, given a mass M, inside what radius is the curvature such that the escape velocity exceeds the speed of light?

FWIW, according to wikipedia, this was calculated from purely Newtonian (classical) physics long before Einstein and Swarzchild came along and put their stamps of approval on it, so to speak.

If you start from known physics and ask a sensible question, then the wording of the posed question implies a path of confirmation or refutation. You can follow that path and hopefully find an answer. Since you started within physics, hopefully you never left physics and your result is indicative of physics.

The starting point for the Planck units is: given we have a complete set of fundamental constants with "mixed" units (like [G] = m^3/(kg s^2), [c] = m/s, etc.), can we juggle them to get constants with "simple" units. And the answer is yes.

In the former, the starting point was physics. In the latter, the starting point is math.
It is often the case that even when your starting point is math, you can find true statements in physics, but that's not guaranteed.

It's worth noting that what we consider the "simple" units is purely due to subjective human and historical reasons. The universe has handed us a set of units that are fundamental in some sense, but we stubbornly hold on to our historical units based on what is easy for us to measure. The universe says, "Length is not fundamental" and humans say, "But we feel like we 'get' lengths."

The math appears to be telling us that something important happens at the Planck scale, there's an abrupt change from classical to quantum.

It's all always quantum. It's all always GR. The "lines" between them are due to human simplifications and limitations.


Angular momentum is quantized, but angular position is not. The rotational momentum comes in integer bumps, but the rotational position can be anything. Indeed, the fact that angular momentum is conserved means that physics is independent of the rotation of your coordinate system. (See Emmy Noether's discovery: every conservation law implies a symmetry of nature) In this case, the Conservation of Angular Momentum (CoAM) implies that physics is the same in all directions. Note that is all directions, regardless of location. Position is no part of it.

It goes: IF AND ONLY IF CoAM is true, THEN physics is unchanged under any rotation of coordinate system.

Since we have never seen even the barest hint of violation of the Law of Conservation of Angular Momentum, that implies that direction is irrelevant. Any violation in 1 is a violation of the other.

That alone kills the "quantized length" argument. If lengths are quantized, then the physics when you look in 1 direction is not the same as the physics in another direction. The quantized lengths impose some kind of grid on things. IDK the shape of that grid, but that's a moot point because the grid alone violates CoAM.

Note that whether or not Angular Momentum is quantized is no part of that argument. Sometimes it works out like that. Something cancels from both sides of an equation that you didn't expect to cancel out.

[If] spacetime is quantised, there's a minimum distance and time. These are constants, right?

That's big if, but the answer is still yes.

Yes but only one can be "fundamental". This is kinda like saying that both one and two electrons are constant so there's nothing special about one electron.

No, it's not like that. It's like ... if we juggled constants to get something with units kg m^2/s^2, then we can interpret that as kinetic energy mv^2. But we're missing the factor of (1/2). So we got an answer that is pretty close, but off by some unknown constant multiple.

*unknown* constant multiple

In my example, that was "only" 1/2. Not too far off, really. But what if that unknown constant was pi^4? Now I'm off by a factor of ~100. I know something about what I was looking for, but not enough to interpret it physically.

[MadMojoMonkey]

Probably there's nothing special about the Planck length, nor the Planck second, rather what's special is the Planck lengthsecond. That's what in quantised. Spacetime.

If this is true, the universe is a 4D ticking clock, and motion is instant between one instant in spacetime to the next, taking 1 Planck second to move 1 Planck length (how can it be slower?), which incidentally is the speed of light (so how can it be faster?), because the Planck second is defined as the time it takes light to move 1 Planck length. So if we're essentially moving from one quanta of spacetime to the next at the speed of light, that's instant. We don't move at the speed of light, but spacetime does. So in this context the Planck lengthsecond is the quanta of energy "dragging" through spacetime in discrete values, which translates into our less-than-c motion through space.

I mean... I really like this thought process.

In a sense, all particles always move at c. but that's a tricky one to unfold, and frankly, you're probably better suited to do so than I.


I'm strongly convinced by my prior argument invoking Noether's Theorem that since CoAM isn't violated, lengths (not spacetime) cannot be quantized.

I'm as strongly convinced by the crystal structure argument as well, but you've not been in the past, so I tried a new tack.

[MadMojoMonkey]

It's sunday morning spliff time, can you tell?

Make every morning Sunday spliff time, IMO.
The world needs more people who think critically about ... literally whatever they're passionate about.
I don't care what avenue anyone takes to find their passion.

OK, that's not true. There's plenty of things I can't get behind, but if you're not doing crimes, you're probably OK.
Probably.