Genady

3 hours ago, Tristan L said:

I use ðe English

This is not English. See English alphabet - Wikipedia.

Plus, the topic of an alphabet is not Physics.

Reminder: this is Physics forum.

49 minutes ago, Airbrush said:

Is "infinity," like a singularity, just an abstract idea?

This question belongs to another forum 😉

23 minutes ago, grayson said:

So, when you take the second derivative of the tensor, what do you get exactly? Is it the length that it bent or is it the length of the curvature itself?

When you take second derivatives of metric, you get rates of its deviation from the flat spacetime.

2 hours ago, grayson said:

how the einstein tensor describes the curvature of spacetime

Curvature is encoded in second derivatives of metric. So, any tensor that depends on second derivatives of metric, describes curvature in some way. Einstein tensor describes curvature in a way that can be related to the spacetime's physical contents.

On 12/18/2023 at 1:05 PM, Airbrush said:

can something finite in mass (or size) expand to an infinite size?  And please explain for me without advanced math.

Let me try to explain it with a bit of algebra.

In an expanding homogeneous isotropic universe, a distance between any two points - let's call them, galaxies - is proportional to a number, \(a(t)\), called scale factor, which increases with time, \(t\). So, for example, if a distance between some two galaxies at some moment is \(D\) then later, when \(a(t)\) is twice as large, the distance between these two galaxies is \(2D\). Thus, this distance increases with time as \(a(t)D\).

If the universe is finite, then there is a largest distance in it, which, just like any other distance, is proportional to \(a(t)\). Let's call it, \(a(t)L\). The only way for the \(a(t)L\) to become infinitely large is that \(a(t)\) becomes infinitely large. But, if \(a(t)\) becomes infinitely large, then distance between any two galaxies, \(a(t)D\), becomes infinitely large.

IOW, all galaxies become infinitely far from each other. We of course know that it isn't so. Thus, either the universe was finite and remains finite, or it was infinite to start with.

The "diagonal argument" can be expressed purely algebraically, without lists, rows, diagonals, and any other "visual aids". I think that such algebraic presentation would eliminate a lot of confusion.

3 hours ago, Airbrush said:

My next question is, can something finite in mass (or size) expand to an infinite size?  And please explain for me without advanced math.  Thank you.

Can anyone explain how mathematicians know that the number TREE3 is finite?

1 hour ago, TheVat said:

https://en.wikipedia.org/wiki/Kruskal's_tree_theorem

Really large number.

 

A really large number is finite by definition.

2 minutes ago, Airbrush said:

can something finite in mass (or size) expand to an infinite size?

No, in a continuous process it cannot.

3 minutes ago, Airbrush said:

Can anyone explain how mathematicians know that the number TREE3 is finite?

What is "number TREE3"?

33 minutes ago, geordief said:

Would probability  equations in QM  normally have a time component?

Here is an example: a superposition of two Hamiltonian eigenstates, \(\psi_1\) and \(\psi_2\), with the energies \(E_1\) and \(E_2\): \(\frac 1 {\sqrt 2}(e^{-iE_1t}\psi_1+e^{-iE_2t}\psi_2)\). The probability is squared modulus of this function, which includes a time component, \((E_1-E_2)t\).

20 minutes ago, geordief said:

I don't "mean" anything.I am asking a question.

You seem to be suggesting that the question is trivial and might as well be considered in macro systems.

I have no way of knowing if that is right or wrong as I don't have the expertise.

I just try to clarify the question because I don't know what they were referring to in the thing you've read. 

Let's consider an example, a particle in a Hamiltonian eigenstate \(\psi(x)\) with energy \(E\). It evolves in time as \(e^{-iEt}\psi(x)\). The probability density for it to be in position \(x\) is \(\bar {\psi}(x) \psi(x)\). This outcome does not depend on time and thus doesn't change if the sign of time is flipped.

5 minutes ago, geordief said:

Is it just that there are some interactions where reversing the time signature in the maths  doesn't change the outcome?(I am fairly sure I have heard this more than a few times)

You mean, like in Newtonian mechanics?

16 minutes ago, geordief said:

I think I have read that there are  circumstances where ,at the quantum level the direction of time  does not apply.

Is this is true ,is it  just for limited circumstances or is it across the board?

 

Do perhaps  quantum systems evolve in time generally  but some do not?

As QFT obeys SR, there is direction of time in the QFT as much as it is in the SR.

3 hours ago, Boltzmannbrain said:

But that's what we were taught in high school.  It turns out there are ways it is allowed.  

The domain of the function \(\frac 1 {1-t}\) excludes \(1\), as the domain of the function \(\frac 1 x\) excludes \(0\):

(Domain of a function - Wikipedia) 

1 hour ago, Boltzmannbrain said:

The universe can reach infinite size in a finite amount of time, mathematically speaking.

Mathematically speaking, it cannot reach infinite size, but it rather increases unboundedly as \(t \rightarrow 1\) from below. At \(t=1\), the formula is undetermined: mathematically speaking, \(\frac 1 0\) is undefined.

31 minutes ago, Boltzmannbrain said:

I think you missed the point.  I was trying to show how you can reach infinite size in a finite amount of time, whatever the formula has to be to reach 14 billion years.

I did not miss that point, but my counterpoint was that it does not work mathematically. It reaches the infinity and momentarily changes to \(- \infty \) and stays negative after that. Distance cannot do this.

P.S. You want the size to reach infinity and to stay infinite. Try another formula.

2 hours ago, StringJunky said:

The universe stops fractalizing beyond a certain scale and becomes homogenous.

 

 

The universe was much more homogeneous just after the Big Bang, as is evidenced by the isotropy of the CMB radiation. It became less homogeneous with time, presumably because of the gravitational clumping. OTOH, expansion works against clumping, and an accelerated expansion even more so. Does anybody know what will happen in the future to the "scale of homogeneity"?

11 minutes ago, Boltzmannbrain said:

If distance is a function of time, the equation 1/(1 - t) reaches infinity in 1 second

and becomes negative after that.

20 minutes ago, Airbrush said:

For this reason

I can't see a reason that connects what comes before and what comes after this phrase. Poetry, perhaps, but not a reason.

2 hours ago, joigus said:

people keep talking about "seeing" tensors

Of course, I "see" multilinear machines taking in vectors and producing numbers.

I also "see" equivalence classes of indexed collections of numbers with certain transformations being the equivalence relations.

55 minutes ago, exchemist said:

Not if all our grandparents had it, surely?

Sure, this is right. 

Then the question is, how much interbreeding would create a stable inherited set throughout the entire population.

The OP question is still open, I think. 1-4% of what? Do we have on average 1 neanderthal gene? 5? 100? Are they genes that we share with neanderthal but with nothing else? Are they "genes" or long chunks of DNA? How long?

Is this just a pop-science number?

2 hours ago, exchemist said:

But my understanding is the 1-4% relates to DNA features found in homo sapiens neanderthalensis but NOT found in homo sapiens sapiens of African origin.

However,

Quote

We all likely have a bit of Neanderthal in our DNA – including Africans who had been thought to have no genetic link to our extinct human relative, a new study finds.

(All modern humans have Neanderthal DNA, new research finds | CNN)

2 hours ago, exchemist said:

it is likely you have some ancestors who were Neanderthals, rather in the way that I have one Welsh great-grandmother.

It is not clear to me how DNA acquired by a direct descent differs from DNA shared because of a common ancestry.

Another question regarding direct descent is about the amount, 1-4%. We have this amount of DNA directly from our grandparents of 5-6 generations back, which is too recent for neanderthals.

Should not this percentage depend on the length of DNA chunks that are compared? In the extreme case, if we compare pieces of one nucleotide long, any organism on Earth shares 100% of its DNS with any other organism on Earth; they all are AT and CG.

28 minutes ago, geordief said:

Can a scenario involving only  quantum objects be modeled using spacetime diagrams and their frames of reference?

Is it ever done? Would there be a need?

I understand that special relativity is used in such scenarios.

 

I've never seen the SR kind of spacetime diagrams used in QM or in QFT. But a rest frame of a particle or a system is often used and selected in such a way that makes calculations easier. In such a frame, some momentum is zero, which simplifies formulas.

24 minutes ago, sethoflagos said:

You seem to be saying that spacetime and a complex 4-D spinor space are the same thing.

To me, any object in a complex space does not have an energy content though its projection in real (non-complex) spacetime will. So they aren't the same thing,,, or are they?

I don't say or imply anything like that and don't have any idea how it seems so. I say that the states, which are spinors, evolve in spacetime. Just like a scalar such as temperature can evolve in spacetime but is not the same as spacetime.

But you don't need to go to spinors to express your concern. Long before Dirac, in the good old Schrödinger equation, the wave functions are complex-valued functions and they represent particle states in a complex vector space. 

In case of the Hamiltonian observable basis, the states, complex functions, are eigenstates, while the energies, real scalars, are eigenvalues. I don't think there is any problem in this distinction.

16 minutes ago, sethoflagos said:

Okay... so I've arbitrarily picked something.

Could such a space be:

... or not.

In the Dirac equation, the evolving objects are spinors. They evolve in spacetime. They constitute states in a spinor space. The equation does not pick any specific basis in this space to expand the spinors as superposition. We are free to choose such a basis and thus such expansion is arbitrary.