Hubble's Law and Star Velocity

[Gege01]

TL;DR Summary

The universe is still expanding faster! The maximum expansion speed can reach the speed of light C! Thereafter, the expansion rate slowed down.

Hubble's Law and Star Velocity
By using Hubble's law and the principle of velocity superposition of relativity, we can deduce when the distance of star from the observation point is (photometric distance)r, its velocity V can be expressed as:
V = Csin (Hr/C)
Therefore, the following conclusions are drawn:

The universe is still expanding faster! The maximum expansion speed can reach the speed of light C! Thereafter, the expansion rate slowed down.
The maximum radius of the universe is 3.14 times that of today.

 

[phyzguy]

Most of what you've said here is incorrect. I highly recommend the attached "Expanding Confusion". Read it carefully and if you have questions, come back and ask.

 

[Dale]

V = Csin (Hr/C)

This equation appears to be dimensionally inconsistent. Where did you get it?

 

[Gege01]

Thank you very much!

 

[Gege01]

This equation appears to be dimensionally inconsistent. Where did you get it?

There is a functional relationship between star velocity and distance:

V=V(r)

 

[Gege01]

There is a functional relationship between star velocity and distance:

V=V(r)
View attachment 245993

It can satisfy Hubble's law :

V(dr’) =Hdr’
The true velocity V(r+dr) of B, measures by viewer of o, is the superposition of the velocity V(r) of the star A and the velocity V(dr') of the star B relative to A.

 

[Gege01]

The relationship between dr’ and dr is:

dr’=γdr
Also, according to the differential definition:

V（r+dr）-V(r)≡dV(r)
We get

After the above formula, wei have

 

[Gege01]

Most of what you've said here is incorrect. I highly recommend the attached "Expanding Confusion". Read it carefully and if you have questions, come back and ask.

Thank you very much!

 

[Dale]

View attachment 245994
It can satisfy Hubble's law :

V(dr’) =Hdr’
The true velocity V(r+dr) of B, measures by viewer of o, is the superposition of the velocity V(r) of the star A and the velocity V(dr') of the star B relative to A.
View attachment 245995

Oh, I see. I thought your Hr was the Hubble radius so Hr/c would have units. But you meant H is the Hubble constant and r the radius so Hr has units of velocity and Hr/c is dimensionless.

Next time please be clear about the meaning of your variables.

 

[Gege01]

Thank you for your reminder.

 

[Dale]

We get

View attachment 245997
After the above formula, wei have

View attachment 245998

Note that the last formula only follows from the previous one for $r\le \frac{\pi}{2} \frac{c}{H}$. So you cannot use it to claim:

The maximum expansion speed can reach the speed of light C! Thereafter, the expansion rate slowed down.

Your claim is based on applying a formula in a region where it is not valid. I am not certain that the rest of the derivation is correct, but at least that part has clear mathematical limitations.

 

[Gege01]

Yes, some of the conclusions are not very rigorous.

 

[Gege01]

However, it is reasonable in the scope of r≤πc/(2H).

 

[Dale]

However, it is reasonable in the scope of r≤πc/(2H).

I am not sure. With none of the variables being clearly defined it is hard to follow. Unless this has been derived in the professional scientific literature (not only Chinese) then it is highly suspect.

 

[Gege01]

Based on the following facts That Hubble's law is valid at short distances and at small speeds:

V(dr') = Hdr'

 

[Gege01]

Although we can't guarantee that Hubble's law holds at large distances and high speeds, we can still assume that special relativity is satisfied.

The true velocity v (r + dr) of b, measured by the observer of o, is the superposition of the velocity v (r) of a star and the velocity v (dr') of B star relative to a.

 

[Ibix]

The relationship between dr’ and dr is:

dr’=γdr

This appears to me to be attempting to relate the distance, $dr$, between two nearby objects as measured by local observers to the distance, $dr'$, measured by observers who see the objects receding at $Hr$. It seems to be simply the special relativistic formula for length contraction, and therefore assumes a single inertial reference frame covering a spatial patch of radius at least $r$. This seems to me unlikely to be valid at cosmological distances.

 

[Dale]

we can still assume that special relativity is satisfied

No we can’t. Special relativity is only satisfied for flat spacetime. The FLRW spacetime is not flat.

The true velocity v (r + dr) of b, measured by the observer of o, is the superposition of the velocity v (r) of a star and the velocity v (dr') of B star relative to a.

What is dr’. What are a, b, and B? Which observer is o?

 

[Gege01]

 

[Gege01]

No we can’t. Special relativity is only satisfied for flat spacetime. The FLRW spacetime is not flat.What is dr’. What are a, b, and B? Which observer is o?

Although FLRW spacetime is not flat，We can still think that special relativity can be satisfied in a very small space-time range.

 

[Gege01]

O is the position of any observer. A is a star whose distance from the observer is r.

 

[Gege01]

B Distance Observer Distance is r+dr

The distance of B measured by point A observer is dr'

 

[Dale]

Although FLRW spacetime is not flat，We can still think that special relativity can be satisfied in a very small space-time range.

Sure, but “the maximum radius of the universe” hardly qualifies as “very small”.

 

[Mordred]

What your seeing as expansion >c is a consequence of separation distance between the observer (us) and the objects beyond Hubble horizon. In point of accuracy the expansion rate is approximately 70 km/s/Mpc far slower than c. If you situate an observer at every Mpc this rate will be the same in our universe today.
Due to Hubble law $v_{resessive}=H_OD$ the greater the distance the greater the resessive velocity the separation distance gives the appearance of exceeding c. However this is merely a consequence of the separation distance and application of Hubble's law nor a true velocity.

If you prefer to apply Hubble's law in terms of radius simply replace D with r. V=Hr. SR isn't needed to apply this as pointed put it's a simple consequence of separation distance and expansion rate.

 

[Gege01]

Sure, but “the maximum radius of the universe” hardly qualifies as “very small”.

When we study specific problems, for example, the distance between A and B is R and r+dr, then, in the range of R to r+dr, it is a small area. Special relativity is satisfied

 

[Mordred]

The radius of the observable universe being roughly 46 billion light years isn't small. Think about how far into the past you are looking at when you see the surface of last scattering at z=1100. Does that sound like something SR handles ?

Edit: I should specify in this scenario as certainly the FRWL metric applies GR and SR. Example being it's light cone and worldline applications.

 

[Gege01]

The radius of the observable universe being roughly 46 billion light years isn't small. Think about how far into the past you are looking at when you see the surface of last scattering at z=1100. Does that sound like something SR handles ?

Edit: I should specify in this scenario as certainly the FRWL metric applies GR and SR. Example being it's light cone and worldline applications.

If the universe is finite but unbounded, it means that the universe is smaller than the observable universe. In this case, galaxies that look far away may be illusions of neighboring galaxies, which are illusions created by light circling the universe. This hypothesis is very difficult to detect because galaxies are different at different ages, or even completely different.

 

[Mordred]

No it doesn't the observable portion is certainly finite as the cosmological event horizon sets an observational limit. The full universe could very well be finite or even infinite.
We simply do not know as we cannot observe or measure the entire universe. We however do know it is far larger than our observable portion.

The mirror effect you described was once studied but observational evidence of objects at extreme distances have different metalicity percentages than objects nearby. Then there is also different temperatures involved. As you approach the CMB the temperature rises.

Observational data doesn't rely solely on distance estimates but also includes thermodynamic changes due to expansion.

 

[Gege01]

No it doesn't the observable portion is certainly finite as the cosmological event horizon sets an observational limit. The full universe could very well be finite or even infinite.
We simply do not know as we cannot observe or measure the entire universe. We however do know it is far larger than our observable portion.

If we use the formula in this post, some data may be changed.

 

[Gege01]

This conclusion is based on the following assumptions:
1. Hubble's law is valid at small distances and velocities.
2. Special relativity can be satisfied in a very small space-time range.
3. There is a functional relationship between star velocity and distance:
V=V (r)

 

[Mordred]

We cross posted see my edits on above post in terms of other data such as metalicity and temperature changes

 

[Mordred]

Your conclusion is in error as your not aware of the extent of the observational data

 

[Gege01]

This conclusion is based on the following assumptions:
1. Hubble's law is valid at small distances and velocities.
2. Special relativity can be satisfied in a very small space-time range.
3. There is a functional relationship between star velocity and distance:
V=V (r)

Although there are other factors affecting the observed data, the above three assumptions are still valid.

 

[pervect]

I would suggest that the OP read something like Ned Wright's Cosmology tutorial. A cosmology textbook would be better, but Ned Wright's tutorial would be a good attempt at start.

Specific quotes from part 2 follows, though I'd recommend starting at part 1.

<<link to part 2>>
<<link to part 1>>

Many Distances
With the correct interpretation of the variables, the Hubble law (v = HD) is true for all values of D, even very large ones which give v > c. But one must be careful in interpreting the distance and velocity. The distance in the Hubble law must be defined so that if A and B are two distant galaxies seen by us in the same direction, and A and B are not too far from each other, then the difference in distances from us, D(A)-D(B), is the distance A would measure to B. But this measurement must be made "now" -- so A must measure the distance to B at the same proper time since the Big Bang as we see now. Thus to determine Dnow for a distant galaxy Z we would find a chain of galaxies ABC...XYZ along the path to Z, with each element of the chain close to its neighbors, and then have each galaxy in the chain measure the distance to the next galaxy at time to since the Big Bang. The distance to Z, D(us to Z), is the sum of all these subintervals:

Dnow = D(us to Z) = D(us to A) + D(A to B) + ... D(X to Y) + D(Y to Z)

And the velocity in the Hubble law is just the change of Dnow per unit time. It is close to cz for small redshifts but deviates for large ones. The space-time diagram below repeats the example from Part 1 showing how a change in point-of-view from observer A to observer B leaves the linear velocity vs. distance Hubble law unchanged:

And another brief quote:

Note that the redshift-velocity law is not the special relativistic Doppler shift law

I won't give the details of the redshift formula at this point, but will refer the original poster (OP) to the article I quoted.

So, there are some important things to notice.

There are many notions of distance in cosmology, and they are not equivalent. Precision is needed to avoid confusion the different notions of distance. I do not believe "photometric distance" is a standard term, though it was the one the Original Poster (OP) asked about. It's not clear to me which of the several possibilities of distance in cosmology might be the same as (or close to) the OP's notion of "photometric distance", as I'm not quite sure what he means. Possibly he could mean luminosity distance.

I am not sure if Ned Wright has any discussion of "luminosity distance". Wiki has some discussion of distance measures in cosmolgoy in <<link>>., though Wiki is not necessarily a reliable source on this topic. Finding a cosmology book would be a much more reliable approach.

The doppler shift law for the "velocity" as used in the Hubble law formula is NOT the formula from special relativity, because neither the distance in Hubble's law nor the velocity in Hubble's law is the same as used in special relativity.

The origin of the difference is that special relativity is a special case of general relativity that applies when space-time is flat. Because the universe is not flat, on the large scales used in cosmololgy, one cannot use special relativity but must use general relativity instead. This is rather similar to the way that if one wants to sail across a pond, one can approximate the pond as being flat, but if one wants to sail across an ocean, one's calculations will have significant errors if one does not account for the curvature of the Earth.

I will also admit that I do not know the correct relationship between Hubble distance and luminosity distance offhand, but if that is the question that the OP would like answered, someone here or on the cosmology forums might be able to give a better answer. But it's not quite clear to me if that is really the question the OP wants to ask.

I think it'd be safest to use the observed redshift factor, z, as the "common denominator" of distances, as that's what's usually measured when one measured the "distance" to a far galaxy.

With this approach, one can find the relationship between redshift factor z and Hubble distance, and also the relationship between redshift factor z and luminosity distance, and come up with a formula that relates Lumionisty distance to Hubble distance.

 

[Mordred]

If you have evidence that counters a theory then that theory becomes invalidated by other evidence.

Cosmology didn't reply solely upon distance estimates. It also encompasses thermodynamic laws and utilizes those laws to validate LCDM.