Redshift z

[modest]

I'm not so sure, CC.

 

I've never read this, but I think surface brightness might stay constant (i.e. be independent of distance) in a static and spatially hyperbolic universe.

 

My reasoning is as follows: the relationship between the apparent brightness of an object, the apparent size of an object, and the redshift of the object is as follows:

[math]\frac{D_L}{D_A}=(1+z)^2[/math]

where D_L is the luminosity distance (the inverse of the apparent luminosity) and D_A is the angular diameter distance (the inverse of the apparent angular diameter).

 

This relationship is true regardless of spatial curvature as supported here:

The two relations are connected by the reciprocity relation, that their ratio is D

_L

/D

_A

=(1+z)

²

where z is the redshift. This relation is independent of spatial curvature, and is a simple consequence of the conservation of photon number.

angular-diameter distance - not, physical, reciprocity relation - Distance, Angular, Diameter, Size, Object, and Universe

In a static Euclidean universe D_L/D_A = 1 for any object at any distance which means the redshift of any object should be:

[math]z = \sqrt{\frac{D_L}{D_A}} - 1[/math]

[math]z = \sqrt{1} - 1[/math]

[math]z = 1 - 1 = 0[/math]

In other words, no object in a static Euclidean universe should have redshift (unless it were to happen by some non-geometric means like tired light).

 

Hyperbolic space affects both D_L and D_A. The luminosity distance is increased (over Euclidean) by hyperbolic space because the object in question appears less bright. Also, the angular diameter distance is increased because the object in question appears smaller in angular diameter. Because both D_L and D_A are increased by what I would assume is the same factor, D_L / D_A should still equal one. In static hyperbolic space one could still solve for z,

[math]z = \sqrt{\frac{D_L}{D_A}} - 1[/math]

[math]z = \sqrt{1} - 1[/math]

[math]z = 1 - 1 = 0[/math]

 

In other words, an object like a star would be both fainter (in total apparent brightness) and take up less room in the night sky. The two factors would cancel such that the surface brightness does not rely on distance.

 

Olbers' paradox would then not be solved by static and spatially hyperbolic geometry.

 

~modest

[quantumtopology]

Hyperbolic space affects both D_L and D_A. The luminosity distance is increased (over Euclidean) by hyperbolic space because the object in question appears less bright. Also, the angular diameter distance is increased because the object in question appears smaller in angular diameter. Because both D_L and D_A are increased by what I would assume is the same factor, D_L / D_A should still equal one. In static hyperbolic space one could still solve for z,

[math]z = \sqrt{\frac{D_L}{D_A}} - 1[/math]

[math]z = \sqrt{1} - 1[/math]

[math]z = 1 - 1 = 0[/math]

 

In other words, an object like a star would be both fainter (in total apparent brightness) and take up less room in the night sky. The two factors would cancel such that the surface brightness does not rely on distance.

 

Hi, Modest

 

First just a small correction, according to the standard model DL/DA = 1 is not correct for z>1.6

 

 

Second, I think that assuming that both D_L and D_A increase by the same factor is not justified at all in hyperbolic space, in fact D_L increases more than D_A .

 

Why is this?

Basically angular or apparent size of objects decreases with distance slower than luminosity because light from an object spreads over a volume bigger than 4/3piR^3 that is the volume where it spreads in Euclidean geometry, now that deviation changes with the cube of the radial distance (R^3) while the magnitude of the deviation of the angular size from the apparent size in a Euclidean geometry changes as a funtion of just the radial distance R, therefore in Hyperbolic space the Luminosity distance tends to be larger than the Angular diameter distance as z increases.

 

Regards

Qtop

[modest]

Hi, Modest

 

First just a small correction, according to the standard model DL/DA = 1 is not correct for z>1.6

 

D_L/D_A = 1 is not correct at any z in the standard model.

 

I said D_L/D_A equals 1 in a static Euclidean space. Space is not static in the standard model. Expansion would increase D_L over Euclidean and decrease D_A.

 

Second, I think that assuming that both D_L and D_A increase by the same factor is not justified at all in hyperbolic space, in fact D_L increases more than D_A .

 

Why is this?

Basically angular or apparent size of objects decreases with distance slower than luminosity because light from an object spreads over a volume bigger than 4/3piR^3 that is the volume where it spreads in Euclidean geometry, now that deviation changes with the cube of the radial distance (R^3) while the magnitude of the deviation of the angular size from the apparent size in a Euclidean geometry changes as a funtion of just the radial distance R, therefore in Hyperbolic space the Luminosity distance tends to be larger than the Angular diameter distance as z increases.

 

I don't follow. Check out the link I gave in the last post. I don't know why you would be considering the volume of a sphere. It is the surface area that is relevant when considering surface brightness.

 

~modest

[quantumtopology]

D_L/D_A = 1 is not correct at any z in the standard model.

 

I said D_L/D_A equals 1 in a static Euclidean space. Space is not static in the standard model. Expansion would increase D_L over Euclidean and decrease D_A.

 

You are right, I shouldn't write posts half-asleep and in a hurry :shrug:

 

I don't follow. Check out the link I gave in the last post. I don't know why you would be considering the volume of a sphere. It is the surface area that is relevant when considering surface brightness.

 

You are right again, I meant to use the area formula 4piR^2, so the change in Luminosity Distance is a function R^2 instead of R^3, the result still makes DL increase more than DA.

The ultimate reason for this is that in hyperbolic space the luminosity spreads over a bigger surface for a given redshift than in a Euclidean space, and the pace at which it departs from flat space is faster than in the case of the departure from Euclidean angular size, where is a function of the distance to the first power , not to the second as in the luminosity.

 

This is only logic if you consider than in Hyperbolic geometry there is a further decrease of magnitude of distant objects that you don't find in Euclidean geometry (thus the solution of Olber's paradox) that is not only due to redshifting but due to the intrinsic properties of hyperbolic distances that are larger than Euclideans for a given z, and the disparity grows with z, being negligible for very low z and tendig to infinite around 14 bly.

 

Regards

Qtop

[quantumtopology]

Let's use some formulas to see things clearer:

 

For Angular diameter distance:

 

and X is different depending on curvature:

 

 

So in hyperbolic space DA departs from DA in Euclidean geometry with the hyperbolic sine of the comoving distance of the object for a given z.

 

For luminosity distance:

 

 

 

 

F can be expressed as a function of z too giving: F =L/4pir^2(1 + z)^2

 

r = fK(z) with

sin f(z); for K = +1;

f(z); for K = 0;

sinh f(z); for K = -1;

 

So we get F =L/4pi*sinhr^2*(1 + z)^2

 

L is assumed constant for a given object

 

We see that DL in a hyperbolic geometry departs departs from DL in Euclidean geometry with the hyperbolic sine of the squared comoving distance from the object.

Notice that the square root of the hyperbolic sine of r^2 is bigger than the hyperbolic sine of r.

 

It's clear that DL increases in a much bigger factor than DA when we switch from Euclidean to Hyperbolic space.

 

Sources:

http://icecube.wisc.edu/~halzen/notes/week1-3.pdf

Angular diameter distance - Wikipedia, the free encyclopedia

 

Regards

Qtop

[quantumtopology]

[math]z = \sqrt{\frac{D_L}{D_A}} - 1[/math]

 

Modest, Thanks for this great description of how z is >0 in a hyperbolic , static scenario. It's much more intuitive and easier to realize with this formula than the way I was trying to (I guess I'm not very good making myself understood in many occasions).

 

It is as easy to proof that DL=DA in Euclidean , static space, than to proof that DL>DA in hyperbolic, static space (like is the case in expanding flat space).

 

 

Regards

Qtop

[quantumtopology]

I have just realized that due to some tricks of notation I made a mistake that invalidates my last three posts, so I have to take them back , sorry.

I'll let you find the error, is quite easy.

 

Modest, you are right about your formula, thanks anyway.

 

Regards

Qtop

[modest]

Yeah, if my back-of-the-envelope chicken scratches are right, both the brightness and the size should be diminished from the Euclidean amount by [math]\frac{R^2 sinh^2 (r/R)}{r^2}[/math] where R is -k^-1/2 and r is the radius. That should be the amount that the surface area of a hyperbolic sphere is greater than the surface area of a Euclidean sphere. Like your image shows,

the luminosity is diminished by the surface area of the sphere, and likewise with angular diameter,

the size of the object is diminished by the size of the sphere. It seems intuitively to me that the two factors should cancel, but honestly this is not something I'm positive of.

 

~modest

[quantumtopology]

Your scratches are right, pal. :shrug:

I got carried away by a reasoning that was not even in agreement with my previous posts about redshift.

Indeed DL=DA too in a hyperbolic static space as in a Euclidean static space, it actually couldn't be any other way as I think of it now.

 

This doesn't mean a great deal for the hyperbolic universe hypothesis, since in it the redshift arises from the transition from Hyperbolic distance to Euclidean distance.

 

Regards

Qtop

[Atlas shrugged]

One question that has always bothered me about Redshift is that all gravitational pull will cause redshift . The original post addresed this I think but I will ask here.

 

So how do we know the redshift that we think is showing it moving away from us is not just the gravity all along the way causing it.

[quantumtopology]

One question that has always bothered me about Redshift is that all gravitational pull will cause redshift . The original post addresed this I think but I will ask here.

 

So how do we know the redshift that we think is showing it moving away from us is not just the gravity all along the way causing it.

 

Cosmologists are quite sure that the cosmological redshift is not a conventional gravitational redshift due to the peculiar way the former relates to distance, and because gravitational redshift is tiny compared to the cosmological at the cosmological distances.

A different issue would be if it is reasonable to ask whether the cosmological redshift has something to do with the geometry of the universe, determined by gravity, as Einstein's theory of General Relativity shows, and in this case the answer in my opinion is affirmative .

 

Regards

Qtop

[quantumtopology]

I found yesterday this website by some guy named Alexander F. Mayer: www.sensible universe.com that has writen a book with some interesting ideas, a good shake of the standard model and a theory that is probably flawed, What do you think? it seems to have been around for a few years so maybe it has already been commented here.

CC, do you find any similarities with what you had in mind? This guy talks about a universe with the shape of a sphere and with redshift due to curvature.

 

Regards

Qtop

[modest]

Is there an online copy of his book or any other of his works? Do you know?

 

~modest

[coldcreation]

Is there an online copy of his book or any other of his works? Do you know?

 

~modest

 

Thanks Qtop, good find. I jusrt downloaded the eBook by Alexander Franklin Mayer (had to download the new version of Adobe Reader too). It looks interesting at first glance. I'm reading it now.

 

Modest, here is the link from where, on the right side of the page, you can download the book in PDF form (free): http://jaypritzker.org/pages/book.html

 

I'll comment on it when I finish reading it.

 

 

CC

[modest]

Thank you. I completely missed that.

 

~modest

[quantumtopology]

It seems like this theory was presented about 5 years ago,by what I have gathered from different forums, and it was torn to pieces quite easily, he presented something called transverse gravitational redshift that showed a poor understanding of relativity theory, seems like the reason he was givn some credit was that he was apparently from Stanford staff and had earned some "pritzker fellow prize" , both things are considered in forums to have been made up, who knows

The book as it is now seems to have been cleaned off some of the early big errors but the main idea remains. Certainly he works hard in the presentation, it looks nice and he writes well enough. There are some that think it all is some kind of hoax or joke for physics students.Perhaps due to his mix of mainstream and antimainstream ideas What do you guys think?

Basically the reason I post about this is that I think he makes some relevant criticism of the standard model, specially the part about the CMB, and the SDSS data.I basically agree with much of chapters 1-8 and 12-15. But he makes a good job of not presenting clearly his theory(actually it is just a preview of the book and he has left out of this preview the bits that were ripped apart in forums in previous years), sometimes one gets the feeling that he is just reinterpreting relativity in his own kind of deceiving way, he presents a few formulas but he does not explain much about how he derives them, he seems to present a static pseudo Einsteinian model without concerns about stability...

 

 

Regards

Qtop

[coldcreation]

I found yesterday this website by some guy named Alexander F. Mayer: www.sensible universe.com that has writen a book with some interesting ideas, a good shake of the standard model and a theory that is probably flawed, What do you think? it seems to have been around for a few years so maybe it has already been commented here.

CC, do you find any similarities with what you had in mind? This guy talks about a universe with the shape of a sphere and with redshift due to curvature.

 

Regards

Qtop

 

The short answer is no. The Mayer redshift is not the same as the cosmological redshift discussed in this thread. The main difference between the two (the Transverse Gravitational Redshift proposed by Mayer and the curved spacetime redshift discussed in this thread) is the transverse aspect. The transverse redshift, perpendicular to the line of sight or propagation of EMR, as I understand, represents a modification of general relativity; inducing an effect (z) that appears to have been falsified or shown to be nonexistent.

 

The redshift discussed in this thread is entirely based on general relativity, with no modification; albeit there is an extension of the concept of local curvature to global curvature, requiring only an interpretation of GR and its astrophysical implications to cosmology. The mechanism involved and the effect (curvature) are identical both locally and globally. In other words, to disprove or falsify the claim would be to disprove or falsify general relativity. Or, it would have to be shown that gravity is a curved spacetime phenomenon locally, but not globally, ie., that mass-energy curves spacetime around massive bodies, but that the mass-energy density has no (or a very small) influence on the curvature of space on the large-scale (globally). The latter is the current view, but since the total mass-energy density of the universe is unknown, it seems premature to discard the idea on that basis.

 

Note here, the above latter argument does not take into consideration the geometric structure of spacetime in the absence of matter (or mass-energy), which according to GR may also have an intrinsic curvature (with a negative sign, in accord with the de Sitter, or anti-de Sitter metric).

 

Why Mayer would propose a transverse redshift, as opposed to one without the apparently needless transverse component, is beyond me. It's a pity too, since besides the transverse z component much of what here writes, e.g., about empirical observations (and his critique of the concordance model, with its inherent big bang) makes sense.

 

 

 

 

CC