I'm interested in this test for real expansion, but I'm unsure how the surface brightness falls off as (1+z)^4.
Most of the literature I've come across deals with the practical problems associated with measuring the actual surface brightness, but not how the above relationship is derived.
Can someone explain (for an interested layperson) how this relationship is derived or a reference to a site that deals with the basics of the Tolman test.

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There was another recent thread where this came up. As I recall, the explanation for the 4 powers of 1+z has to do with the fact that 1+z is both the factor that space has been stretched by, and the factor that clocks have sped up since then. So you get 2 powers of 1+z by virtue of the extra distance the light has had to travel (and the brightness scales like two powers of 1/r, since it's related to the surface over which the energy is shared), and two powers by virtue of how the sense of time has changed. Note that time appears twice, for reasons that are kind of subtle. The first power is easy, because Tolman brightness refers to energy absorbed in a given time interval, and as time speeds up with the age of the universe, this means you have less time to absorb what we mean by 1 second worth of energy. The last power comes in because Tolman brightness is defined in terms of per wavelength interval, and the fact that time is speeding up implies the wavelengths are stretching. Thus you need a wider wavelength interval to catch the same amount of energy. Note that if Tolman had used per frequency bin instead of per wavelength bin, the two powers of 1+z that relate to time would have cancelled each other.

Thanks Ken G.


Here's a snip from:
"THE TOLMAN SURFACE BRIGHTNESS TEST FOR THE REALITY OF THE EXPANSION. I." by Sandage and Lubin

"Tolman (1930, 1934) derived the remarkable result that, in an expanding universe with any arbitrary geometry, the surface brightness of a set of "standard" (identical) objects will decrease by (1 + z)4. One factor of (1 + z) comes from the decrease in the energy of each photon due to the redshift. The second factor comes from the decrease in the number flux per unit time. Two additional factors of (1 + z) come from the apparent increase of area due to aberration."

The part I don't understand is the two additional factors from the apparent increase in area due to aberration.
You cover the redshift and time dilation contributions, but have you included (in your own words) the effect due to aberration and what do they mean by this?

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I think aberration here is another way to say that the flux is spread over a larger area, due to the expansion, i.e., the inverse-square effect in another form. The latter way takes the point of view of the photons, the former, the point of view of the observer. I think you'd really only call it aberration if it was a bound system doing the emitting, so that it doesn't take part in the expansion and ends up looking much larger than it actually is. If it looks larger, it has to be less bright to compensate, for a fixed total energy flux.

Here's a paper on the subject that shows that the universe is not expanding.

http://photoman.bizland.com/bbnh/lernerpaper4.pdf

"If it looks larger, it has to be less bright to compensate, for a fixed total energy flux."

Now this is where I'm confused.
I understand that the surface brightness remains conserved in Euclidean cosmology and falls off as (1+z) in a non-expanding universe with a redshift mechanism, but still can't understand why any emitter would actually look larger- be it an expanding universe or not.
Thanks Ken


"Here's a paper on the subject that shows that the universe is not expanding"

Thanks John. Only just had an opportunity to look at the paper, but it appears to be both relevant and interesting.

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Actually, it would be (1+z)^2 for a redshift mechanism that conserved
photon number. One power of 1+z is from the loss in energy, and the other is from the mapping of the energy into wider wavelength bins.

It would look larger relative to a similar object at a similar distance as the photons have travelled. Maybe it's better to think of it as looking the same size, but having lower integrated flux because the light gets spread out by the expansion. Diluted, if you will.

Because at sufficiently large redshift (z > 1.6 or so depending on the cosmological expansion parameters) the light we observe NOW left at a time sufficiently long ago that the galaxy in question was closer to us THEN than a galaxy at a lower redshift. At very low redshifts, the angular size vs. "distance" is about what you expect, but it eventually deviates at larger z and then turns around because we are looking so far back in time that all galaxies were closer together. So for galaxies of a fixed physical size, their angular sizes at first diminish with increasing redshift, reach a minimum and then become greater at greater redshift. The relationship of angular size vs. redshift is demonstrated at the bottom of this bunch of links of mine, and also here.

I'd be curious to know what current tests are showing relative to which is correct, the Big Bang model or the static model. Since Spaceman Spiff indicates a discontinuity at z > 1.5 or so, I would be most interested in results at lower z's than that.

Yes, that makes sense.
Also, your second reference -section on "Angular Diameter Distance" helped me understand.
I found it a difficult concept to grasp that the 'space' between an emitter and receiver is still expanding while the photon is traveling between them.

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http://arxiv.org/PS_cache/astro-ph/pdf/0210/0210394.pdf

"This short pedagogical paper provides definitions of and equations for the K correction" - David Hogg

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