Is the inverse-square law valid for all cosmological distances?

[StateOfTheEqn]

Among the three Friedmann models k=-1,0,+1, the only model in which the inverse-square law is valid at all distances is that for k=0. In other words, its validity depends on the flatness of space. It is easy to show using the R-W metric with k=+1 that objects get magnified and that effect increases with distance. The magnification not only affects the apparent size but also affects the brightness of 'standard candles'.

Two questions:

(1)Is the value for k=0 in the current consensus model calculated assuming the validity of the inverse-square law? That would obviously create a circular argument.

(2)By affecting the brightness of distant standard candles, can a failure of the inverse-square law account for the apparent acceleration in the expansion of the universe by underestimating the distance of those galaxies producing red-shifts greater than that predicted by the Hubble relation?

 

[Chalnoth]

(1)Is the value for k=0 in the current consensus model calculated assuming the validity of the inverse-square law? That would obviously create a circular argument.

Nope. The measurements we use take General Relativity fully into account. There is no reduction to a Newtonian approximation for cosmological observations.

(2)By affecting the brightness of distant standard candles, can a failure of the inverse-square law account for the apparent acceleration in the expansion of the universe by underestimating the distance of those galaxies producing red-shifts greater than that predicted by the Hubble relation?

It may potentially be the case that if we modify gravity beyond General Relativity, we might be able to explain the cosmic acceleration. Nobody has yet put forward a compelling model for doing so that also fits observation, however. Nevertheless, one of the potential benefits of future weak lensing surveys is that they may be able to distinguish between some sorts of modified gravity and some sort of dark energy.

 

[IsometricPion]

It may potentially be the case that if we modify gravity beyond General Relativity, we might be able to explain the cosmic acceleration. Nobody has yet put forward a compelling model for doing so that also fits observation, however.

I was under the impression that [tex]\Lambda[/tex]http://en.wikipedia.org/wiki/Lambda-CDM_model" (which is just GR with non-zero cosmological constant, [tex]\Lambda[/tex], and appropriate mass-energy boundary conditions) adequately fits the cosmological data gathered so far.

 

[Chalnoth]

I was under the impression that [tex]\Lambda[/tex]http://en.wikipedia.org/wiki/Lambda-CDM_model" (which is just GR with non-zero cosmological constant, [tex]\Lambda[/tex], and appropriate mass-energy boundary conditions) adequately fits the cosmological data gathered so far.

Yes. This is accurate. Some people (unreasonably, I feel) don't like the cosmological constant, and so attempt to propose ways of explaining the acceleration without it.

 

[StateOfTheEqn]

I have attempted to do something of a literature review on the various distance measurements used in cosmology that depend on the inverse-square law. It is often difficult when looking at an equation to see the assumptions contained within. So far, the most enlightening paper I have found is at
http://www.ssl.berkeley.edu/~mlampton/ComovingDistance.pdf

It seems the inverse-square law is deeply embedded in much of the methodology of current cosmology. To me, that seems to be hanging a very heavy weight on a very weak hook.

 

[Chalnoth]

I have attempted to do something of a literature review on the various distance measurements used in cosmology that depend on the inverse-square law. It is often difficult when looking at an equation to see the assumptions contained within. So far, the most enlightening paper I have found is at
http://www.ssl.berkeley.edu/~mlampton/ComovingDistance.pdf

It seems the inverse-square law is deeply embedded in much of the methodology of current cosmology. To me, that seems to be hanging a very heavy weight on a very weak hook.

Why? General Relativity is exceedingly well-tested, and in cosmological studies, many sorts of deviations from General Relativity have been investigated. They simply don't explain any of the apparent observational discrepancies, so as yet there is no reason whatsoever to believe that they are valid.

 

[StateOfTheEqn]

It may potentially be the case that if we modify gravity beyond General Relativity, we might be able to explain the cosmic acceleration. Nobody has yet put forward a compelling model for doing so that also fits observation, however. Nevertheless, one of the potential benefits of future weak lensing surveys is that they may be able to distinguish between some sorts of modified gravity and some sort of dark energy.

The question I was trying to address is this: Could the apparent acceleration in the expansion result from the failure of the inverse-square law to give correct standard-candle distances? In other words, is the acceleration real or the product of false assumption(s) about the validity of the inverse-square law?

 

[StateOfTheEqn]

Why? General Relativity is exceedingly well-tested, and in cosmological studies, many sorts of deviations from General Relativity have been investigated. They simply don't explain any of the apparent observational discrepancies, so as yet there is no reason whatsoever to believe that they are valid.

The cosmological equations derived from GR cover three cases:k=-1,0,+1. The only one consistent with the inverse-square law at all distances is k=0.

To question the validity of the inverse-square law is not the same as challenging the validity of GR.

 

[Chalnoth]

The question I was trying to address is this: Could the apparent acceleration in the expansion result from the failure of the inverse-square law to give correct standard-candle distances? In other words, is the acceleration real or the product of false assumption(s) about the validity of the inverse-square law?

Well, yes, many modifications of gravity to explain the apparent acceleration have been attempted. The ones that are most strongly theoretically-motivated (and they're still pretty speculative even then) don't fit the data.

Our best bet at really examining this in detail is by examining the growth of structure in the universe by use of weak gravitational lensing surveys. If reality is described by General Relativity + dark energy (either cosmological constant or something else), then there are certain relationships that the growth of structure must necessarily follow. Violation of these relationships will be evidence for some sort of deviation from General Relativity.

In general, however, the theoretical bias is strongly against this sort of deviation from General Relativity. We generically expect that General Relativity should be modified at high energies, not low energies: it is extraordinarily difficult to produce a theory of gravity that is modified at low energies (long length scales) without violating solar system tests of gravity.

 

[XilOnGlennSt]

It seems the inverse-square law is deeply embedded in much of the methodology of current cosmology. To me, that seems to be hanging a very heavy weight on a very weak hook.

This has worried me too. The different values for k in the Friedman equations would certainly affect light luminosity at distance, but it seems that there are many other possible factors that could affect light over such long distances, quite outside GR. If there have been any empirical verifications of the inverse-square law at cosmological distances, I would like to know about it.

For that matter, isn't it also possible that red shifting of light could be affected by great distance?

Regarding the need to revise the standard cosmological models; what about the introduction the expansion phase, or the need to introduce dark matter and dark energy. These all smell like fudge factors to me.

Please enlighten me, if you think it possible.

 

[Chronos]

Assuming the laws of physics are time invariant, how would redshift be affected by distance? That appears to invoke a 'special' reference frame.

 

[clamtrox]

So this may be a stupid question... but what is the inverse-square law everyone talks about?

 

[Chalnoth]

So this may be a stupid question... but what is the inverse-square law everyone talks about?

Gravity, as described by Newton, reduces in strength proportional to the inverse square of the distance between objects. General Relativity, the theory of gravity first produced by Einstein, doesn't quite follow this rule for very small or very large distances.

 

[Chalnoth]

I don't completely understand. At small distances, it's a good enough approximation in any case, but we have no evidence to suggest that it holds (or doesn't hold) at large distances. So why would challenging the validity of it be so "blasphemous?"

It's not blasphemous. It's just against the evidence. GR describes remarkably well the large-scale evolution of the universe, and modifications of GR that change how GR behaves on large scales tend to fail to do so.

 

[Whovian]

It's not blasphemous. It's just against the evidence. GR describes remarkably well the large-scale evolution of the universe, and modifications of GR that change how GR behaves on large scales tend to fail to do so.

But GR, as stated already, doesn't actually exhibit the inverse square law for a "nonflat" Universe. So my point was, why would questioning the inverse square law be ... oh ... I absolutely failed, missing the word "not" in the post I quoted. Editing. So I was arguing against my own opinion (for lack of a better word, there are reasons opinion is completely unsuitable here.)

 

[Chalnoth]

But GR, as stated already, doesn't actually exhibit the inverse square law for a "nonflat" Universe.

I'm going to assume that your first sentence here still stands :)

The thing to bear in mind is that for most scales of interest here, the inverse square law holds in General Relativity because the spatial curvature and dark energy contributions are small. It is only at very large scales that there is a noticeable departure. On the scales of galaxies and galaxy clusters, the inverse square law holds exceedingly well.

 

[clamtrox]

Gravity, as described by Newton, reduces in strength proportional to the inverse square of the distance between objects. General Relativity, the theory of gravity first produced by Einstein, doesn't quite follow this rule for very small or very large distances.

So I think they are talking about light intensity decreasing as the inverse of r squared, but I'm not completely sure what the point is there. I think the point missed by original poster is that in GR, there is no well-defined universal length measure. The inverse-square law holds for a certain distance measure, defined precisely to be such that it holds.

 

[XilOnGlennSt]

I thought this thread started talking about the inverse-square law of LIGHT intensity, not gravitational intensity. Are these not distinct?

 

[Chalnoth]

I thought this thread started talking about the inverse-square law of LIGHT intensity, not gravitational intensity. Are these not distinct?

Sorry, yes, they are quite distinct.

The inverse square law for light intensity simply does not hold in an expanding universe, as the light is redshifted, which causes it to lose energy in proportion to (1+z). This means that the effective inverse square law for light in an expanding universe becomes:

[tex]L(z) = {L_0 \over (1+z)r^2}[/tex]

Here [itex]L(z)[/itex] is the observed luminosity, while [itex]L_0[/itex] is what the luminosity would be at r=1, z=0.

 

[XilOnGlennSt]

Thanks for your comments. This seems to have good information on this subject: http://newcosmology.nfshost.com/

 

[clamtrox]

Sorry, yes, they are quite distinct.

The inverse square law for light intensity simply does not hold in an expanding universe, as the light is redshifted, which causes it to lose energy in proportion to (1+z). This means that the effective inverse square law for light in an expanding universe becomes:

[tex]L(z) = {L_0 \over (1+z)r^2}[/tex]

Here [itex]L(z)[/itex] is the observed luminosity, while [itex]L_0[/itex] is what the luminosity would be at r=1, z=0.

Yes, but I guess the relevant bit is that it does hold for the number density of photons, if you define your distance appropriately. There exists a way of measuring distance called "luminosity distance", with the property that the number density of photons decreases as r^-2. This can be nicely defined through some elementary geometry to be
[tex] d_L = \sqrt{\frac{A_O}{\Omega_S}} [/tex]
where [itex] A_O [/itex] is the area of a bundle of light rays at the observer, and [itex] \Omega_S [/itex] is the angle the light rays leave the source.

Then one can additionally correct for the redshift of the photons, and define
[tex] \tilde{d}_L = (1+z) d_L [/tex]
arriving at the "usual" definition.

 

[Chalnoth]

Yes, but I guess the relevant bit is that it does hold for the number density of photons, if you define your distance appropriately. There exists a way of measuring distance called "luminosity distance", with the property that the number density of photons decreases as r^-2. This can be nicely defined through some elementary geometry to be
[tex] d_L = \sqrt{\frac{A_O}{\Omega_S}} [/tex]
where [itex] A_O [/itex] is the area of a bundle of light rays at the observer, and [itex] \Omega_S [/itex] is the angle the light rays leave the source.

Then one can additionally correct for the redshift of the photons, and define
[tex] \tilde{d}_L = (1+z) d_L [/tex]
arriving at the "usual" definition.

Yes, but few people find this sort of distance measure understandable, because it is only defined in this way: so that it has the expected falloff with distance.

You've made me realize, however, that I forgot a factor of (1+z). It should be:

[tex]L(z) = {L_0 \over (1+z)^2 r^2}[/tex]

I forget what the physical explanation is for the second factor of (1+z) is, unfortunately. But the distance [itex]r[/itex] here, properly [itex]D_M[/itex], is quite easy to understand: it's the distance you would obtain if you froze the expansion, bounced some light rays around, and measured the travel time of said light rays.

 

[clamtrox]

I forget what the physical explanation is for the second factor of (1+z) is, unfortunately. But the distance [itex]r[/itex] here, properly [itex]D_M[/itex], is quite easy to understand: it's the distance you would obtain if you froze the expansion, bounced some light rays around, and measured the travel time of said light rays.

I think this is true only in a homogeneous and isotropic universe - not in general.

 

[Chalnoth]

I think this is true only in a homogeneous and isotropic universe - not in general.

I was speaking in the context of a homogeneous, isotropic expanding universe, yes. Things obviously get more complicated if you try to consider more complicated space-times. But the reality is that this is the universe we live in.

 

[clamtrox]

I was speaking in the context of a homogeneous, isotropic expanding universe, yes. Things obviously get more complicated if you try to consider more complicated space-times. But the reality is that this is the universe we live in.

To be fair, having one factor of 1+z less doesn't really make it that much more complicated, now does it?

 

[Chalnoth]

To be fair, having one factor of 1+z less doesn't really make it that much more complicated, now does it?

I don't understand what you mean. In what situation is there only one factor of (1+z)?

 

[clamtrox]

I don't understand what you mean. In what situation is there only one factor of (1+z)?

If you define the formula you gave in terms of luminosity distance instead of D_M, it holds for all spacetimes. Using that, you'd have simply [itex] L(z) = L(0)/D_L^2 [/itex]

 

[Chalnoth]

If you define the formula you gave in terms of luminosity distance instead of D_M, it holds for all spacetimes. Using that, you'd have simply [itex] L(z) = L(0)/D_L^2 [/itex]

Your statement here has nothing to do with my question.

 

[yuiop]

It is easy to show using the R-W metric with k=+1 that objects get magnified and that effect increases with distance. The magnification not only affects the apparent size but also affects the brightness of 'standard candles'.

I can show one situation involving Special Relativity where the inverse square law does not hold, which would as you suggest impinge on the assumed relationship between the brightness of a standard candle and its distance. Let has say we had a hypothetical flat universe and completely ignore gravity. In such a universe and assuming some sort of initial explosion, there would still be a Hubble relationship between the recession velocities of distant galaxies and their distances, with more distant galaxies receding faster. The main difference is that the velocity of the receding galaxies would be a function of the relativistic Doppler Shift rather than the classic Newtonian Doppler shift that is currently assumed. In this flat space model the brightness of distant galaxies would be subject to relativistic aberration or beaming sometimes known as the headlight effect. The observable effect is that the brightness of a receding object with a high relativistic velocity would be much less than a simple application of the inverse square law would imply. In turn distant dim objects would appear to be much further away than the inverse square law suggests. The effect is almost insignificant for objects with low recession velocities and ramps up exponentially for recession velocities that are a significant fraction of the speed of light. Any distant galaxy that is currently assumed to be receding at superluminal speed would always be receding at less than the speed of light in the simplistic relativistic flat space model.

 

[Chalnoth]

The main difference is that the velocity of the receding galaxies would be a function of the relativistic Doppler Shift rather than the classic Newtonian Doppler shift that is currently assumed.

Why do you think that Newtonian Doppler shift is currently assumed?

 

[clamtrox]

Your statement here has nothing to do with my question.

I can see that this discussion is not going to work out. :-)

 

[yuiop]

Why do you think that Newtonian Doppler shift is currently assumed?

Because distant Hubble objects are assumed to be stationary relative to their "local space" and so they are assumed to be not subject to relativistic time dilation which is a function of peculiar velocity relative to local space. Therefore the classic Doppler shift formula which does not include time dilation is used. In a flat-space-no-gravity model a receding object with a high relativistic velocity would have to be time dilated and so the relativistic Doppler equation would have to be used.

What model you use depends on what you measure, but what you measure depends on what model you assume.

 

[Chalnoth]

Because distant Hubble objects are assumed to be stationary relative to their "local space" and so they are assumed to be not subject to relativistic time dilation which is a function of peculiar velocity relative to local space. Therefore the classic Doppler shift formula which does not include time dilation is used. In a flat-space-no-gravity model a receding object with a high relativistic velocity would have to be time dilated and so the relativistic Doppler equation would have to be used.

The peculiar motion of galaxies is generally pretty small compared to relativistic velocities, typically less than 1000 km/sec or so. It can't get much greater because any object that moves rapidly with respect to the expansion quickly catches up to the expansion. The only reason why some things move that fast at all is because they're in the vicinity of some nearby massive object that they're falling towards or in orbit around.

 

[yuiop]

The peculiar motion of galaxies is generally pretty small compared to relativistic velocities, typically less than 1000 km/sec or so. It can't get much greater because any object that moves rapidly with respect to the expansion quickly catches up to the expansion.

I understand that, but in the flat-space-no-gravity model there is no peculiar motion. Peculiar motion only belongs to the Hubble flow idea where Hubble objects are receding at the same velocity as the "local space". The simplistic Special Relativistic interpretation does not consider a vacuum or space itself to be a substance with a measurable velocity, which has etheristic implications.

 

[Chalnoth]

I understand that, but in the flat-space-no-gravity model there is no peculiar motion.

Okay, but that model doesn't describe our universe. So why consider it?