WK: In a space with a non-Euclidian metric, geodetic paths between two points exist, which due to curvature are longer than a virtual straight path.

There is no straight path in that case, but you could say the path length is longer than it would be in flat space.

WK: Along such an extended curved path, the effective speed of light is lower compared with light following the shortest virtual path.

No, the speed is the same but it would take a longer time than in flat space due to the increased length.

WK: The effective light speed reduction leads to an according reduction of the rate, with which information is delivered.

That is the key error in your argument. If the time taken for the light to travel the path is constant then the rate at which it is received will be the same as which it was transmitted hence frequencies will be unaffected.

WK: A possible extension of a light path using mirrors is not a counterexample because such a path is not geodetic.

It is actually, that is the reason for the law of reflection, the point where the image is seen is locally extremal.

Hi George,

Thank you for pointing to the key error. But it is not convincing, because the effective speed of light decreases with the length ratio and  is therefor not constant along the path. The effective speed of light is only visible in the view of the receiver. In the view of the line, the speed is always the same.

Consider a beam of light shining through an aquarium. The speed is different whether the light is in the air outside, the glass or the water inside due to the different refractive indexes, but the frequency at any point is the same.

One way to understand that is to think of wave crests which are emitted at intervals of the period. Each crest takes some time to travel to the point of observation but since that time is the same for consecutive crests, the time between arrivals is the same as at emission.

Two crests are emitted at te1 and te2 and P is the period.

te2 = te1 + P

If the light travel time is T then they are received at tr1 and tr2:

tr1 = te1 + T

tr2 = te2 + T

tr2 - tr1 = te2 - te1 =P

WK: it is not convincing

It is your job to make a convincing argument since you are saying that everyone else (and the trivial maths above) is wrong.

Hi George,

Now we come close to the key.

Imagine your beam of light crossing a very large medium with a gradually increasing refractive index and lets assume there is no dispersion and no absorption. The beam enters the medium with refractive index 1.0  and leaves the medium with refractive index 1.1.

If you compare this beam with the same beam transmitted outside of the medium you will find your crests extended in time due to the gradually reduced group speed in the medium.

In a curved space, the group speed and the phase speed everywhere will stay the same. But the relative path extension may be considered to be equivalent to a gradually modified (vacuum) dielectric constant.

WK: Imagine your beam of light crossing a very large medium with a gradually increasing refractive index and lets assume there is no dispersion and no absorption. The beam enters the medium with refractive index 1.0  and leaves the medium with refractive index 1.1.

OK, no problem.

WK: If you compare this beam with the same beam transmitted outside of the medium you will find your crests extended in time due to the gradually reduced group speed in the medium.

No you won't. For a constant refractive index travelling distance D, the delay is:

T = D / (c/n) = c-1 n D

For a refractive index that varies along the path, use the path integral. If

T = c-1 times the integral from 0 to D of n(l) dl

That delay value T is still constant so my equation in the previous post still applies, the frequency will not be affected.

Hi George,

I think you are right, in our Euclidian space it is not possible to construct any example, which leads to a geometrical red shift.

But this does not disprove the existence of a geometrical red shift due to length extension effects along a geodetic path in a curved geometry. It only shows, that it is not easy finding a plausible example.

The path integral I wrote is valid for any path in any geometry. As long as the path has constant length and the refractive index doesn't vary with time, the delay is constant and the received frequency will be unchanged.

But this is only valid if the relation between time and length is not touched by the curvature effects.

Right, that is the standard cosmological explanation for redshift, the paths are getting longer due to spacetime curvature, also called "expansion". In a static universe with non-zero curvature, the path length could be longer than in a flat geometry but it remains constant so there would be no redshift.

Hi George,

according to the general theory of relativity, red shift may also occur in a static geometry witch curvature caused by gravitation.

Therefore, is the following conclusion acceptable?

The question 'does a geometrical red shift exist' already has been positively answered by the general theory of of relativity. According to this theory, the gravitational dilatation of the space actually can lead to a frequency shift. A space dilatation or distortion in a curved  geometry, produced by the finite size of the space, therefore also can lead to a frequency shift. What remains is a quantitative analysis of vacuum wave propagation with a modified spatial part of the Minkowsky metric.

A preliminary answer therefore can be "Theoretically, a geometric red shift in a static geometry is possible". The final answer requires a profound analysis.

WK: according to the general theory of relativity, red shift may also occur in a static geometry witch curvature caused by gravitation.

As I have explained repeatedly throughout this thread, that is not true. Spatial curvature alone in a static universe alters "angular size distance" but does not cause any redshift.

Hi George,

you only address the long range effect of the "gravitaional red shift".

This kind of red shift indeed does not affect the frequency of light which enters and leaves the curved area, but the short range effect alters the frequecy of light, which only leaves this area. The classical physical reasoning for this red shift is the gravitational potential difference.

But there is also a geometric explanation of the gravitational short range effect, based on the curved space. We therefore also can assume a possible influene of "finite Universe curvature" on wave propagation.

I would like to thank you for your patience in this thread which seems to leave the classical state of art in cosmology. And I think that we are still making progess in this discussion as long as continously new kinds of arguments are taken into account.

Hi Wolfgang,

WK: you only address the long range effect of the "gravitaional red shift".

Yes, I thought that was the topic you were examining, a universe with overall positive curvature on cosmological scales where local effects are averaged out.

WK: This kind of red shift indeed does not affect the frequency of light which enters and leaves the curved area, but the short range effect alters the frequecy of light, which only leaves this area. The classical physical reasoning for this red shift is the gravitational potential difference.

That is a local effect though, I have not bee addressing that as you say though of course we are both aware of "gravitational time dilation".

WK: But there is also a geometric explanation of the gravitational short range effect, based on the curved space

It is based on "curved spacetime" which is an important distinction. Curved space alone doesn't produce redshift.

WK: We therefore also can assume a possible influene of "finite Universe curvature" on wave propagation.

When you revert to discussing the universe as a whole, the same distinction applies.

It has been an interesting discussion but it is important that we keep our terminology consistent, curved space versus curved spacetime is a key factor, as is the distinction between local and global effects.

Hi George,

ok, in my terms, I have not been precise enough. Remaining is the question about a possible analogy of curved space time effects, which produce the gravitational red shift and possible "finite Universe" space time effects. 

Or with other words, do we expect an influence of very tiny corrections of the Minkowsky metric on the wavelength.

(The modification of the metric is extremely tiny, we are using the secant-  instead of the arc- length for any distance mapped on a circle with a radius of about 7Billion light years. )

A possible classical explanation of the effect would be, that due to the tiny deviation from a plane wave geometry, the shape/volume of the wave (and with it the wave length) becomes modified on its extemely long way. 

WK: Or with other words, do we expect an influence of very tiny corrections of the Minkowsky metric on the wavelength.

The answer is no, we don't. Have a look at the linked page. It shows, amongst other aspects, how measurements of distance using triangulation would give a different result in curved space compared to flat. However, as long as that space is not expanding, the length of the path would be constant hence there will be no redshift.

http://scienceblogs.com/startswithabang/2012/07/18/how-big-is-the-entire-universe/

Hi George,

Consider a 6 dimensional space defined as follows:

The original coordinates are: t, x, y, z, the new coordinates are: t, a, u, b, v, c, w.

The coordinate transformation is: a=R*sin(x/R), u=R*cos(x/R), b=R*sin(y/R), v=R*cos(y/R), c=R*sin(z/R), w=R*cos(z/R)

and inversely

x=R*atan2(a,u), y=R*atan2(b,v), z=R*atan2(c,w).

In this space, we consider a photon starting at Point (x0,y0,z0) in direction of z. Then we observe the point (x0+dx,y0,z0) and recognize, that at this point, the z-direction is turned in y-direction by the extremely tiny angle dx/R in respect to the z-direction at Point (x0,y0,z0).

Observing the point (x0,y0+dy,z0) accordingly shows the z-direction altered  in  x-direction by dy/R.

The altered neighboured z-directions (which are geodesics) mean, that on its way in z-direction the photon experiences the area perpendicular to its pointing vector diverging with a rate proportional to (z-z0)dx*dy/R². This means that the energy density of the photon is reduced with a corresponding tiny rate.

This is not a proof, but a very plausible scenario. In the local geometric sense, the space described above is completely flat. Parallel transport around a limited path leads to zero deviation. But neighboured geodesics are not parallel.

Considering the metrics, indeed does not explain this red shift.

WK: The coordinate transformation is: a=R*sin(x/R), u=R*cos(x/R), b=R*sin(y/R), v=R*cos(y/R), c=R*sin(z/R), w=R*cos(z/R)

Those are still just a function three independent variables, x, y and z, so you haven't increased the number of dimensions. However, that is probably irrelevant for the argument.

WK: This means that the energy density of the photon is reduced with a corresponding tiny rate.

If light spreads out over a larger area, consider the usual inverse square law for example, then the number density of photons is decreased but the energy per photon is unchanged hence there is no change in frequency. If you want to express that in 6 dimensions, or any other number, the same will still be true, there is no redshift in your example.

Hi George,

Closing a space in the way that one side is connected to the opposite side requires an excursion to a higher dimension. But you are right with the argument, that the proposed space is not really six dimensional because three dimensions are kept shrinked to length zero.

But for analysing the impact of the geometry on physics, we consider a single photon and analyse what happens to that photon on its way through the proposed space. We also can assume that a single photon travels along a single geodesic line because we do not consider a light source emitting photons in all directions.

If now the single photon has a certain nonzero extension (which is only an assumption) and if within this extension geodesic lines are not parallel, then this must have an impact on the photon.

The fact, that the Riemannian curvature in that space is exactly zero, does not mean that long distance effects are excluded.

A single photon travels along every possible path, look up "sum of histories". There are consequences, notably in the double slit experiment where interference fringes arise, but the colour of the light hitting the screen is the same as was emitted, as long as the path length is constant, there is no redshift.

Hi George

What we are looking for, are just cases in which a red shift occurs after a photon has passed a very long way. If there are diverging geodesics on that path, it would be the same as an expansion of the space which you consider as the only possible source of a red shift.

What remains is therefore proving that diverging geodesics occur in the proposed space. 

WK: If there are diverging geodesics on that path, it would be the same as an expansion of the space

No, it isn't. Think of firing a shotgun compared to a rifle. The various projectiles diverge as well as travelling to the target, but that doesn't alter the distance travelled. If the distance between the gun and target is constant, the kinetic energy of the shot is not reduced and the sound of the gun firing is not affected by Doppler shift just because the shot diverges.

Hi George

The comparison with the shotgun may be applicable on a first view. But diverging geodesics are not analog to individual paths of shotgun particles. 

It is only my assumption that diverging geodesics lead to a dilatation of the surrounding space on a path along a single geodesic.  But proving or disproving  this assumption only requires appropriate mathematics. And we already know that space dilatation actually leads to a red shift.

WK: But diverging geodesics are not analog to individual paths of shotgun particles.

I see no difference though it is just an analogy. The mathematical point remains, redshift is caused by increasing path length, not spatial curvature.

Hi George,

Let us imagine that we virtually divide the space covered by a single light particle into a set of sub-spaces. The sub-spaces then individually follow diverging geodesics. After that, we virtually re-combine the modified subspaces. As a result we get a dilataded space because the distance between the subspaces has been increased. If now that dilataded space is covered by the same light particle as before, we can expect an influence on the frequency.

WK: The sub-spaces then individually follow diverging geodesics

No, they don't "follow" anything, they are spaces through which the light moves.

If each "sub-space" is expanding along the path then the ends are moving apart, that is the current cosmological model. The source and observer are moving apart hence you get redshift similar to the Doppler effect. If each "sub-space" has a constant size then there is no redshift. Spatial curvature produces a magnification effect but not a frequency change.

Hi George

The sub-spaces do not have a constant size, the size increases with the increased distance. The sub spaces always cover the complete volume.

If the sizes are increasing as the light passes through them then you have the standard "Big Bang" model and explanation for redshift.

Hi George,

The light does not pass through the volume, it is transported with the volume. Every single point of the volume follows a geodesic and the geometric shape of all geodesics in the underlying space is absolutely static. In addition, this space is homogenous and isotropic.

WK: The light does not pass through the volume, it is transported with the volume.

That makes no sense for volumes through which light is travelling in different directions. The CMB is isotropic so that is always the situation.

Hi George,

This is the model:

The virtual volume, we are talking about is just the volume covered by a single photon. It moves with the speed of light along geodesic lines and keeps carrying the complete photon.

Because every point in that volume independently follows its own geodesic the volume modifies its shape along non parallel geodesics. If the geodesics diverge, there is a dilation of the volume, the same kind of dilation as we would expect in an expanding space.

The expansion is transverse in that case though. If it doesn't affect the path length, there will still be no redshift. The whole energy of a photon is always transferred in a single interaction, not spread over a widening area.

Hi George,

The mathematical evaluation shows, that the expansion is not only transverse. The longitudinal direction is also affected. 

Not if you are describing just geodesic deviation in a static but curved spacetime. If the path length is increasing space is expanding as described by the usual scale factor.

The path length is not modified. Only the distance between points, wich follow neighbored geodesic lines is increasing. The geodesic lines are associated to one common displacement vector, but start at neighboured points.

The mathematical model with this properties associates an orientation vector to a position and a rotation vector to a displacement. The rotation axis is perpendicular to the orientation and to the rotation vector. A displacement operation in the Euclidian space is then composed by a coordinate transform into the image space, a rotation and the back transform.

A reference point follows a geodesic determined by a given displacement vector. The orientation of the rotation axes of neighboured points in transversal direction, is different. Neigbored points in longitudinal direction have a modified direction of the rotation vector. ("Transverse and longitudinal direction" as well as "different" always means "compared with the situation of the reference point")

WK: The path length is not modified.

In that case, there will be no redshift.

Fact is a dilatation of the volume transported along those geodetic lines. Is there only a red shift, if the cause of the volume dilatation is an expansion of the whole universe?

WK: Is there only a red shift, if the cause of the volume dilatation is an expansion of the whole universe?

No, there will be a redshift if the path length changes for any reason, for example in the Doppler effect. You can understand this if you consider what will happen to the gap between two independent light flashes. Imagine looking through a telescope at a remote lighthouse, what would cause the period of the flashes observed to differ from the period measured at the source? Fourier analysis tells us that the redshift of the light must be the same ratio as the ratio of the periods for the flashes.

We accept, that modified length scaling causes a red shift and in the case we are currently talking about, we consider a modified length scaling within the small volume covered by the electromagnetic energy.

But the topological length extension does not only modify this small volume, it also applies to the larger space between different pulses.

Therefore, the topological space dilatation is homologous to the assumed temporal space extension of the whole universe.

WK: But the topological length extension does not only modify this small volume, it also applies to the larger space between different pulses

In which case you have the standard model, if the distance between lighthouse flashes increases and flashes are neither created nor destroyed, the distance between the source and observer must also be increasing.

Wolfgang, you keep making the same statements just in different words so I don't think this conversation is going anywhere. The bottom line is always the same, if the distance between the source and observer is increasing, there will be a redshift and if it is constant, there will be none. Spatial curvature alone would produce a lensing effect but no redshift.

Ok, I agree that we will not be able to reach a consensus. The concept of a topological space dilatation is something which does not have any analogon within our physical experience. Let mathematicians find out if space dilatation caused by topology has the same physical consequences as space dilatation caused by an expansion of the universe.