Cosmological Redshift and metric expansion

[KJW]

On 11/25/2023 at 4:48 AM, KJW said:

On 11/25/2023 at 12:33 AM, Markus Hanke said:

I also suspect (not sure though) that this would introduce off-diagonal terms into the metric?

No, it doesn't. Only the time coordinates are involved in the coordinate transformation. If the original metric describes a flat three-dimensional space, then the transformed metric will be a scalar function multiple of the Minkowskian metric:

g_uv = ƒ(t) η_uv

Consider:

(ds)² = (c dt)² – a(t)² ((dx)² + (dy)² + (dz)²)

t = ƒ(t') ; x = x' ; y = y' ; z = z'

dt = ∂t/∂t' dt' + ∂t/∂x' dx' + ∂t/∂y' dy' + ∂t/∂z' dz' = dƒ(t')/dt' dt'
dx = ∂x/∂t' dt' + ∂x/∂x' dx' + ∂x/∂y' dy' + ∂x/∂z' dz' = dx'
dy = ∂y/∂t' dt' + ∂y/∂x' dx' + ∂y/∂y' dy' + ∂y/∂z' dz' = dy'
dz = ∂z/∂t' dt' + ∂z/∂x' dx' + ∂z/∂y' dy' + ∂z/∂z' dz' = dz'

dƒ(t')/dt' = a(ƒ(t'))

Solve for ƒ(t'), then let A(t') = dƒ(t')/dt' = a(ƒ(t'))

Then:

(ds)² = A(t')² ((c dt')² – (dx')² – (dy')² – (dz')²)

..............................

For example, let a(t) = k t. Then:

(ds)² = (c dt)² – (k t)² ((dx)² + (dy)² + (dz)²)

dƒ(t')/dt' = k ƒ(t')

ƒ(t') = exp(k t' + C)

where C is an arbitrary constant.

A(t') = k exp(k t' + C)

(ds)² = (k exp(k t' + C))² ((c dt')² – (dx')² – (dy')² – (dz')²)

 

 

Edited November 26, 2023 by KJW

[KJW]

Continuing from my previous post:

 

 

On 11/26/2023 at 9:47 AM, KJW said:

(ds)² = (k exp(k t' + C))² ((c dt')² – (dx')² – (dy')² – (dz')²)

This can be further developed by manipulating the arbitrary constant to obtain:

(ds)² = exp(k (t' – t'₀))² ((c dt')² – (dx')² – (dy')² – (dz')²)

where t'₀ is an arbitrarily chosen value of t' at which the metric is locally Minkowskian

 

 

On 11/26/2023 at 9:47 AM, KJW said:

(ds)² = A(t')² ((c dt')² – (dx')² – (dy')² – (dz')²)

For the general case:

dƒ(t')/dt' = a(ƒ(t'))

1/a(ƒ(t')) dƒ(t')/dt' = 1

Let F(t) be such that:

dF^–1(t)/dt = 1/a(t)

Then:

F^–1(ƒ(t')) = t' – t'₀

where t'₀ is an arbitrarily chosen value of t'

ƒ(t') = F(t' – t'₀)

dƒ(t')/dt' = dF(t' – t'₀)/dt' = A(t' – t'₀)

(ds)² = A(t' – t'₀)² ((c dt')² – (dx')² – (dy')² – (dz')²)

 

 

Edited November 27, 2023 by KJW

[KJW]

If one has a conformally flat metric, the corresponding Friedmann-Lemaître-Robertson-Walker (FLRW) metric can be obtained from it by a coordinate transformation:

(ds)² = A(t)² ((c dt)² – (dx)² – (dy)² – (dz)²)

t = ƒ(t') ; x = x' ; y = y' ; z = z'

dt = ∂t/∂t' dt' + ∂t/∂x' dx' + ∂t/∂y' dy' + ∂t/∂z' dz' = dƒ(t')/dt' dt'
dx = ∂x/∂t' dt' + ∂x/∂x' dx' + ∂x/∂y' dy' + ∂x/∂z' dz' = dx'
dy = ∂y/∂t' dt' + ∂y/∂x' dx' + ∂y/∂y' dy' + ∂y/∂z' dz' = dy'
dz = ∂z/∂t' dt' + ∂z/∂x' dx' + ∂z/∂y' dy' + ∂z/∂z' dz' = dz'

Note that the primed coordinates (t', x', y', z') and unprimed coordinates (t, x, y, z) have reversed roles compared to the earlier posts in this thread.

dƒ(t')/dt' = 1/A(ƒ(t'))

A(ƒ(t')) dƒ(t')/dt' = 1

Let F(t) be such that:

dF^–1(t)/dt = A(t)

Then:

dF^–1(ƒ(t'))/dt' = A(ƒ(t')) dƒ(t')/dt' = 1

F^–1(ƒ(t')) = t' – t'₀

where t'₀ is an arbitrarily chosen value of t'

ƒ(t') = F(t' – t'₀)

Let a(t' – t'₀) = A(ƒ(t'))

Then:

dƒ(t')/dt' = 1/A(ƒ(t')) = dF(t' – t'₀)/dt' = 1/a(t' – t'₀)

And therefore:

(ds)² = (c dt')² – a(t' – t'₀)² ((dx')² + (dy')² + (dz')²)

..............................

For example:

(ds)² = exp(k t)² ((c dt)² – (dx)² – (dy)² – (dz)²)

A(t) = exp(k t)

1/A(ƒ(t')) = dƒ(t')/dt' = exp(–k ƒ(t'))

exp(k ƒ(t')) dƒ(t')/dt' = 1

(1/k) exp(k ƒ(t')) = t' – t'₀

where t'₀ is an arbitrarily chosen value of t'

ƒ(t') = (1/k) ln(k (t' – t'₀))

1/a(t' – t'₀) = dƒ(t')/dt' = 1/(k (t' – t'₀))

a(t' – t'₀) = k (t' – t'₀)

Therefore:

(ds)² = (c dt')² – (k (t' – t'₀))² ((dx')² + (dy')² + (dz')²)

as would be expected from the earlier post.
 

 

Edited December 4, 2023 by KJW

[KJW]

(ds)² = A(t)² ((c dt)² – (dx)² – (dy)² – (dz)²)

For the cosmological redshift, Z, the use of the conformally flat metric simplifies the calculation because the equation of a light-like trajectory in spacetime has the simple form of a straight line. Let the observer be at the origin of the coordinate system (t = 0 ; x = 0 ; y = 0 ; z = 0), and let the emitter of two light-pulses, an infinitesimal interval of time apart, be at x = X > 0 ; y = 0 ; z = 0. Then, the equation of the two light-pulses:

t + x/c = 0 ; y = 0 ; z = 0

and:

t + x/c – dt = 0 ; y = 0 ; z = 0

Thus, the emitter emitted the two light-pulses at:

t = –X/c

and:

t = –X/c + dt

These two light-pulses were observed at:

t = 0

and:

t = dt

Then the cosmological redshift, Z:

Z = (ds(t = 0) / ds(t = –X/c)) – 1 = (A(0) c dt / A(–X/c) c dt) – 1

And therefore:

Z = (A(0) / A(–X/c)) – 1

Note that for a redshift, Z > 0, A(0) > A(–X/c), and for a blueshift, Z < 0, A(0) < A(–X/c)

Also note that the x, y, and z coordinates of the conformally flat metric are the same as for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, and therefore the cosmological redshift, specified in terms of –X/c is unchanged for the FLRW metric.

For example:

(ds)² = exp(k t)² ((c dt)² – (dx)² – (dy)² – (dz)²)

Z = exp(k X/c) – 1

 

 

Edited December 20, 2023 by KJW

[AbstractDreamer]

I not going to pretend to follow the math.  Neither do I want to interrupt where the thread has gone, but I want to bring it back layman speak on my level.
 

On 11/24/2023 at 6:48 PM, KJW said:

The OP enquired as to why it is only space that expands and not time. The answer is that the difference between time expanding with space and time not expanding with space is just a coordinate transformation, which means that there is no physical difference. However, as you correctly point out, the time coordinate of the time-expanding metric doesn't correspond to anything, in particular, not a co-moving clock, whereas the spatial coordinates actually do correspond to the co-moving cosmological fluid.

Earlier in this thread I did already mention that the FLRW metric is "orientated" where time does not expand with space.  I had suspected that a transformation could orientate it differently such that time does expand with space.  And I suspected at the other extreme we can have a solution where space does not expand at all and only time does.

In another thread I asked about a variable "metric of time", and the "rate of flow of time", which was very difficult to conceptualise and it sort of ended there.

My position is that l still maintain the validity of the interpretation that:  non-relative time expansion/contraction is indistinguishable from space expansion/contraction.

When you look up cosmological redshift in wiki there is no "Temporal Redshift" type.  That is, redshift caused by an expanding temporal metric.  It doesn't exist.  Not a single reference, no studies, no papers.

Why?  Just because its too complex and abstract compared to space-expansion-only theory?   I don't believe complexity is a reason for the entire physics community to shy away from such an interpretation.  If it is valid, and no-one has researched into temporal redshift, it can only be because "space expansion" and its universal acceptance has blinded us to the truth that is only one alternative of other interpretations.




 

[Markus Hanke]

7 hours ago, AbstractDreamer said:

I had suspected that a transformation could orientate it differently such that time does expand with space.

You are of course always free to pick different coordinates to describe the same spacetime - which is one of the central insights in GR. However, when you do this you also change the physical meaning of those coordinates. In the standard FLRW metric, the time coordinate is chosen such that it corresponds to a clock that is co-moving with the cosmological medium, meaning it fits well with our own physical clocks here on Earth, and thus the “phenomenology” of the metric corresponds to what we actually observe, without any need for complicated transformations.

You are free to choose a coordinate system where eg tick rates aren’t constant, but then you need to be very careful how you relate the metric to real-world observations, since the t-coordinate no longer corresponds to Earth-bound clocks.

Ultimately it is best to describe the spacetime in terms of geometric properties that are independent of coordinate choice; in the case of FLRW for example, we can say the spacetime is conformally flat, meaning during free fall angles are preserved, but not volumes.

7 hours ago, AbstractDreamer said:

When you look up cosmological redshift in wiki there is no "Temporal Redshift" type.  That is, redshift caused by an expanding temporal metric.  It doesn't exist.  Not a single reference, no studies, no papers.

These aren’t different “theories”, but simply coordinate choices. You’re describing the same spacetime in different coordinates. KJW has given an example how a “time-only” expansion metric could look like. Ultimately you want to choose coordinates that make your calculations as simple as possible, and that’s often ones based on the cosmological medium. But in principle, the choice is yours, so long as they’re related by valid transformations.

[KJW]

13 hours ago, AbstractDreamer said:

And I suspected at the other extreme we can have a solution where space does not expand at all and only time does.

No, it is not possible to coordinate-transform a metric of the form:

(ds)² = c² (dt)² – a(t)² ((dx)² + (dy)² + (dz)²)

to a metric of the form:

(ds)² = α(t)² c² (dt)² – ((dx)² + (dy)² + (dz)²)

In general, a metric of the form:

(ds)² = c² (dt)² – a(t)² ((dx)² + (dy)² + (dz)²)

has non-zero Ricci curvature, whereas a metric of the form:

(ds)² = α(t)² c² (dt)² – ((dx)² + (dy)² + (dz)²)

describes flat spacetime. Note that this metric can be transformed to the Minkowskian metric by the coordinate transformation:

t' = t'(t)    ;    x' = x    ;    y' = y    ;    z' = z

where t'(t) is a solution to the differential equation:

dt'(t)/dt = α(t)

 

 

Edited 9 hours ago by KJW

[AbstractDreamer]

7 hours ago, Markus Hanke said:

In the standard FLRW metric, the time coordinate is chosen such that it corresponds to a clock that is co-moving with the cosmological medium, meaning it fits well with our own physical clocks here on Earth, and thus the “phenomenology” of the metric corresponds to what we actually observe, without any need for complicated transformations.

So by choosing such time coordinates, it also inherits the assumption that the cosmological medium of time is moving uniformly, everywhere and always (an Earth bound clock must tick at the same rate as the rest of the universe now and in the past and in the future).   And yet, our observations are bounded to a infinitesimally small location of the universe (observations from our solar system compared to the size of the universe), and a very small period of time (150/13billion years). 

There is a problem here I cant quite put into words, so I will use bad analogies.  Its like everyone being colour-blind and believing the universe is shades of grey.  You can observe light wavelengths, but you cant observe the colour blue.  It has no physical meaning.   Its like believing gravity is a force before GR modelled spacetime curvature. 

Everything must fit what we observe (of course, to be empirically tested), but we don't acknowledge enough how severe our observations are restricted/limited.  In many areas of science, where and when you perform an observation has no bearing on what is being tested.  In THIS particular case of redshift, when and where you perform an observation is of paramount significance... and we are straight-jacketed into observations from our solar system location (where ever it is in the universe), and observations from our moment in time (a few hundred years).  The limitations of our observations are significant relative to the field of study.

 

7 hours ago, Markus Hanke said:

These aren’t different “theories”, but simply coordinate choices. You’re describing the same spacetime in different coordinates. KJW has given an example how a “time-only” expansion metric could look like. Ultimately you want to choose coordinates that make your calculations as simple as possible, and that’s often ones based on the cosmological medium. But in principle, the choice is yours, so long as they’re related by valid transformations.

But space-expansion IS a theory, as is the more absurd temporal-expansion.   The premise for the theories is from choice of coordinates.
 

 

2 hours ago, KJW said:

No, it is not possible to coordinate-transform a metric of the form:

(ds)² = c² (dt)² – a(t)² ((dx)² + (dy)² + (dz)²)

to a metric of the form:

(ds)² = α(t)² c² (dt)² – ((dx)² + (dy)² + (dz)²)

 

Is this saying there is no transformation that will allow only time to expand and not space?  What is the meaningful consequence of this? 

[Mordred]

3 hours ago, AbstractDreamer said:

So by choosing such time coordinates, it also inherits the assumption that the cosmological medium of time is moving uniformly, everywhere and always (an Earth bound clock must tick at the same rate as the rest of the universe now and in the past and in the future).   And yet, our observations are bounded to a infinitesimally small location of the universe (observations from our solar system compared to the size of the universe), and a very small period of time (150/13billion 

No there is no assumptions due to coordinate choice. 

You already know time dilation is a consequence of spacetime curvature or Relativistic inertia. 

The math and observational evidence shows us that there is no curvature term k=0. So where would you get time dilation ? This has already previously been mentioned.  As massless particles travel at c we can ignore the inertial gamma factor. 

A higher density past the answer either. To go into greater detail if you take 3 time slices say time now, time at the CMB say z=1100. And a slice at say universe age 7 billion years old.  If you describe the geometry of each slice. Each slice has a uniform mass distribution so no slice has a non uniform mass distribution to have a curvature term.

Hint this is the real advantage of the scale factor a. No time slice has any change in geometry or curvature it's simply volume change between slices and density changes as a result of the ideal gas laws

Edited 4 hours ago by Mordred

[AbstractDreamer]

31 minutes ago, Mordred said:

No there is no assumptions due to coordinate choice. 

I don't know the maths at all.  But it seems fundamental to me that if you make a choice, you instantiate something.  When something is instanced, things are set and other settings are rejected.  When you reject other settings, there are fundamental consequences.  These consequences are the assumptions.

If time coordinates are chosen such that earth-bound clocks are comoving with the cosmological medium, that has consequences.  The very choosing of those coordinate forbids a non-relative (non-gravitational) time dilation effect.  That is why FLRW metric forbids temporal redshift, because it was chosen to be orientated that way.  Am I wrong?
 

33 minutes ago, Mordred said:

The math and observational evidence shows us that there is no curvature term k=0. So where would you get time dilation ? This has already previously been mentioned.  As massless particles travel at c we can ignore the inertial gamma factor. 

The observational evidence in this case is dubious solely because of the narrow range of observation relative to the field of study.   We've never made an observation of cosmological redshift from outside of our solar solar system, let alone from a distance where space-expansion or temporal-expansion is significant.   We've never made an observation of cosmological redshift from a time in the past or the future, where spatial or temporal expansion is significant.  I'm not saying any of this is possible.  I'm just saying our sample range of observations is far too narrow to be confident to say our evidence is significant.

We've taken a handful of stones from a beach, and assumed all beaches must be stoney.

As for where would we get time dilation?  Where do we get space-expansion?  Dark energy?  We can make up anything to fit the narrative.

36 minutes ago, Mordred said:

A higher density past the answer either. To go into greater detail if you take 3 time slices say time now, time at the CMB say z=1100. And a slice at say universe age 7 billion years old.  If you describe the geometry of each slice. Each slice has a uniform mass distribution so no slice has a non uniform mass distribution to have a curvature term.

Hint this is the real advantage of the scale factor a. No time slice has any change in geometry or curvature it's simply volume change between slices and density changes as a result of the ideal gas laws 

Advantage for what purpose?  Simplicity and accuracy to fit other observations are similarly limited in their scope?  This again falls foul of confirmation bias.

BTW, someone anonymous is downvoting all my threads.  Not that I care about reputation, but being anonymous and not saying why I'm wrong feels like im being victimised and rather abusive. 

[Mordred]

2 hours ago, AbstractDreamer said:

I don't know the maths at all.  But it seems fundamental to me that if you make a choice, you instantiate something.  When something is instanced, things are set and other settings are rejected.  When you reject other settings, there are fundamental consequences.  These consequences are the assumptions.

If time coordinates are chosen such that earth-bound clocks are comoving with the cosmological medium, that has consequences.  The very choosing of those coordinate forbids a non-relative (non-gravitational) time dilation effect.  That is why FLRW metric forbids temporal redshift, because it was chosen to be orientated that way.  Am I wrong?
 

 

no coordinate choice affects the mass distribution. I could describe the universe in numerous different coordinate choices example Euclidean, spherical cylindrical etc without causing any difference. It is precisely why we use invariance. The mathematics is set up that way so that we do not have any coordinate choice dependency.

2 hours ago, AbstractDreamer said:

As for where would we get time dilation?  Where do we get space-expansion?  Dark energy?  We can make up anything to fit the narrative.

 

you know full well GR fully describes time dilation the FLRW metric is a GR solution.

We don't arbitrarily choose DM and DE as the full explanation those two terms are simply placeholders until we can determine the cause of each. We still can measure their effects through indirect evidence.

2 hours ago, AbstractDreamer said:

BTW, someone anonymous is downvoting all my threads.  Not that I care about reputation, but being anonymous and not saying why I'm wrong feels like im being victimised and rather abusive. 

I rarely give downvotes so its someone else.

As far as sampling range is concerned, redshift is only one of many pieces of evidence of an expanding universe. In point of detail its not even close to the strongest evidence. Its the one most ppl are familiar with but the real evidence comes from our thermodynamic laws in regards to temperature and how it influences the SM model of particles via processes such as BB nucleosynthesis in regards to the CMB.

 One danger of trying to understand cosmology by rote instead of learning the math is that too often you get incorrect information. I will give an example if I looked up hydrogen and its temperature it could form with stability a google search will state 3000 kelvin. However if one knows how to use the Saha equations that would reveal that value equates to 75 % of the potential hydrogen. Hydrogen can start to form as low as 6000 kelvin=25% 4000 kelvin for 50 %. That is just one example.

however knowing this one can study the metallicity of our universe evolution via hydrogen, lithium, deuterium etc. So I just described another piece of evidence for expansion.

In other words were not restricted to redshift to determine if our universe is expanding

. In point of detail we do not rely on redshift in cosmology it is too full of other influences such as gravitational redshift, transverse redshift, Integrated Sache-Wolfe effect, Doppler redshift. etc etc.

We examine all pieces of possible evidence to confirm the accuracy of cosmological redshift. Nor do we use the generic formula everyone sees on google.

https://en.wikipedia.org/wiki/Redshift

this formula only works for nearby objects it loses accuracy as near as one MPC. The full formula includes the influence of the evolution history of matter, radiation and Lambda.

details can be found here

"Distance measures in cosmology"

David W. Hogg

https://arxiv.org/abs/astro-ph/9905116

side note the paper also applies to luminosity distance we also have a different formula for Luminosity distance than what one would google. 

\[H_O dl=(1+z)|\Omega_k|^{-1/2}sinn[\Omega_k^{1/2} \int^z_o\frac{d\acute{z}}{\sqrt{(1+\acute{z})^2\Omega_R+(1+\acute{z}\Omega_m-\acute{z})(2+\acute{z})\Omega_\Lambda}}]\]

What this equation shows is that matter, radiation and Lambda density not only influences expansion rates it also influences redshift and luminosity as well as any curvature term k

 

Edited 3 hours ago by Mordred

[Mordred]

you won't find that equation in a textbook, textbooks only show the basic equations in math speak in this case you would usually see the first order equation this delves into the second order.

just as most textbooks won't show the equation 

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

this shows the expansion rate H varies over time (it will also help to better understand the first equation as well as the Hogg paper I posted.

now as you mentioned DM and DE one line of research is Higgs being responsible. Sterile neutrinos (right hand are heavier than left hand neutrinos  ) antimatter and matter neutrinos. 

so the calculated abundance could fall into range 

\[\Omega_pdmh^2=\frac{G^{3/2}T_0^3h^2}{H_0\sigma v}=\frac{3*10-{27} cm^3s^{-1}}{\sigma v}\]

research is still on going. Just as the equation of state for the Higgs field may explain inflation as well as the cosmological constant.

That should sufficiently show that what really goes on in the professional circles isn't something one can simply google at best that just gives hints 

Edited 3 hours ago by Mordred