Cosmological Redshift and metric expansion

[Markus Hanke]

2 hours ago, AbstractDreamer said:

Is FLRW spacetime not flat because of curvature caused by mass

It’s not flat because when solving the EFE, you start with a universe filled with energy-momentum, which necessarily has a gravitational effect. 

2 hours ago, AbstractDreamer said:

and the scale factor of expansion that changes over time?

The scale factor arises as part of the solution; it’s not a source term that appears in the initial setup.

2 hours ago, AbstractDreamer said:

If we zero the non-flatness effect of gravity AND zero the non-flatness effect of the scale factor, is FLRW spacetime otherwise flat?

If you remove all gravitational sources, you no longer get an FLRW solution - you’d be in a different spacetime.

[AbstractDreamer]

5 hours ago, Markus Hanke said:

It’s not flat because when solving the EFE, you start with a universe filled with energy-momentum, which necessarily has a gravitational effect. 

The scale factor arises as part of the solution; it’s not a source term that appears in the initial setup.

If you remove all gravitational sources, you no longer get an FLRW solution - you’d be in a different spacetime.

Right.  But if you had an FLRW universe devoid of energy momentum, it would still be a valid solution for that universe, if energy momentum could exist.  It just has zero value at the start and at least until the moment of observation.   Sure, that universe would look different to their observers than our universe looks to us.  And sure it probably wouldn't be a solution their would come up with as there is nothing in their universe that would cause curvature, so why would they have a solution that permits it.   But then they could go looking for signs of energy momentum to validate their FLRW solution of a universe that started and still has zero energy momentum.

I don't understand how the cause of spacetime expansion is dependent on energy momentum.  If anything, they are opposing "forces".   I understand how observationally the measurement of spacetime expansion, and the evolution of OUR universe under FLRW is modulated by energy momentum, but the actual mechanic of spacetime expansion - dark energy - does not depend on energy momentum as far as I can understand.

And I accept that even light has energy momentum, so the very presence of redshfted light means energy momentum is present and some degree of non-flat geometry.

Going back to the thought experiment, 

On 5/24/2024 at 3:29 PM, AbstractDreamer said:

Say we observe two redshift galaxies at z=5.    Let's say one galaxy is only spatially expanding away from us, and the other is both spatially and temporally expanding away from us, and all three locations (two galaxies and the observer) are on a spacetime plane that has observably flat geometry.   Could you distinguish which is which?

If dark energy could cause both spatial and temporal expansion, then, in a spacetime geometry of net zero energy-momentum (Minkowski spacetime?)  and over a short period of constant scale factor, could you distinguish how much of the redshift in the wavelength of photons is due to spatial expansion and how much is due to temporal expansion?

 

Edited May 28 by AbstractDreamer

[Mordred]

7 hours ago, Markus Hanke said:

It’s not flat because when solving the EFE, you start with a universe filled with energy-momentum, which necessarily has a gravitational effect. 

The scale factor arises as part of the solution; it’s not a source term that appears in the initial setup.

If you remove all gravitational sources, you no longer get an FLRW solution - you’d be in a different spacetime.

Unfortunately this is rather misleading to understanding the FLRW metric. The FLRW metric has a curvature term K.

K can still be zero and still have expansion for a critically dense universe. Expansion isn't curvature though curvature affects expansion. In our current universe the FLRW metric the stress tensor for energy conservation and applying the cosmological principle is as follows.

Cosmological Principle implies

\[d\tau^2=g_{\mu\nu}dx^\mu dx^\nu=dt^2-a^2t{\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\varphi^2}\]

the Freidmann equations read

\[(\frac{\dot{a}}{a})^2+\frac{k}{a^2}=\frac{8\pi G}{3}\rho\]

for \[\rho=\sum^i\rho_i=\rho_m+\rho_{rad}+\rho_\Lambda\]

\[2\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{k}{a^2}=-8\pi Gp\]

for 

\[p=\sum^ip_i=P_{rad}+p_\Lambda\]

with conservation of the energy momentum stress tensor

\[T^{\mu\nu}_\nu=0\]

\[\dot{p}a^3=\frac{d}{dt}[a^3(\rho+p)]\Rightarrow \frac{d}{dt}(\rho a^3)=-p\frac{d}{dt}a^3\]

\[p=\omega\rho\]

given w=0 \(\rho\propto a^{-3}\) for matter, radiation P=1/3 \(\rho\propto{-3}\), Lambda w=-1.\(p=-\rho\) for k=0

It is the equations of state in the last line that determines expansion and expansion rate. For our current universe we can accurately set k=0 for good approximation. This would actually be a critically dense universe. Our universe however does have a slight curvature term but overall is considered flat.

Another way to see the above is

FLRW Metric equations

\[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\]

\[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\]

\[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\]

\[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\]

setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as 

\[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\]

\[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\]

which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0.

the related GR solution the the above will be the Newton approximation. Shown here

\[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\]

\[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\]

now here you can see where curvature gets applied

\[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\]

 

 

 

In the above expansion rate is determined by the relation of each equation of state (including the curvature term k) to the critical density. That density can be any value it is a calculated density (matter only EFE solution). The actual energy density to critical density ratio is what determines expansion rate. That ratio is affected by the density of matter, radiation and Lambda.

Edited May 28 by Mordred

[Mordred]

A detail I forgot to mention earlier. You can have an energy density with a non zero value (T_(00) ) component of the stress energy momentum tensor but as long as there is no inherent directional flow (scalar field) you can set the stress tensor to zero it becomes the background energy density which you can set for value zero. This determines  geometry. The metric tensor, for the global. When you have permutations ie flow etc you set that in the permutation tensor \( h_{\mu\nu}\) 

Then you add this to the global metric to establish the local metric.

For the FLRW metric you have a non zero global energy density but for purpose of the metric it's set at zero hence the stress tensor is zero. (Also done for conservation laws for symmetry relations) as well as renormalization procedures.

Edited May 28 by Mordred

[Mordred]

Just in case anyone isn't familiar with time derivatives in the above for example \(\dot{a}\) is the recessive velocity term of expansion. When you see two overdots this is a second order derivative for instantaneous acceleration in  usage above the accelerating rate of expansion via \(\dot{a}\)

\[\ddot{a}=\frac{dv}{dt}=\frac{d^2x}{dt^2}\]

 

 

Edited May 29 by Mordred

[Genady]

Typo correction: \[\ddot{a}=\frac{dv}{dt}=\frac{d^2 x}{dt^2}\]

Edited May 29 by Genady

[Mordred]

4 minutes ago, Genady said:

Typo correction:

a¨=dvdt=d2xdt2

 

correction applied and thanks for catching that

[Markus Hanke]

16 hours ago, AbstractDreamer said:

But if you had an FLRW universe devoid of energy momentum, it would still be a valid solution for that universe, if energy momentum could exist.  It just has zero value at the start and at least until the moment of observation.

I’m afraid I still don’t really know what you mean by this. The FLRW solution explicitly depends on the energy-momentum tensor being non-zero - an empty universe where it vanishes won’t have the same geometry as FLRW. You can’t get the FLRW solution from the vacuum field equations.

16 hours ago, AbstractDreamer said:

I don't understand how the cause of spacetime expansion is dependent on energy momentum.

The cause isn’t the presence of energy-momentum, since there also exist vacuum solutions that metrically expand (eg the Kasner metric); in my earlier comment I meant that the presence of energy-momentum implies that the spacetime cannot be flat. And by this I meant Riemann flat, which perhaps I should have stated explicitly.

16 hours ago, AbstractDreamer said:

but the actual mechanic of spacetime expansion - dark energy -

Dark energy is also not the cause of metric expansion; it merely influences the rate at which the expansion happens. The expansion itself happens whether there is DE or not.

16 hours ago, AbstractDreamer said:

in a spacetime geometry of net zero energy-momentum (Minkowski spacetime?)  and over a short period of constant scale factor, could you distinguish how much of the redshift in the wavelength of photons is due to spatial expansion and how much is due to temporal expansion?

But in a spacetime that is globally Minkowski, there is no expansion, and thus no redshift. So I struggle to make sense of your question.

15 hours ago, Mordred said:

Unfortunately this is rather misleading to understanding the FLRW metric. The FLRW metric has a curvature term K.

By curvature I meant Riemann curvature - sorry, I should have made that explicit.

 

Edited May 29 by Markus Hanke

[Mordred]

8 minutes ago, Markus Hanke said:

 

By curvature I meant Riemann curvature - sorry, I should have made that explicit.

 

That makes a lot more sense  thanks for clarifying 

Edited May 29 by Mordred

[Mordred]

I would like to clarify one detail. The vacuum can have numerous connotations. For example the Einstein vacuum (devoid of all particles including virtual) which is the vacuum solution. Is different than the way vacuum is applied in the FLRW metric. In the FLRW metric vacuum is the relation between the kinetic energy and potential energy terms of a scalar field using the scalar field equation of state.

Example w=-1 for the Cosmological constant. This value specifically -1 describes a constant incompressable fluid with negative pressure. Inflation  would have a different value but can slow roll to the value for the cosmological constant (you may see that in the case of Higgs inflation where the cosmological constant term is also considered being due to the Higgs field)

Another detail is that curvature can also mean slightly different things. For example extrinsic and intrinsic curvature. You can also have curvature independent of coordinate or coordinate choice or curvature that has a coordinate dependency. An example of the latter. Is the localized curvature term due to gravity. It is dependent on the location. (Localized anistropy)

Edited May 29 by Mordred

[AbstractDreamer]

Thank you all for being really specific and pedantic in your wordings.  I genuinely need this to help understand with more clarity as I know words are a poor substitute for maths.   I will take some time to absorb all this so I can pose questions that make more sense in terms of real physics and mathematics.

[Mordred]

Your welcome likely tonight I will setup the derivatives of how the equations of state are determined via the ideal gas law relations to help you better understand how thermodynamics are applied.

[Mordred]

Ok this is simply useful to understand the FLRW acceleration equation above in terms of the first law of thermodynamics.

expansion is a homogeneous and isotropic that is maximally symmetric. This expansion is also adiabatic in thermodynamics if you have a container (system)there is no net outflow of energy with the surroundings. This also describes a conserved system. \(T^{\mu\nu}_\mu\). The first law of thermodynamics for an adiabatic expansion is

\[de=pDv=0\]

with \(E=\rho V\) and \(V=\frac{4}{3}\pi a^3\) this becomes

\[d(\rho a^3)+pda^3=0\]

with respect to cosmic time

\[\dot{\rho}+3(\rho+p)\frac{\dot{a}}{a}=0\]

differentiating the first Friedmann equation in the form 

\[\dot{a}^2=\frac{8\pi G\rho a^2}{3}-kc^2\]

gives

\[2\dot{a}\ddot{a}=\frac{8\pi G}{3}(\dot{a}a^2+2a\dot{a})\]

substituting \(\dot{\rho}\) from the first law

\[2\dot{a}\ddot{a}=\frac{8 \pi G}{3}a\dot{a}(-\rho-3p)\]

gives the acceleration equation 

\[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)\]

you can find other forms of the above in different literature its a fairly standard heuristic Newtonian treatment. Its useful to recognize how thermodynamics is used to describe our universe expansion as well as understanding how the equations of state apply which is already above. Hope this helps

 

 

Edited May 30 by Mordred

[Markus Hanke]

13 hours ago, AbstractDreamer said:

Thank you all for being really specific and pedantic in your wordings.  I genuinely need this to help understand with more clarity as I know words are a poor substitute for maths.   I will take some time to absorb all this so I can pose questions that make more sense in terms of real physics and mathematics.

Any time

And your questions are good and valid ones, it’s just that they’re kind of difficult to address without lots of maths references. 

The key concept in this is diffeomorphism invariance - that one can describe the same spacetime/geometry in many different coordinates, without affecting any of the physics. Thus, having purely spatial expansion, and a mix of spatial and temporal expansion in the metric, really can be two different descriptions of the exact same physical situation. This is not intuitively obvious, but mathematically rigorous.

[Mordred]

Just an fyi there are treatments using Heaviside step functions for bounce cosmologies.

https://skim.math.msstate.edu/reprints/ShinKim_Recursive_Heaviside_17.pdf

this is one example though I have seen others its not something I would describe as a main stream approach but its been tried and has some merit.

here is a treatment in regards to QCD once again just an FYI

https://arxiv.org/pdf/hep-th/9603119

Edited June 9 by Mordred