The Friedmann-Lemaître-Robertson-Walker (FLRW) metric used to describe the cosmic spacetime is based on the cosmological principle, which assumes homogeneity and isotropy throughout the Universe. It also adopts free-fall conditions via the selection of a constant lapse function, gtt = 1, regardless of whether or not the chosen energy-momentum tensor Tαβ produces an accelerated expansion. This is sometimes justified by arguing that one may shift the gauge, if necessary, transforming the time dt to a new coordinate dt′g ttdt, thereby re-establishing a unitary value for gt′t′. Previously, we have demonstrated that this approach is inconsistent with the Friedmann equations derived using comoving coordinates. In this paper, we advance this discussion significantly by using the Local Flatness Theorem in general relativity to prove that gtt in FLRW is inextricably dependent on the expansion dynamics via the expansion factor a(t), which itself depends on the equation-of-state in Tαβ. One is therefore not free to choose gtt arbitrarily without ensuring its consistency with the energy-momentum tensor. We prove that the use of FLRW in cosmology is valid only for zero active mass, i.e. ρ + 3p = 0, where ρ and p are, respectively, the total energy density and pressure in the cosmic fluid.
FLRW Metric
Posts
Total:
25
which assumes homogeneity
Dafuq?
That's an ignorant assumption.
-->
@WyIted
The meaning of HOMOGENEITY is the quality or state of being of a similar kind or of having a uniform structure or composition throughout : the quality or state of being homogeneous.
-->
@FLRW
You've gone all Ebuc for a moment.
Will it last.
-->
@WyIted
Assumptions are in the main, ignorance based.
If one knew for certain, then one wouldn't need to make an assumption.
-->
@FLRW
the quality or state of being homogeneous.
I appreciate being called genius but I ain't no homo nigga
-->
@WyIted
Do NOT use N word.
Your slave mind is showing where you think certain sounds mean more than what those sounds mean. Nigga is a term of endearment where as nigger is derived from the Spanish word negra and merely means black person. It is the same as Caucasian and the euphemism wheel has only made it bad and the euphemism likely made it bad for racist reasons
-->
@zedvictor4
You've gone all Ebuc for a moment.
That is precisely what I was thinking. We don't need two of ''em.
Homogenous is not what I would call humanity, I celebrate our diversity, even if it doesn't fit the paradigm of whatever ebuc/FLRW are conjuring. Isotropy? Sounds like flat-earth theory, re-imagined as an AI nightmare.
-->
@WyIted
Dafuq?That's an ignorant assumption.
1. Observational homogeneity is an approximation, not a law
The cosmological principle is an assumption—not an observable truth. Galaxy surveys like SDSS show large-scale structures (voids, walls, filaments) extending hundreds of megaparsecs. So FLRW’s starting point (a smooth universe) already smuggles in an unjustified simplification. It's more of a convenient idealization than a physical reality.
2. Local Flatness ≠ Global Gauge Fixing
The Local Flatness Theorem guarantees that locally you can always find coordinates where spacetime is Minkowskian. But that doesn’t mean globally the lapse function gᵗᵗ must vary with expansion. That’s conflating coordinate artifacts with physical necessity. Gauge choices aren't physical—they’re mathematical conveniences.
3. The Friedmann equations already emerge with gᵗᵗ= 1
In comoving coordinates, setting gᵗᵗ = 1 isn't arbitrary—it reflects free-falling observers who see cosmic time as proper time. The Einstein field equations still work. If the derivation is consistent and predictive, then demanding a variable lapse is a solution in search of a problem.
4. The time coordinate is just a label
Coordinate time can always be rescaled or redefined. Physical observables like redshift, distances, and curvature invariants don’t care about your time label. So making gᵗᵗ vary doesn’t inject new physics—it just rephrases it awkwardly.
5. There’s no contradiction in the active mass condition ρ+3p≠0
It is unsupported by how the field equations are actually derived.
6. The energy-momentum tensor already feeds into curvature—no need to ‘correct’ gᵗᵗ
Einstein’s equations ensure that the influence of
Tμν is already baked into the spacetime geometry. There’s no hidden inconsistency unless you try to force extra structure where none is needed.
7. We observe an accelerating universe and FLRW accommodates it
Using standard FLRW cosmology with dark energy (ΛCDM), we predict an accelerating expansion, which matches supernova and CMB data. If FLRW were invalid under those conditions, this match wouldn’t exist.
8. You don’t get to dictate physical laws by picking a weird coordinate system
Sure, you can choose coordinates that give a variable lapse, but that doesn’t make it necessary. That’s like saying Newton’s laws don’t work because you went into a rotating frame and added fake forces, it’s a bad faith critique of a good model.
9. A dynamic lapse function risks introducing nonphysical degrees of freedom
Letting gᵗᵗ vary without constraint means you need to check whether you’re introducing ghost degrees of freedom, violating energy conditions, or breaking the well-posedness of Einstein’s equations. Most of the time, you’re not adding insight—you’re adding baggage.
10. There are models beyond FLRW—this paper is barking up the wrong tree
If you don’t like homogeneity, use Lemaître–Tolman–Bondi (LTB) or Bianchi models. But trying to "fix" FLRW from within by hacking gᵗᵗ is like trying to give a motorcycle four wheels—it’s no longer what it was, and worse at doing what it was meant for.
Why is Trump cutting funding for scientific research funding if so much has been theorized about our universe.
-->
@FLRW
Isotropic = ..' Isotropy means uniform in all directions and comes from the Greek words isos (equal) and tropos (way). Isotropic materials like glass exhibit the same material properties in all directions, whereas anisotropic materials, like graphite, exhibit different material properties depending on the direction. '...
.." The isotropic vector matrix is four-dimensional and 60-degree coordinated. It provides an omni-rational accounting system that, if arbitrarily accounted on a three-dimensional, 90-degree basis, becomes inherently irrational. The isotropic vector matrix demonstrates the ability of the symmetrically and asymmetrically terminaled, high-frequency energy vectors to accommodate the structuring of any shape "..
Do NOT use N word.
Canadians use the N word less than Americans.
-->
@ebuc
Thanks ebuc, you are the only one that understands me.
Oh sorry, Zed gets me too.
-->
@FLRW
Thanks ebuc, you are the only one that understands me.
You must have a communication problem.
-->
@Shila
Wink, wink !
-->
@FLRW
Wink, wink !
So it’s your eyes that don’t communicate well.
Oh wait, TGSG gets me too.
Actually , all the women entering our Big Bang Club get me too.
Well, that's if they pay for me.
-->
@FLRW
Zed gets me too.
I try to get everyone.
It's basically just accepting that the human condition is an output relative to input thing.
Wherein, will occur similarities and dissimilarities.
Though base systems are inevitably always similar...Notwithstanding developmental and inherent anomalies, such as psychopathy, which nonetheless must be got and either dealt with or avoided.
Though avoidance might result in horrific shower incidences.
-->
@FLRW
FLRW, I have two scenarios, --no idea if true or not-- wherein, our finite, overall Universe is not expanding, nor shrinking. Instead it is mis-perception of what is occurring --ex Hubble expansion and later on the Dark Energy accelerating expansion-- via what I say, may be occurring in every space and time, toroidal based particle of Universe, creating our misperception of a spatially expanding Universe. I dunno
What is space? How does it expand? Like molecular rubber band, or any occupied space substance when heated?
In these two scenarios below, the sine-wave ^v^v^ associated physical reality --Fermionic Matter and Bosonic Forces--- are internal to each space and time torus. In my four line, sequentially numerical, geodesically spiral trajectory pattering, 2D matrice pattern, --see bottom of this page to understand the spiral geoedesic trajectory pattern--- we humans do not observe { quantize } the top/outer { circle plane } line of Gravity, nor the bottom/inner line of Dark Energy { circle plane }.
We only see our physical reality, that, I envision to exist within each particle of Universe, the sine-wave ^v^v^ is --over billions or zillions of years-- and as a resultant of entropic heat death of Universe, that sine-wave of reality is moving either from outer/top Gravity field, toward the inner/bottom Dark Energy field, or vice versa. Not sure which direction would fit our observations of expanding Universe.
What is space? How does it expand? Like molecular rubber band, or any occupied space substance when heated?
In these two scenarios below, the sine-wave ^v^v^ associated physical reality --Fermionic Matter and Bosonic Forces--- are internal to each space and time torus. In my four line, sequentially numerical, geodesically spiral trajectory pattering, 2D matrice pattern, --see bottom of this page to understand the spiral geoedesic trajectory pattern--- we humans do not observe { quantize } the top/outer { circle plane } line of Gravity, nor the bottom/inner line of Dark Energy { circle plane }.
We only see our physical reality, that, I envision to exist within each particle of Universe, the sine-wave ^v^v^ is --over billions or zillions of years-- and as a resultant of entropic heat death of Universe, that sine-wave of reality is moving either from outer/top Gravity field, toward the inner/bottom Dark Energy field, or vice versa. Not sure which direction would fit our observations of expanding Universe.
Graphic 1: The sine-wave ^v^v of our Observed Time/Reality enters the outer Gravity Field, then travels on geodesic spiral around the space and time torus, back to inner Dark Energy Field and invaginates back internal to the inner Dark Energy Field, yet it is a very short angular invagination in, then back out to repeat the same trajectory { spiral pathway } on surface, back to outer Gravity and and invaginates there. This an eternal cyclic phenomena.
The (( )) is birds-eye-view of the abstract circle plane created by the two field lines of OBTime/Reality internal to all space and time tori.
Outer Gravity, two internal realities, Inner Dark Energy......( (( )( )) ).......< cross-bisection of space and time torus.
..Outer Gravity Peak of Convex Curvature.............loops back to self to create the outer surface circle plane space and time torus ( )
...ObTime/Reality.........800 Bl?. Yr. ..................................................................^v^v ...loops back to self internal to torus (( ))
...ObTime/Reality.........200...Bl?. Yr..........................................................^v^v ......loops back to self internal to torus (( ))
..ObTime/Rreality.......100 Bl.? Yr.....................................................^v^v............loops back to self internal to torus (( ))
..ObTime/Reality..........20 Bl.? Yr....................................^v^v....................loops back to self internal to torus (( ))
...ObTime/Reality........10 Bl.? yr.........................^v^v.............................loops back to self internal to torus (( ))
...ObTime/Reality.........5 Ml.? yr...............^v^.....................................loops back to self internal to torus (( ))
...ObTime/Reality......^v^v Dark Inflation period ^v^.............loops back to self internal to torus (( ))
..Inner Dark Energy Peak of Negative Curvature ............loops back to self on inner surface of space and time torus )(
Inflationary Period of Universe { I call Dark Inflation } was developed by ALan Guth in 70's to help explain aspects of Universe
AI... ' Alan Guth's work is primarily known for his inflationary period theory, which proposes a rapid, exponential expansion of the universe in its early moments.This theory helps explain several puzzling aspects of the early universe that are not addressed by the standard Big Bang model.
Graphic 2: Opposite of the above
..Outer Gravity Peak of Curvature.....................loops back to self on outer surface of torus
-......ObR......^v^v......Dark Inflation period..loops back to self inside torus
-......ObR.........5 Ml. yr.............^v^v............................loops back to self **inside** torus
-......ObR........10 Bl. yr.....................^v^v.......................loops back to self **inside** torus
-.....ObR..........20 Bl. Yr...............................^v^v.................loops back to self **inside** torus
-.....ObR.......100 Bl. Yr.............................................^v^v............loops back to self **inside** torus
-.....ObR.........200...Bl. Yr...................................................^v^v ......loops back to self **inside** torus
-.....ObR.........800 Bl. Yr. ............................................................^v^v ...loops back to self **inside** torus
-Inner Dark Energy Peak of Curvature............loops back to self on inner surface of torus
...space...Outer >( (( () )) )< Outer......space
..space.....( reality>((< reality. () reality >))< reality )......space..
...space.....( (( )< Inner >( )) )...space
Yes this takes some imagination, --and the specific geodesic spiral trajectories I present later below---. since im using texticonic charactors to represent the outer and inner circles as well as the two circles planes of reality (( )).
Ok so the geodesics from outer Gravity to Inner Dark Energy surface and then back to outer Gravit repeat is as follows. And yes, the geodesics are interrupted by two invaginations at the peak of outer ( ) and inner curvature )(.
Ok so here is the geodesic pathways that follow the numerically sequential spiral pathway.
1 ( 2p....4 ) 5p....7p ( 8.....10 ) 11p......13p ( 14....16 ) 17p..... so that is 6 geodesics { define spiral surface } in the 18/0 space and time torus.
All of the above stems from this 2D matrice pattern:
........1..........5p...7p.............11p....13p...............17p..........loops around and not co-joined as 17p/1
-
18/0.................6........................12............................18/0.loops around co-joined as one nodal event 18/0.
...............3p.................9............................15.....................loops around and not co-joined as 15/3p
-
...........2p...4..............8....10..................14.....16................loops around and not co-joined as 16/2p
Hope this is clear enough to make sense to others. I re-did stuff here today to improve clarity.