u/Rimurwaz

Let me start with the important part: I'm not a physicist.

I'm someone who's always been fascinated by cosmology and likes coming up with ideas and seeing how far I can take them. I don't have the formal physics or mathematics education necessary to work through something like GR on my own, so I've been using GPT as a tool to help me translate my ideas into mathematics, run numerical tests, find relevant established physics, and point out things I need to learn.

I'm completely aware of the problem with doing that. GPT can be wrong, especially with advanced math, and I can eventually reach a point where I don't know enough to recognize that it's wrong.

I think I've reached that point.

So I'm not posting this as "I solved cosmology." I'm basically putting my cards on the table and asking people who actually understand GR and black-hole physics to tell me whether there's an interesting question here or whether I've made a fundamental mistake.

If the answer is "this doesn't work because X," that's genuinely useful to me. I want to understand why.

The original idea started with the various proposals that a universe could be associated with the interior of a black hole. What I became interested in was a more specific question:

If a parent black hole evolves, could that evolution affect the expansion of its interior geometry?

For a static Schwarzschild black hole,

rₛ = 2GM/c²

where rₛ is the Schwarzschild radius.

If I naively allow M to decrease,

Ṁ < 0 → ṙₛ < 0

although I understand that once M actually varies with time, we're no longer dealing with an exact Schwarzschild spacetime and need a genuinely dynamical metric.

My original intuition was basically:

Ṁ < 0 → ȧ > 0

with a representing an interior cosmological scale factor.

I've since learned that this is probably not the right way to formulate the question. A Schwarzschild interior can be represented as a homogeneous but anisotropic Kantowski–Sachs geometry, so treating it as ordinary FLRW with one scale factor is too simplistic.

A better question seems to be:

Does Ṁ < 0 alter the expansion scalar θ of a physically defined timelike congruence inside a dynamical black-hole geometry, and could that correspond to sustained positive interior volume expansion?

For a four-velocity field u^μ,

θ = ∇_μu^μ

and θ has dimensions of inverse time.

If an effective isotropic expansion rate were meaningful in some appropriate regime, then one could potentially define:

H_eff = θ/3

also with units of inverse time.

I am NOT claiming that this relationship exists. That's actually the part I can't derive and the main reason I'm posting.

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The reason Hawking radiation originally entered the idea was straightforward.

In the standard semiclassical evaporation estimate applied to a Schwarzschild black hole,

Ṁ_Hawking ∝ −1/M²

so I wondered whether the resulting evolution of the parent horizon might correspond to evolution of the interior geometry.

In words:

parent mass decreases → horizon evolves → interior geometry evolves → possibly interior expansion

The last arrow is the conjecture.

I don't currently have a derivation establishing it.

Before getting this far into the GR problem, I tried a much simpler phenomenological experiment: suppose some unknown interior mechanism contributes an additional amount to the cosmological expansion rate. How large could it be before it obviously creates observational problems?

I initially wrote:

H²(a) = H²_ΛCDM(a) + H²_int(a)

where, schematically for flat ΛCDM,

H²_ΛCDM(a) = H₀²[Ωᵣa⁻⁴ + Ωₘa⁻³ + ΩΛ]

and:

H = ȧ/a

I deliberately call the hypothetical term H_int now rather than H_Hawking, because calling it Hawking-driven would assume what I'm trying to establish.

I first tried a constant H_int. Unsurprisingly, a large additional contribution operating throughout cosmic history quickly causes problems with cosmological distance measurements.

So I experimented with an arbitrary late-time transition:

S(a) = 1 / [1 + exp(−k(ln(a) − ln(aₜ)))]

using illustrative values:

aₜ ≈ 0.9

k ≈ 12

Since:

1 + z = 1/a

this corresponds to a transition around:

zₜ ≈ 0.11

There is no black-hole physics behind that logistic function. It's purely a phenomenological test of what happens if a modification is concentrated at low redshift.

During a pre-post audit, we actually found a mistake here.

I originally wrote:

H_int(a) = xH₀S(a)

while defining x as H_int(1)/H₀.

But with aₜ = 0.9 and k = 12,

S(1) ≈ 0.780

rather than 1.

So a corrected definition is:

S̄(a) = S(a)/S(1)

which guarantees:

S̄(1) = 1

and then:

H_int(a) = x₀H₀S̄(a)

where:

x₀ = H_int(1)/H₀

really is the present-day fractional amplitude.

We found another, more important normalization issue too.

If:

Ωᵣ + Ωₘ + ΩΛ = 1

and I simply add H²_int to the standard flat ΛCDM equation, then at a = 1:

H²(1) = H₀² + H²_int(1)

so H(1) is no longer H₀.

One mathematically normalized toy parameterization would instead be:

E(a) = H(a)/H₀

and:

E²(a) = [Ωᵣa⁻⁴ + Ωₘa⁻³ + ΩΛ + x₀²S̄²(a)] / [Ωᵣ + Ωₘ + ΩΛ + x₀²]

which gives:

E(1) = 1

by construction.

I don't claim that's the physically preferred formulation. If H²_int represents an actual additional effective density, it may instead make more sense to modify the closure relation itself. The point is that the original parameterization wasn't properly normalized.

This matters because my earlier toy calculations appeared to give rough upper limits around x ≈ 0.05–0.08.

I'm withdrawing those numbers.

They came from the improperly normalized parameterization and shouldn't be presented as observational constraints until the calculation is redone properly.

That's also a good example of why I'm asking humans who know the mathematics to look at this rather than assuming GPT's calculations are right.

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For the observational sanity checks, I used standard flat-background distance quantities:

D_M(z) = c ∫₀ᶻ dz′/H(z′)

and:

D_H(z) = c/H(z)

and compared quantities such as:

D_M/r_d

and:

D_H/r_d

against representative published BAO measurements.

I looked at BOSS DR12 measurements around:

z = 0.38, 0.51, 0.61

along with approximate structure-growth information from fσ₈.

The BOSS fiducial sound horizon is:

r_d,fid = 147.78 Mpc

I had previously used approximately 147.1 Mpc in some calculations, which was inconsistent.

I also understand that 147.78 Mpc is the BOSS fiducial value, not some universal physical r_d that should simply be imposed on every dataset.

I also used a representative DESI DR1 LRG+ELG measurement at:

z_eff ≈ 0.93

with approximately:

D_M/r_d = 21.71 ± 0.28

D_H/r_d = 17.88 ± 0.35

and an eBOSS Ly-α measurement around:

z ≈ 2.34

with approximately:

D_M/r_d = 37.41 ± 1.86

D_H/r_d = 8.86 ± 0.29

The point of including the higher-redshift measurements was that if this hypothetical effect were primarily late-time, it shouldn't be allowed to quietly alter the entire expansion history.

I also did approximate growth checks using:

f(a) = d ln(D)/d ln(a)

and the approximation:

f ≈ Ωₘ^0.55

along with fσ₈.

I know this isn't sufficient for a real model. A serious calculation would need to solve the perturbation equations directly, and if the proposed mechanism effectively modifies gravity, the usual GR growth treatment may itself change.

The only observational conclusion I'm comfortable keeping is qualitative:

A large extra contribution operating over a broad redshift range causes problems quickly. A sufficiently small effect confined mainly to low redshift is less immediately destructive in these limited tests.

That's NOT a detection.

ΛCDM remained completely viable.

And because of the normalization issue I found, I'm not claiming a numerical constraint on x₀.

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I also checked whether a local underdensity could mimic some low-redshift expansion behavior.

Using:

δ = (ρ − ρ̄)/ρ̄

and the approximate linear relation:

ΔH/H ≈ −(1/3)fδ

with:

Ωₘ ≈ 0.3

and therefore:

f ≈ Ωₘ^0.55 ≈ 0.52

a 10% underdensity:

δ = −0.10

gives:

ΔH/H ≈ +0.017

or roughly:

+1.7%

An earlier estimate I'd made of 3–4% for a 10% underdensity was wrong. Under this approximation, getting that size of enhancement would require something closer to a ~20% underdensity.

So a local underdensity could potentially mimic part of a small low-z expansion deviation.

Whether it could mimic the associated structure-growth behavior is a separate question.

We also caught a sign error in an earlier effective equation-of-state calculation.

For the total effective equation of state associated with the background expansion, the corrected diagnostic expression is:

w_eff(z) = −1 + (1/3) d ln[H²(z)] / d ln(1 + z)

This is not automatically the equation of state of H_int itself.

That mistake didn't affect the distance calculations because those were computed directly from H(z), but it did affect how I had interpreted the expansion behavior.

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One of the exploratory things I did was compare the hypothetical expansion scale with a black-hole light-crossing scale.

For:

rₛ = 2GM/c²

the quantity:

c/rₛ = c³/(2GM)

has units of s⁻¹, the same dimensions as H.

For clarity on dimensional consistency:

[c/rₛ] = (m s⁻¹)/m = s⁻¹

and:

[H] = s⁻¹

So the quantities can be dimensionally compared.

That does NOT mean they are physically equal.

I initially explored the dimensional identification:

H_int ~ c/rₛ

which would imply:

M ~ c³/(2GH_int)

and, if:

H_int = x₀H₀

then:

M ~ c³/(2Gx₀H₀)

For:

H₀ ≈ 70 km s⁻¹ Mpc⁻¹

this gives approximately:

M ~ (8.9 × 10⁵²/x₀) kg

or:

M ~ (4.5 × 10²²/x₀) M☉

For the purely illustrative value x₀ = 0.05:

M ~ 1.8 × 10⁵⁴ kg ≈ 9 × 10²³ M☉

Again, this is not a prediction or derivation.

It's an exploratory dimensional comparison. c/rₛ is an inverse light-crossing time, not the Hawking evaporation rate.

And this is where the biggest problem with the original idea appears.

The standard idealized Schwarzschild evaporation lifetime is:

t_evap = 5120πG²M³/(ℏc⁴)

For M ≈ 1.8 × 10⁵⁴ kg, this is of order:

10¹³⁹ years

and the fractional mass-loss rate is only roughly:

|Ṁ/M| ~ 10⁻¹⁴⁷ s⁻¹

That's absurdly small.

For comparison:

H₀ ≈ 2.27 × 10⁻¹⁸ s⁻¹

so even an illustrative 5% contribution would be:

H_int ≈ 1.13 × 10⁻¹⁹ s⁻¹

If I write a generic proportional relationship:

H_int = C|Ṁ/M|

the units work because both H_int and |Ṁ/M| have units s⁻¹, so C is dimensionless.

But matching those illustrative numbers would require:

C ~ 10¹²⁸

I don't consider "there must therefore be a factor of 10¹²⁸" an explanation. Unless that factor comes naturally out of the geometry, inserting it by hand would just be curve fitting.

I did check whether black-hole physics naturally contains dimensionless quantities remotely that large.

For example:

S_BH/k_B = A/(4ℓ_P²) = 4π(M/m_P)²

For the illustrative mass above, this is of order:

10¹²⁵

which happens to be within roughly three orders of magnitude of 10¹²⁸.

I am not claiming:

C = S_BH/k_B

or that the numerical proximity means anything.

Without a derivation connecting the two, it could easily be coincidence. I'm including it because it was one of the things I checked, not because I'm treating it as evidence.

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This is basically where I've hit the wall.

I don't think the next step should be inventing more functions for H(z), looking for convenient constants, or fitting more cosmological data.

The next step seems like it has to come from GR.

What I'd really like to know is whether something schematically like:

θ = F(M, Ṁ, r_h, T_μν, g_μν, ...)

can actually be derived for an appropriate dynamical black-hole interior.

Here:

- θ is the interior expansion scalar, with units s⁻¹

- M is the evolving mass parameter, in kg

- Ṁ has units kg s⁻¹

- r_h is an appropriate horizon length, in m

- T_μν is the stress-energy tensor

- g_μν is the dimensionless metric tensor in the usual coordinate convention

I'm not proposing that F has any particular form. That's exactly what I don't know.

It seems like the calculation would require a dynamical black-hole metric, possibly something Vaidya-like, a physically meaningful family of interior observers u^μ, calculation of:

θ = ∇_μu^μ

and then determining whether θ has any meaningful dependence on M and Ṁ.

If the result ultimately reduces to something on the order of:

θ ~ |Ṁ/M|

with no enormous naturally generated factor, then ordinary Hawking evaporation for a parent this massive appears hopelessly too weak.

That could simply kill the Hawking-driven version of the idea.

On the other hand, if the geometry produces a fundamentally different relationship, then I'd want to understand what it predicts before doing any more observational fitting.

So the specific questions I'm hoping someone here can help me with are:

1. Is asking how Ṁ affects an interior expansion scalar θ mathematically well-defined for an evaporating black-hole spacetime?

2. What metric would actually be appropriate? Is an evaporating Vaidya-type geometry a reasonable starting point?

3. What physically meaningful timelike congruence u^μ should be used to calculate θ?

4. Is the Kantowski–Sachs interpretation of the Schwarzschild interior useful once M becomes dynamical?

5. Can θ be related to M and Ṁ in a coordinate-independent physically meaningful way?

6. Does the extremely small Hawking mass-loss rate already make a cosmologically significant effect impossible regardless of the interior geometry?

7. Are there known GR or semiclassical results that already answer this question?

8. Have I made any additional mathematical, dimensional, or conceptual mistakes above?

And finally, for complete transparency about the AI involvement:

GPT has been used extensively as a tool while I've worked on this.

It helped me manipulate equations, perform numerical calculations, compare simple phenomenological models with published cosmological measurements, locate relevant concepts, and organize my thoughts.

It has also been wrong.

While reviewing this before posting, we found an incorrect effective-w sign, an incorrect earlier local-underdensity estimate, inconsistent treatment of r_d, an improperly normalized logistic amplitude, and an H₀ normalization problem in the original toy model.

I've corrected or disclosed those above, and I withdrew the numerical x constraint rather than present something I no longer trust.

There could absolutely be more mistakes.

That's actually why I'm here.

I don't know enough GR to independently certify the next stage of the calculation, and continuing to have an LLM derive increasingly complicated geometry that I can't personally verify doesn't seem useful.

I'm asking people who do understand the math to check the idea.

I'm completely okay with the conclusion being:

"No. This fails because of X."

I'm not trying to overturn ΛCDM or established GR. Any version of this idea would have to be consistent with established physics and observations.

What I'm trying to figure out is whether there's a legitimate GR question hiding inside my original intuition:

Can evolution of a parent black hole produce physically meaningful expansion of its interior geometry?

And only if the answer to that is yes:

Could Hawking evaporation have anything whatsoever to do with it?