1 hour ago, Dhillon1724X said:

So the temperature does decrease, but as a structured, geometric evolution, not as simple heat loss.

I’m not sure it would be considered heat loss anyway; you’re converting thermal energy into the mass energy of the chortons. It’s more like a phase transition.

44 minutes ago, Dhillon1724X said:

Before geometry forms, there's no light speed, no spacetime, no causal horizon.

What about after?

This has to be testable, and what you’re describing is happening in the first tiny fraction of a second after the big bang.

35 minutes ago, swansont said:

What about after?

This has to be testable, and what you’re describing is happening in the first tiny fraction of a second after the big bang

In QCF, the universe begins without spacetime, light speed, or causality — only a non-local energy field. Geometry emerges when the local energy density exceeds a critical threshold, forming Chortons: quantized units of curvature. Only after this transition can physical particles like photons exist.

To test this, I redshifted a present-day CMB photon (at 2.725 K) back in time.

The current energy of a CMB photon is:

[math] E_{\text{now}} \approx 1.17 \times 10^{-12} , \text{GeV} [/math]

The Planck energy is:

[math] E_{\text{Planck}} \approx 1.22 \times 10^{19} , \text{GeV} [/math]

Using the standard redshift relation:

[math] 1 + z = \frac{E_{\text{Planck}}}{E_{\text{now}}} \Rightarrow z \approx 1.043 \times 10^{31} [/math]

To convert redshift into time during the radiation-dominated era, I used:

[math] t(z) \approx \frac{1}{2 H_0 \sqrt{\Omega_r}} \cdot \frac{1}{(1 + z)^2} [/math]

With these constants:

[math] H_0 \approx 67.66 , \text{km/s/Mpc} \approx 2.19 \times 10^{-18} , \text{s}^{-1} [/math]
[math] \Omega_r \approx 9.236 \times 10^{-5} [/math]

This gives:

[math] t \approx 2.18 \times 10^{-43} , \text{s} [/math]

This is just after the Planck time:

[math] t_p \approx 5.39 \times 10^{-44} , \text{s} [/math]

So photons were not present at the very beginning — they formed just after spacetime emerged. This supports QCF’s core idea:

[math] \rho(x) \geq \rho_{\text{Planck}} \Rightarrow \chi_{\mu\nu}(x) \neq 0 [/math]
(Geometry turns on after Planck density is crossed)

Then:

[math] \chi_{\mu\nu}(x) \neq 0 \Rightarrow \gamma(x) \text{ can emerge} [/math]
(Photons are permitted only after geometry exists)

Thus, photons are not fundamental to spacetime — they are emergent, structured excitations born after Chortons form the geometric fabric.

3 hours ago, Dhillon1724X said:

In QCF, the universe begins without spacetime, light speed, or causality — only a non-local energy field. Geometry emerges when the local energy density exceeds a critical threshold, forming Chortons: quantized units of curvature. Only after this transition can physical particles like photons exist.

To test this, I redshifted a present-day CMB photon (at 2.725 K) back in time.

The current energy of a CMB photon is:

Enow≈1.17×10−12,GeV

The Planck energy is:

EPlanck≈1.22×1019,GeV

Using the standard redshift relation:

1+z=EPlanckEnow⇒z≈1.043×1031

To convert redshift into time during the radiation-dominated era, I used:

t(z)≈12H0Ωr√⋅1(1+z)2

With these constants:

H0≈67.66,km/s/Mpc≈2.19×10−18,s−1
Ωr≈9.236×10−5

This gives:

t≈2.18×10−43,s

This is just after the Planck time:

tp≈5.39×10−44,s

So photons were not present at the very beginning — they formed just after spacetime emerged. This supports QCF’s core idea:

ρ(x)≥ρPlanck⇒χμν(x)≠0
(Geometry turns on after Planck density is crossed)

Then:

χμν(x)≠0⇒γ(x) can emerge
(Photons are permitted only after geometry exists)

Thus, photons are not fundamental to spacetime — they are emergent, structured excitations born after Chortons form the geometric fabric.

Yes, you’ve made my point. (though your proposal says energy density, so the details might be different)

Your scenario can only be realized very, very early in the universe.

I didn’t mention photons; you brought that up, but how would you show experimentally that photons didn’t exist?

I asked about what happens after spacetime is created?

Why aren’t things attracted to these massive particles?

12 hours ago, swansont said:

Why aren’t things attracted to these massive particles?

In the QCF model, Chortons are not massive particles — they are quantized excitations of curvature, not matter.

They don’t exert force or gravity directly like particles do.
Instead, they collectively build the structure of spacetime curvature.
ach Chorton is a quantum of curvature, forming only when local energy exceeds the Planck threshold in a Planck volume:

[math]∑Eγ≥EPinVP⇒Chorton forms\sum E_\gamma \geq E_P \quad \text{in} \quad V_P \Rightarrow \text{Chorton forms}∑Eγ≥EPinVP⇒Chorton forms[/math]

• These Chortons generate χμν(x)≠0\chi_{\mu\nu}(x) \neq 0χμν(x)=0, i.e. spacetime geometry becomes active.

• When Chorton fields become dense, their curvature fields superpose, and this curvature manifests as gravity — as described by:

  • Newtonian gravity in low curvature limits

  • Full GR behavior in strong field zones

12 hours ago, swansont said:

I asked about what happens after spacetime is created?

I gave you example of photon with proofs that they are created after spacetime.However when spacetime got created then time really started.
I will tell more.

@swansont here you can see paper if you want to
https://zenodo.org/records/16313645

12 hours ago, swansont said:

I didn’t mention photons; you brought that up, but how would you show experimentally that photons didn’t exist?

In the Quantum Chorton Framework (QCF), photons cannot exist before spacetime because they require a defined geometric background to propagate — this is true even in classical physics, where Maxwell’s equations and null geodesics rely on the metric. I reverse-redshifted a present-day CMB photon back to Planck energy and found that it corresponds to [math] t≈2.18×10−43 st \approx 2.18 \times 10^{-43} \, \text{s}t≈2.18×10−43s[/math], just after Planck time. This suggests photons can only emerge after spacetime forms via Chorton activation. Chortons themselves do not attract — they are not massive particles, but quantized curvature units that build spacetime geometry. Gravity, in QCF, arises from the curvature structure they create, and the model recovers both Newtonian and relativistic gravity in the correct limits.

I've been sharing my framework and answering questions, but so far no one has pointed out any major flaws. They're mainly asking how specific things work or requesting evidence — which I welcome — but I'm specifically looking for clear critiques. If there’s something logically inconsistent, mathematically incorrect, or physically unjustified, please point it out directly.

If you have no critiques, feel free to say that too — silence makes it hard to tell whether the theory holds up or just isn’t being understood deeply enough.

3 hours ago, Dhillon1724X said:

I gave you example of photon with proofs that they are created after spacetime.However when spacetime got created then time really started.
I will tell more.

They are created after spacetime? Mmm... So something happens after time...

Your theory is a contradiction fest. Time after time.

Just now, joigus said:

Time after time.

Nah it's just in a time loop of its own. +1

😄

33 minutes ago, joigus said:

They are created after spacetime? Mmm... So something happens after time...

Your theory is a contradiction fest. Time after time.

17 minutes ago, studiot said:

Nah it's just in a time loop of its own. +1

😄

Yes, it may seem like a contradiction at first, but let me explain it more precisely:

When I say "photons are created after spacetime", I mean this in a physical—not purely logical—sense. The creation of spacetime refers to the emergence of the arena in which cause-effect events like the creation of photons can occur. Time, as a dimension, begins with spacetime—but time as a measurable process only becomes meaningful once physical events start happening inside it, such as the appearance of energy or light.
There is no contradiction unless one assumes that “time” must already be ticking before spacetime exists.

There’s no contradiction in what I said. Spacetime means 3D space plus time — it's the full 4D structure. So obviously, when spacetime got created, time as a dimension came with it. I said photons came after that. And when I said ‘time really started’, I meant that’s when physical things started happening — like energy, light, motion. So yeah, time existed as part of spacetime, but it started ticking only when events began. No contradiction, just misunderstanding what I meant by ‘really started’.

I will rather welcome critiques on my theory rather then a mockery.

4 hours ago, Dhillon1724X said:

In the QCF model, Chortons are not massive particles — they are quantized excitations of curvature, not matter.

But if they’re massless particles they must move at the speed of light.

@swansont here you can see paper if you want to

I’m losing patience with the way you continue to ignore our rules regarding relying on links

In the Quantum Chorton Framework (QCF), photons cannot exist before spacetime because they require a defined geometric background to propagate — this is true even in classical physics, where Maxwell’s equations and null geodesics rely on the metric. I reverse-redshifted a present-day CMB photon back to Planck energy and found that it corresponds to t≈2.18×10−43 st≈2.18×10−43st≈2.18×10−43s, just after Planck time. This suggests photons can only emerge after spacetime forms via Chorton activation. Chortons themselves do not attract — they are not massive particles, but quantized curvature units that build spacetime geometry. Gravity, in QCF, arises from the curvature structure they create, and the model recovers both Newtonian and relativistic gravity in the correct limits.

And I asked you what evidence there is to support the notion that photons didn’t exist. Repeating yourself isn’t evidence.

And backtracing the CMB to before the so-called recombination era is somewhat misguided.

Just now, Dhillon1724X said:

like energy

And yet didn't you refer to energy density before everything else ?

And isn't that a loop ?

3 minutes ago, swansont said:

I’m losing patience with the way you continue to ignore our rules regarding relying on links

Sorry sir,
i will never do it again.
Can i share file instead?

1 minute ago, Dhillon1724X said:

Sorry sir,
i will never do it again.
Can i share file instead?

No. I’m not sure why this isn’t registering: material for discussion must be posted. You can’t just point us to a link or an uploaded file.

Just now, swansont said:

No. I’m not sure why this isn’t registering: material for discussion must be posted. You can’t just point us to a link or an uploaded file

OK sir.
I will never do it again.

9 minutes ago, swansont said:

And I asked you what evidence there is to support the notion that photons didn’t exist.

You asked for evidence that photons didn’t exist before spacetime. The evidence comes directly from standard physics — both classical electrodynamics and quantum field theory:

---

1. Maxwell’s equations require a spacetime metric

In curved spacetime, they’re written as:

[math]\nabla_\mu F^{\mu\nu} = \mu_0 J^\nu[/math]

The covariant derivative [math]\nabla_\mu[/math] depends on the metric tensor [math]g_{\mu\nu}[/math].
Without spacetime geometry, the field equations themselves aren’t defined.
So photons — which are solutions of Maxwell’s equations — can’t exist without a spacetime background.

---

2. Photon motion follows null geodesics

[math]ds^2 = g_{\mu\nu} dx^\mu dx^\nu = 0[/math]

Photon worldlines follow [math]ds^2 = 0[/math] — this only makes sense if [math]g_{\mu\nu}[/math] exists.
No metric → no null geodesics → no photon propagation.

---

3. In QFT, photons are excitations of the EM field

The vector potential is written as:

[math]A_\mu(x) = \sum_{\vec{k},\lambda} \left[ \epsilon_\mu^\lambda(\vec{k}) a_{\vec{k},\lambda} e^{-ikx} + \epsilon_\mu^{\lambda *}(\vec{k}) a_{\vec{k},\lambda}^\dagger e^{ikx} \right][/math]

This formalism assumes a spacetime structure with coordinates [math]x^\mu[/math] and a Lorentz-invariant background.
You can’t define the field or its quanta (photons) without that.

---

4. My redshift argument adds consistency

I backtraced a present-day CMB photon to Planck energy using:

[math]E = h\nu = \frac{hc}{\lambda} \quad \text{with} \quad \lambda = \lambda_0 / (1+z)[/math]

And found it reaches Planck scale at [math]t \approx 2.18 \times 10^{-43} , \text{s}[/math], just after Planck time — the era where spacetime geometry (via Chortons in QCF) forms.

So the appearance of photons after spacetime is not just assumed — it follows directly from the structure of the physics we already use.

---

If you believe photons can exist without spacetime, I’d genuinely like to see what framework allows that. Otherwise, this order — spacetime first, photons after — is supported by both classical and quantum theory."

its all i can state for now

9 minutes ago, studiot said:

And yet didn't you refer to energy density before everything else ?

And isn't that a loop ?

Yes, I mentioned energy density before spacetime — but I never said energy existed inside time before spacetime. I said that extreme energy density triggered spacetime formation. That’s not a loop — it’s a cause-effect relation where energy density acts as the condition for geometry to emerge. It’s similar to how, in quantum gravity approaches, spacetime itself is emergent from deeper pre-geometric conditions. There’s no loop unless you assume time existed before spacetime — which I don’t. The energy density I referred to is a pre-spacetime initial condition, not something happening ‘in time.

Just now, Dhillon1724X said:

Yes, I mentioned energy density before spacetime — but I never said energy existed inside time before spacetime. I said that extreme energy density triggered spacetime formation. That’s not a loop — it’s a cause-effect relation where energy density acts as the condition for geometry to emerge. It’s similar to how, in quantum gravity approaches, spacetime itself is emergent from deeper pre-geometric conditions. There’s no loop unless you assume time existed before spacetime — which I don’t. The energy density I referred to is a pre-spacetime initial condition, not something happening ‘in time.

That's playing with words.

You can't have energy density without energy, which you claim emerges later.

I could also say you can't have density without volume and therefore space either.

In my opinion this conundrum is faced rather badly by all attempts to extrapolate backwards, including those claiming a big bang of some sort.

I have never seen one that doesn't start hand waving when pressed.

But then it is only one spectacular failure in a long line of such failures when over ambitious extrapolations are attempted, going back millenia yo mthe Bible trying to set the Earth age by adding up ages of a few named humans.

By the way do you understand the difference between extrapolation and interpolation ?

1 hour ago, Dhillon1724X said:

You asked for evidence that photons didn’t exist before spacetime. The evidence comes directly from standard physics — both classical electrodynamics and quantum field theory:

No, I want evidence - actual evidence - that photons didn’t exist in the early universe. Using your conjecture as evidence is circular reasoning

My redshift argument adds consistency

I backtraced a present-day CMB photon to Planck energy using:

E=hν=hcλwithλ=λ0/(1+z)

And found it reaches Planck scale at t≈2.18×10−43,s, just after Planck time — the era where spacetime geometry (via Chortons in QCF) forms.

So the appearance of photons after spacetime is not just assumed — it follows directly from the structure of the physics we already use.

The redshift argument is based on the thermodynamics of expansion, but recombination is not accounted for here - it would be like extrapolating a temperature of H2O across either or both phase transitions. Forming neutral atoms releases photons, which do not have a thermal spectrum.

5 hours ago, Dhillon1724X said:

I've been sharing my framework and answering questions, but so far no one has pointed out any major flaws. They're mainly asking how specific things work or requesting evidence — which I welcome — but I'm specifically looking for clear critiques. If there’s something logically inconsistent, mathematically incorrect, or physically unjustified, please point it out directly.

Lack of evidence is a VERY clear critique. If your ideas are unsupported, then they're opinions. Hypotheses require a LOT of support before they can ever be thought of as theories.

1 hour ago, swansont said:

The redshift argument is based on the thermodynamics of expansion, but recombination is not accounted for here - it would be like extrapolating a temperature of H2O across either or both phase transitions. Forming neutral atoms releases photons, which do not have a thermal spectrum.

You're mistaking the point. I'm not talking about recombination or the thermal origin of CMB photons. I'm tracing any massless field excitation's energy back through geometric redshift. This is a kinematic argument — not thermodynamic."

"The equation [math]λ=λ0/(1+z)\lambda = \lambda_0 / (1+z)λ=λ0/(1+z)[/math] applies universally to massless particles, regardless of their origin. If you accept general relativity and redshift scaling, then scaling a photon’s energy back in time shows that it asymptotically approaches the Planck energy at [math] t∼tPt \sim t_Pt∼tP[/math]. That matches the QCF threshold for spacetime activation.

1 hour ago, swansont said:

No, I want evidence - actual evidence - that photons didn’t exist in the early universe. Using your conjecture as evidence is circular reasoning

No, there is no direct experimental evidence for what happened before Planck time (t ≈ 10⁻⁴³ s) or near it, because no experiment can probe that regime yet. This is not a failure of theory — it’s a known limit of current physics."

"However, we do have strong indirect support for the redshift scaling of photons all the way back to near-Planck times:

1. Cosmic Microwave Background (CMB) shows photons cooled with universe expansion.

2. Redshift scaling — confirmed by observations of distant galaxies, quasars, and background radiation — follows the equation:
   [math]λ∝a(t)\lambda \propto a(t)λ∝a(t)[/math], and
   [math]E=hcλ⇒E∝1a(t)E = \frac{hc}{\lambda} \Rightarrow E \propto \frac{1}{a(t)}E=λhc⇒E∝a(t)1.[/math]

3. This is confirmed up to redshifts [math]z∼1100z \sim 1100z∼1100[/math] (CMB), and even higher via primordial nucleosynthesis and inflation models.

4. Extrapolating this trend mathematically (as all theories do), photon energies reach [math] EPE_PEP around z∼1031z \sim 10^{31}z∼1031[/math] — corresponding to [math]t∼tPt \sim t_Pt∼tP[/math]. That's the domain of quantum gravity, where our current physics breaks down."

"So while we don't have experiments from t < 10⁻⁴³ s, we do have theoretical predictions built from experimentally verified equations — redshift, thermodynamics, and quantum field theory. QCF follows this structure and places photon emergence after geometry formation, in perfect agreement with what physics currently allows.

1 hour ago, studiot said:

That's playing with words.

You can't have energy density without energy, which you claim emerges later.

I could also say you can't have density without volume and therefore space either.

In my opinion this conundrum is faced rather badly by all attempts to extrapolate backwards, including those claiming a big bang of some sort.

I have never seen one that doesn't start hand waving when pressed.

But then it is only one spectacular failure in a long line of such failures when over ambitious extrapolations are attempted, going back millenia yo mthe Bible trying to set the Earth age by adding up ages of a few named humans.

By the way do you understand the difference between extrapolation and interpolation ?

You're assuming I said energy 'exists' in the way we describe energy within spacetime. I didn’t. I said energy density — meaning a localized quantum potential — is the triggering condition for spacetime to form. That's a pre-geometric condition, not energy in space or time. It's a concept used in many quantum gravity models — including causal set theory, loop quantum gravity, and even some interpretations of string theory."

"You say 'you can't have density without space.' That's true in classical physics. But pre-spacetime physics is not classical. In QCF, density refers to a configuration on a quantum graph — a structure without metric or coordinates. It's like describing potential in a lattice model. So your objection applies to classical models, not mine."

"Also, calling it 'word games' isn’t a counterargument — it’s a dismissal. If you're asking how something can 'happen' without time, then yes, it's hard. But that's exactly why quantum gravity needs new formalisms like mine — where time and space are emergent, not assumed."

"And yes, I do understand extrapolation vs interpolation. Interpolation fits within known data; extrapolation extends beyond it. My model is explicitly a proposal for what's beyond current data, but it’s anchored in formal QFT, GR limits, and redshift equations — so it’s not blind guessing."

"We may not yet have experiments for this domain — no theory does. But that doesn't mean logic and math can't extend our reach. Dismissing all early-universe models as 'failures' because they're ambitious is not critique — it's resignation.

1 hour ago, Phi for All said:

Lack of evidence is a VERY clear critique. If your ideas are unsupported, then they're opinions. Hypotheses require a LOT of support before they can ever be thought of as theories.

Sure, my model isn't experimentally proven — but neither are String Theory, Loop Quantum Gravity, or Causal Set Theory. None of them have direct experimental evidence. Yet they’re still taken seriously because they are mathematically consistent, logically sound, and grounded in established physics. That’s exactly what I’m aiming for.

So if there’s a flaw in my math, logic, or consistency with known physics — point it out. I welcome real critique. But if you're dismissing it solely because we can't yet test Planck-scale physics in a lab, then you're applying a double standard that would also rule out every other quantum gravity approach.

String Theory postulates 10 or 11 dimensions and supersymmetry, yet none of those have been observed. Why is it considered viable if it's still unfalsifiable after decades?"

If testability is your standard, how do you justify ST's continued dominance despite zero experimental confirmation?

Edited July 24Jul 24 by Dhillon1724X

6 hours ago, Dhillon1724X said:

I've been sharing my framework and answering questions, but so far no one has pointed out any major flaws. They're mainly asking how specific things work or requesting evidence — which I welcome — but I'm specifically looking for clear critiques. If there’s something logically inconsistent, mathematically incorrect, or physically unjustified, please point it out directly.

To better understand the clear critiques other members have posted you may want to look up and read about “ontological bootstrapping problem” (using tools that can’t logically exist at the model’s starting point). That may give you valuable insight about the overall problems with the explanations you post.

Also, there is a risk that AI usage reinforces this. The LLM does not contain any non-spacetime model about ”chortons” and will likely fall back on spacetime terms, even if they logically does not apply.

7 minutes ago, Ghideon said:

Also, there is a risk that AI usage reinforces this. The LLM does not contain any non-spacetime model about ”chortons” and will likely fall back on spacetime terms, even if they logically does not apply.

Maybe it can be problem,as i told earlier,to refine my answer and correct the grammer i use LLM sometimes.

But in paper i have precisely checked it.

14 minutes ago, Ghideon said:

To better understand the clear critiques other members have posted you may want to look up and read about “ontological bootstrapping problem” (using tools that can’t logically exist at the model’s starting point). That may give you valuable insight about the overall problems with the explanations you post.

Also, there is a risk that AI usage reinforces this. The LLM does not contain any non-spacetime model about ”chortons” and will likely fall back on spacetime terms, even if they logically does not apply.

In QCF, what I call 'energy density' isn't defined spatially or temporally — it's defined over a pre-geometric structure that behaves more like a graph or adjacency matrix of potential interactions. The Chorton field activates when a certain excitation threshold is reached in this network. So it's not a matter of using energy in spacetime to generate spacetime — it's using localized excitation patterns in a discrete configuration space to cause spacetime to emerge as a relational structure."

"I agree that if I or the LLM ever fall back into describing things in spacetime terms (like 'at a point' or 'collapse in a region'), then that would be a problem — and I appreciate you calling that out. Moving forward, I’ll clarify that those are approximations or emergent descriptions, not fundamental ones. I want QCF to remain firmly pre-geometric at its core."

If you see any specific spot in my paper or posts where I’ve unintentionally slipped into ontologically circular language, I’d really appreciate you pointing it out. That kind of critique helps me improve the model at its foundations.

I asked you a very clear and simple question, in the hope that you would gain something from it.

But you did not answer, instead choosing to hide any response in a flood of meaningless text.

I will now ask another one.

What is the difference between circular language and ontologically circular language ?

2 hours ago, Dhillon1724X said:

You're mistaking the point. I'm not talking about recombination or the thermal origin of CMB photons. I'm tracing any massless field excitation's energy back through geometric redshift. This is a kinematic argument — not thermodynamic."

"The equation λ=λ0/(1+z)λ=λ0/(1+z)λ=λ0/(1+z) applies universally to massless particles, regardless of their origin. If you accept general relativity and redshift scaling, then scaling a photon’s energy back in time shows that it asymptotically approaches the Planck energy at t∼tPt∼tPt∼tP. That matches the QCF threshold for spacetime activation

But the CMB photons came from the so-called recombination epoch, where the photons reached thermal equilibrium, and the redshift is a thermal effect from expansion.

2 hours ago, Dhillon1724X said:

Sure, my model isn't experimentally proven — but neither are String Theory, Loop Quantum Gravity, or Causal Set Theory. None of them have direct experimental evidence. Yet they’re still taken seriously because they are mathematically consistent, logically sound, and grounded in established physics. That’s exactly what I’m aiming for.

But there must be some way, in principle, to get experimental confirmation. Not being able to do it yet is not the same as not being able to do it at all.

7 hours ago, swansont said:

But the CMB photons came from the so-called recombination epoch, where the photons reached thermal equilibrium, and the redshift is a thermal effect from expansion.

Thats the right answer.

So how we can say photons existed before Spacetime?

They didn’t exist.It is my point.

I just traced back photon energy to Planck scales by reversing redshift.They reached Planck scale even before reaching Planck epoch,that means theirs no chance.

15 hours ago, studiot said:

I asked you a very clear and simple question, in the hope that you would gain something from it.

But you did not answer, instead choosing to hide any response in a flood of meaningless text.

I will now ask another one.

What is the difference between circular language and ontologically circular language ?

"Circular language is a rhetorical issue — where definitions or explanations loop back on themselves without adding clarity."

"Ontologically circular language is deeper: it refers to using concepts that depend on an entity (like spacetime), to explain the origin of that very entity. For example, saying 'spacetime was created when curvature occurred' — but curvature requires spacetime to be defined — is ontologically circular."

Before you become too engrossed by the more speculative aspects of your document, perhaps you could reply to the following which I posted earlier:

On 7/21/2025 at 11:30 PM, Dhillon1724X said:

For a massless particle (such as a photon or graviton), which experiences no proper time, the trajectory is instead governed by the null geodesic equation:

[math]\dfrac{d^2 x^\mu}{dλ^2} + \Gamma^\mu_{\nu\sigma} \dfrac{dx^\nu}{dλ} \dfrac{dx^\sigma}{dλ} = 0[/math]

where λ is an affine parameter along the particle’s worldline.

Perhaps you can elaborate on what an affine parameter is, particularly with regards to null geodesics, which cannot be parametrised with proper time or arc length.

1 minute ago, KJW said:

Before you become too engrossed by the more speculative aspects of your document, perhaps you could reply to the following which I posted earlier:

Sorry sir.
Its very hard to manage and balance everything while being a student.
I will reply now.

7 minutes ago, KJW said:

Before you become too engrossed by the more speculative aspects of your document, perhaps you could reply to the following which I posted earlier:

Perhaps you can elaborate on what an affine parameter is, particularly with regards to null geodesics, which cannot be parametrised with proper time or arc length.

An affine parameter is a parameter along a geodesic that preserves the geodesic equation's form. For null geodesics, proper time τ\tauτ is zero, so we can't use it. Instead, we use an affine parameter λ\lambdaλ, which labels points along the path in a way that keeps the motion equation:

[math]d2xμdλ2+Γνσμdxνdλdxσdλ=0\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\nu \sigma} \frac{dx^\nu}{d\lambda} \frac{dx^\sigma}{d\lambda} = 0dλ2d2xμ+Γνσμdλdxνdλdxσ=0[/math]

This ensures the particle’s path remains a true geodesic, even without proper time.