Chapter 55

Gravity and Temporal Momentum

Introduction

In Chapter 51 we established that gravity in TMT couples to temporal momentum density $\rho_{p_T}=\rho_0 c$ through the scalar interaction $\mathcal{L}_{\text{int}}=-(\Phi/2M_{\text{Pl}})T^\mu{}_\mu$. In Chapter 53 we showed that this coupling naturally satisfies the equivalence principle.

This chapter explores the most profound consequence of this coupling: gravity couples to rest mass, not energy. This distinction—which is invisible for non-relativistic systems where rest mass dominates energy—has deep implications for photons, relativistic particles, vacuum energy, and the universality of gravitational interaction. It is the single insight that distinguishes TMT's gravitational sector from both general relativity and all other modified gravity theories.

Why Gravity Couples to $p_T$, Not $E$

The Fundamental Question

In general relativity, gravity couples to the full stress-energy tensor $T_{\mu\nu}$, which includes rest mass, kinetic energy, pressure, binding energy, and field energy—everything. The source of GR gravity is the energy-momentum content of spacetime.

In TMT, the scalar gravitational field $\Phi$ couples only to the trace $T^\mu{}_\mu$, which for matter is:

$$ T^\mu{}_\mu = -\rho_0 c^2 = -\rho_{p_T}\cdot c $$ (55.1)

Why this and not the full $T_{\mu\nu}$? The answer comes from three independent arguments.

Argument 1: The Lorentz Invariance Requirement

The scalar field $\Phi$ is a Lorentz scalar. Therefore, it must couple to a Lorentz-invariant source. Consider two candidates:

Candidate 1: Energy density $\rho_E = T^{00}$. Under a Lorentz boost with factor $\gamma$:

$$ T^{00} \to \gamma^2 T^{00} $$ (55.2)

This is not Lorentz invariant. A Lorentz scalar cannot consistently couple to a Lorentz non-scalar.

Candidate 2: Temporal momentum density $\rho_{p_T}=\rho_0 c$. Under a Lorentz boost:

$$ \rho_{p_T} = n\cdot p_T = (\gamma n_0)\cdot\frac{m_0 c}{\gamma} = n_0 m_0 c = \rho_0 c $$ (55.3)

The $\gamma$ factors cancel exactly: the number density increases by $\gamma$ (Lorentz contraction), while the temporal momentum per particle decreases by $1/\gamma$ (time dilation). The product $\rho_{p_T}=\rho_0 c$ is Lorentz invariant.

Key Result

Lorentz invariance selects $\rho_{p_T}=\rho_0 c$ as the unique consistent source for a scalar gravitational field.

Argument 2: The Tracelessness Theorem

From the null constraint P1 ($ds_6^{\,2}=0$), the 6D stress-energy tensor is traceless (Chapter 51):

$$ T^A{}_A = T^\mu{}_\mu + T^a{}_a = 0 $$ (55.4)

The 4D trace $T^\mu{}_\mu$ is the only Lorentz-invariant contraction of $T_{\mu\nu}$, and it equals (minus) the $S^2$ trace. The scalar $\Phi$ emerges from the $S^2$ sector, so its natural coupling is to $T^\mu{}_\mu=-\rho_0 c^2$.

For a perfect fluid with energy density $\rho$ and pressure $p$:

$$ T^\mu{}_\mu = -\rho c^2 + 3p $$ (55.5)

For non-relativistic matter ($p\ll\rho c^2$): $T^\mu{}_\mu\approx -\rho_0 c^2$.

For radiation ($p=\rho c^2/3$): $T^\mu{}_\mu = -\rho c^2 + \rho c^2 = 0$.

The trace automatically selects rest mass and excludes radiation.

Argument 3: The Monopole Charge Criterion

From Chapter 51, the monopole charge of gravity is $q=0$ (the metric is real and gauge-invariant). This means gravity can be “volume integrated” over $S^2$—it goes THROUGH the scaffolding, not AROUND it. The volume integral projects out only the $\ell=0$ mode on $S^2$, which corresponds to the rest-frame scalar quantity $\rho_0$, not the frame-dependent energy density $T^{00}$.

The Three Arguments Converge

Table 55.1: Three independent reasons gravity couples to $\rho_{p_T}$

Argument	Mechanism	Source
Lorentz invariance	Scalar must couple to scalar	§sec:ch54-why-pT
Tracelessness	$T^A{}_A=0$ selects $T^\mu{}_\mu$	Chapter 51
Monopole charge	$q=0$ allows volume integration	Chapter 51

All three arguments independently require that the scalar gravitational field couple to temporal momentum density, not energy density.

Polar Field Perspective on the Three Arguments

In the polar field variable $u = \cos\theta$, the three arguments for $\rho_{p_T}$ coupling acquire transparent geometric meaning. Recall that $S^2$ integrals factorize in the flat measure $du\,d\phi$ as THROUGH ($u$) $\times$ AROUND ($\phi$) channels (Chapter 12).

Argument 1 (Lorentz invariance) in polar: The temporal momentum $p_T = m_0 c/\gamma$ is the component of 4-momentum in the THROUGH direction of the polar rectangle. Its Lorentz invariance follows from the cancellation $n \cdot p_T = (\gamma n_0)(m_0 c/\gamma) = n_0 m_0 c$, where the boost factor $\gamma$ affects the AROUND (spatial-motion) channel and its inverse affects the THROUGH (temporal) channel—perfect balance between the two polar directions.

Argument 2 (Tracelessness) in polar: The 6D tracelessness $T^A{}_A = 0$ becomes $T_4 + T^u{}_u + T^\phi{}_\phi = 0$ in polar form (Chapter 6). The scalar field $\Phi$ couples to the trace $T^\mu{}_\mu = -(T^u{}_u + T^\phi{}_\phi)$, which is the total temporal participation across both THROUGH and AROUND channels. For radiation: equal THROUGH and AROUND contributions cancel $T_4$, giving $T^\mu{}_\mu = 0$. For massive matter: net THROUGH dominance gives $T^\mu{}_\mu \approx -\rho_0 c^2 \neq 0$.

Argument 3 (Monopole charge $q = 0$) in polar: The scalar gravitational field has monopole charge $q = 0$, meaning it is the unique uniform mode on the polar rectangle—constant in both $u$ and $\phi$:

$$ \Phi(x^\mu, u, \phi) = \Phi(x^\mu) \cdot \frac{1}{4\pi} $$ (55.6)

The volume integral over the polar rectangle projects out this uniform mode:

$$ \int_0^{2\pi} d\phi \int_{-1}^{+1} du\; \Phi = 4\pi \cdot \Phi $$ (55.7)

with the flat measure $du\,d\phi$ (no Jacobian factors). This picks out the rest-frame scalar $\rho_0$, not the frame-dependent $T^{00}$.

Property	Spherical $(\theta, \phi)$	Polar $(u, \phi)$
Gravitational source	$T^\mu{}_\mu$	$-(T^u{}_u + T^\phi{}_\phi)$ (THROUGH + AROUND)
Monopole charge	$q = 0$ (isotropic)	Uniform mode on $[-1,+1] \times [0,2\pi)$
Volume integral	$\int \sin\theta\,d\theta\,d\phi$	$\int du\,d\phi$ (flat, no Jacobian)
$\gamma$ cancellation	Number $\times$ momentum	AROUND boost $\times$ THROUGH $1/\gamma$
Radiation trace	$-\rho c^2 + 3p = 0$	THROUGH + AROUND cancel $T_4$ exactly

The polar perspective reveals a simple principle: gravity couples to the uniform ($\ell = 0$) mode on the polar rectangle because the scalar field itself IS that uniform mode. Only rest mass contributes to this mode; kinetic energy, radiation, and vacuum fluctuations have structured angular dependence that averages to zero over the flat rectangle.

Scaffolding Interpretation

Scaffolding note: The polar field variable $u = \cos\theta$ is a coordinate choice, not a new physical assumption. The THROUGH/AROUND decomposition of the gravitational source is verified identically in both $(\theta, \phi)$ and $(u, \phi)$ coordinates; the polar form simply makes the uniform-mode projection geometrically transparent.

**Figure 55.1:** Temporal momentum on the polar rectangle. **Left:** A massive particle at rest has $v_T = c$ and fills the polar rectangle uniformly (teal shading), giving maximum coupling to the $\ell = 0$ scalar gravitational field $\Phi$. **Right:** A photon has $v_T = 0$ and does not traverse the rectangle—it has zero overlap with the uniform mode and does not source $\Phi$. This is why masslessness IS zero temporal momentum.

Implications for Photons and Light

Photon Temporal Momentum

For a photon traveling at $v=c$, the velocity budget gives:

$$ v_T = c\sqrt{1-v^2/c^2} = c\sqrt{1-1} = 0 $$ (55.8)

Therefore:

$$ \boxed{p_T^{\text{photon}} = 0} $$ (55.9)

Key Result

Photons have zero temporal momentum. A photon moves entirely through ordinary space at $v=c$, with no velocity budget remaining for temporal motion. Masslessness IS zero temporal momentum.

Does Radiation Gravitate?

This raises a critical question: if photons have $p_T=0$ and the scalar field $\Phi$ couples to $\rho_{p_T}$, does radiation gravitate at all?

The answer is nuanced: TMT has two gravitational sectors.

Theorem 55.1 (Dual Gravitational Coupling)

TMT possesses two gravitational couplings:

Tensor coupling: Standard GR coupling to the full stress-energy tensor $T_{\mu\nu}\to g_{\mu\nu}$ (spin-2 graviton).
Scalar coupling: TMT scalar $\Phi$ coupling to the trace $T^\mu{}_\mu=-\rho_{p_T}\cdot c$.

Their gravitational sources differ:

Source Type	Tensor ($T_{\mu\nu}$)	Scalar ($\Phi$)
Non-relativistic matter	Gravitates	Gravitates
Radiation	Gravitates	Does not couple
Vacuum energy	Would gravitate	Does not couple

Proof.

Step 1 (Tensor sector): The Einstein equation $G_{\mu\nu}=8\pi G\,T_{\mu\nu}$ couples to all forms of energy-momentum. For radiation with equation of state $p=\rho c^2/3$, the stress-energy tensor is non-zero: $T^{00}=\rho c^2$, $T^{ij}=(\rho c^2/3)\delta^{ij}$. Radiation gravitates through the tensor sector.

Step 2 (Scalar sector): The scalar coupling $\mathcal{L}_{\text{int}}=-(\Phi/2M_{\text{Pl}})T^\mu{}_\mu$ involves the trace. For radiation:

$$ T^\mu{}_\mu = -\rho c^2 + 3\cdot\frac{\rho c^2}{3} = 0 $$ (55.10)

The trace vanishes identically for radiation, so photons do not source the scalar field.

Step 3 (Vacuum): For vacuum energy with equation of state $p=-\rho_{\text{vac}}c^2$:

$$ T^\mu{}_\mu = -\rho_{\text{vac}}c^2 + 3(-\rho_{\text{vac}}c^2) = -4\rho_{\text{vac}}c^2 $$ (55.11)

However, vacuum fluctuations have $\langle\rho_0\rangle=0$ (no net rest mass), so the temporal momentum density vanishes: $\langle\rho_{p_T}\rangle=0$. Vacuum energy does not source the scalar field.

Step 4: Non-relativistic matter has $p\ll\rho c^2$, giving $T^\mu{}_\mu\approx -\rho_0 c^2\neq 0$. Both tensor and scalar sectors are sourced. □

(See: Part 1 §3.6, Theorem 3.7) □

Detailed Source Comparison

Table 55.2: Gravitational source comparison: GR vs TMT scalar sector

Source Type	GR Source ($T^{00}$)	TMT Scalar ($\rho_{p_T}$)	Difference
Non-rel. matter	$\rho c^2$	$\rho_0 c$	Negligible
Relativistic matter	$\gamma^2\rho c^2$	$\rho_0 c$ (rest frame)	Factor of $\gamma^2$
Radiation ($p=\rho c^2/3$)	$4\rho c^2/3$	0	Complete decoupling
Cosmological vacuum	$\rho_{\text{vac}}$	0	Complete decoupling

How Photons Still Respond to Gravity

Although photons do not source the scalar field, they still respond to gravity through the tensor sector. Light bending, gravitational redshift, and Shapiro time delay all proceed exactly as in GR because these effects come from the metric tensor $g_{\mu\nu}$, not the scalar $\Phi$.

The distinction is between sourcing gravity and responding to gravity:

Table 55.3: Photon interaction with gravity in TMT

Effect	Mechanism	TMT Status
Light bending	Geodesic in $g_{\mu\nu}$	Same as GR
Gravitational redshift	Time component of $g_{\mu\nu}$	Same as GR
Shapiro delay	Spatial curvature of $g_{\mu\nu}$	Same as GR
Gravitational lensing	Full metric effect	Same as GR
Photon as gravity source	$T_{\mu\nu}$ only, not $\Phi$	Different from GR

The Trace Condition and Conformality

The vanishing of $T^\mu{}_\mu$ for radiation is connected to conformal invariance. Maxwell's equations in 4D are conformally invariant, which is equivalent to the statement that the electromagnetic stress-energy tensor is traceless.

In TMT, this has a geometric interpretation: conformal invariance means the electromagnetic field has no “temporal momentum content”—it exists entirely in ordinary 3+1 spacetime with $v_T=0$ for all its quanta.

Cosmological Implications

The decoupling of radiation and vacuum energy from the scalar sector has profound cosmological consequences:

(1) Early universe (radiation-dominated): Radiation gravitates through the tensor sector, so the Friedmann equations are preserved: $H^2=(8\pi G/3)\rho_{\text{rad}}$. Standard Big Bang nucleosynthesis and CMB physics are unaffected.

(2) Late universe (matter-dominated): Non-relativistic matter gravitates through both tensor and scalar sectors. The scalar coupling becomes relevant for the MOND–GR transition at galactic scales.

(3) Vacuum energy: The cosmological constant problem is addressed because vacuum fluctuations have $\langle\rho_{p_T}\rangle=0$ and therefore do not source the scalar field. The enormous vacuum energy predicted by quantum field theory ($\rho_{\text{vac}}\sim M_{\text{Pl}}^4$) does not produce gravitational effects through the scalar sector.

Universality of Gravitational Coupling

Why Everything With Mass Gravitates

In standard physics, the universality of gravity—the fact that all massive objects experience gravitational attraction—is an empirical observation without a deeper explanation. GR elevates it to a postulate (the equivalence principle).

In TMT, universality has a simple explanation:

Key Result

Everything with mass has temporal momentum. Gravity conserves temporal momentum. Therefore gravity is universal.

The chain of reasoning:

(1) From P1 ($ds_6^{\,2}=0$): $v^2+v_T^2=c^2$

(2) Any particle with $v0$, therefore $p_T=m_0 c/\gamma>0$.

(3) Temporal momentum must be conserved (it is a component of the 4-momentum).

(4) The mechanism that enforces this conservation is gravity.

(5) Therefore every massive particle—regardless of charge, color, flavor, or generation—experiences gravity.

Why Gravity Is Weak

The weakness of gravity compared to other forces has a TMT explanation:

Key Result

Gravity is weak because it is a small correction to something already moving at maximum speed.

At rest, all your motion is temporal: $v_T=c$, $p_T=m_0 c$. The gravitational field produces tiny adjustments to this temporal trajectory. Relative to the enormous temporal momentum $m_0 c$, these adjustments are minute:

$$ \frac{\delta p_T}{p_T}\sim\frac{\Phi}{c^2}\sim\frac{GM}{rc^2} \sim 10^{-9}\quad\text{(Earth surface)} $$ (55.12)

Other forces (electromagnetic, strong, weak) operate within the $S^2$ scaffolding through interface physics and can be $\mathcal{O}(1)$ relative to their respective charge-coupling products. Gravity operates across the 4D–3D connection and is inherently perturbative.

Polar Velocity Budget and Gravitational Hierarchy

The velocity budget acquires a transparent THROUGH/AROUND decomposition in the polar field variable $u = \cos\theta$:

$$ |\dot{\vec{x}}|^2 + R^2\!\left(\frac{\dot{u}^2}{1-u^2} + (1-u^2)\,\dot{\phi}^2\right) = c^2 $$ (55.13)

The temporal velocity $v_T$ splits into its polar channels:

Channel	Polar form	Physics	Gravity role
Spatial	$\|\dot{\vec{x}}\|^2$	Kinetic energy	Tensor sector sources
THROUGH	$R^2\dot{u}^2/(1-u^2)$	Mass/radial $S^2$	Scalar $\Phi$ sourced here
AROUND	$R^2(1-u^2)\dot{\phi}^2$	Gauge charge	Gauge forces, not $\Phi$

At rest ($\dot{\vec{x}} = 0$): $v_T = c$, all velocity is on the polar rectangle—maximally THROUGH. The particle fills the uniform ($\ell = 0$) mode, and gravity has full grip.

At $v = c$ (photon): $v_T = 0$, no velocity on the polar rectangle at all. The photon does not traverse the rectangle; it has zero overlap with the uniform mode. Scalar gravity has no grip.

Why gravity is weak, in polar language: Gauge forces couple to structured modes on the rectangle (polynomial $\times$ Fourier modes with $\ell \geq 1$, $m \neq 0$) and can produce $\mathcal{O}(1)$ overlaps with particle wavefunctions. Gravity couples to the unique $\ell = 0$ uniform mode, whose overlap with localized wavefunctions $(1-u^2)^c$ is always suppressed by $\langle u^2 \rangle = 1/3$—the same second-moment factor that governs the coupling hierarchy. The scalar gravitational coupling is further diluted by the $M_{\text{Pl}}$ suppression, making $\delta p_T / p_T \sim \Phi/c^2 \sim 10^{-9}$ at Earth's surface.

Gravitational Time Dilation as Empirical Proof

Gravitational time dilation provides direct empirical evidence for temporal momentum:

(1) If temporal momentum is real, gravity must affect time (to conserve 4D momentum in curved spacetime).

(2) Gravitational time dilation IS the temporal-trajectory adjustment.

(3) This is confirmed to extraordinary precision:

GPS satellites: clocks gain $\sim 38\;\mu\text{s/day}$ relative to ground clocks
Pound–Rebka experiment (1959): gravitational redshift confirmed to 1%
Atomic fountain clocks: relative frequency shift $\Delta f/f\sim 10^{-16}$ per meter of altitude

In standard physics, gravitational time dilation is a mathematical consequence of curved spacetime but lacks a physical explanation for why mass-energy affects time flow. In TMT, it is required: gravity conserves temporal momentum, and temporal momentum IS the rate of time passage.

The Deep Unity

Theorem 55.2 (Deep Unity of Mass, Time, and Gravity)

In TMT, mass, time, and gravity are not three separate concepts but three aspects of a single phenomenon:

$$\begin{aligned} \text{Mass} &= \text{temporal momentum}\;(p_T=m_0 c/\gamma) \\ \text{Gravity} &= \text{conservation of temporal momentum} \\ \text{Time dilation} &= \text{gravitational adjustment of temporal trajectory} \end{aligned}$$ (55.16)

Equivalently:

$$ \boxed{\text{Mass IS temporal momentum.}\quad \text{Gravity IS its conservation.}\quad \text{Time dilation IS its observable effect.}} $$ (55.14)

Proof.

Step 1: From P1, $v^2+v_T^2=c^2$ implies that any particle with \(v

Step 2: From the volume integration criterion (Chapter 51), gravity has monopole charge $q=0$ and couples to $\rho_{p_T}=\rho_0 c$. This coupling enforces temporal momentum conservation: the Lagrangian $\mathcal{L}_{\text{int}}=-(\Phi/2M_{\text{Pl}})T^\mu{}_\mu$ generates equations of motion that conserve the 4D stress-energy tensor including its temporal components.

Step 3: In a gravitational potential $\Phi$, the temporal velocity is modified:

$$ v_T(\Phi) = c\sqrt{1+2\Phi/c^2}\approx c(1+\Phi/c^2) $$ (55.15)

A clock at gravitational potential $\Phi$ ticks at rate $d\tau/dt=\sqrt{1+2\Phi/c^2}\approx 1+\Phi/c^2$, which is the standard gravitational time dilation formula. This IS the observable consequence of temporal momentum conservation.

Step 4: The three statements are therefore equivalent: mass is the magnitude of temporal momentum; gravity is the force that conserves it; time dilation is the observable effect of that conservation. □

(See: Part 1 §3.7.4; Part A) □

Comparison: TMT vs GR vs Newtonian Gravity

Table 55.4: Why gravity affects time: three perspectives

Theory	Why gravity affects time	Why gravity is universal
Newton	No explanation	Postulated
GR	Curved spacetime (math)	Equivalence principle (axiom)
TMT	Conserves temporal momentum	Everything has $p_T$

Chapter Summary

Key Result

Gravity and Temporal Momentum

TMT's scalar gravitational field couples to temporal momentum density $\rho_{p_T}=\rho_0 c$—rest mass, not energy. This is uniquely required by Lorentz invariance, the tracelessness theorem, and the monopole charge criterion. Consequences: (1) photons have $p_T=0$ and do not source scalar gravity, (2) vacuum energy has $\langle\rho_{p_T}\rangle=0$ and does not gravitate through $\Phi$, (3) all massive particles are universal gravitational sources. The universality of gravity is explained: everything with mass has temporal momentum, and gravity conserves it. Gravitational time dilation is the observable proof that temporal momentum is physical. In polar field coordinates ($u = \cos\theta$), $p_T$ is the THROUGH channel momentum; gravity couples to the uniform ($\ell = 0$) mode on the polar rectangle $[-1,+1] \times [0,2\pi)$; photons have zero rectangle traversal ($v_T = 0$); and gravitational weakness traces to the $\langle u^2\rangle = 1/3$ suppression of the uniform mode overlap.

Derivation Chain Summary

Step	Result	Justification	Reference
\endhead 1	$ds_6^{\,2} = 0$ (P1)	Single postulate	Ch 2
2	$v^2 + v_T^2 = c^2$	Null constraint	Ch 5
3	$p_T = m_0 c/\gamma$	Temporal momentum defined	Ch 5
4	$T^A{}_A = 0$ (tracelessness)	From P1	Ch 6
5	Scalar couples to $T^\mu{}_\mu$	Lorentz inv. + tracelessness + $q{=}0$	§sec:ch54-why-pT
6	Photon $p_T = 0$	$v = c \Rightarrow v_T = 0$	Eq. (eq:ch54-photon-pT)
7	Dual coupling (tensor + scalar)	GR tensor $+$ TMT scalar	Thm. thm:P1-Ch54-dual-coupling
8	Deep unity: mass $=$ $p_T$	Gravity conserves $p_T$	Thm. thm:P1-Ch54-deep-unity
9	Polar: $p_T$ = THROUGH channel	$q{=}0$ uniform mode on $[-1,+1] \times [0,2\pi)$	§sec:ch54-polar-three-arguments

Table 55.5: Chapter 54 results summary

Result	Value	Status	Reference
Scalar couples to $\rho_{p_T}$	$\rho_0 c$	PROVEN	§sec:ch54-why-pT
Photon $p_T$	0	PROVEN	Eq. (eq:ch54-photon-pT)
Dual coupling structure	Tensor + Scalar	PROVEN	Thm. thm:P1-Ch54-dual-coupling
Radiation decoupled from $\Phi$	$T^\mu{}_\mu=0$	PROVEN	Thm. thm:P1-Ch54-dual-coupling
Vacuum doesn't gravitate via $\Phi$	$\langle\rho_{p_T}\rangle=0$	PROVEN	§sec:ch54-photons
Deep unity	Mass $=$ $p_T$; gravity $=$ conservation	PROVEN	Thm. thm:P1-Ch54-deep-unity

Verification Code

The mathematical derivations and proofs in this chapter can be independently verified using the formal and computational scripts below.

◆ Lean 4 Formal proof verification Coming Soon ● Python Numerical verification & plots Coming Soon ★ Mathematica Symbolic computation & CAS verification Coming Soon

All verification code is open source. See the complete verification index for all chapters.

Property	Spherical \((\theta, \phi)\)	Polar \((u, \phi)\)
Gravitational source	\(T^\mu{}_\mu\)	\(-(T^u{}_u + T^\phi{}_\phi)\) (THROUGH + AROUND)
Monopole charge	\(q = 0\) (isotropic)	Uniform mode on \([-1,+1] \times [0,2\pi)\)
Volume integral	\(\int \sin\theta\,d\theta\,d\phi\)	\(\int du\,d\phi\) (flat, no Jacobian)
\(\gamma\) cancellation	Number \(\times\) momentum	AROUND boost \(\times\) THROUGH \(1/\gamma\)
Radiation trace	\(-\rho c^2 + 3p = 0\)	THROUGH + AROUND cancel \(T_4\) exactly

Source Type	GR Source (\(T^{00}\))	TMT Scalar (\(\rho_{p_T}\))	Difference
Non-rel. matter	\(\rho c^2\)	\(\rho_0 c\)	Negligible
Relativistic matter	\(\gamma^2\rho c^2\)	\(\rho_0 c\) (rest frame)	Factor of \(\gamma^2\)
Radiation (\(p=\rho c^2/3\))	\(4\rho c^2/3\)	0	Complete decoupling
Cosmological vacuum	\(\rho_{\text{vac}}\)	0	Complete decoupling

Effect	Mechanism	TMT Status
Light bending	Geodesic in \(g_{\mu\nu}\)	Same as GR
Gravitational redshift	Time component of \(g_{\mu\nu}\)	Same as GR
Shapiro delay	Spatial curvature of \(g_{\mu\nu}\)	Same as GR
Gravitational lensing	Full metric effect	Same as GR
Photon as gravity source	\(T_{\mu\nu}\) only, not \(\Phi\)	Different from GR

Channel	Polar form	Physics	Gravity role
Spatial	\(\|\dot{\vec{x}}\|^2\)	Kinetic energy	Tensor sector sources
THROUGH	\(R^2\dot{u}^2/(1-u^2)\)	Mass/radial \(S^2\)	Scalar \(\Phi\) sourced here
AROUND	\(R^2(1-u^2)\dot{\phi}^2\)	Gauge charge	Gauge forces, not \(\Phi\)

Step	Result	Justification	Reference
\endhead 1	\(ds_6^{\,2} = 0\) (P1)	Single postulate	Ch 2
2	\(v^2 + v_T^2 = c^2\)	Null constraint	Ch 5
3	\(p_T = m_0 c/\gamma\)	Temporal momentum defined	Ch 5
4	\(T^A{}_A = 0\) (tracelessness)	From P1	Ch 6
5	Scalar couples to \(T^\mu{}_\mu\)	Lorentz inv. + tracelessness + \(q{=}0\)	§sec:ch54-why-pT
6	Photon \(p_T = 0\)	\(v = c \Rightarrow v_T = 0\)	Eq. (eq:ch54-photon-pT)
7	Dual coupling (tensor + scalar)	GR tensor \(+\) TMT scalar	Thm. thm:P1-Ch54-dual-coupling
8	Deep unity: mass \(=\) \(p_T\)	Gravity conserves \(p_T\)	Thm. thm:P1-Ch54-deep-unity
9	Polar: \(p_T\) = THROUGH channel	\(q{=}0\) uniform mode on \([-1,+1] \times [0,2\pi)\)	§sec:ch54-polar-three-arguments

Result	Value	Status	Reference
Scalar couples to \(\rho_{p_T}\)	\(\rho_0 c\)	PROVEN	§sec:ch54-why-pT
Photon \(p_T\)	0	PROVEN	Eq. (eq:ch54-photon-pT)
Dual coupling structure	Tensor + Scalar	PROVEN	Thm. thm:P1-Ch54-dual-coupling
Radiation decoupled from \(\Phi\)	\(T^\mu{}_\mu=0\)	PROVEN	Thm. thm:P1-Ch54-dual-coupling
Vacuum doesn't gravitate via \(\Phi\)	\(\langle\rho_{p_T}\rangle=0\)	PROVEN	§sec:ch54-photons
Deep unity	Mass \(=\) \(p_T\); gravity \(=\) conservation	PROVEN	Thm. thm:P1-Ch54-deep-unity

Introduction

Why Gravity Couples to \(p_T\), Not \(E\)

The Fundamental Question

Argument 1: The Lorentz Invariance Requirement

Argument 2: The Tracelessness Theorem

Argument 3: The Monopole Charge Criterion

The Three Arguments Converge

Polar Field Perspective on the Three Arguments

Implications for Photons and Light

Photon Temporal Momentum

Does Radiation Gravitate?

Detailed Source Comparison

How Photons Still Respond to Gravity

The Trace Condition and Conformality

Cosmological Implications

Universality of Gravitational Coupling

Why Everything With Mass Gravitates

Why Gravity Is Weak

Polar Velocity Budget and Gravitational Hierarchy

Gravitational Time Dilation as Empirical Proof

The Deep Unity

Comparison: TMT vs GR vs Newtonian Gravity

Chapter Summary

Derivation Chain Summary

Verification Code