Gravity Waves

Coherence-Wave Radiation in PBG

(“gravity waves” without spacetime ripples)

1 What can actually wave in PBG?

Concept	GR wording	PBG wording
Background	spacetime metric $g_{μ ν}$	static phase bias field $Φ_{0} (x)$ produced by all stationary masses
Dynamic dof	tensor perturbation $h_{i j}$	transverse phase-shear packet in the gradient $E_{i} \equiv \partial_{i} Φ$
Propagation law	$◻ h_{i j} = 0$	$γ \partial_{t}^{2} Φ - α \nabla^{2} Φ + β Φ = - ρ_{c o h}$
Polarisation	2 tensor modes + / ×	2 vector shear modes in $E_{i}$ (no extra scalar)

2 Derive the radiative field from first principles

2.1 Action + source

S = \int d^{3} x d t [\frac{1}{2} γ (\partial_{t} Φ)^{2} - \frac{1}{2} α | \nabla Φ |^{2} - \frac{1}{2} β Φ^{2} - Φ ρ_{c o h} (t, x)],

ρ_{c o h} = \sum_{n} ψ_{n}^{2} (t, x) (coherence density of moving modes) .

2.2 Inhomogeneous field equation

γ \partial_{t}^{2} Φ - α \nabla^{2} Φ + β Φ = - ρ_{c o h} .

2.3 Far-zone, high-frequency limit

For wavelengths $ \lambda\ll \ell=1/k=\sqrt{\alpha/\beta}$ (true for LIGO bands), set $β \to 0$ :

γ \partial_{t}^{2} Φ - α \nabla^{2} Φ = - ρ_{c o h} .

Green’s function:

G_{ret} (t, r) = \frac{1}{4 π c} \frac{δ (t - r / c)}{r}, c = \sqrt{α / γ} .

2.4 Quadrupole term survives

Monopole $\dot{Q}$ and dipole ${\ddot{D}}_{i}$ vanish for an isolated system ⇒ leading radiation is the quadrupole:

Φ_{rad} (t, x) = - \frac{1}{8 π c^{3} r} n_{i} n_{j} {\ddot{Q}}_{i j} (t - \frac{r}{c}) .

$Q_{i j} = \int ρ_{c o h} (x_{i} x_{j} - \frac{1}{3} r^{2} δ_{i j}) d^{3} x$ .

3 What actually hits a detector?

Define the shear field

E_{i} = \partial_{i} Φ_{rad} \approx \frac{n_{i} n_{j} n_{k}}{8 π c^{4} r} {\ddot{Q}}_{j k} (t - r / c) .

Because $E_{i}$ is a vector perpendicular to $n$ , it decomposes into two orthogonal, transverse directions—the direct analogues of GR’s $+$ and $\times$ tensors.

No isotropic breathing mode appears: the scalar phase is “eaten” by its gradient.

4 Energy flux and power

Energy density:

u = \frac{1}{2} γ (\partial_{t} Φ)^{2} + \frac{1}{2} α | \nabla Φ |^{2} \approx γ (\partial_{t} Φ)^{2} .

Radiated power (average over angles):

P_{G W}^{PBG} = \frac{γ}{120 π c^{6}} ⟨ {\overset{⃛}{Q}}_{i j} {\overset{⃛}{Q}}_{i j} ⟩ .

Using $c^{2} = α / γ$ and $G \equiv α Q_{M}^{2} / (4 π M^{2})$ , one finds

P_{G W}^{PBG} = \frac{G}{5 c^{5}} ⟨ {\overset{⃛}{Q}}_{i j} {\overset{⃛}{Q}}_{i j} ⟩ (identical to GR) .

5 Predictions & falsifiers

Observable	GR prediction	PBG prediction	Test status
Speed of GW	$c$	$c$ (exact)	GW170817 OK
Polarisation content	+/× only	2 shear modes (+/×) (no breathing)	current data OK
Binary-pulsar energy loss	GR quadrupole rate	Same rate (formula above)	PSR-1913 OK
Echoes from saturation shell	none	ms echoes at $10^{- 3}$ – $10^{- 2}$ strain	LIGO O4 will probe

6 Why echoes?

Inside $r_{s}$ the coherence field saturates, creating a partially reflecting shell one kernel length $ℓ$ outside the would-be horizon.
Outgoing shear packets reflect, pick up a time delay

t_{e c h o} \approx 2 r_{s} / c \ln (r_{s} / ℓ) [ms–scale],

and re-emerge at $\sim 10^{- 3}$ of peak ring-down amplitude. Upcoming stacks of heavy-BBH events (O4/O5) should detect or exclude such echoes.

7 Bottom-line statement

Gravitational waves in PBG are phase-shear pulses of the scalar field’s gradient.
They propagate at $c$ , carry quadrupole power identical to GR, and excite the same two transverse polarisation patterns in laser interferometers.
The only novel, falsifiable difference is a (small) reflected echo sequence set by the universal kernel length $ℓ = \sqrt{α / β}$ . Detect echoes ⇒ support PBG; rule them out below $\sim 10^{- 3}$ ⇒ minimal-parameter PBG shell is falsified.

links → Field Normalisation in Phase-Biased Geometry • Unification from 3 Constants • PBG Black-Hole Shell