II-6. Particle
Physics
RIKEN
Accel.
Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

R = 12H02 and its implications to gravity and cosmology
Y. Akiba ∗1
In the preceding article,1) I report a model (QSTmodel) that can yield the formulas of the standard
model (SM) parameters. The model implies that theproduct of the Hubble constant H0 and the Plancktime tpl is H0 tpl = 2 × (6π)−48 = ϵ0 . This relation isderived as follows.
The spacetime metric of the Hubble expansion ismetric (Sp-metric):
2ds = −dt +e√√()dr22222+r (dθ +sin θdϕ ) .
1− 2rrs

Rtt = 3H02 , Rrr = 3H02 −2rs −2H0 te,
r3rs −2H0 te,
r3
R = Rtt + Rrr + Rθθ + Rϕϕ = 12H02 .

Rθθ = Rϕϕ = 3H02 +−g of this metric

R = 12H02 ,This metric describes the spacetime metric outside avery large spherical mass, e.g., a star or a planet. It is ametric with spherical symmetry, like the Schwartzchild
metric, but with R = 12H02 . Note that Rµν = 0 andR = 0 for the Schwartzchild metic.
One can solve Keplerian moton in this metric. Thesolution implies that the radius r of a circular orbit ofan object around the large mass at the origin increases
with time as√ṙ/r = 2H0 = 1.0104 × 10−10 /yr.
√The factor 2 arises from the fact that the Keplerianmotion is a two dimensional motion. One can showthat the scale expansion√ rate of two-dimensional radius
ρ with R = 12H02 is 2H0 .
This prediction agrees with the observed rate of expansion of the the Moon’s orbit radius rMoon aroundthe Earth2) within the uncertainty of the data.

−g = e3H0 t .

√For t = tpl , we have −g = e3H0 t ≃ 1 + 3H0 tpl . In theQST model, the correspondence relation is√−g ↔ 1 + L̂g3D = 1 + 3ϵ0 .
Thus, we have H0 tpl = ϵ0 . Because H0 = ϵ0 /tpl isa constant, R is a constant. The QST model implies
that R = 12H02 (constant). We discuss the implicationof this equation in general relativity and cosmology
below.
We can generalize the metric to allow local changesof the scale with a constraint R = 12H02 :ds2 = −e2u(x,y,z) dt2 + e2H0 t (dx2 + dy 2 + dz 2 ).
The R of this metric is
R = −(2∆u + 3(∇u)2 )e−2H0 t + 12H02 e−2uṙMoon = 3.82 ± 0.07cm/yr,ṙMoon /rMoon = (0.994 ± 0.0182) × 10−10 /yr.

= 12H02 .

We can show that Coulomb gravity can be derived fromthis metric. When we take the approximation H0 ≃ 0,
we have
R = −2∆u − 3(∇u)2 = 0.

This good agreement indicates that the expansion ofthe lunar-orbit radius is due to Hubble expansion.
We found a few emprical formulas of cosmologicalparameters that are similar to those of the SM parameters. For CMB temperature, we found
()1/2
1
1
11/2
T =−ϵ0.
1+(6π)24π (6π)2

If u is small, we can ignore the (∇u)2 term, and wehave ∆u = 0. In general relativity, g00 ≃ −1 − 2ϕG ,where ϕG is the gravitational potential. As e2u = −g00
and e2u ≃ 1 + 2u, u ≃ ϕG . Therefore,∆ϕG ≃ ∆u = 0.

calc
This formula yields TCMB= 2.7249 K, which agrees
with the obsrved value TCMB = 2.7255 ± 0.0006 K
within the uncertainty of the data. The QST model
can explain the emprical formula.

Thus, the relation R = 0 implies Coulombic gravitational potential under the weak gravity approximation.
If we take into account the fact that H0 ̸= 0, the potential equation for gravity becomes non-linear.
Next, we discuss the eﬀect of R = 12H02 for the motion of an astronomical object. Consider the following
∗1

2H0 t

One can show that the Ricci tensor Rµν and the Ricci
scalar curvature R of this metric are

ds2 = −dt2 + e2H0 t (dx2 + dy 2 + dz 2 ).
The Ricci scalar curvature R andare

2

References
1) Y. Akiba, in this report.
2) B. G. Bills, R. D. Ray, Geophys. Res. Lett. 26, 3045
(1999).

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