Unified Time Scale and Time Geometry:\\ Causal Ordering, Unitary Evolution, and Generalized Entropy

Unied Time Scale and Time Geome y:
Causal O de ing, Uni a y E olu ion, and Gene alized
En opy
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 20, 2025
Abs ac
We p opose a unied ime scale equi alence class igo ously gluing h ee ends
o ela i i y, quan um sca e ing, and in o ma ionholog aphy. Co e scale iden-
i y unies de i a i e o o al sca e ing phase, ela i e s a e densi y, and ace o
Wigne Smi h g oup delay as die en p ojec ions o same objec :
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω), Q(ω) = −iS(ω)†∂ωS(ω), φ =1
2a g de S.
In geome y end, Killing ime, ADM lapse, null geodesic ane pa ame e , and
FRW con o mal ime p o en mu ually escalable wi hin unied equi alence class;
in in o ma ionholog aphy end, aking Tomi aTakesaki modula ow as in insic
ime, con olling gene alized en opy ex emali y on small causal diamonds wi h
QFC/QNEC and ela i e en opy mono onici y, hus de i ing Eins ein equa ions in
semiclassicalholog aphic window. F amewo k ob ains h ee alignmen s: (i) phase
p ope ime equi alence
ϕ= (mc2/ℏ)Rdτ
; (ii) g a i a ional ime delay = g oup
delay ace
∆T=∂ωΦ = T Q
; (iii) FRW edshi = phase hy hm a io
1 +
z=a( 0)/a( e) = [(dϕ/d )e]/[(dϕ/d )0]
. This pape unde h ee axioms o
causal
o de inguni a y e olu ionen opy mono onici y/ex emali y
, es ablishes
exis ence and ane uniqueness o unied ime scale, gi ing ealiza ion schemes o
expe imen al and enginee ing me ology.
Keywo ds:
Time Geome y; Unied Time Scale; Wigne Smi h G oup Delay; Spec-
al Shi Func ion; Killing/ADM/Null/Con o mal/Modula Time; Gene alized En opy;
QFC/QNEC
MSC 2020:
83C45, 81U40, 81T20, 83C57

1 In oduc ion and His o ical Con ex
Role o ime spli s in die en heo ies: gene al ela i i y scales causal s uc u e wi h
p ope ime, quan um heo y gene a es uni a y e olu ion wi h ex e nal pa ame e , in o ma ion
holog aphy iews modula ow as in insic  he mal ime. Ye when h ee end eadings
1
synch onize, s ill lacking igo ous and me ologically measu able common scale. Wigne
and Smi h in oduced de i a i e o phase wi h espec o ene gy in sca e ing heo y
dening ime delay, ace o Wigne Smi h g oup delay ma ix
Q=−iS†∂ωS
equals
de i a i e o o al phase
Φ = a g de S
, making idea o  ime = phase g adien   s
land on expe imen ally eadable ule . Bi manK en o mula cha ac e izing s a e den-
si y change caused by in e ac ion wi h spec al shi unc ion igh ly connec s phase o
sca e ing de e minan wi h spec al geome y, hus de i ing phase de i a i e = ela i e
s a e densi y.
In ela i i y side, edshi /clock a e in s a ic space ime gi en by
g
o Tolman
Eh en es law; lapse
N
in ADM
(3+1)
decomposi ion cha ac e izes a io o coo dina e
ime o p ope ime; o oise coo dina es and
(u, )
in asymp o ically a ex e io p o ide
na u al null ime a inni y; in FRW cosmology
1+z=a( 0)/a( e)
linea izes null geodesics
wi h con o mal ime.
In in o ma ionholog aphy side, Tomi aTakesaki modula heo y endows any (s a e,
algeb a) pai wi h amily o in insic one-pa ame e au omo phisms (modula ow);
ConnesRo elli he mal ime hypo hesis iews his modula ow as physical ime candi-
da e; ela i e en opy mono onici y, QFC, and QNEC bind changes o gene alized en opy
oge he wi h s ess enso cons ain s, connec ing o eld equa ions and econs uc ion
in small causal diamond limi .
Abo e his o ical h eads sugges :
uni ying phaseg oup delayspec al shi 
wi h clock a e edshi ane/con o mal ime and modula imegene alized
en opy in o single scale
, hope ul o ob ain c oss-scale imegeome y amewo k.

2 Model and Assump ions
(A) Causal and Global S uc u e
Le
(M, g)
be s ably causal Lo en zian mani old;
unde global hype bolici y condi ion he e exis smoo h ime unc ion and smoo h de-
composi ion
M∼
=R×Σ
.
(B) Sca e ing and Spec al Shi
On absolu ely con inuous spec al ene gy win-
dow
I⊂R
, sca e ing ma ix
S(ω)
uni a y and smoo h; dene o al phase
Φ = a g de S
,
g oup delay
Q=−iS†∂ωS
. The e exis s spec al shi unc ion
ξ(ω)
and ela i e s a e
densi y
ρ el =−ξ′
; Bi manK en o mula
de S(ω) = exp[−2πi ξ(ω)]
holds.
(C) Unied Scale Iden i y (Co e Assump ion)
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω), φ =1
2Φ.
(D) Bounda y En opy and Modula Flow
Take small causal diamond
Dp,
h ough poin
p
, null gene a o ane pa ame e
λ
; gene alized en opy
Sgen(λ) = A ea(Σλ)
4Gℏ+Sou (λ)
sa ises QFC/QNEC ype inequali ies and ela i e en opy mono onici y; modula
Hamil onian
K=−ln ρ
gene a es modula ow
σs
.
(E) Uni a y E olu ion
On s a e space
H
he e exis s s ongly con inuous uni a y
g oup
U( ) = e−iH
; semiclassical wo ldline limi can ela e
and
τ
h ough phase densi y
(see Sec ion 4.1).

2
3 Unied Time Scale: Deni ion and Th ee Axioms
3.1 Unied Time Scale Equi alence Class
Deni ion 3.1
(Unied Time Scale (Deni ion 3.1))
.
The e exis s equi alence class
[T]∼ {τ, , K,(N, Ni), λnull, u, , η, ω−1, z, smod},
whose membe s mu ually con e ible h ough mono onic escaling and geome ic/en opy
s uc u e, making dynamics local, causally o de ed, and en opy s uc u e simples .
3.2 Th ee Axioms
Axiom 3.1
(Causal O de ing (Axiom I))
.
In local hype bolic domain he e exis s s ic ly
inc easing ime unc ion, making undamen al equa ions local (hype bolic/ s -o de )
o m.
Axiom 3.2
(Uni a y E olu ion (Axiom II))
.
The e exis s s ongly con inuous uni a y
g oup
U( )
; in semiclassical limi phase ime ela ion de e mined by Lag angian s a ion-
a y phase (Sec ion 4.1).
Axiom 3.3
(En opy Mono onici y/Ex emali y (Axiom III))
.
Along null cu amily
{Σλ}
,
Sgen
sa ises ela i e en opy mono onici y and QFC/QNEC mono onici y/con exi y
and akes ex emum unde physical e olu ion; modula ow pa ame e
s
makes o gani-
za ion law o
Sgen
simples .
Theo em 3.1
(Mu ual Implica ion in SemiclassicalHolog aphic Window (Theo em
3.2))
.
Unde small causal diamond limi and ela i e en opy mono onici y/QNEC hold-
ing:
Axiom I
+
Axiom II
⇐⇒
Scale Iden i y
=⇒
Axiom III
=⇒
Eins ein Equa ions
.
P oo in Sec ion 5 and Appendices D/E.

4 Main Resul s (Theo ems and Alignmen s)
4.1 PhaseP ope Time Equi alence
Theo em 4.1
(Wo ldline P incipal Phase (Theo em 4.1))
.
Fo na ow wa e packe o
mass
m
, in semiclassical limi
ϕ=−1
ℏS[γcl] = mc2
ℏZγcl
dτ, dϕ
dτ=mc2
ℏ.
(P oo : wo ldline pa h in eg al s a iona y phase; see Appendix B.)
4.2 G a i a ional Time Delay = G oup Delay T ace
Theo em 4.2
(EikonalSca e ing Alignmen (Theo em 4.2))
.
In geome ic op ics limi
o s a ic o asymp o ically a backg ound,
∆T(ω) = ∂ωΦ(ω) = T Q(ω).
Weak eld limi e u ns o Shapi o delay.
3
4.3 Redshi = Phase Rhy hm Ra io
P oposi ion 4.3
(FRW Phase Exp ession (P oposi ion 4.3))
.
Unde a FRW me ic
como ing obse e s measu e
1 + z=νe
ν0
=
dϕ
d e
dϕ
d 0
=a( 0)
a( e).
(See Appendix C.)
4.4 Fou B idges o GR Time S uc u e
B idge B (Killing TimeClock Ra eRedshi )
In s a ic me ic
ds2=−V(x)c2d 2+
···
s a iona y obse e s ha e
dτ=√Vd
,
√V
is local edshi /clock a e ac o (Tolman
Eh en es ).
B idge C (ADM LapseLocal Clock Ra e)
ADM decomposi ion
ds2=−N2d 2+
hij(dxi+Nid )(dxj+Njd )
; Eule amily o hogonal o slicing sa ises
dτ=Nd
.
B idge D (Null Ane Pa ame e Re a ded/Ad anced/Con o mal Time)
Asymp o ically a ex e io denes o oise
∗
and
u= − ∗, = + ∗
, mono oni-
cally equi alen o null geodesic ane pa ame e ; in FRW
dη= d /a( )
linea izes null
geodesics.
B idge E (Modula TimeEn opy G adien Geome ic Equa ions)
Pa am-
e e
s
o modula ow
σs
p o ides in o ma ion- heo e ic ime; ela i e en opy mono-
onici y and QNEC/QFC bind ex emali y/mono onici y o
∂sSgen
wi h
⟨Tkk⟩
.
4.5 En opic Geome ic Fo m o Eins ein Equa ions
Theo em 4.4
(En opyGeome y (Theo em 4.4))
.
Unde Axiom III and Raychaudhu i
equa ion, on small causal diamond ha e
Rµν −1
2Rgµν + Λgµν = 8πG ⟨Tµν⟩.
(P oo : combining second-o de a ea a ia ion wi h QNEC/ ela i e en opy, see Ap-
pendix D; c . Jacobson and subsequen holog aphic a gumen s.)
4.6 Unied Scale Iden i y (Spec alSca e ingGeome y)
Co olla y 4.5
(Co olla y 4.5)
.
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
ob ained by combining Bi manK en and
T Q=∂ωΦ
(Appendix A).

4
5 P oo s (Key Poin s)
5.1 Theo em 4.1:
Wo ldline ac ion s a iona y phase along imelike geodesic, uc ua-
ions only change quan um p e ac o , p incipal phase gi es
ϕ= (mc2/ℏ)Rdτ
(Appendix
B).
5.2 Theo em 4.2:
Eikonal phase die ence
∆S≃ −ℏω∆T
aligns wi h sca e ing
∂ωΦ = T Q
, yielding
∆T= T Q
; weak eld es e u ns o Shapi o delay.
5.3 P oposi ion 4.3:
Null geodesic and scale ac o gi e
ν∝1/a( )
, edshi as
phase hy hm a io (Appendix C).
5.4 B idges BE:
S a ic clock a e, ADM lapse, null coo dina es, and modula ow
each consis en wi h s anda d conclusions and li e a u e (Appendix E).
5.5 Theo em 4.4:
Raychaudhu i's second-o de a ea a ia ion
∝ −RRkk
combines
wi h QNEC
S′′
ou ≥(2π/ℏ)R⟨Tkk⟩
, ex emali y condi ion
S′
gen(0) = 0
yields enso equa ion
(Appendix D); QFC p o ides s onge mono onici y backg ound.

6 Model Applica ions
A. F equency-Domain Recons uc ion o Sola Sys em Geome ic Delay
Di -
e en ia ing mul i- equency ada echo phase
Φ(ω)
gi es
∆T(ω) = ∂ωΦ = T Q
, compa e
wi h Shapi o delay, can pa allelly s ip plasma dispe sion e m.
B. PhaseG oup Delay Unica ion o G a i a ional Lensing
Image pai
(i, j)
Fe ma po en ial die ence
∆ ij
equals
∂ω[Φi−Φj]
; b oadband elec omagne ic/g a i a ional
wa e join use o Hubble cons an and mass model sys ema ic e o supp ession.
C. Cosmological Phase Rule 
Di ec ly es ima e
a( )
using phase hy hm a io o
pulsa /FRB, a oiding specic spec al line sys ema ics; phase edshi  synch oniza ion
gua an eed by P oposi ion 4.3.
D. Eec i e Time Re ac i e Index Tomog aphy
In e
n = (−g )−1/2
om
spa ial dis ibu ion o
ρ el(ω)
o
Q(ω)
, coope a ing wi h op ical me icFe ma p inciple
o weak eld ime delay imaging.

7 Enginee ing P oposals
1.
On-chip g oup delay omog aphy me ology:
Measu e
S(ω)
in in eg a ed
pho onics and eal- ime compu e
T Q(ω)
, gene a ing equi alen g a i a ional ime
delay map o de ice in e sion and obus design.
2.
Double-heigh ma e wa e s anda d:
COW geome ic a angemen compa e
∆ϕ
and
∆τ
, es linea egime o
∆ϕ= (mc2/ℏ)∆τ
.
3.
B oadband lens g oup delay spec um:
Synch onously  each image a i al
ime delay and dispe sion using
∂ωΦ
, educing ime delay cosmology sys ema ic
e o s.
4.
En opy ligh cone pla o m:
Measu e second-o de de o ma ion and ene gy
ow o
Sou
on con ollable quan um sys em, es QNEC/QFC coecien s and
sa u a ion condi ions.
5


8 Discussion (Risks, Bounda ies, Pas Wo k)
(i) Spec al endpoin s and egula i y:
Scale iden i y equi es
S(ω)
die en iable and
belonging o app op ia e de e minan class; nea esonances and h esholds need con ou
displacemen and ace-class egula iza ion.
(ii) Geome ic op ics and s ong elds:
S ong spin/non-s a ic backg ounds need
gene alized op ical me ic and cohe en anspo ; nea -ho izon egions be e use null
coo dina es and nume ical ay acing.
(iii) En opygeome y assump ion domain:
QNEC has gene al QFT p oo s and
holog aphic p oo s con inuously s eng hening (including la es new p oo app oaches),
bu s ill ad ancing in high cu a u e, s ong quan um g a i y egions.
(i ) GR ime s uc u e unica ion:
Killing/ADM/null/con o mal/modula imes
a e coo dina iza ions o unied scale in die en p ojec ions; Be nalSánchez's global ime
unc ions and ADM olia ion p o ide igo ous ounda ion.

9 Conclusion
Unde h ee axioms o
causal o de inguni a y e olu ionen opy mono onic-
i y/ex emali y
, ob ain unied ime scale equi alence class, aligning mic oscopic (phase/sca e ing),
mesoscopic (g oup delay/ edshi ), and mac oscopic (en opygeome y) h ee-end lan-
guages. Co e conclusions:
ϕ=mc2
ℏZdτ, ∆T(ω) = ∂ωΦ(ω) = T Q(ω),1+z=(dϕ/d )e
(dϕ/d )0
, Gµν+Λgµν = 8πG ⟨Tµν⟩,
and spec alsca e inggeome y scale iden i y
φ′/π =ρ el = (2π)−1 Q
. Time hus
can be cha ac e ized as:
equi alence class o one-dimensional pa ame e making
dynamics local, causali y clea , en opy s uc u e simples
; i s die en names
me ely coo dina es o same objec in die en p ojec ions.

Acknowledgemen s, Code A ailabili y
Thanks o ela ed ex books and li e a u e. Symbolic de i a ions and nume ical sc ip s
o g oup delay ime delay econs uc ion and FRW phase hy hm demons a ion a ail-
able upon eques .

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Encycl. Ma h. Phys.
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93
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93
(2016) 024017.
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QNEC,
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7
A Wigne Smi h G oup Delay and Spec alSca e ing
Geome y Iden i y
A.1 Bi manK en and Spec al Shi
Fo sel -adjoin pai
(H, H0)
wi h ace-class/quasi- ace-class pe u ba ion, spec al shi
unc ion
ξ(ω)
sa ises
de S(ω) = e−2πiξ(ω)⇒1
2π∂ωΦ(ω) = −ξ′(ω) = ρ el(ω).
(See e e ence 3.)
A.2 T ace Iden i y
F om
Q(ω) = −iS†∂ωS
and
T ln S= ln de S
ob ain
T Q(ω) = ∂ωΦ(ω).
Combining A.1 gi es scale iden i y:
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
B PhaseP ope Time Unde Wo ldline Pa h In e-
g al
Fe mi no mal coo dina e expansion along imelike geodesic
γcl
S[γ] = −mc2Zdτ+m
2Zdτ δij ˙yi˙yj+··· .
S a iona y phase gi es
ϕ=−1
ℏS[γcl] = mc2
ℏZdτ, dϕ
dτ=mc2
ℏ.
C Phase Exp ession o Redshi in FRW Cosmology
Fla FRW:
ds2=−d 2+a( )2dx2
. Como ing obse e
uµ= (1,0,0,0)
, pho on eikonal
phase
ϕ
has
kµ=∂µϕ
and
ν=−1
2πkµuµ=1
2π
dϕ
d ∝1
a( )⇒1 + z=(dϕ/d )e
(dϕ/d )0
=a( 0)
a( e).
(Re e ence 7.)
8
D Gene alized En opy Ex emali y/Mono onici y and
Field Equa ions
Le
{Σλ}
be null cu amily h ough
p
. Raychaudhu i:
˙
θ=−1
2θ2−σ2−Rkk
. Second-o de
a ea a ia ion
d2A/dλ20∝ −ZRkk dA
.
QNEC and ela i e en opy mono onici y:
d2Sou /dλ20≥2π
ℏZ⟨Tkk⟩dA
.
Ex emum
S′
gen(0) = 0
combines o gi e
Rkk = 8πG⟨Tkk⟩
, holding o any
kµ
, upg ades
o enso equa ion and gi es
Λ
as in eg a ion cons an .
E Renemen o GR Time B idges
E.1 S a ic Space ime (Killing):
ξµ
is imelike Killing ec o , s a iona y obse e
uµ=ξµ/p−ξ2
, i
g =−V
hen
dτ=√Vd
. (TolmanEh en es empe a u e edshi
law same o m.)
E.2 ADM Lapse:
ds2=−N2d 2+hij(dxi+Nid )(dxj+Njd )
; slicing o hogonal
amily sa ises
dxi+Nid = 0 ⇒dτ=Nd
.
E.3 Null Coo dina es:
Schwa zschild ex e io
∗= +2Mln | /2M−1|
,
u= − ∗
,
= + ∗
; in FRW
dη= d /a( )
.
E.4 Modula Time:
Gi en (algeb a, s a e) pai
(A, ω)
GNS ep esen a ion, modula
ow
σs
in insically denes ime; unde hal -space and small de o ma ions,
K
localizes o
RTkk
, isomo phic wi h ANEC/QNEC, JLMS/ ela i e en opy.
F Shapi o Delay and G oup Delay
Weak eld Schwa zschild ex e io
∆ ≃4GM
c3ln 4 E R
b2+··· ,
consis en wi h equency-domain measu ed
∂ωΦ = T Q
; mul i- equency echo  ing can
sepa a e dispe sion and geome ic e ms.
G Exis ence and Uniqueness o Unied Time Scale
Gi en sca e ing da a
(S(ω))
sa is ying scale iden i y. Dene
− 0=Zω
ω0
1
2πT Q(˜ω) d˜ω=Zω
ω0
φ′(˜ω)
πd˜ω=Zω
ω0
ρ el(˜ω) d˜ω.
In non-degene a e equency window de i a i e posi i e,
(ω)
locally bijec i e; i ano he
˜
sa ises same condi ion, hen
˜
=a +b
(
a > 0
), gi ing ane uniqueness.

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