Optical Metric from Local Information Volume Conservation and the Entropic Derivation of Einstein Equations

Op ical Me ic om Local In o ma ion Volume
Conse a ion and he En opic De i a ion o Eins ein
Equa ions
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
F om he pe spec i e o Quan um Cellula Au oma a (QCA) and in o ma ion
on ology, space ime geome y should be unde s ood as an eme gen ep esen a ion
o he in o ma ion p ocessing capaci y and connec i i y s uc u e o an unde lying
disc e e quan um ne wo k. In his amewo k, we p opose and o malize he p in-
ciple o "Local In o ma ion Volume Conse a ion": in he coa se-g aining p ocess
om disc e e QCA o a con inuous eec i e mani old, he " o al amoun o dis in-
guishable quan um deg ees o eedom capable o being hos ed pe uni coo dina e
olume and pe uni coo dina e ime" wi hin any local olume elemen mus e-
main in a ian . In he s a ic, iso opic case, his p inciple uniquely selec s a class
o dual- ac o scaled op ical me ics
ds2=−n−2(x)c2d 2+n2(x)δijdxidxj,
whe e
n(x)≥1
can be in e p e ed as an eec i e e ac i e index eld induced by
he local in o ma ion p ocessing load.
In he weak eld limi , aking
n(x) = 1 −Φ(x)/c2+O(Φ2/c4)
, whe e
Φ
is he
New onian po en ial, he esul ing line elemen expands as
ds2=−(1 + 2Φ/c2)c2d 2+ (1 −2Φ/c2)δijdxidxj+O(Φ2/c4),
which co esponds exac ly o he gene al ela i is ic weak eld me ic wi h
γ= 1
in
he Pa ame ized Pos -New onian (PPN) o malism. This yields obse ables such
as ligh deec ion and Shapi o delay, wi h a  s -o de deec ion angle coecien
o
4GM/(bc2)
, he eby esol ing he his o ical dicul y o Eins ein's 1911 scala
g a i y heo y (which yielded only hal he deec ion angle) wi hin a scala -eld-
d i en " e ac i e index g a i y" amewo k.
Fu he mo e, his pape p oposes he In o ma ionG a i y Va ia ional P inciple
(IGVP): ega ding he geome ic ac ion as pa o a gene alized en opy unc ional
and he local in o ma ion p ocessing load as a sou ce e m o en anglemen en opy
densi y, we pe o m a ia ion on he o al "in o ma ion en opy" unc ional in he
sense o local equilib ium. D awing on Jacobson's idea o "Eins ein equa ion as
equa ion o s a e," we p o e ha in gene alized sca e ing egions whe e he op ical
me ic is iable, he equilib ium condi ion o he IGVP is equi alen o he Eins ein
eld equa ions wi h an in o ma ion s ess-ene gy enso .
1
Thus, his pape es ablishes b idges on h ee le els: (1) de i ing an expe i-
men ally es able op ical me ic s a ing om QCA uni a y e olu ion and local
in o ma ion olume conse a ion; (2) aligning pe ec ly wi h he PPN s uc u e o
s anda d Gene al Rela i i y in he weak eld limi ; (3) ein e p e ing he Eins ein
equa ions as a "geome icin o ma ion equilib ium condi ion" wi hin he amewo k
o en opic o ces and in o ma ion geome y, p o iding se e al enginee ing p oposals
es able in nume ical QCA and g a i y-analog op ical expe imen s.
Keywo ds:
Quan um Cellula Au oma on; Op ical Me ic; Local In o ma ion Volume
Conse a ion; G a i a ional Lensing; Shapi o Delay; En opic Fo ce; Eins ein Equa ions;
In o ma ion Geome y

1 In oduc ion & His o ical Con ex
Gene al Rela i i y (GR) akes a ou -dimensional mani old
(M, gµν)
wi h Lo en z signa-
u e as he undamen al s age o g a i a ional heo y, in e p e ing g a i y as he cu a-
u e o space ime geome y a he han a long- ange o ce ac ing on a a backg ound.
Obse a ional es s in he weak eld limi (such as pe ihelion p ecession o Me cu y, ligh
deec ion by he Sun, Shapi o delay, e c.) ha e p o ided highly p ecise suppo o his
geome ic g a i a ional na a i e.
On he o he hand, he in o ma ion on ology pe spec i e ep esen ed by "I om
Qubi " posi s ha he physical wo ld is, a i s deepes le el, cons i u ed by quan um in-
o ma ion and i s p ocessing ules, wi h con inuous space ime, ma e elds, and measu e-
men esul s being eme gen p ope ies o some unde lying disc e e in o ma ion s uc u e.
In his ein, he Quan um Cellula Au oma on (QCA) is iewed as a na u al model uni y-
ing quan um elds and disc e e causal s uc u es: dening ni e-dimensional local Hilbe
spaces on disc e e la ices, ealizing global ime e olu ion h ough ni e-neighbo hood,
spa ially homogeneous uni a y e olu ion ules, and subsequen ly eme ging s anda d el-
a i is ic eld equa ions in he con inuum limi .
As ea ly as 1911, when discussing he inuence o g a i y on ligh , Eins ein p oposed
a scala g a i y model whe e he g a i a ional po en ial
Φ
ac ed only as a scala eld
aec ing he local speed o ligh , de i ing a ligh deec ion angle o
2GM/(bc2)
o ligh
passing he Sun, which is hal o he co ec esul la e gi en by Gene al Rela i i y. This
disc epancy can be aced o he model modi ying only he empo al componen
g00
o
he me ic while igno ing he cu a u e con ibu ion o he spa ial componen s
gij
. The
comple e Gene al Rela i i y, in he s a ic, weak eld limi , gi es a line elemen w i able
as
ds2=−(1 + 2Φ/c2)c2d 2+ (1 −2γΦ/c2)(dx2+dy2+dz2),
compa ible wi h expe imen s only when he PPN pa ame e
γ= 1
.
In li e a u e s udying g a i a ional lensing and wa e p opaga ion, he concep o "op-
ical me ic" has g adually o med: o s a ic space imes, he p ojec ion o null geodesics
can be equi alen o a geodesic p oblem on a h ee-dimensional Riemann mani old wi h
me ic
γij =−g−1
00 gij
, he eby in oducing an eec i e e ac i e index
N
e
(x)
o s udy
ligh deec ion and wa e p opaga ion in weak g a i a ional elds. Howe e , such op ical
me ics a e usually ew i es s a ing om a known
gµν
, a he han cons ain s de i ed
om a mo e undamen al in o ma ion o compu a ional s uc u e.
2
On he o he hand, Jacobson's wo k showed ha i one assumes he en opy o any lo-
cal Rindle ho izon is p opo ional o i s a ea and equi es he Clausius ela ion
δQ =TdS
o hold o all local ho izons, he Eins ein eld equa ions can be in e p e ed as an "equa-
ion o s a e," linking ene gy ux o cu a u e in some he modynamic limi . Ve linde
subsequen ly p oposed ha g a i y could be iewed as an en opic o ce o posi ion-
dependen in o ma ion, de i ing New on's law o g a i y and i s ela i is ic gene aliza-
ion wi hin a holog aphic amewo k. Al hough hese wo ks p o ided inspi a ion o
he "en opic o igin o g a i y," hei o iginal o mula ions usually assumed con inuous
space ime, local he mal equilib ium, and holog aphic sc een s uc u es, wi hou di ec ly
combining wi h he disc e e QCA amewo k and cons ain s on local in o ma ion p o-
cessing capaci y; hey also ace a ious c i icisms and sugges ions o modica ion.
The goal o his pape is o uni y hese h eads om a QCA pe spec i e:
1. S a ing om QCA uni a i y and local in o ma ion p ocessing densi y, p opose
he p inciple o "Local In o ma ion Volume Conse a ion"; 2. P o e ha in he s a ic,
iso opic case, his p inciple uniquely selec s a class o dual- ac o scaled op ical me ics
ds2=−n−2c2d 2+n2δijdxidxj
, whe e
n(x)
is gi en by he local in o ma ion p ocessing
load; 3. P o e ha in he weak eld limi , his me ic is equi alen o he linea ized
me ic o Gene al Rela i i y in PPN o m, hus yielding co ec ligh deec ion and
Shapi o delay; 4. Inspi ed by Jacobson and Ve linde, in e p e he geome ic ac ion as an
in o ma iongeome ic en opy unc ional, p opose he In o ma ionG a i y Va ia ional
P inciple (IGVP), and p o e i s local equilib ium condi ion is equi alen o he s anda d
Eins ein eld equa ions.
In his sense, g a i y is no longe a "con inuous geome ic backg ound added o QCA,"
bu an eec i e op ical geome y ha he QCA ne wo k is o ced o adop o main ain
global uni a i y and in o ma ion olume conse a ion unde non-uni o m local in o ma-
ion p ocessing loads.

2 Model & Assump ions
This sec ion cons uc s he modeling amewo k om disc e e QCA o con inuous op ical
me ics and lis s explici assump ions.
2.1 1. QCA Uni e se and Local In o ma ion P ocessing Densi y
Assume he unde lying uni e se is desc ibed by a QCA objec
U
QCA
= (Λ,H
cell
, U, ω0),
whe e:

Λ⊂Z3
is he se o h ee-dimensional la ice si es wi h la ice spacing
a
;

Each cell ca ies a ni e-dimensional Hilbe space
H
cell
∼
=Cd
;

The global Hilbe space is he quasi-local enso p oduc
H=Nx∈ΛH
cell
;

Time e olu ion is ealized by a uni a y ope a o
U
wi h ni e p opaga ion adius
Ra
:
ρn+1 =UρnU†,
3
whe e
ρn
is he global s a e a he
n
- h disc e e ime s ep;

ω0
is he gi en ini ial s a e o amily o ini ial s a es.
Gi en a cu o scale
L≫a
,
T≫∆
, we can coa se-g ain he QCA, dening con-
inuous coo dina es
xµ= (c , x)
, and in oducing an eec i e eld heo y desc ip ion.
The uni a i y o he QCA ensu es he conse a ion o he o al sys em Hilbe space di-
mension and on Neumann en opy, bu locally, quan um en anglemen and in o ma ion
p ocessing loads can be highly non-uni o m.
Dene he dimensionless local in o ma ion p ocessing densi y eld
ρ
in o
(x)∈[0,1),
ep esen ing he p opo ion o he compu a ional budge consumed by "in insic e o-
lu ion" such as in e nal spins, local en anglemen , and eedback loops, ela i e o he
maximum acuum budge , pe uni coo dina e olume and pe uni coo dina e ime.
Co espondingly, in oduce he local in o ma ion conges ion ac o
n(x) = 1
1−αρ
in o
(x)≥1,
whe e
α∈(0,1)
is a coupling cons an o scale no maliza ion.
n(x) = 1
co esponds o
acuum (no ex a load), and
n(x)>1
co esponds o egions o high in o ma ion load.
In he con inuum limi o he QCA, he maximum g oup eloci y o he ee ligh cone
is gi en by he la ice spacing and ime s ep
c=a/∆ .
When local in o ma ion load inc eases, he budge a ailable o ex e nal signal p opaga-
ion dec eases, mani es ing as a educ ion in eec i e p opaga ion speed, i.e.,
ex
(x)< c
.
2.2 2. Local In o ma ion Volume Conse a ion and Scaling Fac-
o s
Conside a small ou -dimensional coo dina e olume elemen
d4x= d d3x.
In QCA coa se-g aining, his co esponds o he e olu ion o a ce ain ni e clus e o
cells
Ω⊂Λ
o e a ni e numbe o ime s eps. Due o QCA uni a i y, he numbe o dis-
inguishable quan um s a es wi hin
Ω
is cons ained by he local Hilbe space dimension
and en anglemen s uc u e, bu should no change unde coo dina e epa ame iza ion.
To cha ac e ize me ic de o ma ion, in oduce local scaling ac o s o ime and space:

The ela ion be ween physical p ope ime and coo dina e ime is
dτ=η (x) d ;

The ela ion be ween physical leng h elemen and coo dina e leng h elemen is
dℓ=ηx(x)|dx|.
4
Then he physical ou - olume elemen is
dV
phys
4= dτd3ℓ=η (x)η3
x(x) d d3x.
Dene he "in o ma ion capaci y" pe uni coo dina e olume pe uni coo dina e ime
as he maximum loga i hmic numbe o dis inguishable quan um s a es
C(x)
. Unde in-
o ma ion on ology, assume he
C(x)
gi en by he QCA local s uc u e in coa se-g aining
is p opo ional o he physical ou - olume elemen bu supp essed by he local in o ma-
ion load
ρ
in o
(x)
:
C(x)∝η (x)η3
x(x)
n(x).
He e
n(x)
cha ac e izes he comp ession o a ailable e olu ion s eps pe uni physical
ime. To ensu e local in o ma ion capaci y emains in a ian unde pu e coo dina e
epa ame iza ion, we in oduce:
Axiom 2.1
(Local In o ma ion Volume Conse a ion)
.
In he mapping om QCA o
con inuous mani old, o any sucien ly small mani old segmen , he local in o ma ion
capaci y sa ises
η (x)η3
x(x)∝n(x),
and degene a es o Minkowski alues
η =ηx=n= 1
in he pe u ba i e weak eld
limi .
In he s a ic, iso opic, and weak eld cases ocused on in his pape , ac ual obse a-
ional cons ain s mainly come om ligh deec ion and ime delay, which a e sensi i e
p ima ily o he me ic s uc u e in he

subspace. To his end, we u he adop he
ollowing simpli ying assump ions.
Hypo hesis 2.2
(Iso opy Simplica ion)
.
A he conside ed scale, he local in o ma ion
load is spa ially iso opic, i.e.,
ηx(x) = ηs(x),
whe e
ηs
scales all h ee spa ial di ec ions uni o mly, and
n(x)
depends only on he
po en ial unc ion
Φ(x)
.
Hypo hesis 2.3
(2D Ligh Cone Con o mali y)
.
Requi e he a ea elemen o any adial
ligh cone in he
( , )
wo-dimensional subspace o emain con o mally in a ian , he eby
p ese ing he "causal budge " conse a ion o null geodesics in he QCA. This is equi -
alen o equi ing he de e minan o he 2D sub-me ic o be aec ed only by an o e all
scale ac o unde coo dina e ans o ma ion.
In he case o a s a ic, sphe ically symme ic me ic
ds2=−A( )c2d 2+B( )(d 2+ 2dΩ2),
he de e minan o he me ic in he
( , )
subspace is
de g( , )=−A( )B( ).
2D Ligh Cone Con o mali y combined wi h Local In o ma ion Volume Conse a ion
yields:
P oposi ion 2.4.
Unde he abo e se ings,
A( )B( ) = 1
.
5

In ui i ely, his is because: i ime is s e ched by a ac o
η
due o in o ma ion load,
hen o keep he ligh cone a ea and he "disc e e s ep shell" o null geodesics in he
QCA in a ian , he adial di ec ion mus scale by
η =η−1
, hus
A∝η2
,
B∝η2
gi ing
AB =η2
η2
= 1
. This conclusion will be de i ed mo e sys ema ically ia Liou ille- ype
a gumen s in Appendix A.
Thus we can choose he pa ame e iza ion
A( ) = n−2( ), B( ) = n2( ),
ob aining a amily o op ical me ics
ds2=−n−2( )c2d 2+n2( )d 2+ 2dΩ2.
This is wha we e m he "In o ma ionOp ical Me ic."
2.3 3. Ma ching In o ma ion Re ac i e Index wi h New onian
Po en ial
To eco e he New onian g a i a ional po en ial a mac oscopic scales, conside he
geodesic equa ion o a slow es pa icle in his me ic. Le he gene al o m o he me ic
be
ds2=−(1 + 2Φ/c2)c2d 2+ (1 −2γΦ/c2)δijdxidxj,
whe e
γ= 1
in PPN o m co esponds o Gene al Rela i i y. In he weak eld, slow
speed limi , he dispe sion ela ion de e mined by he ime componen gi es he eec i e
po en ial
Φ
, and pa icle accele a ion sa ises
d2x/d 2≈ −∇Φ.
Expand he In o ma ionOp ical Me ic o  s o de :
n(x) = 1 + ϵ(x),|ϵ| ≪ 1,
hen
n−2= (1 + ϵ)−2≈1−2ϵ, n2≈1+2ϵ.
Thus he line elemen becomes
ds2≈ −(1 −2ϵ)c2d 2+ (1 + 2ϵ)δijdxidxj.
Compa ing wi h he PPN o m, aking
ϵ(x) = −Φ(x)/c2
yields
g00 =−(1 + 2Φ/c2), gij = (1 −2Φ/c2)δij,
which is he weak eld me ic o Gene al Rela i i y wi h
γ= 1
. We can hus dene:
Deni ion 2.5
(In o ma ion Re ac i e Index)
.
Le he New onian po en ial
Φ(x)
sa is y
∇2Φ = 4πGρ
. The in o ma ion conges ion ac o is dened as
n(x)=1−Φ(x)/c2.
In g a i a ional po en ial wells whe e
Φ<0
, we ha e
n(x)>1
, co esponding o egions
wi h highe local in o ma ion load and educed eec i e ligh speed.

6
3 Main Resul s (Theo ems and Alignmen s)
Unde he abo e model and assump ions, he main esul s o his pape can be summa-
ized in he ollowing heo ems and co olla ies.
Theo em 3.1
(Local In o ma ion Volume Conse a ion
⇒
Op ical Me ic)
.
In a s a ic,
sphe ically symme ic, weak eld egion ob ainable by QCA coa se-g aining, i he ollow-
ing a e sa ised:
1. The unde lying e olu ion is a local uni a y QCA;
2. The Local In o ma ion Volume Conse a ion axiom holds;
3. Iso opy Simplica ion and 2D Ligh Cone Con o mali y hypo heses hold;
hen he me ic allowed in he con inuum limi mus belong o he In o ma ionOp ical
Me ic amily
ds2=−n−2( )c2d 2+n2( )(d 2+ 2dΩ2),
whe e
n( )≥1
is a scala eld de e mined by he local in o ma ion p ocessing densi y.
Fu he mo e, in he weak eld limi , equi ing he exis ence o a New onian po en ial
Φ( )
such ha slow pa icle mo ion sa ises
d2x/d 2≈ −∇Φ
uniquely de e mines
n( ) = 1 −Φ( )/c2+O(Φ2/c4).
Theo em 3.2
(PPN Alignmen o In o ma ionOp ical Me ic)
.
Unde he se ing o
Theo em 1, he line elemen expands in he limi
|Φ|/c2≪1
as
ds2=−(1 + 2Φ/c2)c2d 2+ (1 −2Φ/c2)(d 2+ 2dΩ2) + O(Φ2/c4).
Thus, in PPN no a ion,
γ= 1,
au oma ically yielding:
1. The weak eld deec ion angle o ligh passing a poin mass
M
wi h impac pa-
ame e
b
:
∆θ=4GM
bc2+OG2M2
b2c4;
2. Shapi o delay:
∆
Shapi o
∝(1 + γ)GM
c3ln 4 E R
b2,
whe e
1 + γ= 2
, ully consis en wi h s anda d Gene al Rela i i y.
Thus, wi hin he amewo k o a scala e ac i e index eld, conside ing he coo dina ed
scaling o empo al and spa ial componen s sa is ying
A( )B( )=1
na u ally a oids he
his o ical dicul y o adi ional scala g a i y heo ies yielding only hal he deec ion
angle.
Theo em 3.3
(In o ma ionG a i y Va ia ional P inciple and Eins ein Equa ions)
.
Le
he o al "In o ma ion En opy" unc ional be dened as
S
o
[g, Ψ] = S
geom
[g] + S
in o
[g, Ψ],
whe e:
7

Ψ
ep esen s ma e and in o ma ion deg ees o eedom;

The geome ic pa akes he o m o JacobsonIye Wald ype geome ic en opy,
i.e., unde app op ia e no maliza ion
δS
geom
[g] = 1
4GZM
√−g(Rµν −1
2Rgµν )δgµν d4x;

The a ia ion o he in o ma ion pa denes he in o ma ion s ess-ene gy enso
T
in o
µν =−2
√−g
δS
in o
δgµν .
I we equi e local equilib ium o exis on Rindle small neighbo hoods seen by all local
obse e s, such ha o any compac ly suppo ed me ic a ia ion
δS
o
= 0,
hen his condi ion is equi alen o
Rµν −1
2Rgµν = 8πG T
in o
µν .
In he mac oscopic limi , iden i ying
T
in o
µν
wi h he s anda d s ess-ene gy enso
Tµν
eco e s he usual Eins ein eld equa ions.

4 P oo s
This sec ion p o ides p oo ou lines o he main heo ems abo e; de ailed calcula ions
a e placed in he Appendices.
4.1 P oo o Theo em 1 (Ou line)
1. **2D Ligh Cone and Liou ille In a iance** In he geome ic op ics limi o he QCA,
conside adial ligh ays in he
( , )
subspace. Each disc e e ligh ay can be iewed as
a Hamil onian ow on he phase space
( , ;p , p )
, whe e Liou ille's heo em gua an ees
he in a iance o he phase space olume elemen
d d dp dp
. Coa se-g aining o a
con inuous me ic, equi ing he a ea elemen o he null geodesic shell in
( , )
subspace
p−de g( , )d d
o be p opo ional o he co esponding QCA s ep shell implies ha
de g( , )
scales only by an o e all ac o . In he s a ic, sphe ically symme ic me ic
ds2=−A( )c2d 2+B( )d 2+. . . ,
we ha e
de g( , )=−A( )B( )
. Ma ching he QCA s ep shell wi h he con inuous
ligh cone a ea elemen p o es
A( )B( ) =
cons . Requi ing he me ic o degene a e
o Minkowski (
A→1, B →1
) a inni y
→ ∞
, he cons an mus be 1, yielding
A( )B( ) = 1
.
2. **Iso opy and Scaled Pa ame e iza ion** Unde he assump ion o 3D spa ial
iso opy, assigning
B( )
uni o mly o adial and angula pa s yields
ds2=−A( )c2d 2+B( )(d 2+ 2dΩ2).
8
F om
A( )B( )=1
, he wo unc ions can be exp essed by a single scala
n( )
:
A( ) = n−2( ), B( ) = n2( ).
3. **Uniqueness o Ma ching wi h New onian Po en ial** Expand he line elemen
o  s o de :
n( ) = 1 + ϵ( )⇒A( )≈1−2ϵ( ), B( )≈1+2ϵ( ).
The geodesic equa ion in he slow limi can be w i en as
d2x/d 2=−c2
2∇g00 +O( 2/c2).
Subs i u ing
g00 =−A( )≈ −(1 −2ϵ)
, we ge
d2x/d 2≈ −∇(c2ϵ).
To ma ch New on's law
d2x/d 2=−∇Φ
, he unique possibili y is
c2ϵ( ) = Φ( ),
i.e.,
ϵ( ) = Φ( )/c2
. By he con en ion
Φ<0
, o keep
n≥1
, edene
ϵ( ) = −Φ( )/c2
,
de e mining
n( )=1−Φ( )/c2.
This comple es he ou line o Theo em 1. Full phase space de i a ion is in Appendix
A.
4.2 P oo o Theo em 2 (Ou line)
1. **PPN Fo m Expansion** Subs i u ing
n( ) = 1 −Φ( )/c2
in o he In o ma ionOp ical Me ic and expanding o  s o de gi es
g00 =−n−2≈ −(1 + 2Φ/c2), gij =n2δij ≈(1 −2Φ/c2)δij.
Compa ing wi h s anda d PPN no a ion, we ead
γ= 1
.
2. **Op ical Me ic and Eec i e Re ac i e Index** Fo any s a ic space ime, he
3D op ical me ic is dened as
γij =−g−1
00 gij.
Null geodesic p ojec ions on 4D space ime a e equi alen o geodesics on his op ical 3D
mani old. In he In o ma ionOp ical Me ic:
γij =n4( )δij.
Thus he eec i e e ac i e index o ligh in 3D space is
N
e
( ) = n2( ) = 1−Φ( )/c22≈1−2Φ( )/c2.
9
[2] T. Jacobson, "The modynamics o Space ime: The Eins ein Equa ion o S a e,"
Physical Re iew Le e s 75, 12601263 (1995).
[3] C. M. Will, "The Con on a ion be ween Gene al Rela i i y and Expe imen ," Li ing
Re iews in Rela i i y 17, 4 (2014).
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a ing p inciples o gene al ela i i y," Ame ican Jou nal o Physics 71, 770773
(2003).
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G a i a ional Lensing," Classical and Quan um G a i y 25, 235009 (2008).
[6] E. P. Ve linde, "On he O igin o G a i y and he Laws o New on," Jou nal o High
Ene gy Physics 2011(4), 029 (2011).
[7] S. Gao, "Is G a i y an En opic Fo ce?" En opy 13, 936948 (2011).
[8] T. Fabe and M. Visse , "Combining o a ion cu es and g a i a ional lensing: How
o measu e he equa ion o s a e o da k ma e in he galac ic halo," Mon hly No ices
o he Royal As onomical Socie y 372, 136150 (2006).
[9] I. Kle ogiannis, "Eme gen space ime om pu ely andom s uc u es," Physical
Re iew Resea ch 4, 033004 (2022).
[10] A. Kobakhidze, "G a i y is no an En opic Fo ce," Physical Re iew D 83, 021502
(2011).
[11] R. M. Wald, "Black Hole En opy is Noe he Cha ge," Physical Re iew D 48,
R3427R3431 (1993).
[12] G. ' Hoo ,
The Cellula Au oma on In e p e a ion o Quan um Mechanics
,
Sp inge , 2016.
[13] P. A ighi, S. Facchini, and M. Fo e s, "Disc e e Lo en z co a iance o quan um
walks and quan um cellula au oma a," New Jou nal o Physics 16, 093007 (2014).
[14] T. A. B un, H. A. Ca e e , and A. Ambainis, "Quan um walks d i en by many
coins," Physical Re iew A 67, 052317 (2003).
[15] L. Fidkowski e al., "A Quan um Cellula Au oma on F amewo k o Symme y
P o ec ed Topological Phases," 2025.
[16] W. Ja ed e al., "Weak g a i a ional lensing in da k ma e and plasma mediums o
wo mhole-like s a ic ae he solu ion," Eu opean Physical Jou nal C 82, 1044 (2022).
[17] R. K. Solanki, "Ko le Space ime in Iso opic S a ic Coo dina es," a Xi :2103.10002
(2021).
[18] J. No on, "The his o ical ounda ion o Eins ein's gene al heo y o ela i i y," PhD
Thesis, Uni e si y o Pi sbu gh (1981).
[19] A. Ö gün, "Deec ion angle o pho on h ough da k ma e by black holes in Ein-
s einMaxwelldila on g a i y," Ad ances in High Ene gy Physics 2019, 19 (2019).
16

A Liou ille-Type De i a ion o Local In o ma ion Vol-
ume Conse a ion and
A( )B( )=1
Conside he es ic ion o a s a ic, sphe ically symme ic me ic o he
( , )
subspace:
ds2
( , )=−A( )c2d 2+B( )d 2.
Le he conjuga e momen a be
p =g ˙
=−A( )c2˙
, p =g ˙ =B( ) ˙ ,
whe e he do deno es die en ia ion wi h espec o an ane pa ame e
λ
. The Hamil-
onian cons ain o null geodesics sa ises
H=1
2gµνpµpν= 0,
which on he
( , )
subspace is
H=−1
2A−1( )p2
/c2+1
2B−1( )p2
= 0.
Dene he phase space olume elemen
dΓ = d d dp dp .
Hamil on's equa ions p ese e
dΓ
, which is he con en o Liou ille's heo em. Fo he
null geodesic shell
H= 0
, he olume elemen can be w i en as
dΓ|H=0 = d d dp dϕ,
whe e
ϕ
is an angula a iable pa ame e izing he shell. Since QCA uni a i y equi es
ha he "s ep densi y" co esponding o he null geodesic shell in he coa se-g aining map
emains in a ian , he scaling ac o o
p−de g( , )d d
ela i e o
d d
mus emain
cons an .
Specically,
q−de g( , )=pA( )B( ).
I
A( )B( )
we e allowed o a y wi h
, pa ial a ia ion could be abso bed by coo dina e
edeni ion
→ ′( , ), → ′( )
, bu he "s ep shell densi y" on he null geodesic shell
would no longe depend solely on local in o ma ion load, bu would mix in pu e coo dina e
deg ees o eedom. To ensu e "in o ma ion olume" eec s only physical load in he
QCA and no coo dina e choice, we equi e
A( )B( ) =
cons .
A inni y
→ ∞
, equi ing he me ic o app oach Minkowski implies
A(∞) =
B(∞) = 1
, so he cons an mus be 1, yielding
A( )B( ) = 1.
Combined wi h he iso opy assump ion, his can be w i en as
A( ) = n−2( ), B( ) = n2( ).
17
B Calcula ion o Ligh Deec ion unde In o ma ion
Op ical Me ic
Conside ligh p opaga ion in a plane, choosing he plane
θ=π/2
, wi h line elemen
ds2=−n−2( )c2d 2+n2( )d 2+ 2dφ2.
Null geodesics sa is y
ds2= 0
, and Killing symme ies gi e conse ed quan i ies
E=−g ˙
=n−2( )c2˙
, L =gφφ ˙φ=n2( ) 2˙φ.
He e
E
and
L
a e "ene gy" and "angula momen um" espec i ely. The null geodesic
condi ion gi es
0 = −n−2c2˙
2+n2˙ 2+n2 2˙φ2.
Elimina ing
˙
, ˙φ
yields he adial equa ion
˙ 2+L2
n4 2=E2
c2n4.
Dene impac pa ame e
b=Lc/E
, and le
u(φ)=1/ (φ)
, ans o ming he adial
equa ion o
du
dφ2
+u2=1
b2n−4( (u)).
In he weak eld limi , aking
n( )=1−Φ( )/c2,Φ( ) = −GM/ ,
hen
n−4( )≈1−4Φ( )/c2= 1 + 4GM/( c2).
This gi es he pe u ba ion equa ion
d2u
dφ2+u=2GM
c2b2.
The solu ion is
u(φ) = sin φ
b+GM
c2b2(1 + cos φ) + O(G2M2/(b3c4)).
When
φ
a ies om
−π/2−δ
o
π/2 + δ
, ligh a els om inni y o inni y, equi ing
u(φ)→0
. Sol ing o he deec ion angle gi es
∆φ= 2δ=4GM
bc2.
This ma ches he s anda d Gene al Rela i i y calcula ion exac ly.
Ano he mo e concise de i a ion uses he pa axial app oxima ion and in eg al o m
unde he op ical me ic
∆θ≈Z∞
−∞ ∇⊥N
e
( ) dl,
He e
N
e
=n2≈1+2GM/( c2)
, in eg a ing along he unpe u bed line
=√b2+z2
also yields
4GM/(bc2)
.
18
C Va ia ional Calcula ion o In o ma ionG a i y Va i-
a ional P inciple
Le he geome ic en opy unc ional be
S
geom
[g] = 1
4GZH
d2Σκ,
whe e
H
is he local ho izon c oss-sec ion and
κ
is he su ace g a i y. Jacobson and sub-
sequen wo k showed ha he co esponding olume unc ional a ia ion can be w i en
as
δS
geom
=1
4GZM
(Rµν −1
2Rgµν )δgµν√−gd4x.
The in o ma ion en opy unc ional akes he o m
S
in o
[g, Ψ] = ZM
s
in o
(g, Ψ)√−gd4x,
whe e
s
in o
is he local in o ma ion en opy densi y, depending on he me ic and ma e 
in o ma ion deg ees o eedom in he QCA. I s a ia ion is
δS
in o
=Z∂s
in o
∂gµν δgµν +∂s
in o
∂ΨδΨ√−gd4x+Zs
in o
δ√−gd4x.
Using
δ√−g=−1
2√−ggµν δgµν ,
his can be ea anged as
δS
in o
=−1
2ZT
in o
µν δgµν√−gd4x+Z∂s
in o
∂ΨδΨ√−gd4x,
whe e we dene
T
in o
µν =−2∂s
in o
∂gµν +s
in o
gµν.
The o al en opy a ia ion is
δS
o
=1
4GZ(Rµν −1
2Rgµν )δgµν√−gd4x−1
2ZT
in o
µν δgµν√−gd4x+. . . .
Requi ing
δS
o
= 0
o any compac ly suppo ed me ic a ia ion
δgµν
and assuming
ma e elds sa is y hei espec i e Eule Lag ange equa ions, we mus ha e
Rµν −1
2Rgµν = 8πG T
in o
µν .
In he mac oscopic limi , iden i ying
T
in o
µν
wi h he s anda d s ess-ene gy enso
Tµν
eco e s he usual Eins ein eld equa ions. This de i a ion demons a es he equi a-
lence be ween IGVP and s anda d ac ion o ms, while endowing geome ic en opy and
in o ma ion en opy wi h explici in o ma iongeome ic meaning.
19