F om O de o En opy: How G a i y Shapes Time’s
A ow
A s ep-by-s ep, een- iendly walk h ough o en opy, coa se-g aining, and g a i y
Ma hew J. Hall & GPT-5 Thinking
ORCID: 0009-0001-7066-2558
Da e: Oc obe 8, 2025
(Uni s: c=ℏ=kB= 1, signa u e (−,+,+,+))
Abs ac
En opy is no “messiness”; i is a p ecise coun o possibili ies. In his companion o
F om Eins ein’s Equa ions o Time You Can Feel, we show how mic oscopic dynamics plus
coa se-g aining p oduce he Second Law, why g a i y bends he ules owa d clumping, and
how black holes o ce us o upg ade en opy o include geome y. Each s ep pai s ma h
wi h plain English and a eason i ma e s. The punchline: he a ow o ime comes
om low-en opy ini ial condi ions and g ows unde dynamics, while g a i y
ansla es a ea in o en opy, linking ma e , mo ion, and geome y.
Con en s
1 Symbols a a Glance 1
2 S ep 1: Wha is En opy? S= ln Ω 2
3 S ep 2: In o ma ion Fo m S=−Rρln ρ2
4 S ep 3: Liou ille’s Theo em and Why En opy Needs Coa se-G aining 2
5 S ep 4: The Second Law (Coa se-G ained) 2
6 S ep 5: G a i y Changes he Game (Clumping and Redshi ) 3
7 S ep 6: Black Hole En opy and Tempe a u e 3
8 S ep 7: A ea Inc ease om Raychaudhu i (Classical A ow wi h G a i y) 3
9 S ep 8: The Gene alized Second Law (GSL) 3
10 S ep 9: Cosmology and he A ow 4
11 Resul s: One A ow, Two Engines 4
12 F equen ly Asked (Teen) Ques ions 4
1 Symbols a a Glance
De ini ion
•S: en opy. kB: Bol zmann cons an (se o 1 he e).
•Ω: numbe o mic os a es compa ible wi h mac oscopic da a.
•ρ: p obabili y densi y (classical phase space o quan um s a e).
1
•H[ρ]=−Rρln ρ: Gibbs/Shannon en opy (classical).
•ˆρ: quan um densi y ma ix; S=−T (ˆρln ˆρ) ( on Neumann).
•Γ: phase space; Liou ille low p ese es olume.
•θ, σµν, ωµν: expansion, shea , o ici y o a cong uence.
•A: a ea; SBH =A
4G: Bekens ein–Hawking en opy (wi h c=ℏ=kB= 1).
2 S ep 1: Wha is En opy? S= ln Ω
S= ln Ω.(1)
Ma h Plain English Why his s ep?
S= ln Ω
Coun how many mic oscopic
a angemen s i wha you
know mac oscopically.
Tu ns “diso de ” in o coun ing.
Mo e ways o be = highe S.
3 S ep 2: In o ma ion Fo m S=−Rρln ρ
S[ρ] = −ZΓ
ρ(x) ln ρ(x) dx(classical), S =−T (ˆρln ˆρ) (quan um).(2)
Ma h Plain English Why his s ep?
En opy o a dis ibu ion
Unce ain y abou he exac
mic os a e.
Le s us ack en opy when sys-
ems a e mixed, measu ed, o
en angled.
4
S ep 3: Liou ille’s Theo em and Why En opy Needs Coa se-
G aining
dρ
d ={ρ, H} ⇒ phase-space olume is p ese ed. (3)
Ma h Plain English Why his s ep?
Liou ille low p ese es olume
Mic oscopic e olu ion is e-
e sible.
To ge an a ow, we a e age
(coa se-g ain) o e de ails we
can’ ack.
Takeaway
Second Law needs a iewpoin : Once we blu mic oscopic de ails ( ini e esolu ion,
noise, igno ance), ypical e olu ions mo e p obabili y in o mo e nume ous mac os a es
⇒S
inc eases.
5 S ep 4: The Second Law (Coa se-G ained)
∆Scg ≥0 ( o ypical e olu ions and mac oscopic pa i ions). (4)
2
Ma h Plain English Why his s ep?
So coa se cells g ows
De ails sc amble;
mac os a es wi h mo e
mic os a es domina e.
Explains i e e sibili y in a
wo ld whose undamen al laws
a e e e sible.
6
S ep 5: G a i y Changes he Game (Clumping and Redshi )
In a g a i a ional ield wi h imelike Killing ec o ξµ,
Tp−ξµξµ= cons (Tolman law).(5)
Ma h Plain English Why his s ep?
T edshi s in g a i y
Ho e deepe down so ha
equilib ium holds globally.
Shows how geome y eshapes
he mal balance; se s up black
hole he modynamics.
7 S ep 6: Black Hole En opy and Tempe a u e
SBH =A
4G, TH=κ
2π,(6)
whe e Ais ho izon a ea and κis su ace g a i y.
Ma h Plain English Why his s ep?
S∝a ea, T∝su ace g a i y
Black holes beha e like he -
modynamic objec s.
Geome y ca ies en opy.
G a i y is ied o in o ma ion.
8
S ep 7: A ea Inc ease om Raychaudhu i (Classical A ow
wi h G a i y)
Fo null geodesic cong uences (ligh ays) wi h expansion θand no caus ic,
dθ
dλ=−1
2θ2−σµνσµν −Rµν kµkν.(7)
Ma h Plain English Why his s ep?
Focusing o ligh like lows
Unde ene gy condi ions,
ho izons end o g ow.
Classical “a ea heo em” mi -
o s Second Law:
Aho izon
doesn’ dec ease.
9 S ep 8: The Gene alized Second Law (GSL)
∆(Sou side +SBH)≥0.(8)
Ma h Plain English Why his s ep?
Ma e en opy + ho izon en-
opy doesn’ dec ease
Losing s u in o a black hole
s ill espec s a bigge Second
Law.
Uni ies ma e in o ma ion
wi h geome ic in o ma ion.
3
10 S ep 9: Cosmology and he A ow
Snow ≫Sea ly,(low-en opy ini ial s a e).(9)
Ma h Plain English Why his s ep?
Huge phase space oday s
ea ly uni e se
The Big Bang s a ed in a
e y special (low
S
) con igu-
a ion.
The a ow o ime e lec s
bounda y condi ions + ypical
e olu ion o la ge Ω.
11 Resul s: One A ow, Two Engines
Takeaway
1.
S a is ical engine: Mic oscopic e e sibili y + coa se-g aining
⇒
ypical
S
inc eases
(Second Law).
2.
Geome ic engine: G a i y links a ea and en opy; ho izons g ow and adia e; he
GSL ex ends he modynamics o space ime i sel .
Toge he hey u n low-en opy beginnings in o a pe sis en a ow. G a i y doesn’ b eak
he modynamics; i comple es i .
12 F equen ly Asked (Teen) Ques ions
•“I undamen al laws a e e e sible, why does my co ee cool down?”
Because you don’ ack e e y molecule. Once you a e age o e de ails, he mos likely
di ec ion is owa d mac os a es ha ha e mo e mic os a es.
•“Why do black holes ha e en opy a all?”
Because di e en ma e s a es can lead o he same inal black hole— o ou side obse e s
hey’ e indis inguishable. The a ea measu es hose hidden possibili ies.
•“So wha se s he a ow o ime?”
A e y low-en opy s a ing poin o he uni e se, plus dynamics ha gene ically climb o
highe en opy.
C edi s & License
This handou is pa o Ma hew J. Hall’s educa ional se ies on ime, g a i y, and dynamics. C ea ed
wi h assis ance om GPT-5 Thinking. Licensed CC BY 4.0.
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