Spacetime Dimensionality from De Sitter Entropy

Home , Causal patch

arXiv:1106.4548v3 [hep-th] 13 Feb 2018 iesos u eieo neetfrtepossible the for interest of spatial regime three Our have cosmological to small favored dimensions. a entropically with is asymp- an universe constant that note Sitter proceed), this we de assump- as In totically out justifiable spelled fairly be energy. certain (to tions under of argue, amount will given we a principle: maximum for have entropic must entropy an configuration phase by spacetime ultimate final understood this the whether (partly) ask be may can One con- cosmological [10]. small stant a with Sitter by de or asymptotically [8] dynamics gas [9]. principle brane entropic of some be context invoking can the spacetime in of made could dimensionality attempts dimensions the Other understand spatial [7]. size to less) macroscopic a (or into grown light three have shed only also may how early cosmology our on gas to String similar of [6]. creation one Universe quantum prefers the universes that inflationary argued String be in context can braneworld it the can Theory, In creation 5]. [4, of cosmological quantified probabilities be competing relative various in- whose which among scenarios, in one frameworks is [3]. particular flation Sitter which present de during may approximately inflation, One was of geometry epoch an spacetime with began verse the understand to dimensionality. way the spacetime only observed whether the wonder is may Principle One Anthropic with [2]. Theory, universes dimensions String of various by multitude humongous sharpened a is allows issue which an- This than large other [1]. three explanations thropic has defies Universe space of observable dimensions the that fact The Introduction 1 nteohrhn,telt hs forUies is Universe our of phase late the hand, other the On Uni- the that holds cosmology of view standard The pr Keywords: holographic de radiation. the of Hawking/Unruh on of description horizon, constituents stretched causal-patch the a c the to cosmological up on small equilibrium a relies with conclusion universes Sitter The de that argue We b a-lnkIsiu ¨rGaiainpsi (Albert-E für Gravitationsphsyik Max-Planck-Institut ) PCTM IESOAIYFO ESTE ENTROPY SITTER DE FROM DIMENSIONALITY SPACETIME eSte pc;etoy pctm dimensions spacetime entropy; space; Sitter de a eateto hoeia hsc,Uiest fDaa D Dhaka, of University Physics, Theoretical of Department ) -al mmnuidaaeu [email protected]. [email protected], E-mail: rhdMomen Arshad a nti-nttt,A ¨hebr ,146Potsdam-Golm, 14476 Mühlenberg 1, Am instein-Institut), n aiu Rahman Rakibur and nil,ado oeasmtosaottentr fgravity of nature the about assumptions some on and inciple, 1 ntn r nrpclyfvrdt aetresaildime spatial three have to favored entropically are onstant itrsae hr dca bevr xeinelclthe local experience observers fiducial where space, Sitter n( In lnklength, Planck fplrcodntso a on coordinates polar of geodesically topol- with a curvature to ogy constant metric of spacetime the complete extend analytically can h bevri h einwihi nfl aslcon- causal 0 full in namely is her, which with region tact the is observer the oio tadistance a at horizon l bevrcnacs h nied itrsae an space: Sitter de entire the at observer access can observer gle clcntn s = Λ as: constant ical parameter ble iglrt at singularity ubro pta dimensions, spatial of number o h aeo clarity. of sake the for M where n w uhosreswl eogt ieetcausal different to belong will observers such two and that in observer-dependent is ewl s h aua units: natural the Thereof use Entropy will & We Space Sitter de 2 Su- yields approximation. gravity low-energy some quantum as of that pergravity and theory de- infrared, is underlying the gravity the in that Relativity General assumes by one scribed if follows restriction a ielk edsccnb hsnt eat be to chosen be can geodesic time-like a erci the in metric ds P e swiedw h ( the down write us Let 2 d S ≡ = )saeiedmnin,ti est nt the unity to sets this dimensions, spacetime 1) + d d ~ × − Ω c − d 2 (1 R − 1 l 1 1 r − P − where , steln lmn on element line the is 1 xeine h rsneo nevent an of presence the experiences 0 = static H r hc oee a ecridaround carried be may however which , H l P 2 1 = r aa10,Bangladesh 1000, haka srltdt h pstv)cosmolog- (positive) the to related is ≡ 2 ) coordinates: /H r dt b d r − ersnsatpdlorigins antipodal represents 0 = 2 √ 1 1 = sacodnt riat One artifact. coordinate a is + 1 2 G de d ≤ ~ d ( d /H 1 c shr.Hwvr osin- no However, -sphere. d 1-iesoa eSitter de +1)-dimensional − − r − 3 h cua ac”of patch” “causal The . dr d n h lnkmass, Planck the and , any H ≤ c s2 is , 1) 2 = 2 H r 1 S 2 /H bevrfollowing observer ~ d 2 − h apparent The . + ≤ = 1 h horizon The . n h Hub- the and , r d k 2 B ≤ d Ω = 0 Such 10. d 2 Germany − r G 1 nsions. , 0, = 1. = rmal and (1) patches. While the isometry group for de Sitter space A global notion of temperature is not meaningful in is SO(d +1, 1), the manifest symmetries of the causal curved spacetime [14]. Instead, one may need to intro- patch are SO(d) rotations plus translation in t. The duce operationally meaningful local concepts of tem- remaining d compact and d non-compact generators perature and thermal equilibrium [19]. It was shown in displace an observer from one causal patch to another. Ref. [13] that the unique invariant locally Minkowskian In what follows we will restrict all attention to a state of quantum fields in de Sitter space has exactly single causal patch, àla [4, 11]. As regions that are the temperature given by (2). We will therefore con- out of causal contact with a particular observer have sider only local thermal equilibrium with temperature no operational meaning to her, the observer should con- T (r) of the physical degrees of freedom (DOF) acces- sider the physics inside her horizon as complete, with- sible to a FIDO at a radial position r. What DOFs out making reference to any other region. Without loss does the thermal radiation contain, i.e., what are the of generality, this we can choose to be the “southern” constituents of Hawking/Unruh radiation? Postponing causal patch, where the Killing vector ∂t is time-like justification until later, let us assume that the accessi- and future-directed, so that time evolution is well de- ble number of DOFs, D, does not depend on r. fined. We imagine that the causal patch is filled with The PI can ask a FIDO at radial position r to mea- “fiducial observers” (FIDOs), each of whom is at rest sure the local entropy density, and receive the result relative to the static coordinate system, i.e. each is d +1 located at a fixed r and fixed values of the angular σ(r)= D a(d)[T (r)]d, (4) d variables. The only geodesic observer is the FIDO at r = 0, whom we call the “principal investigator” (PI). where a(d) the radiation constant per DOF in d spa- The PI can send a request to any other FIDO to per- tial dimensions− is given by form certain local measurements and report the results, − d/2 which the PI will eventually receive after waiting for a ω(d) 2π a(d)= d ζ(d + 1)Γ(d + 1), ω(d) d , (5) finite amount of time. (2π) ≡ Γ 2 The PI at r = 0 detects a thermal radiation with ω(d) being the surface area of the boundary of a unit a temperature TGH = H/2π the Gibbons-Hawking d-ball. The PI can take the volume integral of (4) to temperature of de Sitter space− [12]. More generally, compute the total thermal entropy of the causal patch: a FIDO at a radial position r, whose Killing orbit has a proper acceleration a = H2r/√1 H2r2, detects a rc dr rd−1 − S = ω(d) σ(r). (6) local 2 2 thermal bath with an effective temperature [13]: 0 √1 H r Z − 1 H T (r)= H2 + a2 = , (2) This integral can be expressed in terms of hyperge- 2π 2π√1 H2r2 ometric functions by the variable redefinition: x p − H2r2, and the use of the integral representation [20]:≡ which is just the Gibbons-Hawking temperature multi- z α−1 α plied by a Tolman factor [14]. Using an Unruh-like de- dx x z β = (1 z) 2F1(α + β, 1; α + 1; z), tector [15], the FIDO can indeed discover a thermal ra- 1−β 0 (1 x) α − diation with the temperature T (r). This effect is real, Z − (7) and can also be understood as pure Unruh effect as- which holds for R(α) > 0. Thereby the PI finds that sociated with Rindler motion in the global embedding the total entropy amounts to Minkowski space [16]. The local temperature, however, d d−1 blows up at the horizon r =1/H. A way to regularize d +1 √1 ǫ 1 S = D ω(d) a(d) − this divergence is to consider a “stretched horizon” [17], d2 2π √ǫ that extends from the mathematical horizon (by some 1 d+2 2F1 2 , 1; 2 ; 1 ǫ , (8) Planck length) up to some rc < 1/H. The thickness × − may well be a physical reality, originating possibly from where ǫ is a positive number defined as quantum fluctuations [18]. The temperature measured 2 2 2 H at the stretched horizon is then large but finite: ǫ 1 H rc = . (9) ≡ − 2πTc H Tc T (r = rc)= < , (3) 2 2 Note that ǫ 1, because we consider H/2π to be much ≡ 2π 1 H rc ∞ ≪ − smaller than Tc MP, which is necessary for a semi- which sets a cutoff value for thep temperature. It is nat- classical treatment∼ to be valid. The total entropy (8), ural to identify the cutoff with the Planck scale. In our which clearly diverges in the limit ǫ 0, is rendered → case it will indeed turn out that Tc MP. large but finite by the stretched horizon. ∼

2 Now, there is an entropy associated with de Sitter In the limit ǫ 0 or A , thanks to Eq. (11), S/A → 1 → ∞ horizon, known as the Gibbons-Hawking entropy [12], reaches the value 4 for any space dimensionality. 1 which is 4 of the horizon area, A, in Planck units: Similarly, the PI can define the total “energy” of the causal patch as follows. d−1 1 A 1 MP rc −1 SGH = d−1 = ω(d) . (10) dr rd 4 lP 4 H E = ω(d) ρ(r), (16) 2 2 0 √1 H r To see its possible connection with the total entropy (8) Z − of the causal patch, we note that the hypergeometric where ρ(r) = D a(d)[T (r)]d+1 is the local energy den- function appearing in the latter can be written as [20]: sity that a FIDO at a radial position r reports to the PI. Again, using the integral representation (7) one finds 1 d+2 d 2F1 2 , 1; 2 ; 1 ǫ [1+ δ(d, ǫ)] , (11) d − ≡ d 1 H √1 ǫ 1 d − E = D ω(d) a(d) − . (17) 2πd 2π √ǫ where the function δ is such that it vanishes in the limit ǫ 0. Thanks to Eq. (11) and some properties of the gamma→ function, one can rewrite the entropy (8) as Expressing √ǫ in terms of the cutoff temperature Tc through the relation (9), one arrives at a rather d+3 d−1 D Γ ζ(d + 1) Tc counter-intuitive conclusion: the “energy” scales like S = ω(d) 2 + ... , (12) d−1 (d 1)(√π )d+3 H (1/H) A. This is due to the extra factor of H " − # appearing∼ in Eq. (17). More explicitly, where the ellipses stand for subleading terms. Their A 2 H-dependencies differ from that of the leading term, E = D a(d)(1 ǫ)d/ T d. (18) 2πd − c which mimics the area law (10) for de Sitter entropy. Now we invoke the holographic principle, which en- Finally, using the expression (13) for Tc, one finds after tails that the leading term in (12) should be identified some simplifications that the total “energy” is given by with the Gibbons-Hawking entropy (10) of the de Sit- ter horizon [21]1. This relates the cutoff temperature 1 d+3 d−1 Tc and the Planck mass MP in the following way: A d 1 (d 1)(√π ) E = − (1 ǫ)d/2 − , 4 d +1 − 4 D Γ d+3 ζ(d + 1) 1 " 2 # d+3 d−1 Tc (d 1)(√π ) (19) = − . (13) D d+3 That the total “energy” (19) follows an area law just MP "4 Γ 2 ζ(d + 1)# like the total entropy was also noticed in Ref. [25] for Similar relations show up in the brick wall model, pro- a box of ideal gas kept near a de Sitter horizon. It pounded in [22], and subsequently used for dS3 in [23]. is therefore natural to consider the quantity E as an Note that the cutoff Tc is independent of the Hubble attribute of the de Sitter horizon. parameter H, as expected, but depends on the num- Our total “energy” E is unique and well defined ber of DOFs in a way that is in complete accordance in the following sense. As soon as the de Sitter en- with the results of [24]. With some reasonable assump- tropy (14) is taken to be finite, we must forgo the sym- tions on D, the right-hand side of Eq. (13) is (1). metry of different causal patches [11]. In the given O This sets Tc MP. Then, it is easy to see that the causal patch, all the FIDOs are on equal footing in ∼ thickness of the stretched horizon is (lP). With the that they all follow time-like trajectories, are in causal identification (13), one can now makeO explicit the area contact with one another, and of course experience the dependence of the entropy: same causal horizon. Any quantity to be attributed to the entire causal patch or to the horizon itself must A d 1 d/2 1 d+2 S = − (1 ǫ) 2F1 , 1; ; 1 ǫ , not depend on which FIDO, be her the PI or not, is 4 d − 2 2 − assigned the job of defining it. In other words, all the (14) FIDOs must agree upon the value of any such quan- where the parameter ǫ depends on A as follows tity. Now that any FIDO can learn about the results 2 d−1 of local measurements performed by any other FIDO, 1 8 D Γ d+3 ζ(d + 1) ǫ = 2 . (15) they all will have identical sets of data for the density 4π2 A (d 1) π3/2 Γ d distributions σ(r) and ρ(r), and therefore will agree " − 2 # 1 Defining the entropy of the causal patch as (6) and identifying it with the Gibbons-Hawking entropy (10) has also been suggested by N. Kaloper in a talk titled “Inflation and leaky cans”. We thank L. Sorbo for pointing this out.

3 upon their respective volume integrals S and E. This (when they are mutually charged), or there is no way is not the case if in the definition (16) one inserts a to distinguish them as different constituent particles redshift factor (as was suggested in Ref. [25]), which of the radiation. At any rate, that photons and gravi- itself depends on the position of the FIDO assigned to tons could be the sole constituents of the radiation may define the quantity. not seem so surprising given that these are the natural Eqs. (14) and (19) can be viewed as relations among DOFs to consider at low energy. three extensive properties of the horizon, namely S, E Thus one can assume, quite justifiably, that for a and A. Dividing Eq. (14) by (19), one can also write small cosmological constant the Hawking/Unruh radi- 1 ation will be a gas of photons and gravitons, whose − S d +1 4 D Γ d+3 ζ(d + 1) d 1 interactions are negligible. In (d + 1) spacetime di- = 2 d+3 mensions photon has (d 1) DOFs, while graviton has E d " (d 1)(√π ) # − 1 (d+1)(d 2). The total− number of DOFs is therefore 1 d+2 2 − 2F1 2 , 1; 2 ; 1 ǫ . (20) × − D D(d) = (d 1)+ 1 (d + 1)(d 2). (22) It is clear, in view of Eq. (11), that in the limit ǫ 0, ≡ − 2 − → the ratio S/E is finite, although both S and E diverge. Let us consider universes that evolve from some ini- For ǫ 1, Eq. (20) can be approximated as tial state into a final state of a de Sitter space with ≪ 1 a small cosmological constant, like our Universe [10]. D d+3 d−1 d +1 4 Γ 2 ζ(d + 1) E The asymptotically de Sitter universes are assumed to S +3 . ≈ d 1 (d 1)(√π )d MP have different dimensionalities, but the same values − " − # (21) of the fundamental constants, which all can be set to The virtue of Eq. (21) is that it allows one to compare unity (we do not consider scenarios in which the Planck the entropies of de Sitter spaces with different d, for a mass MP may depend on spacetime dimensionality). given E in Planckian units. It does not make sense to Now for any given value of the characteristic quantity compare the horizon areas of two spaces with different E, one can use Eq. (21) to formally consider S as a dimensionalities, and indeed A does not appear in (21). function of the number of space dimensions d. Is there any particular value of d that is favored entropically? 3 Is 4D Spacetime Entropically Favored? In the regime 2 d 10, when d is treated as a continuous variable,≤ its≤ function S/E has an abso- lute maximum at d 2.97. An upper bound on d Let us first note that the Hawking/Unruh radiation is ≈ (approximately) thermal. Then, in order for the den- is essential since the function increases monotonically sity distributions σ(r) and ρ(r) to be smooth, the con- as S/E d/2πe for large d. Explicitly, the respective values∼ for d =2, 3, 4,... 10 are approximately stituent particles of the radiation cannot have mass p m & H. This also means that the accessible number of 1.096, 1.495, 1.458, 1.425, 1.409, 1.404, 1.407, 1.414, and DOFs does not depend on r. If the Hubble parameter 1.425. For a given value of the “energy” E, the en- H is smaller than the mass of the lightest massive DOF, tropy is therefore maximum for d = 3. In other words, only strictly massless particles would contribute to the a universe whose final configuration is a de Sitter phase DOFs constituting the radiation. Known difficulties with a small cosmological constant is entropically fa- with massless higher spins [26] then make it natural to vored to have three spatial dimensions. consider only particles of spin s 2. If there is a massless spin-1 particle, no other particle≤ can be charged un- Acknowledgement der this, because otherwise interactions would render the radiation non-thermal. While the Hawking/Unruh We are grateful to G. Gibbons, C. Hull, A. Iglesias, effect is very fundamental and takes place in all di- E. Joung, M. Kleban, D. Klemm, L. Lopez, M. Porrati, mensions, a massless chargeless fermion can exist only A. Sagnotti, L. Sorbo and M. Taronna for useful discus- in some particular dimensions. This rules out spin- sions. AM and RR are thankful for the kind hospitality 1/2 and spin-3/2 particles as non-generic. Scalars are respectively of the HECAP section of ASICTP, Trieste, also ruled out, since there is no symmetry to assure where part of this work was done, and the Centre of their masslessness. So we are left only with spin-1 and Theoretical Physics at Tomsk State Pedagogical Uni- spin-2 particles, whose masslessness can be guaranteed versity during the conference QFTG’14. At the time by gauge invariance. Now, there is one and just one of this work, RR was supported in part by Scuola Nor- massless spin 2, namely the graviton [26]. More than male Superiore, by INFN and by the ERC Advanced one vector particle is not a possibility, because they ei- Investigator Grant no. 226455 “Supersymmetry, Quan- ther confine and cease to exist as long-range particles tum Gravity and Gauge Fields” (SUPERFIELDS).

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