text: (cid:112) (cid:80) { } and Suppose X L is a set of independent random variables c l l=1 ∈ (a + b)2 + 4c > b + . such that for each l, X [a , b ] with probability one. If b(cid:112) S = L X , Hoeffdingl’s inequl alilty [16] states that for every For every positive integer m, let Sm be deﬁned as above. Then u smallle=r1or iequal to ES, we have − (cid:80) ml→im∞ (x,y)∈mcoanxv(Sm)(x + y) = a b + (a + b)2 + 4c. Pr (S ≤ u) ≤ e− Ll2=(1ES(b−l−u)a2l)2 . { } It is easy to see that the sequences above satisfy the If X ij is a set of ij independ(cid:34)ent Ber(cid:35)noulli(p) random l l=1 ≤ ≤(cid:88) assumptions of Lemma 5. Thus variables, then for every l, 0 X 1, and l √ lim sup CS(m−I)E ≤ 5 + 4(cid:15) − 1, (1 − (cid:15))ij ≤ pij = E ij X . →∞ m l m (cid:18) (cid:19) (cid:32)l=1 (cid:33) Therefore, for all but ﬁnitely√many m, Thus(cid:88)Hoeffding’s inequality implies (cid:88) ≤ − C(m−) ( 5 + 4(cid:15) 1)m. (cid:98)(1−(cid:15))ij(cid:99) ij S IE ij − − ≤ − pk(1 p)ij k = Pr X (1 (cid:15))ij k l k=(cid:0)0 (cid:1) l=1 ≤ − − VII. CONCLUSION e 2(p 1+(cid:15))2ij. In this paper, we present a new model for cooperation and Since 2(cid:88)m ≤ (cid:18)2mi, (cid:19)(cid:18) (cid:19) (cid:88) (cid:18) (cid:19) study its beneﬁts in the case of the encoders of a DM-MAC. i (cid:98) − (cid:99) Speciﬁcally, we present channels for which the gain in sum- 2m 2m 2m (1 (cid:15))ij ij − − capacity is “almost” exponential in the cooperation rate. The (cid:88) pk(1 p)ij k i j k CF model can be generalized to other network settings, and i,j=f(m) k=0 its study is subject to future work. ≤ 2m − − 2m(i+j)e 2(p 1+(cid:15))2ij ACKNOWLEDGMENT (cid:88) i,j=f(m) This material is based upon work supported by the National 2m Science Foundation under Grant Numbers CCF-1321129, = e(i+j)mln2−2(p−1+(cid:15))2ij. CCF-1018741, CCF-1038578, and CNS-0905615. i,j=f(m) APPENDIX A If we deﬁne h : Z2 → R as PROOF OF LEMMA 2 − − h(i, j) = (i + j)m ln 2 2(p 1 + (cid:15))2ij, We use the probabilistic method [15]. We assign a probabil- × ≥ ity to every 2m 2m (0, 1)-matrix and show that the probability then for j f(m), of a matrix having both properties is positive for sufﬁciently − h(i + 1, j) h(i, j) large m; hence, there exists at least one such matrix. Fix − − (cid:15)b >i∼id0B, earnndoullelit(Bp)m,(cid:110)w=her(ebi1j)−2i,mj(cid:15)=<1 bpe<a1r.aLndetom(cid:111) matrix with =≤ mm llnn 22(cid:18)− 22((pp − 11 ++ (cid:15)(cid:15)))22jf(m) (cid:19) ij m − − ⊆ X | | ≥ = f(m) ln 2 2(p 1 + (cid:15))2 . Γm = (cid:26)S : S (m), S f(m)(cid:27) f(m) ∈ By assumption, For every S, T Γ , deﬁne the event m m | × ∩ | lim = 0, E(m) = (S | T|)| |1Bm ≤ 1 − (cid:15) . m→∞ f(m) (cid:18) (cid:91) S,T (cid:19) S T so there exists M such that for all m > M , 1 1 It follows m 2 − < (p 1 + (cid:15))2. Pr (cid:88)E(m)(cid:16) (cid:17) f(m) ln 2 ≥ ∈ S,T Therefore, for m > M and y f(m), h is decreasing with S,T Γ 1 ≤ (cid:88) Pr E(m) rmes>pecMt toani.dAi s≥hf(isms)y, mwme eatlrsiochwavitehhr(eis,pje+ct1t)o−iha(in,dj)j,<fo0r. S,T 1 = S,T∈Γ Pr (|(S × T)(cid:18)∩ 1 (cid:19)| ≤ (1 − (cid:15)) |S||T|) Ti,hju≥s (cid:88)hf(ims a).dHeecnrecaesifnogr mfun>ctiMon1,in i and j for m > M1 and S(cid:88),T∈Γ (cid:88) Bm 2m − − (cid:98)(1−(cid:15))|S||T|(cid:99) | || | e(i+j)mln2 2(p 1+(cid:15))2ij S T − | || |− = S,T(cid:88)∈Γ (cid:18)k=0(cid:19)(cid:18) (cid:19) k (cid:88)pk((cid:18)1 (cid:19)p) S T k i,j=f(≤m)(2m − f(m) + 1)2 e2mf(m)ln2−2(p−1+(cid:15))2(f(m))2 (cid:98) − (cid:99) 2m 2m 2m (1 (cid:15))ij ij − − < e2m(1+f(m))ln2−2(p−1+(cid:15))2(f(m))2 = pk(1 p)ij k. i,j=f(m) i j k=0 k = e2(f(m))2((1+f(1m))f(mm) ln2−2(p−1+(cid:15))2). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: Suppose

text: X

text: L

text: is

text: a

text: set

text: of

text: independent

text: random

text: variables

text: c

text: l

text: (a

text: b

text: such

text: that

text: for

text: each

text: l,

text: X

text: [a

text: b

text: with

text: probability

text: one.

text: If

text: S

text: L

text: X

text: Hoeffdingl’s

text: inequl

text: alilty

text: states

text: that

text: for

text: every

text: For

text: every

text: positive

text: integer

text: m,

text: let

text: Sm

text: be

text: deﬁned

text: as

text: above.

text: Then

text: u

text: iequal

text: to

text: ES,

text: we

text: have

text: ml→im∞

text: (x,y)∈mcoanxv(Sm)(x

text: y)

text: a

text: b

text: (a

text: Pr

text: (S

text: u)

text: e−

text: It

text: is

text: easy

text: to

text: see

text: that

text: the

text: sequences

text: above

text: satisfy

text: the

text: If

text: X

text: ij

text: is

text: a

text: set

text: of

text: ij

text: random

text: l

text: assumptions

text: of

text: Lemma

text: Thus

text: variables,

text: then

text: for

text: every

text: l,

text: X

text: and

text: l

text: lim

text: sup

text: CS(m−I)E

text: pij

text: E

text: ij

text: X

text: m

text: l

text: m

text: Therefore,

text: for

text: all

text: but

text: ﬁnitely√many

text: m,

text: inequality

text: implies

text: C(m−)

text: ij

text: S

text: IE

text: ij

text: p)ij

text: k

text: Pr

text: X

text: k

text: l

text: VII.

text: CONCLUSION

text: e

text: In

text: this

text: paper,

text: we

text: present

text: a

text: new

text: model

text: for

text: cooperation

text: and

text: Since

text: study

text: its

text: beneﬁts

text: in

text: the

text: case

text: of

text: the

text: encoders

text: of

text: a

text: DM-MAC.

text: i

text: Speciﬁcally,

text: we

text: present

text: channels

text: for

text: which

text: the

text: gain

text: in

text: sum-

text: ij

text: capacity

text: is

text: “almost”

text: exponential

text: in

text: the

text: cooperation

text: rate.

text: The

text: p)ij

text: k

text: i

text: j

text: k

text: CF

text: model

text: can

text: be

text: generalized

text: to

text: other

text: network

text: settings,

text: and

text: i,j=f(m)

text: its

text: study

text: is

text: subject

text: to

text: future

text: work.

text: ACKNOWLEDGMENT

text: i,j=f(m)

text: This

text: material

text: is

text: based

text: upon

text: work

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: under

text: Grant

text: Numbers

text: and

text: i,j=f(m)

text: APPENDIX

text: A

text: If

text: we

text: deﬁne

text: h

text: R

text: as

text: PROOF

text: OF

text: LEMMA

text: h(i,

text: j)

text: (i

text: j)m

text: ln

text: We

text: use

text: the

text: probabilistic

text: method

text: We

text: assign

text: a

text: probabil-

text: ity

text: to

text: every

text: and

text: show

text: that

text: the

text: probability

text: then

text: for

text: j

text: f(m),

text: of

text: a

text: matrix

text: having

text: both

text: properties

text: is

text: positive

text: for

text: sufﬁciently

text: h(i

text: j)

text: h(i,

text: j)

text: large

text: m;

text: hence,

text: there

text: exists

text: at

text: least

text: one

text: such

text: matrix.

text: Fix

text: matrix

text: with

text: mm

text: llnn

text: ij

text: m

text: X

text: f(m)

text: ln

text: Γm

text: S

text: (m),

text: S

text: f(m)

text: By

text: assumption,

text: For

text: every

text: S,

text: T

text: Γ

text: deﬁne

text: the

text: event

text: m

text: m

text: lim

text: E(m)

text: (S

text: T|)|

text: m→∞

text: f(m)

text: S,T

text: S

text: T

text: so

text: there

text: exists

text: M

text: such

text: that

text: for

text: all

text: m

text: M

text: It

text: follows

text: m

text: (p

text: Pr

text: f(m)

text: ln

text: S,T

text: Therefore,

text: for

text: m

text: M

text: and

text: y

text: f(m),

text: h

text: is

text: decreasing

text: with

text: S,T

text: Γ

text: Pr

text: E(m)

text: rmes>pecMt

text: toani.dAi

text: s≥hf(isms)y,

text: mwme

text: S,T

text: S,T∈Γ

text: Pr

text: (|(S

text: |S||T|)

text: Ti,hju≥s

text: a).dHeecnrecaesifnogr

text: i

text: and

text: j

text: for

text: m

text: and

text: Bm

text: S

text: T

text: k

text: S

text: T

text: k

text: f(m)

text: ij

text: p)ij

text: k.

text: i,j=f(m)

text: i

text: j

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: { | ≤ ≤ | | ≤ } In (c), if the set j 1 j k, S C is empty, then j there’s nothing to prove. Otherwise, it’s a nonempty subset of {1, . . . ,(cid:88)k} and thus(cid:88)has a minimum, which we call j∗. Then k α = α j (cid:88) j | |≤ ∗ j: Sj C j=j k | | − = (cid:88) Sj (p(xj) p(xj−1)) ∗ j=j (cid:88)k | \ | − | | = S S p(x ) S ∗ p(x ∗− ) j j+1 j j j 1 ∗ j=j = (cid:88) p(x) − |S ∗|p(x ∗− ) Figure 3. The sets S1 and S2 (gray area), and their convex hulls. j j 1 x∈Sj∗ ≤ p(x). The intersection of S and S (cid:112)consists of all pairs (x, x) such x∈Sj∗ (cid:18) (cid:19) that 0 ≤ x ≤ x0 wher1e 2 (cid:88) − By (11) of Lemma 3, a b + (a + b)2 + 4c ≤ 1 (cid:18) − H (X) −(cid:19)1 x(cid:112)0 = 2 . x∈Sj∗ p(x) 1 − lologg||SXj∗|| 1 log −|X| Note that since b, c and ab + c are positive2,c ≤ 1 − H (X) 1 (a + b)2 + 4c < a + b + , − 1 |X| , b 1 log|XC| log log so 0 < x <(cid:16)a+ c. The c(cid:17)onvex hull of(cid:16)S con(cid:17)sists of all pairs since |S ∗| ≤ C. (x, y) suc0h that 0b≤ x ≤ y and 1 j c − ≤ c APPENDIX D a + x x + x y a + x , PROOF OF LEMMA 5 b 0 0 b 0 and the convex hul(cid:16)l of S consi(cid:17)sts of a(cid:16)ll pairs(cid:17)(x, y) such that Prior to proving Lemma 5, we state and prove the following ≤ ≤ 2 0 y x and lemma. c − ≤ c Lemma 6. Suppo(cid:112)se a, b, and c are three real numbers such x0x + a + b x0 y a + b x0. that b, c, a + b, and ab + c are positive, and ∪ Note that conv (S ) conv (S ) is the region bounded by the c axes y = 0, x= 01, and the pie2cewise linear function (a + b)2 + 4c > b + . Let S be the set of(cid:26)a(llxp−aiars)((yx+, yb))suc≤h cthb,at x, y ≥ 0, and h(x) = xx00−−xxaa00−−ccbb xx +− a(xa0++−acbcb−)xcb0 0x0≤<xx≤≤xa0,+ cb. − ≤ ≥ (x + b)(y a) c. Since 2x a + c by assumption, 0 b − − If x0 is the unique positive solution to the equation x0 a cb ≥ −x0 − . − x x a c (x a)(x + b) = c, 0 0 b This means the slope of h is decreasing, or equivalently, h is a then concave function. Thus conv (S ) ∪ conv (S ) is convex. But 1 2 max (x + y) = 2x . ⊆ ∪ ⊆ ∈ 0 (x,y) conv(S) S conv(S1) conv(S2) conv(S), so Proof: Since ∪ conv(S) = conv(S ) conv(S ). − − − − 1 2 (x a)(y + b) (x + b)(y a) = (a + b)(x y) This implies ∪ and a+b > 0, the set S can be written as S = S S (Figure 1 2≤ ≤ max (x + y) = 2x . 3), where S1 is the set of all pairs (x, y) such that 0 x y (x,y)∈conv(S) 0 and − ≤ (x + b)(y a) c, Using this lemma, we may prove Lemma 5. There exists ≤ ≤ ≥ and S2 is the set of all pairs (x, y) such that 0 y x and a positive M such that for every m M, − ≤ (x a)(y + b) c. b , c , a + b , a b + c > 0 m m m m m m m ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: In

text: (c),

text: if

text: the

text: set

text: j

text: j

text: k,

text: S

text: C

text: is

text: empty,

text: then

text: j

text: there’s

text: nothing

text: to

text: prove.

text: Otherwise,

text: it’s

text: a

text: nonempty

text: subset

text: of

text: and

text: a

text: minimum,

text: which

text: we

text: call

text: j∗.

text: Then

text: k

text: α

text: α

text: j

text: j

text: j:

text: Sj

text: C

text: j=j

text: k

text: Sj

text: (p(xj)

text: j=j

text: S

text: S

text: p(x

text: S

text: p(x

text: j

text: j

text: j

text: j

text: j=j

text: p(x)

text: |S

text: ∗|p(x

text: Figure

text: The

text: sets

text: and

text: (gray

text: area),

text: and

text: their

text: convex

text: hulls.

text: j

text: j

text: x∈Sj∗

text: p(x).

text: The

text: intersection

text: of

text: S

text: and

text: S

text: of

text: all

text: pairs

text: (x,

text: x)

text: such

text: x∈Sj∗

text: that

text: x

text: By

text: of

text: Lemma

text: a

text: b

text: (a

text: H

text: (X)

text: x∈Sj∗

text: p(x)

text: lologg||SXj∗||

text: log

text: −|X|

text: Note

text: that

text: since

text: b,

text: c

text: and

text: ab

text: c

text: are

text: H

text: (X)

text: (a

text: a

text: b

text: |X|

text: b

text: log|XC|

text: log

text: log

text: so

text: x

text: c.

text: The

text: hull

text: of

text: all

text: pairs

text: since

text: |S

text: C.

text: (x,

text: y)

text: that

text: x

text: y

text: and

text: j

text: c

text: c

text: APPENDIX

text: D

text: a

text: x

text: x

text: x

text: y

text: a

text: x

text: PROOF

text: OF

text: LEMMA

text: b

text: b

text: and

text: the

text: convex

text: of

text: S

text: of

text: y)

text: such

text: that

text: Prior

text: to

text: proving

text: Lemma

text: we

text: state

text: and

text: prove

text: the

text: following

text: y

text: x

text: and

text: lemma.

text: c

text: c

text: Lemma

text: a,

text: b,

text: and

text: c

text: are

text: three

text: real

text: numbers

text: such

text: a

text: b

text: y

text: a

text: b

text: that

text: b,

text: c,

text: a

text: b,

text: and

text: ab

text: c

text: are

text: positive,

text: and

text: Note

text: that

text: conv

text: (S

text: conv

text: (S

text: is

text: the

text: region

text: bounded

text: by

text: the

text: c

text: axes

text: y

text: x=

text: and

text: the

text: linear

text: function

text: (a

text: b

text: Let

text: S

text: be

text: the

text: set

text: yb))suc≤h

text: cthb,at

text: x,

text: y

text: and

text: h(x)

text: xx

text: cb.

text: (x

text: b)(y

text: a)

text: c.

text: Since

text: a

text: c

text: by

text: assumption,

text: b

text: If

text: is

text: the

text: unique

text: positive

text: solution

text: to

text: the

text: equation

text: a

text: cb

text: x

text: x

text: a

text: c

text: (x

text: a)(x

text: b)

text: c,

text: b

text: This

text: means

text: the

text: slope

text: of

text: h

text: is

text: decreasing,

text: or

text: equivalently,

text: h

text: is

text: a

text: then

text: concave

text: function.

text: Thus

text: conv

text: (S

text: conv

text: (S

text: is

text: convex.

text: But

text: max

text: (x

text: y)

text: (x,y)

text: conv(S)

text: S

text: conv(S),

text: so

text: Proof:

text: Since

text: conv(S)

text: conv(S

text: conv(S

text: (x

text: a)(y

text: b)

text: (x

text: b)(y

text: a)

text: (a

text: b)(x

text: y)

text: This

text: implies

text: and

text: a+b

text: the

text: set

text: S

text: can

text: be

text: written

text: as

text: S

text: S

text: S

text: (Figure

text: 2≤

text: max

text: (x

text: y)

text: where

text: is

text: the

text: set

text: of

text: all

text: pairs

text: (x,

text: y)

text: such

text: that

text: x

text: y

text: (x,y)∈conv(S)

text: and

text: (x

text: b)(y

text: a)

text: c,

text: Using

text: this

text: lemma,

text: we

text: may

text: prove

text: Lemma

text: There

text: exists

text: and

text: is

text: the

text: set

text: of

text: all

text: pairs

text: (x,

text: y)

text: such

text: that

text: y

text: x

text: and

text: a

text: positive

text: M

text: such

text: that

text: for

text: every

text: m

text: M,

text: (x

text: a)(y

text: b)

text: c.

text: b

text: c

text: a

text: b

text: a

text: b

text: c

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: 12 state, but also to the superconducting state. We then calculated the critical temperature T , and found that it c decreases as the anisotropy becomes large. It is very interesting to note that a real-world iron-based superconductor with the same property has been reported recently [? ]. We also computed the optical conductivities in both of the x- and y-directions. We estimated the energy gap ω g from the conductivity in the y-direction and found that ω /T increases as the anisotropy becomes large, and hence g c that ω is larger than 8T . As it is much larger than the BCS prediction 3.54T , the strong coupling eﬀect for the g c c holographic superconductor model is not broken by the anisotropy, but rather enhanced. On the other hand, the conductivity in the x-direction never drops to zero even at the suﬃciently low temperature. This indicates that a pseudogap appears much like in the p-wave superconductors [? ? ? ], and we found that the magnitude of the pseudogap seems to increase as the anisotropy becomes large. It will be interesting if a peudogap appears generally in an anisotropic model of holographic superconductors, without depending on details of holographic models. Acknowledgments We are grateful to T. Okamura for helpful discussion. We also thank M. Miura for enlightening us on the experiment [? ]. The work of K. M. was supported in part by JSPS KAKENHI Grant Number 23740200. [1] J. Maldacena, “The Large N limit of superconformal ﬁeld theories and supergravity” Adv. Theor. Math. Phys. 2, 231-252 (1998); O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, “Large N ﬁeld theories, string theory and gravity” Phys. Rep. 323, 183-386 (2000). [2] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, “Building an AdS/CFT superconductor”, Phys. Rev. Lett. 101 031601 (2008). [3] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, “Holographic Superconductors”, JHEP 0812 015 (2008). [4] K. Maeda, T. Okamura, and J. Koga, “Inhomogeneous charged black hole solutions in asymptotically anti-de Sitter spacetime”, Phys. Rev. D 85 066003 (2012). [5] G. T. Horowitz, J. E. Santos, and D. Tong, “Optical Conductivity with Holographic Lattices,” JHEP 07 168 (2012). [6] G. T. Horowitz, J. E. Santos, and D. Tong, “Further Evidence for Lattice-Induced Scaling,” JHEP 11 102 (2012). [7] G. T. Horowitz and J. E. Santos, “General Relativity and the Cuprates,” [arXiv:1302.6586 [hep-th]]. [8] S. S. Gubser and S. S. Pufu, “The gravity dual of a p-wave superconductor”, JHEP 0811 033 (2008). [9] M. M. Roberts and S. A. Hartnoll, “Pseudogap and time reversal breaking in a holographic superconductor”, JHEP 0808 035 (2008). [10] J. A. Hutasoit, G. Siopsis, J. Therrien, “Conductivity of Strongly Coupled Striped Superconductor”, JHEP 1401 132 (2014). [11] N. Iizuka and K. Maeda, “Study of Anisotropic Black Branes in Asymptotically anti-de Sitter”, JHEP 1207 129 (2012). [12] N. Iizuka, N. Kundu, P. Narayan, and S. P. Trivedi, “Holographic Fermi and Non-Fermi Liquids with Transitions in Dilaton Gravity”, JHEP 1201 094 (2012). [13] S. Kachru, X. Liu, and M. Mulligan, “Gravity Duals of Lifshitz-like Fixed Points”, Phys. Rev. D 78 106005 (2008). [14] M. Taylor, “Non-relativistic holography”, arXiv: 0812.0530[hep-th]. [15] K. Goldstein, S. Kachru, S. Prakash, and S. P. Trivedi, “Holography of Charged Dilaton Black Holes”, JHEP 1008 078 (2010). [16] K. Goldstein, N. Iizuka, S. Kachru, S. Prakash, S. P. Trivedi, and A. Westphal, “Holography of Dyonic Dilaton Black Branes”, JHEP 10 027 (2010). [17] N. Iizuka, S. Kachru, N. Kundu, P. Narayan, N. Sircar, and S. P. Trivedi, “Bianchi Attractors: A Classiﬁcation of Extremal Black Brane Geometries”, JHEP 1207 193 (2012). [18] D. Mateos and D. Trancanelli, “The anisotropic N=4 super Yang-Mills plasma and its instabilities”, Phys. Rev. Lett. 107 101601 (2011). [19] D. Mateos and D. Trancanelli, “Thermodynamics and Instabilities of a Strongly Coupled Anisotropic Plasma”, JHEP 1107 054 (2011). [20] I. R. Klebanov and E. Witten, “AdS/CFT correspondence and symmetry breaking”, Nucl. Phys. B 556 89 (1999). [21] M. Miura et al, “Anisotropy and Superconducting Properties of BaFe (As − P ) Films with Various Phosphorus Con- 2 1 x x 2 tents”, Appl. Phys. Express 6 093101 (2013). ##LTLine## ##LTLine## ##LTLine##

text: state,

text: but

text: also

text: to

text: the

text: superconducting

text: state.

text: We

text: then

text: calculated

text: the

text: critical

text: temperature

text: T

text: and

text: found

text: that

text: it

text: c

text: decreases

text: as

text: the

text: anisotropy

text: becomes

text: large.

text: It

text: is

text: very

text: interesting

text: to

text: note

text: that

text: a

text: real-world

text: iron-based

text: superconductor

text: with

text: the

text: same

text: property

text: has

text: been

text: reported

text: recently

text: We

text: also

text: computed

text: the

text: optical

text: conductivities

text: in

text: both

text: of

text: the

text: x-

text: and

text: y-directions.

text: We

text: estimated

text: the

text: energy

text: gap

text: ω

text: g

text: from

text: the

text: conductivity

text: in

text: the

text: y-direction

text: and

text: found

text: that

text: ω

text: /T

text: increases

text: as

text: the

text: anisotropy

text: becomes

text: large,

text: and

text: hence

text: g

text: c

text: that

text: ω

text: is

text: larger

text: than

text: As

text: it

text: is

text: much

text: larger

text: than

text: the

text: BCS

text: prediction

text: the

text: strong

text: coupling

text: eﬀect

text: for

text: the

text: g

text: c

text: c

text: holographic

text: superconductor

text: model

text: is

text: not

text: broken

text: by

text: the

text: anisotropy,

text: but

text: rather

text: enhanced.

text: On

text: the

text: other

text: hand,

text: the

text: conductivity

text: in

text: the

text: x-direction

text: never

text: drops

text: to

text: zero

text: even

text: at

text: the

text: suﬃciently

text: low

text: temperature.

text: This

text: indicates

text: that

text: a

text: pseudogap

text: appears

text: much

text: like

text: in

text: the

text: p-wave

text: superconductors

text: and

text: we

text: found

text: that

text: the

text: magnitude

text: of

text: the

text: pseudogap

text: seems

text: to

text: increase

text: as

text: the

text: anisotropy

text: becomes

text: large.

text: It

text: will

text: be

text: interesting

text: if

text: a

text: peudogap

text: appears

text: generally

text: in

text: an

text: anisotropic

text: model

text: of

text: holographic

text: superconductors,

text: without

text: depending

text: on

text: details

text: of

text: holographic

text: models.

text: Acknowledgments

text: We

text: are

text: grateful

text: to

text: T.

text: Okamura

text: for

text: helpful

text: discussion.

text: We

text: also

text: thank

text: M.

text: Miura

text: for

text: enlightening

text: us

text: on

text: the

text: experiment

text: The

text: work

text: of

text: K.

text: M.

text: was

text: supported

text: in

text: part

text: by

text: JSPS

text: KAKENHI

text: Grant

text: Number

text: J.

text: Maldacena,

text: “The

text: Large

text: N

text: limit

text: of

text: superconformal

text: ﬁeld

text: theories

text: and

text: supergravity”

text: Adv.

text: Theor.

text: Math.

text: Phys.

text: O.

text: Aharony,

text: S.

text: Gubser,

text: J.

text: Maldacena,

text: H.

text: Ooguri,

text: and

text: Y.

text: Oz,

text: “Large

text: N

text: ﬁeld

text: theories,

text: string

text: theory

text: and

text: gravity”

text: Phys.

text: Rep.

text: S.

text: A.

text: Hartnoll,

text: C.

text: P.

text: Herzog,

text: and

text: G.

text: T.

text: Horowitz,

text: “Building

text: an

text: AdS/CFT

text: superconductor”,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: A.

text: Hartnoll,

text: C.

text: P.

text: Herzog,

text: and

text: G.

text: T.

text: Horowitz,

text: “Holographic

text: Superconductors”,

text: JHEP

text: K.

text: Maeda,

text: T.

text: Okamura,

text: and

text: J.

text: Koga,

text: “Inhomogeneous

text: charged

text: black

text: hole

text: solutions

text: in

text: asymptotically

text: anti-de

text: Sitter

text: spacetime”,

text: Phys.

text: Rev.

text: D

text: G.

text: T.

text: Horowitz,

text: J.

text: E.

text: Santos,

text: and

text: D.

text: Tong,

text: “Optical

text: Conductivity

text: with

text: Holographic

text: Lattices,”

text: JHEP

text: G.

text: T.

text: Horowitz,

text: J.

text: E.

text: Santos,

text: and

text: D.

text: Tong,

text: “Further

text: Evidence

text: for

text: Lattice-Induced

text: Scaling,”

text: JHEP

text: G.

text: T.

text: Horowitz

text: and

text: J.

text: E.

text: Santos,

text: “General

text: Relativity

text: and

text: the

text: Cuprates,”

text: [hep-th]].

text: S.

text: S.

text: Gubser

text: and

text: S.

text: S.

text: Pufu,

text: “The

text: gravity

text: dual

text: of

text: a

text: p-wave

text: superconductor”,

text: JHEP

text: M.

text: M.

text: Roberts

text: and

text: S.

text: A.

text: Hartnoll,

text: “Pseudogap

text: and

text: time

text: reversal

text: breaking

text: in

text: a

text: holographic

text: superconductor”,

text: JHEP

text: J.

text: A.

text: Hutasoit,

text: G.

text: Siopsis,

text: J.

text: Therrien,

text: “Conductivity

text: of

text: Strongly

text: Coupled

text: Striped

text: Superconductor”,

text: JHEP

text: N.

text: Iizuka

text: and

text: K.

text: Maeda,

text: “Study

text: of

text: Anisotropic

text: Black

text: Branes

text: in

text: Asymptotically

text: anti-de

text: Sitter”,

text: JHEP

text: N.

text: Iizuka,

text: N.

text: Kundu,

text: P.

text: Narayan,

text: and

text: S.

text: P.

text: Trivedi,

text: “Holographic

text: Fermi

text: and

text: Non-Fermi

text: Liquids

text: with

text: Transitions

text: in

text: Dilaton

text: Gravity”,

text: JHEP

text: S.

text: Kachru,

text: X.

text: Liu,

text: and

text: M.

text: Mulligan,

text: “Gravity

text: Duals

text: of

text: Lifshitz-like

text: Fixed

text: Points”,

text: Phys.

text: Rev.

text: D

text: M.

text: Taylor,

text: “Non-relativistic

text: holography”,

text: arXiv:

text: K.

text: Goldstein,

text: S.

text: Kachru,

text: S.

text: Prakash,

text: and

text: S.

text: P.

text: Trivedi,

text: “Holography

text: of

text: Charged

text: Dilaton

text: Black

text: Holes”,

text: JHEP

text: K.

text: Goldstein,

text: N.

text: Iizuka,

text: S.

text: Kachru,

text: S.

text: Prakash,

text: S.

text: P.

text: Trivedi,

text: and

text: A.

text: Westphal,

text: “Holography

text: of

text: Dyonic

text: Dilaton

text: Black

text: Branes”,

text: JHEP

text: N.

text: Iizuka,

text: S.

text: Kachru,

text: N.

text: Kundu,

text: P.

text: Narayan,

text: N.

text: Sircar,

text: and

text: S.

text: P.

text: Trivedi,

text: “Bianchi

text: Attractors:

text: A

text: Classiﬁcation

text: of

text: Extremal

text: Black

text: Brane

text: Geometries”,

text: JHEP

text: D.

text: Mateos

text: and

text: D.

text: Trancanelli,

text: “The

text: anisotropic

text: super

text: Yang-Mills

text: plasma

text: and

text: its

text: instabilities”,

text: Phys.

text: Rev.

text: Lett.

text: D.

text: Mateos

text: and

text: D.

text: Trancanelli,

text: “Thermodynamics

text: and

text: Instabilities

text: of

text: a

text: Strongly

text: Coupled

text: Anisotropic

text: Plasma”,

text: JHEP

text: I.

text: R.

text: Klebanov

text: and

text: E.

text: Witten,

text: “AdS/CFT

text: correspondence

text: and

text: symmetry

text: breaking”,

text: Nucl.

text: Phys.

text: B

text: M.

text: Miura

text: et

text: al,

text: “Anisotropy

text: and

text: Superconducting

text: Properties

text: of

text: BaFe

text: (As

text: P

text: Films

text: with

text: Various

text: Phosphorus

text: Con-

text: x

text: x

text: tents”,

text: Appl.

text: Phys.

text: Express

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 (cid:18) (cid:19) (cid:18) (cid:19) (cid:48) (cid:48) − λ − (cid:48) Ω2 − z2Φ 2 − 2q2L2Ψ2 aˆ − B aˆ + a = 0, (3.8) y 1 z y g L2 z2 y (cid:18) (cid:48)(cid:19)− λa − aˆ a −y y = 0, (3.9) y 1 z g (cid:48) (cid:18) (cid:19) (cid:18)(cid:48) − (cid:48) − (cid:48) λ (cid:19) z2Φ a ζ A B + − ζ + y = 0, (3.10) ty 1 z ty L2 (cid:48) (cid:48) − λ (cid:48) Ω2 − z2Φ 2 − 2q2L2Ψ2 (cid:48) aˆ − + B aˆ + a + 2αΦ ηˆ = 0, (3.11) x 1 z x g L2 z2 x (cid:18) (cid:19) (cid:48) − λa − aˆ a −x x = 0, (3.12) x 1 z g − (cid:48) − λ − 2 − (cid:48) Ω2 − αe (A+B)ζ ηˆ − A ηˆ + η tx = 0, (3.13) 1 z z g g (cid:18) (cid:19) (cid:48) − λη − ηˆ η − = 0, (3.14) 1 z g (cid:48) (cid:48) − (cid:48) (cid:48) λ z2Φ a − ζ A + B + − ζ + x 2αgηˆ = 0, (3.15) tx 1 z tx L2 which are written, near the horizon, in the form of the eigenvalue equationsas         − (cid:48) 0 Ω2/g (1) 0 aˆ aˆ (cid:48) y y  1/g (1) 0 0 a = λ a , (3.16) y y 0 0 0 ζ ζ    ty   ty   − (cid:48)       0 Ω2/g (1) 0 0 0  aˆ  aˆ   (cid:48)   x   x  1/g (1) 0 0 0 0 a a − (cid:48) (cid:48) x x 0 0 0 Ω2/g (1) β/g (1) ηˆ = λ ηˆ , (3.17) (cid:48) 0 0 1/g (1) 0 0 η η 0 0 0 0 0 ζ ζ tx tx ≡ − − where β αe A(1) B(1). Here we impose the ingoing boundary condition at the horizon, which corresponds to the retarded Green’s function in the dual ﬁeld theory. Then the eigenvectors of Eq. (??) which satisfy this boundary condition are derived as         0 iΩ 0 , 1 , (3.18) 1 0 (cid:48) with the corresponding eigenvalues being given by λ = 0 and λ = iΩ/g (1), respectively. We numerically solve Eqs.(??)–(??), by substituting each of these eigenvalues λ and taking the corresponding eigenvectors in Eq. (??) as the initial condtion. We thus obtain in this way two linearly independent solutions. Similarly, the eigenvectors of Eq. (??) satisfying the ingoing boundary condition are found tobe                0   0  iΩ       0 0 1 0 , iΩ , 0 , (3.19) 1 1 0 − Ω2β 1 0 0 where the eigenvalue corresponding to the ﬁrst eigenvector is given by λ = 0, and the eigenvalues for the other two (cid:48) are found to be degenerate and given by λ = iΩ/g (1). Again, we substitute each eigenvalue into Eqs. (??)–(??) and impose the initial condition as required by the corresponding eigenvectors in Eq. (??), to obtain three linearly independent solutions. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: λ

text: Ω2

text: aˆ

text: B

text: aˆ

text: a

text: y

text: z

text: y

text: g

text: y

text: λa

text: aˆ

text: a

text: −y

text: y

text: y

text: z

text: g

text: λ

text: a

text: ζ

text: A

text: B

text: ζ

text: y

text: ty

text: z

text: ty

text: λ

text: Ω2

text: aˆ

text: B

text: aˆ

text: a

text: 2αΦ

text: ηˆ

text: x

text: z

text: x

text: g

text: x

text: λa

text: aˆ

text: a

text: −x

text: x

text: x

text: z

text: g

text: λ

text: Ω2

text: αe

text: (A+B)ζ

text: ηˆ

text: A

text: ηˆ

text: η

text: tx

text: z

text: z

text: g

text: g

text: λη

text: ηˆ

text: η

text: z

text: g

text: λ

text: a

text: ζ

text: A

text: B

text: ζ

text: x

text: 2αgηˆ

text: tx

text: z

text: tx

text: which

text: are

text: written,

text: near

text: the

text: horizon,

text: in

text: the

text: form

text: of

text: the

text: eigenvalue

text: equationsas

text: aˆ

text: aˆ

text: y

text: y

text: 0

text: a

text: λ

text: a

text: y

text: y

text: ζ

text: ζ

text: ty

text: ty

text: aˆ

text: aˆ

text: x

text: x

text: a

text: a

text: x

text: x

text: β/g

text: ηˆ

text: λ

text: ηˆ

text: η

text: η

text: ζ

text: ζ

text: tx

text: tx

text: where

text: β

text: αe

text: Here

text: we

text: impose

text: the

text: ingoing

text: boundary

text: condition

text: at

text: the

text: horizon,

text: which

text: corresponds

text: to

text: the

text: retarded

text: Green’s

text: function

text: in

text: the

text: dual

text: ﬁeld

text: theory.

text: Then

text: the

text: eigenvectors

text: of

text: Eq.

text: which

text: satisfy

text: this

text: boundary

text: condition

text: are

text: derived

text: as

text: iΩ

text: with

text: the

text: corresponding

text: eigenvalues

text: being

text: given

text: by

text: λ

text: and

text: λ

text: iΩ/g

text: respectively.

text: We

text: numerically

text: solve

text: Eqs.(??)–(??),

text: by

text: substituting

text: each

text: of

text: these

text: eigenvalues

text: λ

text: and

text: taking

text: the

text: corresponding

text: eigenvectors

text: in

text: Eq.

text: as

text: the

text: initial

text: condtion.

text: We

text: thus

text: obtain

text: in

text: this

text: way

text: two

text: linearly

text: independent

text: solutions.

text: Similarly,

text: the

text: eigenvectors

text: of

text: Eq.

text: satisfying

text: the

text: ingoing

text: boundary

text: condition

text: are

text: found

text: tobe

text: iΩ

text: iΩ

text: Ω2β

text: where

text: the

text: eigenvalue

text: corresponding

text: to

text: the

text: ﬁrst

text: eigenvector

text: is

text: given

text: by

text: λ

text: and

text: the

text: eigenvalues

text: for

text: the

text: other

text: two

text: are

text: found

text: to

text: be

text: degenerate

text: and

text: given

text: by

text: λ

text: iΩ/g

text: Again,

text: we

text: substitute

text: each

text: eigenvalue

text: into

text: Eqs.

text: and

text: impose

text: the

text: initial

text: condition

text: as

text: required

text: by

text: the

text: corresponding

text: eigenvectors

text: in

text: Eq.

text: to

text: obtain

text: three

text: linearly

text: independent

text: solutions.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Contents II.ntroduction 2 ICI.onformal gravity and the static, spherically symmetric metric 3 IKIIe. rr metric in conformal gravity 8 AK. err metric in fourth-order gravity 10 BT. he separability of the Hamilton-Jacobi equation 14 IGV.eodesic motion and the Flyby Anomaly 21 VC.onclusions 30 Acknowledgments 31 References 31 I. INTRODUCTION In recent years alternative theories of gravity have become progressively more popular in the scientiﬁc literature due to their ability to account for astrophysical observations without resorting to dark matter (DM) and dark energy (DE). In fact, despite recent experimental and observational data supporting the case for DM (for example, see [1], [2]), there is still no conclusive evidence about the actual origin of this component of the Universe. Similarly, the observation of an accelerated expansion of the Universe prompted cosmologists to introduce a DE component within the framework of General Relativity (GR) and standard cosmology. On the other hand, alternative theories of gravity (such as MOND [3]-[4], TeVeS [5], NGT [6], and others) have built the case for a possible paradigm shift by avoiding the exotic DM/DE components and by introducing possible modiﬁcations to standard gravity (for reviews see [7], [8], [9], and references therein). This proposed paradigm shift is not so diﬀerent from what Einstein himself did in 1915 by extending Newtonian gravity into GR. For instance, Einstein’s explanation of the anomalies in the rate of precession of the planet Mercury, using GR instead of Newton’s law of gravitation, virtually eliminated a possible “dark matter” component in the Solar System (the proposed existence of an un- 2

text: Contents

text: II.ntroduction

text: ICI.onformal

text: gravity

text: and

text: the

text: static,

text: spherically

text: symmetric

text: metric

text: IKIIe.

text: rr

text: metric

text: in

text: conformal

text: gravity

text: AK.

text: err

text: metric

text: in

text: fourth-order

text: gravity

text: BT.

text: he

text: separability

text: of

text: the

text: Hamilton-Jacobi

text: equation

text: IGV.eodesic

text: motion

text: and

text: the

text: Flyby

text: Anomaly

text: VC.onclusions

text: Acknowledgments

text: References

text: I.

label: section

text: INTRODUCTION

text: In

text: recent

text: years

text: alternative

text: theories

text: of

text: gravity

text: have

text: become

text: progressively

text: more

text: popular

text: in

text: the

text: scientiﬁc

text: literature

text: due

text: to

text: their

text: ability

text: to

text: account

text: for

text: astrophysical

text: observations

text: without

text: resorting

text: to

text: dark

text: matter

text: (DM)

text: and

text: dark

text: energy

text: (DE).

text: In

text: fact,

text: despite

text: recent

text: experimental

text: and

text: observational

text: data

text: supporting

text: the

text: case

text: for

text: DM

text: (for

text: example,

text: see

text: there

text: is

text: still

text: no

text: conclusive

text: evidence

text: about

text: the

text: actual

text: origin

text: of

text: this

text: component

text: of

text: the

text: Universe.

text: Similarly,

text: the

text: observation

text: of

text: an

text: accelerated

text: expansion

text: of

text: the

text: Universe

text: prompted

text: cosmologists

text: to

text: introduce

text: a

text: DE

text: component

text: within

text: the

text: framework

text: of

text: General

text: Relativity

text: (GR)

text: and

text: standard

text: cosmology.

text: On

text: the

text: other

text: hand,

text: alternative

text: theories

text: of

text: gravity

text: (such

text: as

text: MOND

text: TeVeS

text: NGT

text: and

text: others)

text: have

text: built

text: the

text: case

text: for

text: a

text: possible

text: paradigm

text: shift

text: by

text: avoiding

text: the

text: exotic

text: DM/DE

text: components

text: and

text: by

text: introducing

text: possible

text: modiﬁcations

text: to

text: standard

text: gravity

text: (for

text: reviews

text: see

text: and

text: references

text: therein).

text: This

text: proposed

text: paradigm

text: shift

text: is

text: not

text: so

text: diﬀerent

text: from

text: what

text: Einstein

text: himself

text: did

text: in

text: by

text: extending

text: Newtonian

text: gravity

text: into

text: GR.

text: For

text: instance,

text: Einstein’s

text: explanation

text: of

text: the

text: anomalies

text: in

text: the

text: rate

text: of

text: precession

text: of

text: the

text: planet

text: Mercury,

text: using

text: GR

text: instead

text: of

text: Newton’s

text: law

text: of

text: gravitation,

text: virtually

text: eliminated

text: a

text: possible

text: “dark

text: matter”

text: component

text: in

text: the

text: Solar

text: System

text: (the

text: proposed

text: existence

text: of

text: an

text: un-


text: (cid:101) (cid:101) where the functions R(r), R(r), Θ(θ), and Θ(θ) are deﬁned in Eqs. (41)-(42), while the auxiliary quantities and functions are described in Eqs. (28) and (30).11 It should be noted that our fourth-order equations of motion above are very similar to the analogous equations of motion in Kerr-de Sitter spacetimes ([42], [43], [44]), once the CG parameter k is connected to the cosmological constant Λ by setting k = Λ/3. In fact, our Eqs. (44)-(47) have the same structure, for example, of Eqs. (14)-(17) in Ref. [43], since (cid:101) (cid:101) the procedure used for the separation of variables is the same. However, these two groups of equations diﬀer in the deﬁnition of the auxiliary functions, ∆ , ∆ and ∆ , ∆ , respectively r θ r θ (and the additional function χ = 1 + a2Λ is used only in the Kerr-de Sitter case), so they 3 are not completely equivalent. As in the second-order case, following Eqs. (44)-(47), it is evident that the problem of solving the equations of geodesic motion has been reduced to one of quadratures. In the next section, we will apply directly our results to the particular case of the so-called Flyby Anomaly (FA) in order to check if our CG fourth-order solutions can explain this (cid:101) gravitational puzzle. K As a ﬁnal point of this section, we note that the new, fourth-order Carter’s constant K (as well as the second-order ) can be obtained explicitly, in terms of r or θ, by combining together Eqs. (41) and (42) with Eqs. (44) and (45): (cid:101) (cid:101) (cid:101) − K − [(r2 + a2) E aL ]2 − = (cid:101)∆rp2r + z δ1r2 (48) −(cid:101)∆r (aE sin θ L csc θ)2 = ∆ p2 + z + δ a2 cos2 θ ; 4th-order θ θ 1 ∆ θ − K − [(r2 + a2) E aL ]2 − = ∆p2 + z δ r2 r ∆ 1 − = p2 + (aE sin θ L csc θ)2 + δ a2 cos2 θ ; 2nd-order. θ z 1 (cid:101) Q Q The alternative form of Carter’s constant ( or ) can be obtained using Eq. (39), or the equivalent second-order relation. (cid:104) (cid:105) 11 Comparing the fourth-order equations with the respective second-order equations of the standard theory, (cid:102) (cid:101) (cid:101) (cid:101) (cid:101) it can be noted that the CG equations are obtained from the GR equations, by performing the following → → (cid:101)→ ± → ∓ − − substitutions: M M, ∆ ∆ , Σ Σ, 2Mr ∆ (r2 a2)∆ and, in some appropriate r r θ places, by replacing a factor of 1 with ∆ . θ 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: the

text: functions

text: R(r),

text: R(r),

text: Θ(θ),

text: and

text: Θ(θ)

text: are

text: deﬁned

text: in

text: Eqs.

text: while

text: the

text: auxiliary

text: quantities

text: and

text: functions

text: are

text: described

text: in

text: Eqs.

text: and

text: It

text: should

text: be

text: noted

text: that

text: our

text: fourth-order

text: equations

text: of

text: motion

text: above

text: are

text: very

text: similar

text: to

text: the

text: analogous

text: equations

text: of

text: motion

text: in

text: Kerr-de

text: Sitter

text: spacetimes

text: once

text: the

text: CG

text: parameter

text: k

text: is

text: connected

text: to

text: the

text: cosmological

text: constant

text: Λ

text: by

text: setting

text: k

text: In

text: fact,

text: our

text: Eqs.

text: have

text: the

text: same

text: structure,

text: for

text: example,

text: of

text: Eqs.

text: in

text: Ref.

text: since

text: the

text: procedure

text: used

text: for

text: the

text: separation

text: of

text: variables

text: is

text: the

text: same.

text: However,

text: these

text: two

text: groups

text: of

text: equations

text: diﬀer

text: in

text: the

text: deﬁnition

text: of

text: the

text: auxiliary

text: functions,

text: and

text: respectively

text: r

text: θ

text: r

text: θ

text: (and

text: the

text: additional

text: function

text: χ

text: is

text: used

text: only

text: in

text: the

text: Kerr-de

text: Sitter

text: case),

text: so

text: they

text: are

text: not

text: completely

text: equivalent.

text: As

text: in

text: the

text: second-order

text: case,

text: following

text: Eqs.

text: it

text: is

text: evident

text: that

text: the

text: problem

text: of

text: solving

text: the

text: equations

text: of

text: geodesic

text: motion

text: has

text: been

text: reduced

text: to

text: one

text: of

text: quadratures.

text: In

text: the

text: next

text: section,

text: we

text: will

text: apply

text: directly

text: our

text: results

text: to

text: the

text: particular

text: case

text: of

text: the

text: so-called

text: Flyby

text: Anomaly

text: (FA)

text: in

text: order

text: to

text: check

text: if

text: our

text: CG

text: fourth-order

text: solutions

text: can

text: explain

text: this

text: gravitational

text: puzzle.

text: K

text: As

text: a

text: ﬁnal

text: point

text: of

text: this

text: section,

text: we

text: note

text: that

text: the

text: new,

text: fourth-order

text: Carter’s

text: constant

text: K

text: (as

text: well

text: as

text: the

text: second-order

text: can

text: be

text: obtained

text: explicitly,

text: in

text: terms

text: of

text: r

text: or

text: θ,

text: by

text: combining

text: together

text: Eqs.

text: and

text: with

text: Eqs.

text: and

text: K

text: E

text: aL

text: z

text: (aE

text: sin

text: θ

text: L

text: csc

text: z

text: δ

text: θ

text: θ

text: θ

text: θ

text: K

text: E

text: aL

text: z

text: δ

text: r

text: (aE

text: sin

text: θ

text: L

text: csc

text: δ

text: θ

text: θ

text: z

text: Q

text: Q

text: The

text: alternative

text: form

text: of

text: Carter’s

text: constant

text: or

text: can

text: be

text: obtained

text: using

text: Eq.

text: or

text: the

text: equivalent

text: second-order

text: relation.

text: Comparing

text: the

text: fourth-order

text: equations

text: with

text: the

text: respective

text: second-order

text: equations

text: of

text: the

text: standard

text: theory,

text: it

text: can

text: be

text: noted

text: that

text: the

text: CG

text: equations

text: are

text: obtained

text: from

text: the

text: GR

text: equations,

text: by

text: performing

text: the

text: following

text: substitutions:

text: M

text: M,

text: Σ

text: Σ,

text: and,

text: in

text: some

text: appropriate

text: r

text: r

text: θ

text: places,

text: by

text: replacing

text: a

text: factor

text: of

text: with

text: θ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ where the CG γ parameter is split into a γ , due to the contributions to the gravitational potential of the individual stars in the galaxy being studied (stars of reference mass equal to the solar mass M(cid:12)), and a cosmological parameter γ , due to the contributions of all 0 the other galaxies in the Universe. The third parameter κ is considered to be of purely cosmological origin. By performing a “kinematical approach” to CG, in previous publications ([10], [11]), we have shown a diﬀerent way to compute the CG parameters, obtaining values which diﬀer by a few orders of magnitude from those in Eq. (12): × − − × − − γ = 1.94 10 28 cm 1, κ = 6.42 10 48 cm 2 . (13) Using either the values in Eq. (12) or those in Eq. (13), inside Eqs. (7)-(11), one always obtains very small corrections to the dynamics of celestial bodies moving within the Solar System. Therefore, CG corrections were always considered negligible [27] at the Solar System level, not aﬀecting in any measurable way the motion of planets, satellites, etc. The eﬀect of CG would only be appreciable at the galactic and cosmological scales, where CG eﬀectively replaces DM and DE. One could argue that since the CG potential in Eq. (9) contains both a linear and a quadratic term in r, the eﬀects of distant stars, galaxies, etc., should be manifest in our Solar System, thus giving us a way to conﬁrm or rule out Conformal Gravity as a viable gravitational theory. However, CG eﬀects from distant sources only show themselves as “tidal forces” on solar-system bodies, and these tidal eﬀects are completely undetectable in our Solar System ([27], [28]). The above analysis suggests that CG corrections to the dynamics of the Solar System are essentially negligible and that CG eﬀects are of importance only over larger distance scales. However, these considerations were only based on the static, spherically symmetric CG metric, while many gravitational sources in the Universe are rotating bodies that might even possess electric/magnetic charge. In the rest of this paper we will concentrate our eﬀorts on the Kerr metric in CG, i.e., on the case of a stationary, axially symmetric rotating system without any charge. The study of solutions for charged bodies goes beyond the purpose of the current work, also because 7

text: where

text: the

text: CG

text: γ

text: parameter

text: is

text: split

text: into

text: a

text: γ

text: due

text: to

text: the

text: contributions

text: to

text: the

text: gravitational

text: potential

text: of

text: the

text: individual

text: stars

text: in

text: the

text: galaxy

text: being

text: studied

text: (stars

text: of

text: reference

text: mass

text: equal

text: to

text: the

text: solar

text: mass

text: and

text: a

text: cosmological

text: parameter

text: γ

text: due

text: to

text: the

text: contributions

text: of

text: all

text: the

text: other

text: galaxies

text: in

text: the

text: Universe.

text: The

text: third

text: parameter

text: κ

text: is

text: considered

text: to

text: be

text: of

text: purely

text: cosmological

text: origin.

text: By

text: performing

text: a

text: “kinematical

text: approach”

text: to

text: CG,

text: in

text: previous

text: publications

text: we

text: have

text: shown

text: a

text: diﬀerent

text: way

text: to

text: compute

text: the

text: CG

text: parameters,

text: obtaining

text: values

text: which

text: diﬀer

text: by

text: a

text: few

text: orders

text: of

text: magnitude

text: from

text: those

text: in

text: Eq.

text: γ

text: cm

text: κ

text: cm

text: Using

text: either

text: the

text: values

text: in

text: Eq.

text: or

text: those

text: in

text: Eq.

text: inside

text: Eqs.

text: one

text: always

text: obtains

text: very

text: small

text: corrections

text: to

text: the

text: dynamics

text: of

text: celestial

text: bodies

text: moving

text: within

text: the

text: Solar

text: System.

text: Therefore,

text: CG

text: corrections

text: were

text: always

text: considered

text: negligible

text: at

text: the

text: Solar

text: System

text: level,

text: not

text: aﬀecting

text: in

text: any

text: measurable

text: way

text: the

text: motion

text: of

text: planets,

text: satellites,

text: etc.

text: The

text: eﬀect

text: of

text: CG

text: would

text: only

text: be

text: appreciable

text: at

text: the

text: galactic

text: and

text: cosmological

text: scales,

text: where

text: CG

text: eﬀectively

text: replaces

text: DM

text: and

text: DE.

text: One

text: could

text: argue

text: that

text: since

text: the

text: CG

text: potential

text: in

text: Eq.

text: contains

text: both

text: a

text: linear

text: and

text: a

text: quadratic

text: term

text: in

text: r,

text: the

text: eﬀects

text: of

text: distant

text: stars,

text: galaxies,

text: etc.,

text: should

text: be

text: manifest

text: in

text: our

text: Solar

text: System,

text: thus

text: giving

text: us

text: a

text: way

text: to

text: conﬁrm

text: or

text: rule

text: out

text: Conformal

text: Gravity

text: as

text: a

text: viable

text: gravitational

text: theory.

text: However,

text: CG

text: eﬀects

text: from

text: distant

text: sources

text: only

text: show

text: themselves

text: as

text: “tidal

text: forces”

text: on

text: solar-system

text: bodies,

text: and

text: these

text: tidal

text: eﬀects

text: are

text: completely

text: undetectable

text: in

text: our

text: Solar

text: System

text: The

text: above

text: analysis

text: suggests

text: that

text: CG

text: corrections

text: to

text: the

text: dynamics

text: of

text: the

text: Solar

text: System

text: are

text: essentially

text: negligible

text: and

text: that

text: CG

text: eﬀects

text: are

text: of

text: importance

text: only

text: over

text: larger

text: distance

text: scales.

text: However,

text: these

text: considerations

text: were

text: only

text: based

text: on

text: the

text: static,

text: spherically

text: symmetric

text: CG

text: metric,

text: while

text: many

text: gravitational

text: sources

text: in

text: the

text: Universe

text: are

text: rotating

text: bodies

text: that

text: might

text: even

text: possess

text: electric/magnetic

text: charge.

text: In

text: the

text: rest

text: of

text: this

text: paper

text: we

text: will

text: concentrate

text: our

text: eﬀorts

text: on

text: the

text: Kerr

text: metric

text: in

text: CG,

text: i.e.,

text: on

text: the

text: case

text: of

text: a

text: stationary,

text: axially

text: symmetric

text: rotating

text: system

text: without

text: any

text: charge.

text: The

text: study

text: of

text: solutions

text: for

text: charged

text: bodies

text: goes

text: beyond

text: the

text: purpose

text: of

text: the

text: current

text: work,

text: also

text: because


text: 1. Introduction The anti-de Sitter/conformal ﬁeld theories (AdS/CFT) correspondence [?, ?, ?] has provided us a useful theoretical method to study the strongly coupled systems in various ﬁelds of physics. In recent years, it has been widely applied to study the holographic model of superconductors [?, ?, ?]. It is indicated that the bulk AdS black hole becomes unstable and scalar hair condenses as one tunes the temperature of black hole. Besides the bulk AdS black hole spacetime, since the AdS soliton conﬁguration has the same boundary topology as the Ricci ﬂat black hole and the AdS space in the Poincar´e coordinates, a superconductor phase dual to an AdS soliton conﬁguration without backreaction was studied in Ref. [?]. It was shown that if one increases a chemical potential µ to the AdS soliton, there is a second order phase transition at a critical value µ beyond which the charge scalar ﬁeld turns on, c even at zero temperature. Considering the backreaction of the matter ﬁelds on the soliton geometry, a ﬁrst order phase transition occurs for the same chemical potential as the backreaction is strong enough [?]. When thinking about the higher order correction to the Maxwell ﬁeld, it is of interest to investigate the eﬀect of the Born- Infeld electrodynamics [?, ?, ?, ?, ?, ?] on the insulator/superconductor transition. The Born-Infeld electrodynamics, which was proposed in 1934 to avoid the inﬁnite self-energies for charged point particles arising in Maxwell theory [?], displays good physical properties including the absence of shock waves and birefringence. It was (cid:18) (cid:113) (cid:19) found that the Born-Infeld electrodynamics is the only possible non-linear version of electrodynamics that is invariant under electromagnetic duality transformation [?]. − The Lagrangian density for Born-Infeld theory is L = 1 1 1 + b2FabFab with BI b2 2 F 2 = F F µν and this Lagrangian will reduce to the Maxwell case as the coupling µν parameter b approaches zero. In the limit of the probe approximation, the Born- Infeld factor b has nothing to do with the insulator/superconductor transition [?]. In this paper we will extend the previous works by considering the full backreaction of the matter ﬁelds with the Born-Infeld electrodynamics on the soliton geometry. On the other hand, the entanglement entropy has become a useful quantity for studying some properties in quantum ﬁeld theories and in many-body physics re- cently [?, ?, ?, ?, ?, ?, ?]. As is well known, dividing a given quantum system into A A a subsystem and its complement, the entanglement entropy of is deﬁned as the von Neumann entropy. In quantum many-body physics, the entanglement entropy allows one to distinguish new topological phases and characterize critical points. In the light of the AdS/CFT correspondence, the entanglement entropy may provide us new insights into the quantum structure of spacetime [?, ?]. Ryu and Takayanagi [?, ?] have provided a proposal to compute the entanglement entropy of CFTs from the minimal area surface in gravity side. This proposal provides a simple and ele- gant way to calculate the entanglement entropy of a strongly coupled system which – 1 – ##LTLine## ##LTLine## ##LTLine##

text: Introduction

text: The

text: anti-de

text: Sitter/conformal

text: ﬁeld

text: theories

text: (AdS/CFT)

text: correspondence

text: has

text: provided

text: us

text: a

text: useful

text: theoretical

text: method

text: to

text: study

text: the

text: strongly

text: coupled

text: systems

text: in

text: various

text: ﬁelds

text: of

text: physics.

text: In

text: recent

text: years,

text: it

text: has

text: been

text: widely

text: applied

text: to

text: study

text: the

text: holographic

text: model

text: of

text: superconductors

text: It

text: is

text: indicated

text: that

text: the

text: bulk

text: AdS

text: black

text: hole

text: becomes

text: unstable

text: and

text: scalar

text: hair

text: condenses

text: as

text: one

text: tunes

text: the

text: temperature

text: of

text: black

text: hole.

text: Besides

text: the

text: bulk

text: AdS

text: black

text: hole

text: spacetime,

text: since

text: the

text: AdS

text: soliton

text: conﬁguration

text: has

text: the

text: same

text: boundary

text: topology

text: as

text: the

text: Ricci

text: ﬂat

text: black

text: hole

text: and

text: the

text: AdS

text: space

text: in

text: the

text: Poincar´e

text: coordinates,

text: a

text: superconductor

text: phase

text: dual

text: to

text: an

text: AdS

text: soliton

text: conﬁguration

text: without

text: backreaction

text: was

text: studied

text: in

text: Ref.

text: It

text: was

text: shown

text: that

text: if

text: one

text: increases

text: a

text: chemical

text: potential

text: µ

text: to

text: the

text: AdS

text: soliton,

text: there

text: is

text: a

text: second

text: order

text: phase

text: transition

text: at

text: a

text: critical

text: value

text: µ

text: beyond

text: which

text: the

text: charge

text: scalar

text: ﬁeld

text: turns

text: on,

text: c

text: even

text: at

text: zero

text: temperature.

text: Considering

text: the

text: backreaction

text: of

text: the

text: matter

text: ﬁelds

text: on

text: the

text: soliton

text: geometry,

text: a

text: ﬁrst

text: order

text: phase

text: transition

text: occurs

text: for

text: the

text: same

text: chemical

text: potential

text: as

text: the

text: backreaction

text: is

text: strong

text: enough

text: When

text: thinking

text: about

text: the

text: higher

text: order

text: correction

text: to

text: the

text: Maxwell

text: ﬁeld,

text: it

text: is

text: of

text: interest

text: to

text: investigate

text: the

text: eﬀect

text: of

text: the

text: Born-

text: Infeld

text: electrodynamics

text: on

text: the

text: insulator/superconductor

text: transition.

text: The

text: Born-Infeld

text: electrodynamics,

text: which

text: was

text: proposed

text: in

text: to

text: avoid

text: the

text: inﬁnite

text: self-energies

text: for

text: charged

text: point

text: particles

text: arising

text: in

text: Maxwell

text: theory

text: displays

text: good

text: physical

text: properties

text: including

text: the

text: absence

text: of

text: shock

text: waves

text: and

text: birefringence.

text: It

text: was

text: found

text: that

text: the

text: Born-Infeld

text: electrodynamics

text: is

text: the

text: only

text: possible

text: non-linear

text: version

text: of

text: electrodynamics

text: that

text: is

text: invariant

text: under

text: electromagnetic

text: duality

text: transformation

text: The

text: Lagrangian

text: density

text: for

text: Born-Infeld

text: theory

text: is

text: L

text: with

text: BI

text: F

text: F

text: F

text: µν

text: and

text: this

text: Lagrangian

text: will

text: reduce

text: to

text: the

text: Maxwell

text: case

text: as

text: the

text: coupling

text: µν

text: parameter

text: b

text: approaches

text: zero.

text: In

text: the

text: limit

text: of

text: the

text: probe

text: approximation,

text: the

text: Born-

text: Infeld

text: factor

text: b

text: has

text: nothing

text: to

text: do

text: with

text: the

text: insulator/superconductor

text: transition

text: In

text: this

text: paper

text: we

text: will

text: extend

text: the

text: previous

text: works

text: by

text: considering

text: the

text: full

text: backreaction

text: of

text: the

text: matter

text: ﬁelds

text: with

text: the

text: Born-Infeld

text: electrodynamics

text: on

text: the

text: soliton

text: geometry.

text: On

text: the

text: other

text: hand,

text: the

text: entanglement

text: entropy

text: has

text: become

text: a

text: useful

text: quantity

text: for

text: studying

text: some

text: properties

text: in

text: quantum

text: ﬁeld

text: theories

text: and

text: in

text: many-body

text: physics

text: re-

text: cently

text: As

text: is

text: well

text: known,

text: dividing

text: a

text: given

text: quantum

text: system

text: into

text: A

text: A

text: a

text: subsystem

text: and

text: its

text: complement,

text: the

text: entanglement

text: entropy

text: of

text: is

text: deﬁned

text: as

text: the

text: von

text: Neumann

text: entropy.

text: In

text: quantum

text: many-body

text: physics,

text: the

text: entanglement

text: entropy

text: allows

text: one

text: to

text: distinguish

text: new

text: topological

text: phases

text: and

text: characterize

text: critical

text: points.

text: In

text: the

text: light

text: of

text: the

text: AdS/CFT

text: correspondence,

text: the

text: entanglement

text: entropy

text: may

text: provide

text: us

text: new

text: insights

text: into

text: the

text: quantum

text: structure

text: of

text: spacetime

text: Ryu

text: and

text: Takayanagi

text: have

text: provided

text: a

text: proposal

text: to

text: compute

text: the

text: entanglement

text: entropy

text: of

text: CFTs

text: from

text: the

text: minimal

text: area

text: surface

text: in

text: gravity

text: side.

text: This

text: proposal

text: provides

text: a

text: simple

text: and

text: ele-

text: gant

text: way

text: to

text: calculate

text: the

text: entanglement

text: entropy

text: of

text: a

text: strongly

text: coupled

text: system

text: which

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:18) (cid:19) (cid:90) (cid:34) (cid:32) (cid:114) (cid:33) √ √ − 12 (cid:3) − 1 − b2F abF S = d5x g R + + d5x g 1 1 + ab L2 b2 2 −|∇ − | − | | Ψ iqAΨ 2 m2 Ψ 2 , (2.1) where L is the radius of AdS spacetime, g is the determinant of the metric, q and m are respectively the charge and the mass of the scalar ﬁeld, and b is the Born-Infeld coupling parameter. Since we are interested in including the backreaction, we will choose the ansatz of the geometry of the AdS soliton with the form (cid:2) (cid:3) dr2 − ds2 = + r2 eC(r)dt2 + dx2 + dy2 + eA(r)B(r)dχ2 . (2.2) r2B(r) Without loss of generality, we set L = 1 in this paper. As in the usual AdS soliton, to get a smooth geometry at the tip r , χ should be periodic with the period 0 − 4πe A(r0)/2 Γ = (cid:48) , (2.3) r2B (r ) 0 0 and B(r) vanishes at the tip of the soliton. The electromagnetic ﬁeld and the scalar ﬁeld can be taken as A = φ(r), ψ = ψ(r). (2.4) t The equations of motion under the above ansatz can be obtained as follows – 3 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: abF

text: S

text: g

text: R

text: g

text: ab

text: Ψ

text: iqAΨ

text: Ψ

text: where

text: L

text: is

text: the

text: radius

text: of

text: AdS

text: spacetime,

text: g

text: is

text: the

text: determinant

text: of

text: the

text: metric,

text: q

text: and

text: m

text: are

text: respectively

text: the

text: charge

text: and

text: the

text: mass

text: of

text: the

text: scalar

text: ﬁeld,

text: and

text: b

text: is

text: the

text: Born-Infeld

text: coupling

text: parameter.

text: Since

text: we

text: are

text: interested

text: in

text: including

text: the

text: backreaction,

text: we

text: will

text: choose

text: the

text: ansatz

text: of

text: the

text: geometry

text: of

text: the

text: AdS

text: soliton

text: with

text: the

text: form

text: Without

text: loss

text: of

text: generality,

text: we

text: set

text: L

text: in

text: this

text: paper.

text: As

text: in

text: the

text: usual

text: AdS

text: soliton,

text: to

text: get

text: a

text: smooth

text: geometry

text: at

text: the

text: tip

text: r

text: χ

text: should

text: be

text: periodic

text: with

text: the

text: period

text: Γ

text: (r

text: and

text: B(r)

text: vanishes

text: at

text: the

text: tip

text: of

text: the

text: soliton.

text: The

text: electromagnetic

text: ﬁeld

text: and

text: the

text: scalar

text: ﬁeld

text: can

text: be

text: taken

text: as

text: A

text: φ(r),

text: ψ

text: ψ(r).

text: t

text: The

text: equations

text: of

text: motion

text: under

text: the

text: above

text: ansatz

text: can

text: be

text: obtained

text: as

text: follows

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The Eﬀect of Network and Infrastructural Variables on SPDY’s Performance Yehia Elkhatib The School of Computing & Communications, Lancaster University y.elkhatib@lancaster.ac.uk Gareth Tyson EECS, Queen Mary, University of London gareth.tyson@eecs.qmul.ac.uk Michael Welzl Department of Informatics, University of Oslo michawe@ifi.uio.no May 26, 2020 Abstract HTTP is a successful Internet technology on top of which a lot of the web resides. However, limitations with its current speciﬁcation, i.e. HTTP/1.1, have encouraged some to look for the next generation of HTTP. In SPDY, Google has come up with such a proposal that has growing community acceptance, especially after being adopted by the IETF HTTPbis-WG as the basis for HTTP/2.0. SPDY has the potential to greatly improve web experience with little deployment overhead. However, we still lack an understanding of its true poten- tial in diﬀerent environments. This paper seeks to resolve these issues, oﬀering a comprehensive evaluation of SPDY’s performance using ex- tensive experiments. We identify the impact of network characteristics and website infrastructure on SPDY’s potential page loading beneﬁts, ﬁnding that these factors are decisive for SPDY and its optimal deploy- ment strategy. Through this, we feed into the wider debate regarding HTTP/2.0, exploring the key aspects that impact the performance of this future protocol. 1 Introduction Web pages are constantly increasing in complexity and size. The HTTP Archive reports that the global average web page size has surpassed 1MB in April 2012 [1]. By the start of July 2013, visiting one of the top 1000 1

text: The

label: title

text: Eﬀect

text: of

text: Network

text: and

text: Infrastructural

text: Variables

text: on

text: SPDY’s

text: Performance

text: Yehia

label: author

text: Elkhatib

text: The

text: School

text: of

text: Computing

text: Communications,

text: Lancaster

text: University

text: y.elkhatib@lancaster.ac.uk

text: Gareth

text: Tyson

text: EECS,

text: Queen

text: Mary,

text: University

text: of

text: London

text: gareth.tyson@eecs.qmul.ac.uk

text: Michael

text: Welzl

text: Department

text: of

text: Informatics,

text: University

text: of

text: Oslo

text: michawe@ifi.uio.no

text: May

text: Abstract

text: HTTP

label: abstract

text: is

text: a

text: successful

text: Internet

text: technology

text: on

text: top

text: of

text: which

text: a

text: lot

text: of

text: the

text: web

text: resides.

text: However,

text: limitations

text: with

text: its

text: current

text: speciﬁcation,

text: i.e.

text: have

text: encouraged

text: some

text: to

text: look

text: for

text: the

text: next

text: generation

text: of

text: HTTP.

text: In

text: SPDY,

text: Google

text: has

text: come

text: up

text: with

text: such

text: a

text: proposal

text: that

text: has

text: growing

text: community

text: acceptance,

text: especially

text: after

text: being

text: adopted

text: by

text: the

text: IETF

text: HTTPbis-WG

text: as

text: the

text: basis

text: for

text: SPDY

text: has

text: the

text: potential

text: to

text: greatly

text: improve

text: web

text: experience

text: with

text: little

text: deployment

text: overhead.

text: However,

text: we

text: still

text: lack

text: an

text: understanding

text: of

text: its

text: true

text: poten-

text: tial

text: in

text: diﬀerent

text: environments.

text: This

text: paper

text: seeks

text: to

text: resolve

text: these

text: issues,

text: oﬀering

text: a

text: comprehensive

text: evaluation

text: of

text: SPDY’s

text: performance

text: using

text: ex-

text: tensive

text: experiments.

text: We

text: identify

text: the

text: impact

text: of

text: network

text: characteristics

text: and

text: website

text: infrastructure

text: on

text: SPDY’s

text: potential

text: page

text: loading

text: beneﬁts,

text: ﬁnding

text: that

text: these

text: factors

text: are

text: decisive

text: for

text: SPDY

text: and

text: its

text: optimal

text: deploy-

text: ment

text: strategy.

text: Through

text: this,

text: we

text: feed

text: into

text: the

text: wider

text: debate

text: regarding

text: exploring

text: the

text: key

text: aspects

text: that

text: impact

text: the

text: performance

text: of

text: this

text: future

text: protocol.

text: Introduction

text: Web

text: pages

text: are

text: constantly

text: increasing

text: in

text: complexity

text: and

text: size.

text: The

text: HTTP

text: Archive

text: reports

text: that

text: the

text: global

text: average

text: web

text: page

text: size

text: has

text: surpassed

text: in

text: April

text: By

text: the

text: start

text: of

text: July

text: visiting

text: one

text: of

text: the

text: top


text: sites incurs, on average, 1246kB of web page resources over 100 separate requests [1]. Such growth has been fueled by the emergence of advanced web-based services (Web 2.0, SaaS cloud services, etc.), enhanced client device capabilities (JavaScript browser runtimes [24], display), and increased downlink speeds [4]. This growing complexity, however, can dramatically slow down page retrieval. Unfortunately, this has negative consequences [26], and very real ones in the case of commercial websites. It has been found that most users cannot tolerate in excess of 2 seconds of page load delay [21], and that increments of just 100ms on shopping websites can decrease sales by 1% [19]. The converse is similarly true: decreasing delay can have a powerful enhancing eﬀect, with Google claiming to have increased ad revenue by 20% through cutting 500ms from load times. To mitigate page load times, various extensions to HTTP have been proposed. However, in practice, little progress has been made, with many web servers, proxies and browsers being slow to adopt these new tweaks (e.g. pipelining [11]). In light of these observations, some have proposed developing a new web protocol. Such eﬀorts include Microsoft’s Speed+Mobility [30] and HTML5 Websockets; most prominent, however, is Google’s SPDY [3]. This has already begun to see deployment by prominent organisations such as Google, Twitter, Akamai and Facebook, whilst also being adopted as the base for HTTP/2.0 by the HTTPbis Working Group. Despite this, we still possess a limited understaning of its behaviour, overheads and performance: does it oﬀer a fundamental improvement or just further tweaking? In an attempt to answer the above question, a small number of early stage studies have explored the topic. They oﬀer a range of results, with some claiming signiﬁcant gains and (curiously) others claiming rather neg- ative results. This report seeks to resolve these issues, by analysing the circumstances under which SPDY can improve page load times, and the ones where the opposite is true. To achieve this, we perform a large-scale evaluation of SPDY using an emulated testbed. Based on NPN negotiation handshakes, we found out that the number of top 10,000 Alexa sites adopting SPDY was 208 on October 14th 2012, rising to 271 on April 23rd 2013. Using some of these websites, we execute a large number of probes to measure the performance of SPDY in the wild. Conﬁrming our suspicions, we ﬁnd highly variable results between diﬀerent websites and samples: SPDY has the potential to both beneﬁt and damage page load times. Motivated by this, we perform a large body of controlled experiments in our local testbed to understand the reasons behind these performance variations. We identify the website types and network characteristics that SPDY thrives under, as well as how these beneﬁts vary based on provider-side infrastructural decisions. To our knowledge, this is the ﬁrst eﬀort to oﬀer such insights. The rest of the report is organized as follows. Section provides back- ground and highlights related work. Section describes our measurement 2

text: sites

text: incurs,

text: on

text: average,

text: of

text: web

text: page

text: resources

text: over

text: separate

text: requests

text: Such

text: growth

text: has

text: been

text: fueled

text: by

text: the

text: emergence

text: of

text: advanced

text: web-based

text: services

text: (Web

text: SaaS

text: cloud

text: services,

text: etc.),

text: enhanced

text: client

text: device

text: capabilities

text: (JavaScript

text: browser

text: runtimes

text: display),

text: and

text: increased

text: downlink

text: speeds

text: This

text: growing

text: complexity,

text: however,

text: can

text: dramatically

text: slow

text: down

text: page

text: retrieval.

text: Unfortunately,

text: this

text: has

text: negative

text: consequences

text: and

text: very

text: real

text: ones

text: in

text: the

text: case

text: of

text: commercial

text: websites.

text: It

text: has

text: been

text: found

text: that

text: most

text: users

text: cannot

text: tolerate

text: in

text: excess

text: of

text: seconds

text: of

text: page

text: load

text: delay

text: and

text: that

text: increments

text: of

text: just

text: on

text: shopping

text: websites

text: can

text: decrease

text: sales

text: by

text: The

text: converse

text: is

text: similarly

text: true:

text: decreasing

text: delay

text: can

text: have

text: a

text: powerful

text: enhancing

text: eﬀect,

text: with

text: Google

text: claiming

text: to

text: have

text: increased

text: ad

text: revenue

text: by

text: through

text: cutting

text: from

text: load

text: times.

text: To

text: mitigate

text: page

text: load

text: times,

text: various

text: extensions

text: to

text: HTTP

text: have

text: been

text: proposed.

text: However,

text: in

text: practice,

text: little

text: progress

text: has

text: been

text: made,

text: with

text: many

text: web

text: servers,

text: proxies

text: and

text: browsers

text: being

text: slow

text: to

text: adopt

text: these

text: new

text: tweaks

text: (e.g.

text: pipelining

text: In

text: light

text: of

text: these

text: observations,

text: some

text: have

text: proposed

text: developing

text: a

text: new

text: web

text: protocol.

text: Such

text: eﬀorts

text: include

text: Microsoft’s

text: Speed+Mobility

text: and

text: Websockets;

text: most

text: prominent,

text: however,

text: is

text: Google’s

text: SPDY

text: This

text: has

text: already

text: begun

text: to

text: see

text: deployment

text: by

text: prominent

text: organisations

text: such

text: as

text: Google,

text: Twitter,

text: Akamai

text: and

text: Facebook,

text: whilst

text: also

text: being

text: adopted

text: as

text: the

text: base

text: for

text: by

text: the

text: HTTPbis

text: Working

text: Group.

text: Despite

text: this,

text: we

text: still

text: possess

text: a

text: limited

text: understaning

text: of

text: its

text: behaviour,

text: overheads

text: and

text: performance:

text: does

text: it

text: oﬀer

text: a

text: fundamental

text: improvement

text: or

text: just

text: further

text: tweaking?

text: In

text: an

text: attempt

text: to

text: answer

text: the

text: above

text: question,

text: a

text: small

text: number

text: of

text: early

text: stage

text: studies

text: have

text: explored

text: the

text: topic.

text: They

text: oﬀer

text: a

text: range

text: of

text: results,

text: with

text: some

text: claiming

text: signiﬁcant

text: gains

text: and

text: (curiously)

text: others

text: claiming

text: rather

text: neg-

text: ative

text: results.

text: This

text: report

text: seeks

text: to

text: resolve

text: these

text: issues,

text: by

text: analysing

text: the

text: circumstances

text: under

text: which

text: SPDY

text: can

text: improve

text: page

text: load

text: times,

text: and

text: the

text: ones

text: where

text: the

text: opposite

text: is

text: true.

text: To

text: achieve

text: this,

text: we

text: perform

text: a

text: large-scale

text: evaluation

text: of

text: SPDY

text: using

text: an

text: emulated

text: testbed.

text: Based

text: on

text: NPN

text: negotiation

text: handshakes,

text: we

text: found

text: out

text: that

text: the

text: number

text: of

text: top

text: Alexa

text: sites

text: adopting

text: SPDY

text: was

text: on

text: October

text: rising

text: to

text: on

text: April

text: Using

text: some

text: of

text: these

text: websites,

text: we

text: execute

text: a

text: large

text: number

text: of

text: probes

text: to

text: measure

text: the

text: performance

text: of

text: SPDY

text: in

text: the

text: wild.

text: Conﬁrming

text: our

text: suspicions,

text: we

text: ﬁnd

text: highly

text: variable

text: results

text: between

text: diﬀerent

text: websites

text: and

text: samples:

text: SPDY

text: has

text: the

text: potential

text: to

text: both

text: beneﬁt

text: and

text: damage

text: page

text: load

text: times.

text: Motivated

text: by

text: this,

text: we

text: perform

text: a

text: large

text: body

text: of

text: controlled

text: experiments

text: in

text: our

text: local

text: testbed

text: to

text: understand

text: the

text: reasons

text: behind

text: these

text: performance

text: variations.

text: We

text: identify

text: the

text: website

text: types

text: and

text: network

text: characteristics

text: that

text: SPDY

text: thrives

text: under,

text: as

text: well

text: as

text: how

text: these

text: beneﬁts

text: vary

text: based

text: on

text: provider-side

text: infrastructural

text: decisions.

text: To

text: our

text: knowledge,

text: this

text: is

text: the

text: ﬁrst

text: eﬀort

text: to

text: oﬀer

text: such

text: insights.

text: The

text: rest

text: of

text: the

text: report

text: is

text: organized

text: as

text: follows.

text: Section

text: provides

text: back-

text: ground

text: and

text: highlights

text: related

text: work.

text: Section

text: describes

text: our

text: measurement


text: Connect Send Wait Receive SSL 1 W 0.8 o of To 00..46 HTTPS ati 0.2 R 0 1 W 0.8 o T 0.6 Y of 0.4 PD Ratio 0. 20 S 128256512102420483072Ba40n96dw51i2d0t6h1 4(4K8b19p2s1)02401228814336163841843220480 Figure 5: HAR Times for diﬀerent Bandwidth values (YouTube, RTT=150ms, PLR=0%). 20 0 W -20 o T n in -40 o cti -60 u d e R -80 % -100 Twitter YouTube imgur -120 0 0.5 1 1.5 2 2.5 3 Packet Loss Ratio (%) Figure 6: Eﬀect of Packet Loss (RTT=150ms, BW=1Mbps). Linux kernel ﬁrewall with a stochastic proportional packet processing rule between 0 and 3%4. Figure 6 presents the results. Immediately, we see that SPDY is far more adversely aﬀected by packet loss than HTTPS is. This has been anticipated in other work [29] but never before tested. It is also contrary to what has been reported in the SPDY white paper [2], which states that SPDY is better able to deal with loss. The authors suggest because SPDY sends fewer packets, the negative eﬀect of TCP backoﬀ is mitigated. We ﬁnd that SPDY does, indeed, send fewer packets (upto 49% less due to TCP connection reuse). However, SPDY’s multiplexed connections persist far longer compared to HTTPS. Thus, a lost packet in a SPDY connection has a more profound setback on the long term ≈ 4Packet loss in US mobile networks is reported to be as low as 0.2% [6] and as high as 1.9% [16], but is considerably higher in other countries; e.g. 2-3% in many European ≥ countries and 3% in China, Russia and several South American states [16]. It is also quite high for WiFi [25]. We therefore consider 0–3% to be an appropriate parameter range. 14 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Connect

font: default

text: Send

text: Wait

text: Receive

text: SSL

text: W

text: o

text: of

text: To

text: HTTPS

text: ati

text: R

text: W

text: o

text: T

text: Y

text: of

text: PD

text: Ratio

text: S

text: Figure

text: HAR

label: caption

text: Times

text: for

text: diﬀerent

text: Bandwidth

text: values

text: (YouTube,

text: W

text: o

text: T

text: n

text: in

text: o

text: cti

text: u

text: d

text: e

text: R

text: Twitter

text: YouTube

text: imgur

text: Packet

text: Loss

text: Ratio

text: Figure

text: Eﬀect

text: of

text: Packet

text: Loss

text: Linux

text: kernel

text: ﬁrewall

text: with

text: a

text: stochastic

text: proportional

text: packet

text: processing

text: rule

text: between

text: and

text: Figure

text: presents

text: the

text: results.

text: Immediately,

text: we

text: see

text: that

text: SPDY

text: is

text: far

text: more

text: adversely

text: aﬀected

text: by

text: packet

text: loss

text: than

text: HTTPS

text: is.

text: This

text: has

text: been

text: anticipated

text: in

text: other

text: work

text: but

text: never

text: before

text: tested.

text: It

text: is

text: also

text: contrary

text: to

text: what

text: has

text: been

text: reported

text: in

text: the

text: SPDY

text: white

text: paper

text: which

text: states

text: that

text: SPDY

text: is

text: better

text: able

text: to

text: deal

text: with

text: loss.

text: The

text: authors

text: suggest

text: because

text: SPDY

text: sends

text: fewer

text: packets,

text: the

text: negative

text: eﬀect

text: of

text: TCP

text: backoﬀ

text: is

text: mitigated.

text: We

text: ﬁnd

text: that

text: SPDY

text: does,

text: indeed,

text: send

text: fewer

text: packets

text: (upto

text: less

text: due

text: to

text: TCP

text: connection

text: reuse).

text: However,

text: SPDY’s

text: multiplexed

text: connections

text: persist

text: far

text: longer

text: compared

text: to

text: HTTPS.

text: Thus,

text: a

text: lost

text: packet

text: in

text: a

text: SPDY

text: connection

text: has

text: a

text: more

text: profound

text: setback

text: on

text: the

text: long

text: term

text: loss

text: in

text: US

text: mobile

text: networks

text: is

text: reported

text: to

text: be

text: as

text: low

text: as

text: and

text: as

text: high

text: as

text: but

text: is

text: considerably

text: higher

text: in

text: other

text: countries;

text: e.g.

text: in

text: many

text: European

text: countries

text: and

text: in

text: China,

text: Russia

text: and

text: several

text: South

text: American

text: states

text: It

text: is

text: also

text: quite

text: high

text: for

text: WiFi

text: We

text: therefore

text: consider

text: to

text: be

text: an

text: appropriate

text: parameter

text: range.

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text: TCP throughput than it would in any of HTTPS’ ephemeral connections, the vast majority of which do not last beyond the TCP slow start phase [27]. Furthermore, packet loss in SPDY aﬀects all following requests and responses that are multiplexed over the same TCP connection. In contrast, a packet loss in one of the parallel HTTPS connections would not aﬀect the other connections, neither concurrent nor subsequent (assuming HTTP pipelining is not used, which is commonly the case). In essence, HTTPS ‘spreads the risk’ across multiple TCP connections. On average, we found that SPDY’s throughput is aﬀected by packet loss up to 7 times more than HTTPS (all experiments were performed using the default Linux CUBIC congestion avoidance algorithm). It is also important to note that the probability of packet loss is higher in SPDY. According to [12], the probability of experiencing loss increases in proportion to the position of the packet in a burst chain. Hence, the chance of experiencing a packet tail drop is much higher for longer lived connections such as SPDY’s. Thus, not only does SPDY react badly to packet loss, the chance of it experiencing loss is also higher. This is eﬀectively highlighted in Figure 6; imgur, which has the longest transfer time (by far), exhibits extremely poor performance under packet loss. Finally, these results indicate that SPDY may not perform that well in mobile settings, one of its key target environments [31]. Whilst both SPDY’s high delay and low bandwidth support is desirable in this environment, the beneﬁts can be undone by relatively low levels of packet loss (e.g. 0.5%). 6 Eﬀect of Infrastructural Decisions The previous section has investigated the performance of SPDY under diﬀerent network conditions between a single client and server. However, our original crawling of the Alexa Top 10k highlighted a tendency for providers to implement a practice known as domain sharding. This is the process of distributing page resources across multiple domains (servers), allowing browsers to open more parallel connections to download page resources. Figure 7 presents a CDF of the number of shards we discovered. We ﬁnd that apart from front-less websites (such as media.tumblr.com and aka- maihd.net), all websites employ some degree of domain sharding. Here, we choose to deep dive into the practice of domain sharding to understand the implications of this infrastructural design choice on SPDY. 6.1 Number of Shards To inspect the impact of sharding, we recreate the earlier experimen- tal setup but mirror the webpages across multiple servers, as occurs in real setups. We consider 7 shards, i.e. servers, an appropiate upper limit here as our measurements ﬁnd that 70 of the top 100 Alexa websites have 7 or fewer shards. Each shard is conﬁgured as in Section . We then distribute the webpage resources across these servers. We perform retrievals using con- 15

text: TCP

text: throughput

text: than

text: it

text: would

text: in

text: any

text: of

text: HTTPS’

text: ephemeral

text: connections,

text: the

text: vast

text: majority

text: of

text: which

text: do

text: not

text: last

text: beyond

text: the

text: TCP

text: slow

text: start

text: phase

text: Furthermore,

text: packet

text: loss

text: in

text: SPDY

text: aﬀects

text: all

text: following

text: requests

text: and

text: responses

text: that

text: are

text: multiplexed

text: over

text: the

text: same

text: TCP

text: connection.

text: In

text: contrast,

text: a

text: packet

text: loss

text: in

text: one

text: of

text: the

text: parallel

text: HTTPS

text: connections

text: would

text: not

text: aﬀect

text: the

text: other

text: connections,

text: neither

text: concurrent

text: nor

text: subsequent

text: (assuming

text: HTTP

text: pipelining

text: is

text: not

text: used,

text: which

text: is

text: commonly

text: the

text: case).

text: In

text: essence,

text: HTTPS

text: ‘spreads

text: the

text: risk’

text: across

text: multiple

text: TCP

text: connections.

text: On

text: average,

text: we

text: found

text: that

text: SPDY’s

text: throughput

text: is

text: aﬀected

text: by

text: packet

text: loss

text: up

text: to

text: times

text: more

text: than

text: HTTPS

text: (all

text: experiments

text: were

text: performed

text: using

text: the

text: default

text: Linux

text: CUBIC

text: congestion

text: avoidance

text: algorithm).

text: It

text: is

text: also

text: important

text: to

text: note

text: that

text: the

text: probability

text: of

text: packet

text: loss

text: is

text: higher

text: in

text: SPDY.

text: According

text: to

text: the

text: probability

text: of

text: experiencing

text: loss

text: increases

text: in

text: proportion

text: to

text: the

text: position

text: of

text: the

text: packet

text: in

text: a

text: burst

text: chain.

text: Hence,

text: the

text: chance

text: of

text: experiencing

text: a

text: packet

text: tail

text: drop

text: is

text: much

text: higher

text: for

text: longer

text: lived

text: connections

text: such

text: as

text: SPDY’s.

text: Thus,

text: not

text: only

text: does

text: SPDY

text: react

text: badly

text: to

text: packet

text: loss,

text: the

text: chance

text: of

text: it

text: experiencing

text: loss

text: is

text: also

text: higher.

text: This

text: is

text: eﬀectively

text: highlighted

text: in

text: Figure

text: imgur,

text: which

text: has

text: the

text: longest

text: transfer

text: time

text: (by

text: far),

text: exhibits

text: extremely

text: poor

text: performance

text: under

text: packet

text: loss.

text: Finally,

text: these

text: results

text: indicate

text: that

text: SPDY

text: may

text: not

text: perform

text: that

text: well

text: in

text: mobile

text: settings,

text: one

text: of

text: its

text: key

text: target

text: environments

text: Whilst

text: both

text: SPDY’s

text: high

text: delay

text: and

text: low

text: bandwidth

text: support

text: is

text: desirable

text: in

text: this

text: environment,

text: the

text: beneﬁts

text: can

text: be

text: undone

text: by

text: relatively

text: low

text: levels

text: of

text: packet

text: loss

text: (e.g.

text: Eﬀect

text: of

text: Infrastructural

text: Decisions

text: The

text: previous

text: section

text: has

text: investigated

text: the

text: performance

text: of

text: SPDY

text: under

text: diﬀerent

text: network

text: conditions

text: between

text: a

text: single

text: client

text: and

text: server.

text: However,

text: our

text: original

text: crawling

text: of

text: the

text: Alexa

text: Top

text: highlighted

text: a

text: tendency

text: for

text: providers

text: to

text: implement

text: a

text: practice

text: known

text: as

text: domain

text: sharding.

text: This

text: is

text: the

text: process

text: of

text: distributing

text: page

text: resources

text: across

text: multiple

text: domains

text: (servers),

text: allowing

text: browsers

text: to

text: open

text: more

text: parallel

text: connections

text: to

text: download

text: page

text: resources.

text: Figure

text: presents

text: a

text: CDF

text: of

text: the

text: number

text: of

text: shards

text: we

text: discovered.

text: We

text: ﬁnd

text: that

text: apart

text: from

text: front-less

text: websites

text: (such

text: as

text: media.tumblr.com

text: and

text: aka-

text: maihd.net),

text: all

text: websites

text: employ

text: some

text: degree

text: of

text: domain

text: sharding.

text: Here,

text: we

text: choose

text: to

text: deep

text: dive

text: into

text: the

text: practice

text: of

text: domain

text: sharding

text: to

text: understand

text: the

text: implications

text: of

text: this

text: infrastructural

text: design

text: choice

text: on

text: SPDY.

text: Number

text: of

text: Shards

text: To

text: inspect

text: the

text: impact

text: of

text: sharding,

text: we

text: recreate

text: the

text: earlier

text: experimen-

text: tal

text: setup

text: but

text: mirror

text: the

text: webpages

text: across

text: multiple

text: servers,

text: as

text: occurs

text: in

text: real

text: setups.

text: We

text: consider

text: shards,

text: i.e.

text: servers,

text: an

text: appropiate

text: upper

text: limit

text: here

text: as

text: our

text: measurements

text: ﬁnd

text: that

text: of

text: the

text: top

text: Alexa

text: websites

text: have

text: or

text: fewer

text: shards.

text: Each

text: shard

text: is

text: conﬁgured

text: as

text: in

text: Section

text: We

text: then

text: distribute

text: the

text: webpage

text: resources

text: across

text: these

text: servers.

text: We

text: perform

text: retrievals

text: using

text: con-