arXiv:1804.00964v1 [hep-th] 3 Apr 2018
Einstein Double Field Equations S TEPHEN A NGUS , K YOUNGHO C HO
AND
J EONG -H YUCK PARK
Department of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, KOREA
[email protected]
[email protected]
[email protected]
Abstract Upon treating the whole closed string massless sector as stringy graviton fields, Double Field Theory may evolve into Stringy Gravity, i.e. the stringy augmentation of General Relativity. Equipped with an O(D, D) covariant differential geometry beyond Riemann, we spell out the definition of the EnergyMomentum tensor in Stringy Gravity and derive its on-shell conservation law from doubled general covariance. Equating it with the recently identified stringy Einstein curvature tensor, all the equations of motion of the closed string massless sector are unified into a single expression, GAB = 8πGTAB , which we dub the Einstein Double Field Equations. As an example, we study the most general D = 4 static, asymptotically flat, spherically symmetric, ‘regular’ solution, sourced by the stringy Energy-Momentum tensor which is nontrivial only up to a finite radius from the center. Outside this radius, the solution matches the known vacuum geometry which has four constant parameters. We express these as volume integrals of the interior stringy Energy-Momentum tensor and discuss relevant energy conditions.
One must be prepared to follow up the consequence of theory, and feel that one just has to accept the consequences no matter where they lead. Paul Dirac Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. Steven Weinberg
Contents 1 Introduction
2
2 GR before DFT
5
2.1
Energy-Momentum tensor and Einstein Field Equations in GR . . . . . . . . . . . . . . . .
5
2.2
Regular spherically symmetric solution in D = 4 GR . . . . . . . . . . . . . . . . . . . . .
8
3 Einstein Double Field Equations
12
3.1
Review of DFT as Stringy Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2
Stringy Energy-Momentum tensor & Einstein Double Field Equations . . . . . . . . . . . . 25 3.2.1
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Further-generalized Lie derivative, Leξ
36
5 Regular spherical solution to Einstein Double Field Equations
39
5.1
Most general D = 4 spherical ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2
Solving the Einstein Double Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3
Energy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Conclusion
61
1
1 Introduction The Einstein-Hilbert action is often referred to as ‘pure’ gravity, as it is formed by the unique two-derivative scalar curvature of the Riemannian metric. Minimal coupling to matter follows unambiguously through the usual covariant derivatives, ▽µ = ∂µ + γµ + ωµ ,
ρ γµσ = 12 gρτ (∂µ gστ + ∂σ gµτ − ∂τ gµσ ) ,
λ e ), ωµpq = ep ν (∂µ eνq − γµν λq
(1.1) which ensures covariance under both diffeomorphisms and local Lorentz symmetry. In the words of ChengNing Yang, symmetry dictates interaction. The torsionless Christoffel symbols of the connection and the spin connection are fixed by the requirement of compatibility with the metric and the vielbein. The existence of Riemann normal coordinates supports the Equivalence Principle, as the Christoffel symbols vanish pointwise. Needless to say, in General Relativity (GR), the metric is privileged to be the only geometric and thus gravitational field, on account of the adopted differential geometry a la Riemann, while all other fields are automatically categorized as additional matter. String theory may put some twist on this Riemannian paradigm. First of all, the metric is merely one segment of closed string massless sector which consists of a two-form gauge potential, Bµν , and a scalar dilaton, φ, in addition to the metric, gµν . A genuine stringy symmetry called T-duality then converts one to another [1, 2]. Namely, the closed string massless sector forms multiplets of O(D, D) T-duality. This may well hint at the existence of Stringy Gravity as an alternative to GR, which takes the entire massless sector as geometric and therefore gravitational. In recent years this idea has been realized concretely through the developments of so-called Double Field Theory (DFT) [3–8]. The relevant covariant derivative has been identified [9, 10] and reads schematically, ¯A , DA = ∂A + ΓA + ΦA + Φ
(1.2)
¯A where ΓA is the DFT version of the Christoffel symbols for generalized diffeomorphisms, while ΦA and Φ are the two spin connections for the twofold local Lorentz symmetries, Spin(1, D−1) × Spin(D−1, 1). They are compatible with, and thus formed by, the closed string massless sector, containing in particular the H-flux (H = dB). The doubling of the spin group implies the existence of two separate locally inertial frames for left and right closed string modes, respectively [11]. In a sense, it is a prediction of DFT (and also Generalized Geometry [12]) that there must in principle exist two distinct kinds of fermions [13]. The DFTChristoffel symbols constitute DFT curvatures: scalar and ‘Ricci’. The scalar curvature naturally defines 2
the pure DFT Lagrangian in analogy with GR. However, in Stringy Gravity the Equivalence Principle is generically broken [13, 14]: there exist no normal coordinates in which the DFT-Christoffel symbols would vanish pointwise. This should not be a surprise since, strictly speaking, the principle holds only for a point particle and does not apply to an extended object like a string, which is subject to ‘tidal forces’ via coupling to the H-flux. Beyond the original goal of reformulating supergravities in a duality-manifest framework, DFT turns out to have quite a rich spectrum. It describes not only the Riemannian supergravities but also various nonRiemannian theories in which the Riemannian metric cannot be defined [15], such as non-relativistic Newton– Cartan or ultra-relativistic Carroll gravities [16], the Gomis–Ooguri non-relativistic string [17, 18], and various chiral theories including the one by Siegel [19]. Without resorting to Riemannian variables, supersymmetrizations have been also completed to the full order in fermions, both on target spacetime [20, 21] and on worldsheet [22]. Combining the scalar and ‘Ricci’ curvatures, the DFT version of the Einstein curvature, GAB , which is identically conserved, DA GA B = 0, and generically asymmetric, GAB 6= GBA , has been identified [23]. Given this identification, it is natural to anticipate the ‘Energy-Momentum’ tensor in DFT, say TAB , which
should counterbalance the stringy Einstein curvature through the Einstein Double Field Equations, i.e. the equations of motion of the entire closed string massless sector as the stringy graviton fields, GAB = 8πG TAB ,
(1.3)
where G (without any subscript index) denotes Newton’s constant. For consistency, the stringy EnergyMomentum tensor should be asymmetric, TAB 6= TBA , and conserved, DA T A B = 0, especially on-shell, i.e. up to the equations of motion of the additional matter fields.
In order to compare the ‘gravitational’ aspects of DFT and GR, circular geodesic motions around the most general spherically symmetric solution to ‘GAB = 0’ have been studied in [24] for the case of D = 4. While the solution was a re-derivation of a previously known result in the supergravity literature [25], the new interpretation was that it is the ‘vacuum’ solution to DFT, with the right-hand side of (1.3) vanishing: it is analogous to the Schwarzschild solution in GR. The DFT spherical vacuum solution turns out to have four (or three, up to a radial coordinate shift) free parameters, in contrast to the Schwarzschild geometry which possesses only one free parameter, i.e. mass. With these extra free parameters, DFT modifies GR 3
at ‘short’ scales in terms of the dimensionless parameter R/(M G), i.e. the radial distance normalized by the mass times Newton’s constant. For large R/(M G), DFT converges to GR, but for finite R/(M G) they differ generically. It is an intriguing fact that the dark matter and dark energy problems all arise from astronomical observations at smaller R/(M G) . 107 , corresponding to long distance divided by far heavier mass [14, 24]. Such a ‘uroboros’ spectrum of R/(M G) is listed below in natural units.
The purpose of the present paper is twofold: i) to propose the definition of the stringy Energy-Momentum tensor which completes the Einstein Double Field Equations spelled out in (1.3), and ii) to analyze the most general spherically symmetric D = 4 ‘regular’ solution which will teach us the physical meanings of the free parameters appearing in the vacuum solution of [24, 25]. The rest of the paper is organized as follows. – In section 2 we collect some known features of GR, such as the general properties of the EnergyMomentum tensor and the most general spherically symmetric (Schwarzschild type) regular solution to the undoubled Einstein Field Equations. We shall double-field-theorize them in the following sections. – We start section 3 by reviewing DFT as Stringy Gravity. We then consider coupling to generic matter fields, propose the definition of the stringy Energy-Momentum tensor, and discuss its properties including the conservation law. Some examples will follow. – In section 4 we devise a method to address isometries in the vielbein formulation of Stringy Gravity. We generalize the known generalized Lie derivative one step further, to a ‘further-generalized Lie derivative’, which acts not only on O(D, D) vector indices but also on all the Spin(1, D−1) × Spin(D−1, 1) local Lorentz indices.
– Section 5 is devoted to the study of the most general, asymptotically flat, spherically symmetric, static ‘regular’ solution to the D = 4 Einstein Double Field Equations. We postulate that the stringy Energy-Momentum tensor is nontrivial only up to a finite cutoff radius, rc . While we recover the vacuum solution of [24] for r > rc , we derive integral expressions for its constant parameters in terms of the stringy Energy-Momentum tensor for r < rc , and discuss relevant energy conditions. – We conclude with our summary and comments in section 6. 4
2 GR before DFT In this section, after reviewing the Energy-Momentum tensor and Einstein Field Equations in GR, we discuss the most general static spherically symmetric regular solution in four-dimensional spacetime.
2.1 Energy-Momentum tensor and Einstein Field Equations in GR Let us consider the Einstein–Hilbert action coupled to generic matter, Z √ 1 dDx −g 16πG R + Lmatter ,
(2.1)
where Lmatter is the Lagrangian for various matter fields, which we denote by Υa . The Energy-Momentum tensor is conventionally defined by
where
δ δg µν
√ −2 δ( −gLmatter ) δLmatter Tµν := √ = −2 + gµν Lmatter , µν −g δg δgµν
(2.2)
is the standard functional derivative which is best computed from the infinitesimal variation of
the Lagrangian density induced by δgµν , as δ
√
√ √ δLmatter −gLmatter + 21 −g δgµν gµν Lmatter = −g δgµν + total derivatives , δgµν
(2.3)
where the disregarded total derivatives appear generically, since Lmatter includes covariant derivatives of the form ▽µΥa , and the connections contain the derivatives of the metric. The right-hand side of (2.3)
matter , as the functional ‘coefficient’ of then gives, or operationally defines, the functional derivative, δLδg µν √ µν −g δg . It is also useful to note that if we vary not the Lagrangian density but the weightless scalar,
Lmatter , covariant total derivatives of the form ‘▽µ J µ ’ will appear and should be neglected, δLmatter = δgµν
δLmatter + covariant total derivatives . δgµν
Of course, when Lmatter involves fermions, √ −gLfermion = |e| ρ¯(γ µ ▽µ ρ + mρ) ,
(2.4)
(2.5)
one should consider a priori the variation of the vielbein rather than the metric, ρ δγµν = 12 gρσ ▽µ δgσν + ▽ν δgµσ − ▽σ δgµν ,
δgµν = 2e(µ a δeν)a ,
−1 −1 δωµab = eµ c e−1 [a ▽c] δeνb − e[b ▽c] δeνa + e[a ▽b] δeνc .
5
(2.6)
Up to the fermionic equations of motion,
we note that
γ µ ▽µ ρ + mρ ≡ 0 ,
(2.7)
▽λ ρ¯γ λµν ρ ≡ −2 ρ¯γ [µ ▽ν] ρ + ▽[µ ρ¯γ ν] ρ .
(2.8)
Here the equivalence symbol, ‘≡’, denotes on-shell equality which holds up to the equations of motion of the matter fields. Using this relation and disregarding any total derivatives (≃), we derive the variation of √ the fermionic Lagrangian, −gLfermion (2.5), given by h i δ |e| ρ¯(γ µ ▽µ ρ + mρ) ≃ 21 |e| δeµp ▽(µ ρ¯γ p) ρ − ρ¯γ (µ ▽p) ρ = − 14 |e| δgµν ▽(µ ρ¯γν) ρ − ρ¯γ(µ ▽ν) ρ . (2.9)
This result shows that even for fermions, the definition of the Energy-Momentum tensor through the ‘metric’ variation (2.2) is in a sense still valid, leading to the following ‘symmetric’ contribution, fermion Tµν = 21 ▽(µ ρ¯γν) ρ − ρ¯γ(µ ▽ν) ρ .
(2.10)
That is to say, the Energy-Momentum tensor is always symmetric in GR.1 In fact, this can be shown in a more general setup. The arbitrary variation of the vielbein decomposes into two parts, δeµ a = eµb δeν (a eb)ν + δeν [a eb]ν eµb = 12 eaν δgµν + δeν [a eb]ν eµb ,
(2.11)
of which the last term can be viewed as the infinitesimal local Lorentz transformation of the vielbein. Since Lmatter is supposed to be a singlet of local Lorentz symmetry, using (2.11) the generic variation of Lmatter can be written as δLmatter =
δLmatter δeµ a δeµ a
δLmatter + δΥa + · · · = 12 δgµν δΥa
δLmatter eaν δeµ a
+ δ ′ Υa
δLmatter + · · · . (2.12) δΥa
Here the dots, ‘· · · ’, denote any disregarded covariant total derivatives; δ′ Υa is the arbitrary variation of the matter field supplemented by the infinitesimal local Lorentz rotation, δeν [a eb]ν . Further,
δLmatter δΥa
corre-
sponds to the Euler–Lagrange equation for each matter field, Υa . Thus, the Energy-Momentum tensor (2.2) is ‘on-shell’ equivalent to δLmatter δ(eLmatter ) δ(eLmatter ) δLmatter 1 −1 1 Tµν = − 2 e eµa + eνa + eνa = − 2 eµa + gµν Lmatter . δea ν δea µ δea ν δea µ (2.13) 1
c.f. (3.81), (3.82) for Stringy Gravity, i.e. DFT.
6
Now for the Einstein–Hilbert term, it is useful to remember that the variation of the Riemann curvature is given by covariant total derivatives, ρ ρ ) − ▽ν (δγµσ ), δRρ σµν = ▽µ (δγνσ
(2.14)
ρ ν δR = δ(gµν Rµν ) = δgµν Rµν + ▽ρ (gµν δγνµ − gρµ δγνµ ),
(2.15)
√ √ √ √ ρ ν δ( −gR) = δgµν −g(Rµν − 12 gµν R) + ∂ρ ( −gg µν δγνµ − −gg ρµ δγνµ ).
(2.16)
such that
and
Bringing this all together, while disregarding any surface integral, the arbitrary variation of the action reads δ
Z
≡
dDx Z
√ −g
√ dDx −g
1 16πG R
+ Lmatter
µν 1 16πG δg
Rµν − 21 gµν R − 8πG Tµν
δLmatter + δ ′ Υa δΥa
(2.17) .
Clearly the equation of motion of the metric leads to the (undoubled) Einstein Field Equations, Rµν − 21 gµν R = 8πGTµν , which are equivalent to
Rµν = 8πG Tµν −
1 λ D−2 gµν T λ
(2.18)
.
(2.19)
In particular, when the variation is caused by diffeomorphisms, i.e. δgµν = Lξ gµν = −2▽(µ ξ ν) , the action
should be invariant. From (2.17), we get, up to the equations of motion, integrals, Z √ 0 ≡ dDx ∂µ ξ µ −g
1 16πG R
+ Lmatter
≡
1 8πG
Z
dDx
√
δLmatter δΥa
≡ 0, and up to surface
−gξν ▽µ Rµν − 12 gµν R − 8πG T µν .
(2.20)
This should hold for arbitrary ξ µ , e.g. highly localized delta-function-type vector fields. Therefore the Einstein curvature and the Energy-Momentum tensor should be conserved off-shell and on-shell, respectively, ▽µ (Rµν − 21 gµν R) = 0 ,
▽µ T µν ≡ 0 .
(2.21)
As is well known, the former can also be obtained directly from the Bianchi identity, ▽[λ Rµν]ρσ = 0.
7
For any Killing vector satisfying the isometric condition, Lξ gµν = ▽µ ξν + ▽ν ξµ = 0 ,
(2.22)
the conservation of the Energy-Momentum tensor (2.21) implies the existence of a conserved Noether current,
√ √ √ ∂µ ( −gT µ ν ξ ν ) = −g▽µ (T µν ξν ) ≡ −gT µν ▽(µ ξν) ≡ 0 ,
(2.23)
which in turn defines a Noether charge, Q[ξ] :=
Z
dxD−1
√
−gT t µ ξ µ .
(2.24)
2.2 Regular spherically symmetric solution in D = 4 GR Here we derive the most general, asymptotically flat, spherically symmetric, static, regular solution to the D = 4 Einstein Field Equations. We require the metric and Energy-Momentum tensor to be spherically symmetric, (2.25)
Lξa Tµν = 0 ,
Lξa gµν = 0 ,
with three Killing vectors, ξaµ , a = 1, 2, 3, corresponding to the usual angular momentum differential operators, ξ1 = sin ϕ∂ϑ + cot ϑ cos ϕ∂ϕ ,
ξ2 = − cos ϕ∂ϑ + cot ϑ sin ϕ∂ϕ ,
ξ3 = −∂ϕ .
(2.26)
They satisfy the so(3) commutation relation,
X ǫabc ξc . ξa , ξb =
(2.27)
c
It follows from (2.25) that gϑϕ = gϕϑ = 0 ,
gϕϕ = sin2 ϑgϑϑ ,
Tϑϕ = Tϕϑ = 0 ,
Tϕϕ = sin2 ϑTϑϑ .
(2.28)
Furthermore, without loss of generality, we can put gtr = 0, gϑϑ = r 2 , and set the metric to be diagonal, utilizing diffeomorphisms (see e.g. [26]), ds2 = −B(r)dt2 + A(r)dr 2 + r 2 dΩ2 , 8
(2.29)
where we put as shorthand notation, dΩ2 := dϑ2 + sin2 ϑ dϕ2 .
(2.30)
The nonvanishing Christoffel symbols are then exhaustively, B′ 2B
t = γt = γtr rt
γttr =
,
B′ 2A
r = γrr
,
A′ 2A
, (2.31)
r =−r , γϑϑ A
r = − sin2 ϑ r , γϕϕ A
ϑ = γϑ = γrϑ ϑr
ϑ = − sin ϑ cos ϑ , γϕϕ
ϕ ϕ γrϕ = γϕr =
ϕ ϕ γϑϕ = γϕϑ = cot ϑ ,
1 r
,
1 r
,
and subsequently the Ricci curvature, Rµν , becomes diagonal, with components Rtt =
B ′′ 2A
−
Rϑϑ = 1 +
B′ 4A
r 2A
A′ A
A′
+
−
A
B′ B
B′ B
+
−
1 A
1 B′ r A
′′
B Rrr = − 2B +
,
B′ 4B
A′ A
+
B′ B
+
1 A′ r A
, (2.32)
Rϕϕ = sin2 ϑ Rϑϑ .
,
Since both the Ricci curvature and the metric are diagonal, the Einstein Field Equations imply that the Energy-Momentum tensor must also be diagonal, thus fixing Ttr = 0, Tµν = diag Ttt , Trr , Tϑϑ , Tϕϕ = sin2 ϑ Tϑϑ .
(2.33)
Now the conservation of the Energy-Momentum tensor, ▽µ T µ ν , boils down to a single equation: d B′ 2 r T r − T ϑϑ + (T r r ) + T rr − T tt = 0 . dr r 2B
(2.34)
The Einstein Field Equations, or (2.19), reduce to B ′′ 2A
−
Rtt
=
Rrr
B = − 2B +
Rϑϑ =
′′
1+
B′ 4A
B′ 4B r 2A
A′ A
+
A′ A
A′ A
+
−
B′ B
B′ B B′ B
+
+
−
= −4πGB T t t − T r r − 2T ϑ ϑ ,
1 B′ r A
= −4πGA T t t − T r r + 2T ϑ ϑ ,
1 A′ r A
= −4πGr 2 T t t + T r r ,
1 A
9
(2.35)
which are linearly equivalent to d dr d dr
r 1−
1 A
ln(AB) =
B ′′ B
−
B′ 2B
A′ A
=
A′ A
rA′ A2 B′ B
+
+
B′ B
+1−
1 A
= −8πGr 2 T t t ,
= −8πGAr(T t t − T r r ) , 1 r
−
A′ A
−
B′ B
= 16πGAT ϑ ϑ .
The first equation can be integrated to give Z r 1 1 = −G dr ′ 4πr ′2 T t t (r ′ ) , 2r 1 − A 0 where we have assumed ‘regularity’ at the origin, 1 = 0. lim r 1 − r→0 A This fixes the function A(r),
1
A(r) = for which we have defined M (r) := −
1−
Z
r
2GM (r) r
(2.36)
,
dr ′ 4πr ′2 T t t (r ′ ) .
(2.37)
(2.38)
(2.39)
(2.40)
0
The regularity condition (2.44) is then equivalent to lim M (r) = 0 .
r→0
(2.41)
Furthermore, the positive energy (density) condition implies T tt ≥ 0, such that, owing to the convention of
the mostly plus signature of the metric (2.29), M (r) is generically positive. Similarly, assuming the ‘flat’ boundary condition at infinity, lim A(r)B(r) = 1 ,
r→∞
the second equation in (2.36) can be integrated to fix B(r), Z ∞ t ′ 2GM (r) ′ ′ ′ r ′ dr r A(r ) T t (r ) − T r (r ) , B(r) = 1 − exp 8πG r r 10
(2.42)
(2.43)
such that the metric takes the final form: 2GM (r) dr2 2 −2∆(r) ds = −e + r 2 dΩ2 . 1− dt2 + (r) r 1 − 2GM r
(2.44)
Here M (r) is given by (2.40) and ∆(r) is defined by ∆(r) := 4πG
Z
∞
dr
′
r
r ′ T r (r ) − T t t (r ′ ) r ′ 1−
2GM (r ′ ) r′
.
(2.45)
Finally, we show that, up to the first and the second relations in (2.36) and their solutions, (2.39), (2.43), the third relation is equivalent to the conservation of the Energy-Momentum tensor (2.34). For this, we solve for T r r and T r r − T t t from the first and the second relations in (2.36), T rr =
1 8πG
B′ ABr
+
1 Ar 2
−
1 r2
T r r − T tt =
,
1 8πGAr
A′ A
+
B′ B
,
(2.46)
and substitute these two expressions into the left-hand side of the conservation relation (2.34), to obtain d dr
(T r r ) +
=
h
1 8πGAr
2 r B ′′ B
T r r − T ϑϑ + −
B′
2B
A′ A
+
B′ 2B
B′ B
T rr − T tt −
1 r
A′ A
−
B′ B
i
(2.47)
− 16πGAT ϑ ϑ .
This result clearly establishes the equivalence between the Energy-Momentum conservation equation (2.34) and the third relation in (2.36). Some comments are in order. – When the matter is localized up to a finite radius rc , such that outside this radius, r > rc , we have Tµν (r) = 0 and ∆(r) = 0, we recover the Schwarzschild solution, in which the mass agrees with the ADM mass [27] and from (2.40) is further given by the volume integral, Z ∞ Z rc 2 t dr 4πr 2 T t t (r) . dr 4πr T t (r) = − M =−
(2.48)
0
0
However, this differs from the Noether charge (2.24) of the time translational Killing vector, Z ∞ dr 4πr 2 e−∆(r) T t t (r) , M 6= −Q[∂t ] = − 0
11
(2.49)
since from (2.44) the integral measure is nontrivial, √ −g = e−∆(r) r 2 sin ϑ 6= r 2 sin ϑ .
(2.50)
This discrepancy and its remedy by an extra surface integral are rather well known, see [28–32]. – If there is a spherical void in which Tµν = 0 for r1 < r < r2 , both M (r) and ∆(r) become constant inside the void as M (r) = M (r1 ) and ∆(r) = ∆(r2 ). After a constant rescaling of the time, tnew = e−∆(r2 ) t, the local geometry inside the void coincides precisely with the Schwarzschild solution. We note that the mass is determined through the integral over 0 < r < r1 only and is independent of the matter distribution outside, r > r2 . While this is certainly true in Newtonian gravity (namely the iron sphere theorem), if we solved the vacuum Einstein Field Equations with vanishing Energy-Momentum tensor inside the void, we would merely recover the Schwarzschild geometry in accordance with Birkhoff’s theorem. Nevertheless it would be hard to conclude that the constant mass parameter is unaffected by the outer region. – The radial derivative of B(r) amounts to the gravitational acceleration for a circular geodesic [24], r
dϑ dt
2
1 dgtt (r) GM (r) r 1 ′ =− = 2B = + 4πGrT r (r) e−2∆(r) . 2 dr r2
(2.51)
Again, inside a void or the outer vacuum region, we may absorb the constant factor of e−2∆(r) into the rescaled time, and thus recover the Keplerian acceleration, r
dϑ dt
2
=
GM . r2
(2.52)
3 Einstein Double Field Equations In this section we first give a self-contained review of DFT as Stringy Gravity, following which we propose the DFT, or stringy, extensions of the Energy-Momentum tensor and the Einstein Field Equations.
12
3.1 Review of DFT as Stringy Gravity We review DFT following the geometrically logical—rather than historical—order: i) conventions, ii) the doubled-yet-gauged coordinate system with associated diffeomorphisms, iii) the field content of stringy gravitons, iv) DFT extensions of the Christoffel symbols and spin connection, and v) covariant derivatives and curvatures. For complementary aspects, we refer readers to [33–35] as well as [36, 37]. • Symmetries and conventions
The built-in symmetries of Stringy Gravity are as follows. – O(D, D) T-duality – DFT diffeomorphisms – Twofold local Lorentz symmetries,2 Spin(1, D−1) × Spin(D−1, 1). We shall use capital Latin letters, A, B, . . . , M, N, . . . for the O(D, D) vector indices, while unbarred small Latin letters, p, q, . . . or Greek letters, α, β, . . . will be used for the vectorial or spinorial indices of Spin(1, D−1), respectively. Similarly, barred letters denote the other Spin(D−1, 1) representations: ¯ . . . (spinorial). In particular, each vectorial index can be freely lowered or p¯, q¯, . . . (vectorial) and α ¯ , β, raised by the relevant invariant metric, JAB
0 1 , = 1 0
ηpq = diag(− + + · · · +) ,
η¯p¯q¯ = diag(+ − − · · · −) .
(3.1)
• Doubled-yet-gauged coordinates and diffeomorphisms
By construction, functions admitted to Stringy Gravity are of special type. Let us denote the set of all the functions in Stringy Gravity by F = { Φi }, which should include not only physical fields but also local sym-
metry parameters. First of all, each function, Φi (x), has doubled coordinates, xM , M = 1, 2, · · · , D+D, as its arguments. Not surprisingly, the set is closed under addition, product and differentiation such that, if Φi , Φj ∈ F and a, b ∈ R, then a Φi + b Φj ∈ F , 2
Φi Φj ∈ F ,
In the most general case, the two spin groups can have different dimensions [15].
13
∂A Φi ∈ F ,
(3.2)
and hence Φi is C ∞ . The truly nontrivial property of F is that every function therein is invariant under a special class of translations: for arbitrary Φi , Φj , Φk ∈ F,
∆M = Φj ∂ M Φ k ,
Φi (x) = Φi (x + ∆) ,
(3.3)
where ∆M is said to be derivative-index-valued. We emphasize that this very notion is only possible thanks to the built-in O(D, D) group structure, whereby the invariant metric can raise the vector index of the partial derivative, ∂ M = J M N ∂N . It is straightforward to show3 that the above translational invariance is equivalent to the so-called ‘section condition’,
∂M ∂ M Φi = 0 ,
∂M Φi ∂ M Φj = 0 ,
(3.4)
which is of practical utility. From (3.3), we infer that ‘physics’ should be invariant under such a shift of ∆M = Φj ∂ M Φk . This observation further suggests that the doubled coordinates may be gauged by an equivalence relation [38], xM ∼ xM + ∆ M ,
∆M ∂M = 0 .
(3.5)
Diffeomorphisms in the doubled-yet-gauged spacetime are then generated (actively) by the generalized Lie derivative, Lˆξ , which was introduced initially by Siegel [4], and also later by Hull and Zwiebach [6]. Acting on an arbitrary tensor density, TM1 ···Mn ∈ F, with weight ω, it reads Lˆξ TM1 ···Mn := ξ N ∂N TM1 ···Mn + ω∂N ξ N TM1 ···Mn +
n X i=1
(∂Mi ξN − ∂N ξMi )TM1 ···Mi−1 N Mi+1 ···Mn . (3.6)
Thanks to the section condition, the generalized Lie derivative forms a closed algebra, h i ˆ ˆ Lζ , Lξ = Lˆ[ζ,ξ]C ,
where the so-called C-bracket is given by 1 ˆ M ˆξ ζ M = ζ N ∂N ξ M − ξ N ∂N ζ M + 1 ξ N ∂ M ζN − 1 ζ N ∂ M ξN . [ζ, ξ]M = L ξ − L ζ C 2 2 2 3
(3.7)
(3.8)
Consider the power series expansion of Φi (x + s∆) around s = 0, where we have introduced a real parameter, s ∈ R.
The linear-order term gives ∂M Φi ∂ M Φj = 0, which in turn, after replacing Φi and Φj by ∂L Φ and ∂N Φ, implies that ∂L ∂ M Φ∂M ∂ N Φ = 0. Consequently, ∂M ∂ N Φ is a nilpotent matrix and thus must be traceless, ∂M ∂ M Φ = 0 [39].
14
Along with this expression, it is worthwhile to note the ‘sum’, Lˆζ ξ M + Lˆξ ζ M = ∂ M (ζN ξ N ) .
(3.9)
Further, if the parameter of the generalized Lie derivative, ξ M , is ‘derivative-index-valued’, the first two terms on the right-hand side of (3.6) are trivial. Moreover, if this parameter is ‘exact’ as ξ M = ∂ M Φ, the generalized Lie derivative itself vanishes identically. Now, the closure (3.7) implies that the generalized Lie derivative is itself diffeomorphism-covariant: δξ (Lˆζ TM1 ···Mn ) = Lˆζ (δξ TM1 ···Mn ) + Lˆδξ ζ TM1 ···Mn = Lˆζ Lˆξ TM1 ···Mn + LˆLˆξ ζ TM1 ···Mn
(3.10)
= Lˆξ Lˆζ TM1 ···Mn + Lˆ[ζ,ξ]C+Lˆξ ζ TM1 ···Mn = Lˆξ (Lˆζ TM1 ···Mn ) , ˆ M = 1 ∂ M (ζN ξ N ) , which where in the last step, from (3.8), (3.9), we have used the fact that [ζ, ξ]M C + Lξ ζ 2
is exact and hence null as a diffeomorphism parameter. However, if the tensor density carries additional
Spin(1, D−1) × Spin(D−1, 1) indices, e.g. TM p¯qαα¯ , its generalized Lie derivative is not local-Lorentz-
covariant. Hence the generalized Lie derivative is covariant for doubled-yet-gauged diffeomorphisms but not for local Lorentz symmetries. We shall fix this limitation in section 4 by further generalizing the generalized Lie derivative. In contrast to ordinary Riemannian geometry, the infinitesimal one-form, dxM , is not (passively) diffeomor-
phism covariant in doubled-yet-gauged spacetime, δxM = ξ M ,
δdxM = dξ M = dxN ∂N ξ M 6= (∂N ξ M − ∂ M ξN )dxN .
(3.11)
Furthermore, it is not invariant under the coordinate gauge symmetry shift, dxM 6= d(xM +∆M ). However,
if we gauge dxM explicitly by introducing a derivative-index-valued gauge potential, AM , DxM := dxM − AM ,
AM ∂M = 0 ,
15
(3.12)
we can ensure both the diffeomorphism covariance and the coordinate gauge symmetry invariance, δxM = ξ M ,
δAM = ∂ M ξN (dxN − AN )
=⇒
δ(DxM ) = (∂N ξ M − ∂ M ξN )DxN ;
δxM = ∆M ,
δAM = d∆M
=⇒
δ(DxM ) = 0 . (3.13)
Utilizing the gauged infinitesimal one-form,
DxM ,
it is then possible to define the duality-covariant ‘proper
length’ in doubled-yet-gauge spacetime [14, 15], and construct associated sigma models such as for the point particle [24, 41], bosonic string [39, 40], Green-Schwarz superstring [22], exceptional string [42, 43], etc. With the decomposition of the doubled coordinates, xM = (˜ xµ , xν ), in accordance with the form of the O(D, D) invariant metric, JM N (3.1), the section condition reads ∂˜µ ∂µ = 0. Thus up to O(D, D) rotations, the section condition is generically solved by setting ∂˜µ = 0, removing the dependence on x ˜µ
coordinates. It follows that AM = Aλ ∂ M xλ = (Aµ , 0) and hence the x ˜µ coordinates are indeed gauged,
DxM = (d˜ xµ − Aµ , dxν ).
• Stringy graviton fields from the closed string massless sector
The O(D, D) T-duality group is a fundamental structure in Stringy Gravity. All the fields therein must assume one representation of it, such that the O(D, D) covariance is manifest. The stringy graviton fields consist of the DFT dilaton, d, and DFT metric, HM N . The former gives the
integral measure in Stringy Gravity after exponentiation, e−2d , which is a scalar density of unit weight. The latter is then, by definition, a symmetric O(D, D) element: HM N = HN M ,
HK L HM N JLN = JKM .
(3.14)
Combining JM N and HM N , we acquire a pair of symmetric projection matrices, PM N = PN M = 12 (JM N + HM N ) ,
PL M PM N = PL N ,
P¯M N = P¯N M = 12 (JM N − HM N ) ,
P¯L M P¯M N = P¯L N ,
16
(3.15)
which are orthogonal and complete, PL M P¯M N = 0 ,
PM N + P¯M N = δM N .
(3.16)
P¯M N = V¯M p¯V¯N q¯η¯p¯q¯ ,
(3.17)
Further, taking the “square roots” of the projectors, PM N = VM p VN q ηpq ,
we acquire a pair of DFT vielbeins, which satisfy four defining properties: VM p V M q = ηpq ,
V¯M p¯V¯ M q¯ = η¯p¯q¯ ,
VM p V¯ M q¯ = 0 ,
VM p VN p + V¯M p¯V¯N p¯ = JM N , (3.18)
such that (3.15) and (3.16) hold. Essentially,
(VM p , V¯M p¯),
when viewed as a (D + D) × (D + D) matrix,
diagonalizes JM N and HM N simultaneously into ‘diag(η, +¯ η )’ and ‘diag(η, −¯ η )’, respectively. The pres-
ence of twofold vielbeins as well as spin groups are a truly stringy feature, as it indicates two distinct locally
inertial frames existing separately for the left-moving and right-moving closed string sectors [11], and may be a testable prediction of Stringy Gravity in itself [13]. It is absolutely crucial to note that DFT [3, 4, 8] and its supersymmetric extensions [20–22] are formulatable in terms of nothing but the very fields satisfying precisely the defining relations (3.14), (3.18). The most general solutions to the defining equations turn out to be classified by two non-negative integers, (n, n ¯ ). With 1 ≤ i, j ≤ n and 1 ≤ ¯ı, ¯ ≤ n ¯ , the DFT metric is of the most general form [15], HM N
=
¯ ¯ı −H µσ Bσλ + Yiµ Xλi − Y¯¯ıµ X λ
H µν
¯ ¯ı Y¯ ν Bκρ H ρν + Xκi Yiν − X κ ¯ı
ρ i B ¯ ¯ı ¯ρ Kκλ − Bκρ H ρσ Bσλ + 2X(κ λ)ρ Yi − 2X(κ Bλ)ρ Y¯ı
(3.19)
where i) H and K are symmetric, but B is skew-symmetric, i.e. H µν = H νµ , Kµν = Kνµ , Bµν = −Bνµ ; ¯ ν¯ı = 0, Kµν Y ν = Kµν Y¯¯ν = 0 ; ii) H and K admit kernels, H µν Xνi = H µν X j
¯ ν¯ı = δµ ν . iii) a completeness relation must be met, H µρ Kρν + Yiµ Xνi + Y¯¯ıµ X It follows from the linear independence of the kernel eigenvectors that Yiµ Xµj = δi j ,
¯µ¯ = δ¯ı ¯ , Y¯¯ıµ X
¯ µ¯ = Y¯¯ıµ Xµj = 0 , Yiµ X 17
H ρµ Kµν H νσ = H ρσ ,
Kρµ H µν Kνσ = Kρσ .
With the section choice ∂˜µ = 0 and the parameter decomposition ξ A = (ξ˜µ , ξ ν ), the generalized Lie derivative, Lˆξ HM N , reduces to the ordinary (i.e. undoubled) Lie derivative, Lξ , plus B-field gauge symmetry, ¯ ¯ı , ¯ ¯ı = Lξ X δX µ µ
δXµi = Lξ Xµi , δH µν = Lξ H µν ,
δY¯¯ν = Lξ Y¯¯ν ,
δYjν = Lξ Yjν ,
(3.20)
δBµν = Lξ Bµν + ∂µ ξ˜ν − ∂ν ξ˜µ .
δKµν = Lξ Kµν ,
Only in the case of (n, n ¯ ) = (0, 0) can Kµν and H µν be identified with the (invertible) Riemannian metric and its inverse. The (0, 0) Riemannian DFT metric then takes the rather well-known form, HM N
=
gµν
−gµλ Bλτ
Bσκ gκν
gστ − Bσκ g κλ Bλτ
and the corresponding DFT vielbeins read VM p =
√1 2
ep µ
eν q ηqp + Bνσ ep σ
,
V¯M p¯ =
√1 2
,
e¯p¯µ e¯ν q¯η¯q¯p¯ + Bνσ e¯p¯σ
,
(3.21)
(3.22)
where eµ p and e¯µ p¯ are a pair of Riemannian vielbeins for the common Riemannian metric, eµ p eνp = −¯ eµ p¯e¯ν p¯ = gµν .
(3.23)
With the non-vanishing determinant, g = det gµν 6= 0, the DFT dilaton can be further parametrized by e−2d =
√
−g e−2φ .
(3.24)
In this way, the stringy gravitons may represent the conventional closed string massless sector, {gµν , Bµν , φ}. Other cases of (n, n ¯ ) 6= (0, 0) are then generically non-Riemannian, as the Riemannian metric cannot
be defined. They include (1, 0) or (D − 1, 0) for non-relativistic Newton–Cartan or ultra-relativistic Carroll gravities [16], (1, 1) for the Gomis–Ooguri non-relativistic string [17, 18], and various chiral theories, e.g. [19].
18
For later use, it is worth noting that the two-indexed projectors generate in turn a pair of multi-indexed projectors, PABC DEF := PA D P[B [E PC] F ] +
2 P P [E P F ]D PM M −1 A[B C]
,
P¯ABC DEF := P¯A D P¯[B [E P¯C] F ] +
2 P¯ P¯ [E P¯ F ]D P¯M M −1 A[B C]
,
(3.25)
satisfying P¯ABC DEF P¯DEF GHI = P¯ABC GHI .
PABC DEF PDEF GHI = PABC GHI ,
(3.26)
They are symmetric and traceless in the following sense: PABCDEF = PDEF ABC ,
PABCDEF = PA[BC]D[EF ] ,
P AB PABCDEF = 0 ,
P¯ABCDEF = P¯DEF ABC ,
P¯ABCDEF = P¯A[BC]D[EF ] ,
P¯ AB P¯ABCDEF = 0 .
(3.27)
• Covariant derivatives with stringy Christoffel symbols and spin connections
The ‘master’ covariant derivative in Stringy Gravity,
¯A , DA = ∂A + ΓA + ΦA + Φ
(3.28)
is equipped with the stringy Christoffel symbols of the diffeomorphism connection [9], ΓCAB = 2 P ∂C P P¯ −4
[AB]
+ 2 P¯[A D P¯B] E − P[A D PB] E ∂D PEC
1 P P D PM M −1 C[A B]
+
1 P¯ P¯ D P¯M M −1 C[A B]
∂D d + (P ∂ E P P¯ )[ED] ,
and the spin connections for the twofold local Lorentz symmetries [10], ¯ A¯pq¯ = Φ ¯ A[¯pq¯] = V¯ B p¯∇A V¯B q¯ . Φ
ΦApq = ΦA[pq] = V B p ∇A VBq ,
(3.29)
(3.30)
In the above, we set ∇A := ∂A + ΓA , 19
(3.31)
which, ignoring any local Lorentz indices, acts explicitly on a tensor density with weight ω as ∇C TA1 A2 ···An := ∂C TA1 A2 ···An − ωT Γ
B
BC TA1 A2 ···An
+
n X i=1
ΓCAi B TA1 ···Ai−1 BAi+1 ···An .
(3.32)
The stringy Christoffel symbols (3.29) can be uniquely determined by requiring three properties: i) full compatibility with all the stringy graviton fields, DA P¯BC = ∇A P¯BC = 0 ,
DA PBC = ∇A PBC = 0 ,
(3.33)
DA d = ∇A d = − 21 e2d ∇A (e−2d ) = ∂A d + 12 ΓB BA = 0 , which implies, in particular, DA JBC = ∇A JBC = 0 ,
ΓABC = −ΓACB ;
(3.34)
ii) a cyclic property (traceless condition), ΓABC + ΓBCA + ΓCAB = 0 ,
(3.35)
which makes ∇A compatible with the generalized Lie derivative (3.6) as well as the C-bracket (3.8), such that we may freely replace the ordinary derivatives therein by ∇A , Lˆξ (∂) = Lˆξ (∇) ,
[ζ, ξ]C (∂) = [ζ, ξ]C (∇) ;
(3.36)
P¯ABC DEF ΓDEF = 0 ,
(3.37)
iii) projection constraints, PABC DEF ΓDEF = 0 , which ensure the uniqueness. Unlike the Christoffel symbols in GR, there exist no normal coordinates where the stringy Christoffel symbols would vanish pointwise. The Equivalence Principle holds for the point particle but not for the string [13, 14]. 20
Once the stringy Christoffel symbols are fixed, the spin connections (3.30) follow immediately from the compatibility with the DFT vielbeins, DA VBp = ∇A VBp + ΦAp q VBq = ∂A VBp + ΓAB C VCp + ΦAp q VBq = 0 ,
(3.38)
¯ A¯p q¯V¯B q¯ = ∂A V¯B p¯ + ΓAB C V¯C p¯ + Φ ¯ A¯p q¯V¯B q¯ = 0 . DA V¯B p¯ = ∇A V¯B p¯ + Φ The master derivative is also compatible with the two sets of local Lorentz metrics and gamma matrices, DA ηpq = 0 ,
DA η¯p¯q¯ = 0 ,
DA (γ p )α β = 0 ,
DA (¯ γ p¯)α¯ β¯ = 0 ,
(3.39)
such that, as in GR, ¯ A¯pq¯ = −Φ ¯ A¯qp¯ , Φ
ΦApq = −ΦAqp ,
ΦA α β = 14 ΦApq (γ pq )α β ,
¯ A α¯ ¯ = 1 Φ ¯ pq¯(¯ Φ γ p¯q¯)α¯ β¯ . β 4 A¯ (3.40)
The master derivative (3.28) acts explicitly as ¯ N p¯q¯TM p α q¯α¯ + Φ ¯ N α¯ ¯TM p α p¯β¯ DN TM p α p¯α¯ = ∇N TM p α p¯α¯ + ΦN p q TM q α p¯α¯ + ΦN α β TM p β p¯α¯ + Φ β = ∂N TM p α p¯α¯ − ωΓL LN TM p α p¯α¯ + ΓN M L TLp α p¯α¯ + ΦN p q TM q α p¯α¯ + ΦN α β TM p β p¯α¯ ¯ N p¯q¯TM p α q¯α¯ + Φ ¯ N α¯ ¯ TM p α p¯β¯ . +Φ β (3.41) Unsurprisingly the master derivative is completely covariant for the twofold local Lorentz symmetries. The characteristic of the master derivative, DA , as well as ∇A is that they are actually ‘semi-covariant’ under doubled-yet-gauged diffeomorphisms: the stringy Christoffel symbols transform as
i h ¯ CAB F DE − δ F δ D δ E ∂F ∂[D ξE] , δξ ΓCAB = Lˆξ ΓCAB + 2 (P + P) C A B δξ ΦApq = Lˆξ ΦApq + 2PApq DEF ∂D ∂[E ξF ] ,
(3.42)
¯ A¯pq¯ = Lˆξ Φ ¯ A¯pq¯ + 2P¯A¯pq¯DEF ∂D ∂[E ξF ] , δξ Φ
such that DA and ∇A are not automatically diffeomorphism-covariant, e.g. δξ ∇C TA1 ···An
n X ¯ CA BDEF ∂D ∂E ξF TA ···A BA ···An . ˆ 2(P+P) = Lξ ∇C TA1 ···An + 1 i i−1 i+1 i=1
21
(3.43)
Nevertheless, the potentially anomalous terms are uniquely given, or controlled, by the multi-indexed projectors, as seen in (3.42) and (3.43), such that they can be easily projected out. Completely covariantized derivatives include [9] PC D P¯A1 B1 · · · P¯An Bn ∇D TB1 ···Bn ,
P¯C D PA1 B1 · · · PAn Bn ∇D TB1 ···Bn ,
P AB P¯C1 D1 · · · P¯Cn Dn ∇A TBD1 ···Dn ,
P¯ AB PC1 D1 · · · PCn Dn ∇A TBD1 ···Dn
(divergences) ,
P AB P¯C1 D1 · · · P¯Cn Dn ∇A ∇B TD1 ···Dn ,
P¯ AB PC1 D1 · · · PCn Dn ∇A ∇B TD1 ···Dn
(Laplacians) , (3.44)
which can be freely pulled back by the DFT vielbeins, with Dp = V p DA and Dp¯ = V¯ A p¯DA , to A
Dp Tq¯1 ···¯qn ,
Dp T p q¯1 ···¯qn ,
Dp¯Tq1 ···qn ,
Dp¯T p¯q1 ···qn ,
Dp D p Tq¯1 ···¯qn ,
Dp¯D p¯Tq1 ···qn . (3.45)
J A,
In particular, for a weightless vector, it is useful to note ∂A e−2d J A = ∇A e−2d J A = e−2d ∇A J A .
(3.46)
Furthermore, from (3.42), the following modules of the spin connections are completely covariant under diffeomorphisms: P¯A B ΦBpq ,
¯ B p¯q¯ , PA B Φ
ΦA[pq V A r] ,
¯ A[¯pq¯V¯ A r¯] , Φ
ΦApq V Ap ,
(3.47)
¯ A¯pq¯V¯ A¯p . Φ
Consequently, acting on Spin(1, D−1) spinors, ρα , ψpα¯ , or Spin(D−1, 1) spinors, ρ′α¯ , ψp′α¯ , the completely covariant Dirac operators are, with respect to both diffeomorphisms and local Lorentz symmetries [10, 20], γ p Dp ρ ,
γ p Dp ψp¯ ,
Dp¯ρ ,
Dp¯ψ p¯ ,
γ¯ p¯Dp¯ρ′ ,
γ¯ p¯Dp¯ψp′ ,
Dp ρ′ ,
Dp ψ ′p . (3.48)
For a Spin(1, D−1) × Spin(D−1, 1) bi-fundamental spinorial field or the Ramond–Ramond potential,
C α α¯ , a pair of completely covariant nilpotent derivatives, D+ and D− , can be defined [49] (c.f. [50]), D± C := γ p Dp C ± γ (D+1) Dp¯C γ¯p¯ , 22
2 C = 0, D±
(3.49)
¯ A . Specifically, the R–R field strength is given by F = D+ C. where, with (3.40), DA C = ∂A C + ΦA C − C Φ Finally, for a Yang–Mills vector potential, AM , the completely covariant field strength reads [13] (c.f. [51]) Fp¯q := V M p V¯ N q¯ ∇M AN − ∇N AM − i [AM , AN ] .
(3.50)
Upon Riemannian backgrounds (3.22), (3.24), the spin connections (3.51) reduce explicitly to V¯ A p¯ΦApq =
√1 e ¯µ 2 p¯ √1 2
ΦA[pq V A r] = ΦApq V Ap =
√1 2
ωµpq + 12 Hµpq ,
ω[pqr] + 16 Hpqr ,
(epµ ωµpq − 2eq ν ∂ν φ) ,
¯ A¯pq¯ = V ApΦ
√1 ep µ 2
¯ A[¯pq¯V¯ A r¯] = Φ ¯ A¯pq¯V¯ A¯p = Φ
√1 2
√1 2
ω ¯ µ¯pq¯ + 12 Hµ¯pq¯ ,
ω ¯ [¯pq¯r¯] + 16 Hp¯q¯r¯ ,
(3.51)
(¯ ep¯µ ω ¯ µ¯pq¯ − 2¯ eq¯ν ∂ν φ) ,
where, generalizing (1.1), we have ωµpq = ep ν (∂µ eνq − γµλ ν eλq ), ω ¯ µ¯pq¯ = e¯p¯ν (∂µ e¯ν q¯ − γµλ ν e¯λ¯q ), and ▽µ := ∂µ + γµ + ωµ + ω ¯µ ,
▽µ eν p = 0 ,
▽µ ηpq = 0 ,
▽µ e¯ν q¯ = 0,
▽µ η¯p¯q¯ = 0 ,
▽λ gµν = 0 . (3.52)
• Curvatures: stringy Einstein tensor
The semi-covariant Riemann curvature in Stringy Gravity is defined by [9] SABCD :=
1 2
RABCD + RCDAB − ΓE AB ΓECD ,
(3.53)
where ΓABC are the stringy Christoffel symbols (3.29) and RABCD denotes their “field strength”, RCDAB = ∂A ΓBCD − ∂B ΓACD + ΓAC E ΓBED − ΓBC E ΓAED .
(3.54)
Crucially, by construction, it satisfies symmetric properties and an algebraic “Bianchi” identity,4
SABCD = SCDAB = S[AB][CD] , 4
SA[BCD] = 0 .
As an alternative to direct verification, the Bianchi identity can also be shown using (3.7), (3.36) and the relation [52] 0=
h
i Lˆζ , Lˆξ − Lˆ[ζ,ξ]C
∂→∇
TA1 A2 ···An =
n X i=1
23
6SAi [BCD] ζ B ξ C TA1 ···Ai−1 D Ai+1 ···An .
(3.55)
Furthermore, just like the Riemann curvature in GR (2.14), it transforms as ‘total’ derivatives under the arbitrary variation of the stringy Christoffel symbols,5 δSABCD = ∇[A δΓB]CD + ∇[C δΓD]AB .
(3.56)
In particular, it is ‘semi-covariant’ under doubled-yet-gauged diffeomorphisms, ¯ D][AB] EF G ∂E ∂F ξG . ¯ B][CD] EF G ∂E ∂F ξG + 2∇[C (P+P) δξ SABCD = Lˆξ SABCD + 2∇[A (P+P)
(3.57)
In DFT there is no completely covariant four-indexed ‘Riemann’ curvature [9, 54], which is in a sense consistent with the absence of ‘normal’ coordinates for strings. The completely covariant ‘Ricci’ and scalar curvatures are then, with SAB = SBA = S C ACB , Sp¯q := V A p V¯ B q¯SAB ,
S(0) := P AC P BD − P¯ AC P¯ CD SABCD = Spq pq − Sp¯q¯p¯q¯ .
(3.58)
These completely covariant curvatures contain both HM N and d, as is the case for the connection, ΓLM N (3.29).
The DFT metric alone cannot generate any covariant curvature c.f. [52]. It is worth noting the identities Sprq¯r = Sp¯rq¯r¯ = 12 Sp¯q ,
Spq pq + Sp¯q¯p¯q¯ = 0 ,
Spqp¯q¯ = 0 ,
Sp¯pqq¯ = 0 ,
(3.59)
(¯ γ p¯Dp¯)2 ε′ + Dp D p ε′ = − 14 Sp¯q¯p¯q¯ε′ = 81 S(0) ε′ ,
(γ p Dp )2 ε + Dp¯D p¯ε = − 14 Spq pq ε = − 81 S(0) ε ,
(3.60) and the commutation relations [12, 53] [Dp , Dq¯]T p = Sp¯q T p ,
[Dq¯, Dp ]T q¯ = Sp¯q T q¯ ,
[γ p Dp , Dq¯]ε = 12 Sp¯q γ p ε ,
[¯ γ q¯Dq¯, Dp ]ε′ = 12 Sp¯q γ q¯ε′ .
(3.61)
Combining the ‘Ricci’ and the scalar curvatures, it is possible to construct the stringy ‘Einstein’ tensor which is covariantly conserved [23], GAB := 4V[A p V¯B] q¯Sp¯q − 21 JAB S(0) , 5
∇A GAB = 0 .
(3.62)
Eq.(3.56) can be generalized to include torsion, such that the ‘1.5’ formalism works in the full-order supersymmetric extensions
of DFT [20, 21], where the connection becomes torsionful, Γ[ABC] 6= 0.
24
From (3.44), this conservation law is completely covariant. Note also that in general, GAB 6= GBA and ∇B GAB 6= 0 . However, we may symmetrize the stringy Einstein tensor, still preserving the conservation law, by multiplying the DFT metric from the right,
(GH)AB = (GH)BA := GAC HC B = −4V(A p V¯B) q¯Sp¯q − 21 HAB S(0) ,
∇A (GH)AB = 0 . (3.63)
Restricting to Riemannian backgrounds (3.22), (3.24), we have explicitly, i h Sp¯q = 21 ep µ e¯q¯ν Rµν + 2▽µ (∂ν φ) − 14 Hµρσ Hν ρσ + 21 ▽ρ Hρµν − (∂ ρ φ)Hρµν , S(0) = R + 4✷φ − 4∂µ φ∂ µ φ −
1 λµν 12 Hλµν H
(3.64)
.
In particular, the upper left D×D diagonal block of (GH)AB contains the undoubled Einstein tensor in GR, 1 µν g Hρστ H ρστ . (3.65) (GH)µν = Rµν − 12 gµν R + 2▽µ (∂ ν φ) − 2gµν (✷φ − ∂σ φ∂ σ φ) − 14 H µρσ H ν ρσ + 24
3.2 Stringy Energy-Momentum tensor & Einstein Double Field Equations We now consider Stringy Gravity coupled to generic matter fields, in analogy to GR (2.1), Z i h 1 S(0) + Lmatter , e−2d 16πG
(3.66)
Σ
where Lmatter is the O(D, D) symmetric Lagrangian of the matter fields, Υa , equipped with the completely covariantized derivatives, DM . Some examples will follow below in subsection 3.2.1. The integral is taken
over a section, Σ. We seek the variation of the above action which is induced by the arbitrary transformations of all the fields, δd, δPAB , δP¯AB , δVAp , δV¯A¯p , and δΥa . They are subject to the following algebraic relations, originating from the defining properties the stringy graviton fields, (3.14), (3.15), (3.18), δPAB = −δP¯AB = 21 δHAB = 2P(A C P¯B) D δPCD = 2V¯(A p¯VB) q V¯ C p¯δVCq , δVAp = V¯A q¯V¯ C q¯δVCp + (δVC[p V C q] )VA q ,
V¯ C q¯δVCp = −V C p δV¯C q¯ ,
δV¯A¯p = VA q V C q δV¯C p¯ + (δV¯C[¯p V¯ C q¯] )V¯A q¯ . (3.67) 25
Similarly to the variation of the ordinary vielbein in GR (2.11), the second terms in the variations of the DFT vielbeins, δVAp and δV¯A¯p above, can be inversely traded with the Spin(1, D−1) × Spin(D−1, 1) local Lorentz transformations of the matter fields with the infinitesimal parameters, VC[p δV C q] and V¯C[¯p δV¯ C q¯] .
Firstly, as is known [9], the pure Stringy Gravity term transforms, from (3.33), (3.55), (3.56), (3.58), as δ e−2d S(0)
= 4e−2d δP AB VA p V¯B q¯Sp¯q − 12 δdS(0) + ∂A 2e−2d P AC P BD − P¯ AC P¯ BD δΓBCD
= 4e−2d V¯ A¯q δVA p Sp¯q − 12 δd S(0)
+ total derivative , (3.68)
of which the total derivative can be ignored in the variation of the action. Secondly, in a similar fashion to (2.12), the (local-Lorentz-symmetric) matter Lagrangian transforms, up to total derivatives (≃), as δLmatter δLmatter δLmatter δLmatter + δV¯A p¯ ¯ p¯ + δd + δΥa p δVA δd δΥa δVA δLmatter δLmatter δLmatter δLmatter − VAp ¯ q¯ + δ ′ Υa . ≃ V¯ B q¯δVB p V¯A¯q + δd p δVA δd δΥa δVA
δLmatter ≃ δVA p
(3.69)
This suggests the following two definitions, 1 Kp¯q := 2
δL δLmatter matter , VAp ¯ q¯ − V¯A¯q δVA p δ VA
T(0) :=
e2d
δ e−2d Lmatter , × δd
(3.70)
both of which will constitute the conserved Energy-Momentum tensor in Stringy Gravity, see (3.81). Eq.(3.69) then reads δLmatter −2d −2d A¯ q p ′ ¯ . δ e Lmatter ≃ e −2V δVA Kp¯q + δdT(0) + δ Υa δΥa
(3.71)
It is worthwhile to note that, for the restricted cases of Lmatter in which the DFT vielbeins are absent and only the projectors are present, we have δLmatter = δPAB
δLmatter δLmatter δLmatter δLmatter + δΥa + δP¯AB ¯ , + δd δPAB δd δΥa δPAB
and, from (3.67), the above definition of Kp¯q reduces to δLmatter δLmatter ¯ − Kp¯q = VAp VB q¯ . δPAB δP¯AB 26
(3.72)
(3.73)
Now, collecting all the results of (3.67), (3.68) and (3.70), the variation of the action (3.66) reads, disregarding any surface integrals, δ
Z
=
e−2d Σ
Z
h
e−2d Σ
1 16πG S(0)
+ Lmatter
i
δLmatter 1 ¯ A¯ q δV p (S − 8πGK ) − 1 δd(S ′ A (0) − 8πGT(0) ) + δ Υa p¯ q p¯ q 4πG V 8πG δΥa
(3.74) .
All the equations of motion are then given by
Sp¯q = 8πGKp¯q ,
S(0) = 8πGT(0) ,
δLmatter ≡ 0, δΥa
(3.75)
where ‘≡’ is used to denote the on-shell equations for the matter fields. Specifically, when the variation is generated by doubled-yet-gauged diffeomorphisms, we have δξ d = − 21 e2d Lˆξ e−2d = − 12 DA ξ A ,
δξ Υa = Lˆξ Υa ,
(3.76)
and, from DB VAp = 0 (3.38),
δξ VAp = Lˆξ VAp = ξ B ∇B VAp + 2∇[A ξB] V B p = −ξ B ΦBp q VAq + 2∇[A ξB] V B p ,
(3.77)
which implies V¯ A¯q δξ VA p = 2D[A ξB] V¯ A¯q V Bp ,
δξ PAB = Lˆξ PAB = 4P¯(A C PB) D D[C ξD] .
(3.78)
Substituting these results into (3.74), utilizing the invariance of the action under doubled-yet-gauged diffeomorphisms while neglecting surface terms, we achieve a crucial result, Z δLmatter 1 0 = e−2d 8πG ξ B D A 4V[A p V¯B] q¯(Sp¯q − 8πGKp¯q ) − 12 JAB (S(0) − 8πGT(0) ) + δ′ Υa . δΥa Σ (3.79) This leads to the definitions of the off-shell conserved stringy Einstein curvature tensor (3.62) from [23], GAB = 4V[A p V¯B] q¯Sp¯q − 21 JAB S(0) ,
27
DA GAB = 0
(off-shell) ,
(3.80)
and separately the on-shell conserved Energy-Momentum tensor in Stringy Gravity, TAB := 4V[A p V¯B] q¯Kp¯q − 12 JAB T(0) ,
DA T AB ≡ 0
(on-shell) .
(3.81)
Note6 TAB 6= TBA and DB T AB 6= 0 . However, like (GH)AB = (GH)BA (3.63), we may symmetrize the
stringy Energy-Momentum tensor,
(T H)AB = (T H)BA := GAC HC B = −4V(A p V¯B) q¯Kp¯q − 12 HAB T(0) ,
DA (T H)AB ≡ 0 .
(3.82)
GAB and TAB each have D 2 + 1 components, given by V A p V¯ B q¯GAB = 2Sp¯q ,
V A p V¯ B q¯TAB = 2Kp¯q ,
GA A = −DS(0) ,
T A A = −DT(0) , (3.83)
respectively. The equations of motion of the DFT vielbeins and the DFT dilaton are unified into a single expression, the Einstein Double Field Equations, GAB = 8πGTAB ,
(3.84)
which is naturally consistent with the central idea that Stringy Gravity treats the entire closed string massless sector as geometrical stringy graviton fields. From (3.76), (3.78) and (3.81), if we contract the stringy Energy-Momentum tensor with an O(D, D) vector, its divergence reads DA T A B ξ B ≡ TAB D A ξ B = −2V Ap V¯ B q¯Kp¯q Lˆξ PAB + T(0) Lˆξ d .
(3.85)
Therefore, if ξ A is a DFT-Killing vector satisfying the DFT-Killing equations [23], Lˆξ PAB = 4P¯(A C PB) D ∇[C ξD] = 0 ,
Lˆξ d = − 21 ∇A ξ A = 0 ,
the contraction T A B ξ B gives an on-shell conserved Noether current (from (3.46)), ∂A e−2d T A B ξ B = e−2d DA T A B ξ B ≡ 0 , 6
(3.86)
(3.87)
Although we use the same conventional letter symbols, no component of GAB or TAB coincides precisely with that of the
undoubled Einstein and Energy-Momentum tensors in GR, c.f. (2.2), (3.65).
28
and the corresponding Noether charge, Q[ξ] =
Z
e−2d T t A ξ A ,
(3.88)
Σ
where the superscript index, t, denotes the time component for a chosen section. It is worthwhile to note that the alternative contraction with the symmetrized Energy-Momentum tensor, (T H)A B ξ B , is not conserved even if ξ A is a Killing vector.
Through contraction with the DFT vielbeins, the conservation law of the Energy-Momentum tensor decomposes into two separate formulae, DA T AB V¯B q¯ = 2Dp K p q¯ − 21 Dq¯T(0) ≡ 0 .
DA T AB VBp = −2Dq¯Kp q¯ − 21 Dp T(0) ≡ 0 ,
(3.89)
Restricting to Riemannian backgrounds (3.22), (3.24), with ▽µ = ∂µ + γµ + ωµ + ω ¯ µ (3.52), we have √ q¯µ e ▽µ Kp¯q − 2∂µ φ Kp¯q + 12 Hµp q Kqq¯ − 0 ≡ DA T A p = − 2¯ =
√1 ep ν 2
1 √ e µ∂ T 2 2 p µ (0)
▽µ Kν µ − 2∂µ φ Kν µ + 12 Hνλµ K λµ − 12 ∂ν T(0) ,
and 0 ≡ DA T A q¯ = =
√
2epµ ▽µ Kp¯q − 2∂µ φ Kp¯q + 21 Hµ¯q r¯Kp¯r −
√1 e ¯ν 2 q¯
(3.90)
1 √ e¯ µ ∂ T 2 2 q¯ µ (0)
(3.91)
▽µ K µ ν − 2∂µ φ K µ ν + 21 Hνλµ K λµ − 21 ∂ν T(0) .
Thus, the conservation law reduces to the following two sets of equations, ∇µ K(µν) − 2∂ µ φ K(µν) + 12 Hν λµ K[λµ] − 12 ∂ν T(0) ≡ 0 , ∇µ e−2φ K[µν] ≡ 0 .
(3.92) (3.93)
In fact, for the above computations, we first put, c.f. (3.64), Kp¯q ≡ 12 ep µ e¯q¯ν Kµν
⇐⇒
Kµν ≡ 2eµ p e¯ν q¯Kp¯q ,
(3.94)
and then let the Greek indices of Kµν be raised by the Riemannian metric (3.23), gµν = ep µ epν = −¯ ep¯µ e¯p¯ν : K µ ν = g µρ Kρν = 2epµ e¯ν q¯Kp¯q ;
Kµ ν = g νρ Kµρ = −2eµ p e¯ν q¯Kp¯q . 29
(3.95)
It follows that
VA p V¯B q¯Kp¯q
and
=
1 4
−K µν
K µ σ + K µλ Bλσ
−Kρ ν − Bρκ K κν
Kρσ + Bρκ K κλ Bλσ + Bρ κ Kκσ + Kρλ B λ σ
TAB = 4V[A p V¯B] q¯Kp¯q − 21 JAB T(0) =
,
(3.96)
K (µλ) gλσ + K [µλ] Bλσ − 21 δµ σ T(0)
−K [µν]
−gρκ K (κν) − Bρκ K [κν] − 12 δρ ν T(0)
K[ρσ] + Bρκ K [κλ] Bλσ + Bρ κ K(κσ) + K(ρλ) B λ σ
.
(3.97)
The Einstein Double Field Equations (3.84) reduce, upon Riemannian backgrounds (3.22), (3.64), to Rµν + 2▽µ (∂ν φ) − 41 Hµρσ Hν ρσ = 8πGK(µν) , = 16πGe−2φ K[µν] , ▽ρ e−2φ Hρµν
R + 4✷φ − 4∂µ φ∂ µ φ −
1 λµν 12 Hλµν H
(3.98) (3.99)
= 8πGT(0) .
(3.100)
These also imply the two reduced conservation laws, (3.92) and (3.93), as DA GAB = 0 is an off-shell
identity. Explicitly, we have ▽µ ▽ν e−2φ H λµν = 21 ▽µ , ▽ν e−2φ H λµν = 12 Rλ [ρµν] e−2φ H ρµν + R[µν] e−2φ H λµν = 0 , (3.101)
which implies the second conservation law (3.93). On the other hand, solving Rµν and R from (3.98) and (3.100) respectively, we get 0 = ▽µ (Rµν − 12 gµν R) = 8πG ▽µ K(µν) − 2∂ µ φK(µν) + 12 Hν ρσ K[ρσ] − 12 ∂ν T(0) −2 Rµν + 2▽µ ∂ν φ − 41 Hµρσ Hν ρσ − 8πGK(µν) ∂ µ φ
+ 14 e2φ ▽µ e−2φ Hµρσ − 16πGe−2φ K[ρσ] + 31 H µρσ ∂[µ Hρσν] , 30
(3.102)
where we have used the identity ✷∂ν φ− ▽ν ✷φ = Rνρ ∂ ρ φ . The last three lines in (3.102) vanish separately
up to (3.98), (3.99), and the closedness of the H-flux. Therefore, we recover the first conservation law (3.92) as the vanishing of the first line on the right-hand side of the equality above. 3.2.1
Examples
Here we list various matter fields coupled to Stringy Gravity and write down their contributions to the stringy Energy-Momentum tensor (3.81), (3.97). • Cosmological constant
In Stringy Gravity, the cosmological constant term is given by a constant, ΛDFT , times the integral measure, e−2d [9], 1 −2d (S (0) 16πG e
− 2ΛDFT ) .
(3.103)
The corresponding Energy-Momentum tensor is 1 JAB ΛDFT , TAB = − 8πG
such that Kp¯q = 0 and T(0) =
(3.104)
1 4πG ΛDFT .
• Scalar field
A free scalar field in Stringy Gravity is described by, e.g. [13], LΦ = − 12 HM N ∂M Φ∂N Φ − 12 m2Φ Φ2 = 21 (P¯ M N − P M N )∂M Φ∂N Φ − 21 m2Φ Φ2 .
(3.105)
It is straightforward to see, with ∂p ≡ V A p ∂A , ∂q¯ ≡ V¯ A q¯∂A , Kp¯q = ∂p Φ∂q¯Φ ,
T(0) = HM N ∂M Φ∂N Φ + m2Φ Φ2 = −2LΦ .
For Riemannian backgrounds (3.22), (3.24), and the section choice ∂˜µ = 0, we have ∂p = ∂q¯ =
√1 e ¯ µ∂ , 2 q¯ µ
(3.106) √1 ep µ ∂µ , 2
and Kµν = K(µν) = ∂µ Φ∂ν Φ ,
K[µν] = 0 .
In particular, each diagonal component of Kµν is non-negative, as Kµµ = (∂µ Φ)2 .
31
(3.107)
• Spinor field
Fermionic spinor fields are described by [20] ¯ p Dp ψ + mψ ψψ ¯ , Lψ = ψγ
(3.108)
where7 ψ¯ = ψ † A, A = A† , and (γ p )† = −Aγ p A−1 . Under arbitrary variations of the stringy graviton fields and the spinor, the fermionic kinetic term transforms, up to total derivatives (‘≃’), as (c.f. [20, 21]) ¯ A DA ψ δ e−2d ψγ
¯ p Dq¯ψ + 1 e−2d δΓABC ψγ ¯ A γ BC ψ ≃ e−2d δVAp V¯ A¯q ψγ 4 ¯ pq − 2δd ψ¯ γ B DB ψ +e−2d δψ¯ − 41 VAp δV A q ψγ
(3.109)
¯ B δψ + 1 VAp δV A q γ pq ψ . −e−2d DB ψγ 4
In the full-order supersymmetric extensions of DFT [20, 21], the variation of the stringy Christoffel symbols, δΓABC , vanishes automatically, which realizes the ‘1.5 formalism’. However, in the present example, we do not consider any supersymmetry nor quartic fermionic terms. Instead, we proceed, with δΓ[ABC] = 0, to obtain 1 −2d δΓABC 4e
¯ A γ BC ψ = 1 e−2d P AB δΓABC ψγ ¯ Cψ ψγ 2 ¯ A ψ (Dp δVA p − 2∂A δd) + 1 e−2d ψγ ¯ p ψ DA δV A p = 12 e−2d ψγ 2 ¯ A ψ + ψγ ¯ A DA ψ − e−2d δV Ap D(A ψγ ¯ p) ψ + ψγ ¯ (A Dp) ψ , ≃ e−2d δd DA ψγ (3.110)
7
In the specific case of four-dimensional spacetime with Minkowskian signature [− + ++], we may set 1
(γ p )† = −Aγ p A−1 ,
A† = A
=⇒
(Aγ p1 p2 ···pn )† = (−1) 2 n(n+1) Aγ p1 p2 ···pn ,
(γ p )T = −Cγ p C −1 ,
C T = −C
=⇒
(Cγ p1 p2 ···pn )T = −(−1) 2 n(n+1) Cγ p1 p2 ···pn .
1
We may also, if desired, identify A with C and use the Majorana, i.e. real representation of the gamma matrices. Our analysis also holds for Majorana spinors satisfying ψ † A = ψ T C .
32
and derive the final form of the variation of the fermionic part of the Lagrangian, c.f. an analogous expression in GR (2.9), ¯ A DA ψ + mψ ψψ ¯ δ e−2d ψγ ≃
1 −2d δVA p V¯ A¯q 2e
¯ p Dq¯ψ − Dq¯ψγ ¯ pψ ψγ
¯ pq +e−2d δψ¯ − δd ψ¯ − 41 VA[p δV A q] ψγ
γ B DB ψ + mψ ψ
¯ B − mψ ψ¯ δψ − δd ψ + 1 VA[p δV A q] γ pq ψ . −e−2d DB ψγ 4
(3.111)
This result is quite satisfactory: unlike (3.109), the hermiticity is now manifest, as the first line on the right-hand side is by itself real while the second and the third are hermitian conjugate to each other. Further, as discussed in the general setup (3.69), the infinitesimal local Lorentz rotation of the spinor field by VA[p δV A q] can be absorbed into the equation of motion for the matter field through δψ ′ = δψ + 41 VA[p δV A q] γ pq ψ. The variation of the DFT dilaton can be also absorbed in the same manner. Comparing (3.71) and (3.111), we obtain ¯ p Dq¯ψ − Dq¯ψγ ¯ p ψ) , Kp¯q = − 41 (ψγ
T(0) ≡ 0 .
(3.112)
Upon Riemannian backgrounds (3.22), fermions thus provide a nontrivial example of asymmetric Kµν , 1 ¯ µ ▽ ψ − ▽ ψγ ¯ µ ψ) 6= Kνµ . (ψγ Kµν := 2eµ p e¯ν q¯Kp¯q = − 2√ ν ν 2
(3.113)
Finally, if we redefine the field in terms of a spinor density, χ := e−d ψ, with weight ω = 12 , the DFT dilaton decouples from the Lagrangian completely, ¯ p Dp ψ + mψ ψψ ¯ = χγ e−2d Lψ = e−2d ψγ ¯ p Dp χ + mψ χχ ¯ .
(3.114)
Like fundamental strings, c.f. (3.122), the weightful spinor field χ couples only to the DFT vielbeins (or gµν and Bµν for Riemannian backgrounds) [13]. In this case, T(0) = 0 holds off-shell.
• Ramond–Ramond sector
The R–R sector of the critical superstring has the kinetic term [21, 49] (c.f. [50, 55, 56]) LRR = 12 Tr(F F¯ ) , 33
(3.115)
where F = D+ C is the R–R field strength given by the nilpotent differential operator D+ (3.49) acting on the Spin(1, 9) × Spin(9, 1) bi-spinorial R–R potential C α α¯ ; F¯ = C¯ −1 F T C is the charge
conjugation of F; and the trace is taken over the Spin(1, 9) spinorial indices. This formalism is ‘democratic’, as in [57], and needs to be supplemented by a self-duality relation, γ (11) F ≡ F .
(3.116)
The R–R sector contributes to the stringy Energy-Momentum tensor, from (25) of [21] as well as (3.3) of [49], by Kp¯q = − 41 Tr(γp F γ¯q¯F¯ ) ,
(3.117)
T(0) ≡ 0 .
Upon Riemannian reduction, Kµν = 2eµ p e¯ν q¯Kp¯q is generically asymmetric, Kµν 6= Kνµ , which can
be verified explicitly after taking the diagonal gauge of the twofold local Lorentz symmetries and then
expanding the R–R potential C α β¯ in terms of conventional p-form fields [49]. The asymmetry is also
consistent with the observation that the O(D, D)-covariant nilpotent differential operator D+ (3.49)
reduces to the H-twisted exterior derivative, dH = d + H(3) ∧ , such that the B-field contributes to the R–R kinetic term (3.115) in a nontrivial manner.
• Point particle
We consider the doubled-yet-gauged particle action from [24] and write the corresponding Lagrangian density using a Dirac delta function, Z e−2d Lparticle = dτ e−1 Dτ y A Dτ y B HAB (x) − 41 m2 e δD x − y(τ ) ,
(3.118)
d M y (τ ) − AM is the gauged infinitesimal one-form (3.12). Integrating the above where Dτ y M = dτ R −2d over a section, Σ e Lparticle , one can recover precisely the action in [24]. Note that in Stringy
Gravity, the Dirac delta function itself should satisfy the section condition and meet the defining property
It follows straightforwardly that Kp¯q = −
Z
Z
Σ
Φ(x)δD(x − y) = Φ(y) .
dτ 2e−1 (Dτ y)p (Dτ y)q¯ e2d(x) δD x − y(τ ) , 34
(3.119)
T(0) = 0 ,
(3.120)
where, naturally, (Dτ y)p = Dτ y A VAp and (Dτ y)q¯ = Dτ y A V¯A¯q . Upon Riemannian backgrounds (3.22), (3.24), with ∂˜µ ≡ 0 and the on-shell value of the gauge con-
d d ν nection, AM = (Aµ , 0) ≡ ( dτ y˜µ − Bµν dτ y , 0) [24], we have p
q¯
Kµν = 2eµ e¯ν Kp¯q =
Z
−1
dτ 2e
dy ρ dy σ δD x − y(τ ) e2φ √ , gµρ gνσ dτ dτ −g
(3.121)
which is symmetric, Kµν = Kνµ , as one may well expect for the point particle. Each diagonal comρ 2 ponent is non-negative, Kµµ ≥ 0, as gµρ dy ≥ 0. dτ • String
In a similar fashion, the doubled-yet-gauged bosonic string action [40][39] gives Z i h √ −2d 1 e Lstring = 4πα′ d2 σ − 12 −hhαβ Dα y A Dβ y B HAB (x) − ǫαβ Dα xA AβA δD x − y(σ) ,
(3.122)
and hence Kp¯q =
1 4πα′
Z
√ d2 σ −hhαβ (Dα y)p (Dβ y)q¯ e2d(x) δD x − y(σ) ,
T(0) = 0 .
(3.123)
Upon reduction to Riemannian backgrounds (3.22), (3.24), the on-shell value of the gauge connection is AM ˜µ − Bµν ∂α y ν + α = (Aαµ , 0) ≡ (∂α y p
q¯
Kµν = 2eµ e¯ν Kp¯q = −
1 2πα′
Z
√1 ǫα β gµν ∂β y ν , 0) −h
[39], and we have
2φ D √ ρ σ δ x − y(τ ) e αβ αβ √ ∂α y ∂β y −hh + ǫ d σ gµρ gνσ , −g (3.124) 2
which is generically asymmetric, Kµν 6= Kνµ , due to the B-field. It is easy to see in lightcone gauge that the diagonal components of Kµν , or gµρ ∂+ y ρ gµσ ∂− y σ , are not necessarily positive.
The above analysis further generalizes to the doubled-yet-gauged Green-Schwarz superstring [22], Kp¯q =
1 4πα′
Z
√ d2 σ −hhαβ Παp Πβ q¯ e2d(x) δD x − y(σ) ,
T(0) = 0 ,
(3.125)
M − AM − iθγ ¯ M ∂α θ − iθ¯′ γ¯M ∂α θ ′ is the supersymmetric extension of Dα y M . where ΠM α α = ∂α y
35
ν M e Upon Riemannian reduction with ΠM α = (Παµ , Πα ), the on-shell value of the gauge connection Aα e αµ − Bµν Πνα + √1 ǫα β gµν Πν ≡ 0 [22], and the corresponding Kµν is similarly asymmetric, sets Π −h
p
q¯
1 Kµν = 2eµ e¯ν Kp¯q = − 2πα ′
β
Z
2
d σ gµρ gνσ
√
αβ
−hh
+ǫ
αβ
Πρα Πσβ
δD x − y(τ ) e2φ √ . (3.126) −g
The asymmetry, Kµν 6= Kνµ , is a genuine stringy property.
4 Further-generalized Lie derivative, Leξ
In analogy with GR, the notion of isometries in Stringy Gravity can be naturally addressed through the generalized Lie derivative (3.6), which can also, from (3.36), be expressed using DFT-covariant derivatives (3.32), leading to the DFT-Killing equations (3.86) [23], ! D ¯ C Lˆ∂ξ PAB = Lˆ∇ ξ PAB = 4P(A PB) ∇[C ξD] = 0 ,
1 A ! Lˆ∂ξ d = Lˆ∇ ξ d = − 2 ∇A ξ = 0 ,
(4.1)
where the final equalities with ‘ ! ’ hold only in the case of an isometry. However, in the vielbein formulation of Stringy Gravity this result needs to be further generalized, as one should be able to construct Killing equations for the DFT vielbeins. Using the fact that the DFT vielbeins are covariantly constant, DA VBp = DA V¯B p¯ = 0 (3.38), their generalized Lie derivatives can be related to the generalized Lie derivatives of the projectors, Lˆξ VAp = P¯A B V C p Lˆξ PBC − ξ B ΦBpq + 2D[p ξq] VA q ,
¯ B p¯q¯ + 2D[¯p ξq¯] V¯A q¯ . Lˆξ V¯A¯p = PA B V¯ C p¯ Lˆξ P¯BC − ξ B Φ
(4.2)
These expressions are quite instructive, as we can arrange them as
ξ B DB VAp + 2D[A ξB] V B p + 2D[p ξq] VA q = P¯A B Lˆξ PBC V C p , ξ B DB V¯A¯p + 2D[A ξB] V¯ B p¯ + 2D[¯p ξq¯] V¯A q¯ = PA B
Lˆξ P¯BC V¯ C p¯ .
(4.3)
Motivated by the expressions of the left-hand sides above, we propose to generalize the generalized Lie derivative one step further by constructing a further-generalized Lie derivative, Leξ , which can act on an 36
arbitrary tensor density carrying O(D, D) and Spin(1, D−1) × Spin(D−1, 1) indices as Leξ TM p¯p α β α¯ β¯ := ξ N DN TM p¯p α β α¯ β¯ + ωDN ξ N TM p¯p α β α¯ β¯ + 2D[M ξN ] T N p¯p α β α¯ β¯
+2D[p ξq] TM q p¯α β α¯ β¯ + 12 D[r ξs] (γ rs )α δ TM p¯p δ β α¯ β¯ − 12 D[r ξs] (γ rs )δ β TM p¯p α δ α¯ β¯ ¯
¯
γ r¯s¯)α¯ δ¯TM p¯p α β δ β¯ − 21 D[¯r ξs¯] (¯ γ r¯s¯)δ β¯ TM p¯p α β α¯ δ¯ . +2D[¯p ξq¯] TM p q¯α β α¯ β¯ + 12 D[¯r ξs¯] (¯ (4.4) In short, the further-generalized Lie derivative comprises the original generalized Lie derivative and additional infinitesimal local Lorentz rotations given by the terms ξ A ΦApq + 2D[p ξq] = 2∂[p ξq] + Φr¯pq ξ r¯ + 3Φ[pqr] ξ r , (4.5) ¯ A¯pq¯ + 2D[¯p ξq¯] = 2∂[¯p ξq¯] + Φ ¯ rp¯q¯ξ r + 3Φ ¯ [¯pq¯r¯] ξ r¯ , ξAΦ such that, for example, Leξ TM p¯p αα¯ = Lˆξ TM p¯p αα¯ + ξ N ΦN pq + 2D[p ξq] TM q p¯αα¯ + ¯ N p¯q¯ + 2D[¯p ξq¯] TM p q¯αα¯ + + ξN Φ
1 4
1 4
ξ N ΦN rs + 2D[r ξs] (γ rs )α β TM p¯p β α¯
r¯s¯ α¯ ¯ ¯ N r¯s¯ + 2D[¯r ξs¯] (¯ ξN Φ γ ) β¯TM p¯p αβ .
(4.6)
The further-generalization is also equivalent to replacing the ordinary derivatives in the original generalized Lie derivative by master derivatives, Lˆ∂ → LˆD , and adding the local Lorentz rotations, 2D[p ξq] and 2D[¯p ξq¯] . ξ
ξ
It is then crucial to note that the further-generalized Lie derivative is completely covariant for both
the twofold local Lorentz symmetries and the doubled-yet-gauged diffeomorphisms, as follows. The local Lorentz covariance is guaranteed by the use of the master derivatives everywhere in the definition (4.4). The diffeomorphism covariance also holds, since the original generalized Lie derivative and the additional local Lorentz rotations (4.5) are separately diffeomorphism covariant, from (3.10) and (3.51). For example, the further-generalized Lie derivative acting on a Spin(1, D−1) spinor field, ψ α , reads Leξ ψ = ξ M DM ψ + 12 D[p ξq] γ pq ψ = ξ p¯Dp¯ψ + 21 γ p Dp (γ q ξq ψ) + 12 γ q ξq (γ p Dp ψ) − 37
1 2
(Dp ξ p )ψ .
(4.7)
As expected, or directly seen from (3.48), this expression is completely covariant under both the local Lorentz and the doubled-yet-gauged diffeomorphism transformations. It is advantageous to use Leξ because of its local Lorentz covariance; whereas, in contrast, Lˆξ is not locally Lorentz covariant.
The isometry of the DFT vielbeins is then characterized by the vanishing of their further-generalized
Lie derivatives, which is then, with (4.1), (4.3), equivalent to nothing but the isometry of the projectors, as Leξ VAp = P¯A B Lˆξ PBC V C p = Lˆξ PAC V C p ,
Moreover, gamma and charge conjugation matrices,
Leξ V¯A¯p = PA B Lˆξ P¯BC V¯ C p¯ = Lˆξ P¯AC V¯ C p¯ .
(γ p )α
β,
(¯ γ p¯)α¯
(4.8) ¯ e β¯, Cαβ , Cα ¯ β¯ , are all compatible with Lξ ,
Leξ γ p = ξ N DN γ p + 2D [p ξ q] γq + 21 D[r ξs] [γ rs , γ p ] = 0 ,
(4.9)
Leξ γ¯ p¯ = ξ N DN γ¯ p¯ + 2D [¯p ξ q¯] γ¯q¯ + 12 D[¯r ξs¯] [¯ γ r¯s¯, γ¯p¯] = 0 , Leξ Cαβ = − 12 D[p ξq] Cγ pq + (γ pq )T C
Leξ C¯α¯ β¯ = − 12 D[¯p ξq¯] C¯ γ¯ p¯q¯ + (¯ γ p¯q¯)T C¯
αβ
α ¯ β¯
= 0, (4.10) = 0.
The further-generalized Lie derivative is closed by the C-bracket (3.8) and twofold local Lorentz rotations, h i Leζ , Leξ = Le[ζ,ξ]C + ωpq (ζ, ξ) + ω ¯ p¯q¯(ζ, ξ) , (4.11) of which the two infinitesimal local Lorentz rotation parameters are specified by ζ M and ξ M as
ωpq (ζ, ξ) = −ωqp (ζ, ξ) = (Dp¯ζp − Dp ζp¯)(D p¯ξq − Dq ξ p¯) − (Dp¯ζq − Dq ζp¯)(D p¯ξp − Dp ξ p¯) , ω ¯ p¯q¯(ζ, ξ) = −¯ ωq¯p¯(ζ, ξ) = (Dp ζp¯ − Dp¯ζp )(D p ξq¯ − Dq¯ξ p ) − (Dp ζq¯ − Dq¯ζp )(D p ξp¯ − Dp¯ξ p ) .
38
(4.12)
The closure essentially boils down to8 h
i Leζ , Leξ VM p = Le[ζ,ξ]C VM p + ωpq (ζ, ξ)VM q ,
h
which can easily be verified using (3.7) and (4.8).
i Leζ , Leξ V¯M p¯ = Le[ζ,ξ]C V¯M p¯ + ω ¯ p¯q¯(ζ, ξ)V¯M q¯ ,
(4.13)
5 Regular spherical solution to Einstein Double Field Equations In this section we derive the most general, spherically symmetric, asymptotically flat, static, Riemannian, regular solution to the D = 4 Einstein Double Field Equations. Hereafter, we fix the section as ∂˜µ ≡ 0, adopt spherical coordinates, {t, r, ϑ, ϕ}, and focus on the Riemannian parametrizations (3.21)–(3.24).
5.1 Most general D = 4 spherical ansatz The spherical symmetry in D = 4 Stringy Gravity is characterized by three Killing vectors, ξaN , a = 1, 2, 3, which form an so(3) algebra through the C-bracket, [ξa , ξb ]C =
X
ǫabc ξc .
c
8
Note, e.g. with Tp = T A VAp , h i h h i i h i eζ , Leξ VAp Leζ , Leξ Tp = Leζ , Leξ T A VAp = Leζ , Leξ T A VAp + T A L =
e[ζ,ξ] VAp + ωpq (ζ, ξ)VA q = Le[ζ,ξ] T A VAp + ωpq (ζ, ξ)T q Le[ζ,ξ]C T A VAp + T A L C C
= Le[ζ,ξ]C Tp + ωpq (ζ, ξ)T q .
39
(5.1)
With ∂˜µ ≡ 0, the so(3) Killing vectors, ξaN = (ξ˜aµ , ξaν ), decompose explicitly into ξ˜a = ξ˜aµ dxµ and
ξa = ξaν ∂ν [24],
ξ˜1 = ξ˜2 =
cos ϕ sin ϑ hdt sin ϕ sin ϑ hdt
ξ˜3 = 0 ,
+ B(r)dr ,
ξ1 = sin ϕ∂ϑ + cot ϑ cos ϕ∂ϕ ,
+ B(r)dr ,
(5.2)
ξ2 = − cos ϕ∂ϑ + cot ϑ sin ϕ∂ϕ , ξ3 = −∂ϕ ,
where h is constant and B(r) is an arbitrary function of the radius. The three of ξaµ ∂µ ’s are the standard (undoubled) so(3) angular momentum operators. In terms of the further-generalized Lie derivative, the spherical symmetry of the stringy graviton fields implies Leξa VAp = 0 ,
Leξa V¯A¯p = 0 ,
Leξa PAB = 0 ,
such that the DFT-Killing equations (4.1) are satisfied, PA C P¯B D (∇C ξaD − ∇D ξaC ) = 0
⇐⇒
Leξa P¯AB = 0 ,
Leξa d = 0 ,
(P ∇)A (P¯ ξa )B = (P¯ ∇)B (P ξa )A ,
(5.3)
∇A ξaA = 0 . (5.4)
On Riemannian backgrounds (3.21), (3.22), (3.24), the above DFT-Killing equations reduce to Lξa gµν = 0 ,
Lξa Bµν + ∂µ ξ˜aν − ∂ν ξ˜aµ = 0 ,
Lξa φ = 0 .
(5.5)
In addition to the spherical symmetry we also require the static condition, such that all the fields are timeindependent, ∂t ≡ 0. By utilizing ordinary diffeomorphisms we can set gtr ≡ 0 [26], and hence without
loss of generality we can put the Riemannian metric into the diagonal form [24]
ds2 = e2φ(r) −A(r)dt2 + A−1 (r)dr 2 + A−1 (r)C(r) dΩ2 ,
(5.6)
B(2) = B(r) cos ϑ dr ∧ dϕ + h cos ϑ dt ∧ dϕ , where dΩ2 = dϑ2 + sin2 ϑdϕ2 and B(2) = 21 Bµν dxµ ∧ dxν . This ansatz solves the spherical DFT-Killing
equations, or (5.5), with four unknown radial functions, A(r), B(r), C(r), φ(r), and one free constant, h. 40
It differs slightly from the rather well known ansatz in GR (2.29), but accords with the analytic solution in [24]. The H-flux then corresponds to the most general spherically symmetric three-form,9 H(3) = dB(2) = B(r) sin ϑ dr ∧ dϑ ∧ dϕ + h sin ϑ dt ∧ dϑ ∧ dϕ ,
Lξa H(3) = 0 .
(5.7)
The above ansatz reduces to the flat Minkowskian spacetime if and only if A = 1, C = r 2 , B = φ = h = 0. It is worth expanding the DFT integral measure, and its integration over 0 ≤ ϑ < π and 0 ≤ ϕ < 2π, on the Riemannian background,
Z Z
ϑ ϕ
√ e−2d = e−2φ −g = e2φ A−1 C sin ϑ = R2 sin ϑ , e−2d ≡
Z
0
π
dϑ
Z
2π
(5.8)
dϕ e−2d = 4πe2φ A−1 C = 4πR2 ,
0
where R denotes the so-called ‘areal radius’, R := eφ
p
C/A .
(5.9)
We further let the Energy-Momentum tensor, i.e. matter, be spherically symmetric,
which, with (3.70) and (5.3), decomposes into
Leξa TAB = 0 , Leξa T(0) = 0 .
Leξa Kp¯q = 0 ,
(5.10)
(5.11)
The latter implies that T(0) (r) is another radial function, while the former gives, using the convention Kp¯q = 12 ep µ e¯q¯ν Kµν (3.94), Lξa Kµν = 0 ,
(5.12)
which follows from the generic expression of the further-generalized Lie derivative acting on Kp¯q , i o o n h n Leξ Kp¯q = 41 ep µ e¯q¯ν 2Lξ Kµν + 2∂[µ ξ˜ρ] + Lξ (B − g)µρ g ρσ Kσν − 2∂[ν ξ˜ρ] + Lξ (B + g)νρ gρσ Kµσ ,
(5.13)
9
In terms of Cartesian coordinates, x1 = r sin ϑ cos ϕ, x2 = r sin ϑ sin ϕ, x3 = r cos ϑ, we have sin ϑ dϑ ∧ dϕ = 21 ǫijk (xi /r 3 ) dxj ∧ dxk .
41
together with the isometry condition (5.5). Combining these results, we arrive at the final form of Kµν ,
Kµν
Ktt (r) D(r) − E(r) = 0 0
D(r) + E(r)
0
0
Krr (r)
0
0
0
Kϑϑ (r)
F (r) sin ϑ
0
−F (r) sin ϑ
Kϑϑ (r) sin2 ϑ
,
(5.14)
such that there are six radial functions, Ktt (r), Krr (r), Kϑϑ (r), D(r), E(r) and F (r), with K(tr) = D(r) ,
K[tr] = E(r) ,
Kϑϕ = −Kϕϑ = F (r) sin ϑ .
(5.15)
In particular, it includes anti-symmetric components which induce a two-form, K(2) := 12 K[µν] dxµ ∧ dxν = E(r)dt ∧ dr + F (r) sin ϑdϑ ∧ dϕ .
(5.16)
This is a novel feature of Stringy Gravity which is not present in GR. For later use, it is worthwhile to note, from (3.88), (3.97), that T t A ξ A = (Kt t − 12 T(0) )ξ t + g tt K(tr) ξ r + K [tr] Brϕ ξ ϕ − K [tr] ξ˜r
(5.17)
= ξ t (Kt t − 21 T(0) ) − ξ r e−2φ A−1 D − e−4φ E(ξ ϕ B cos ϑ − ξ˜r ) , and Kϑ ϑ = e−2φ AC −1 Kϑϑ = Kϕ ϕ ,
e−2d K ϑϕ = e−2φ AC −1 F ,
both of which depend on the radius, r, only.
42
(5.18)
5.2 Solving the Einstein Double Field Equations Having prepared the most general spherically symmetric static ansatz, (5.6), (5.14), we now proceed to solve the Einstein Double Field Equations (3.98), (3.99), (3.100). Subtracting the ‘trace’ of (3.98) from (3.100) and employing the differential form notation of (5.7) and (5.16), we focus on the three equivalent equations ✷φ − 2∂µ φ∂ µ φ +
1 µνρ 12 Hµνρ H
= 4πG(T(0) − Kµ µ ) ,
(5.19)
Rµν + 2▽µ (∂ν φ) − 14 Hµρσ Hν ρσ = 8πGK(µν) , − ⋆ d ⋆ e−2φ H(3) = 16πGe−2φ K(2) .
(5.20) (5.21)
We will assume that the stringy Energy-Momentum tensor is nontrivial only up to a finite radius, rc , and thus vanishes outside this radius: TAB = 0
if
r ≥ rc .
(5.22)
That is to say, matter is localized only up to the finite ‘cutoff’ radius, rc , in a spherically symmetric manner. We emphasize that we never force the H-flux nor the gradient of the string dilaton to be trivial outside a finite radius: this would have been the case if we had viewed them as extra matter, but in the current framework of Stringy Gravity, they are part of the stringy graviton fields, on the same footing as the Riemannian metric, gµν . Their profiles are dictated by the Einstein Double Field Equations only. The strict localization of the matter (5.22) motivates us to restrict spacetime to be asymptotically ‘flat’ (Minkowskian) at infinity, by imposing the following boundary conditions [24], lim A = 1 ,
r→∞
lim r −2 C = 1 ,
r→∞
lim φ = 0 ,
r→∞
lim A′ = 0 ,
lim A′′ = 0 ,
r→∞
r→∞
lim C ′ C −1/2 = 2 ,
r→∞
lim φ′ = 0 ,
lim C ′′ = 2 ,
(5.23)
r→∞
lim φ′′ = 0 .
r→∞
r→∞
The vacuum expectation value of the string dilaton at infinity, or lim e−2φ = 1, is our conventional normalr→∞
ization, as we have the Newton constant, G, at our disposal as a separate free parameter in the master action of Stringy Gravity coupled to matter (3.66). The conditions of (5.22) and (5.23) should enable us to recover the previously acquired, most general, spherically symmetric, asymptotically flat, static vacuum solution to D = 4 Stringy Gravity [24] (c.f. [25]) outside the cutoff radius, r ≥ rc .
43
In addition, we postulate that matter and hence the spacetime geometry are ‘regular’ and ‘non-singular’ at the origin, r = 0. We require lim A′ C = 0 ,
lim C = 0 ,
r→0
lim φ′ C = 0 ,
r→0
(5.24)
r→0
of which the first is a natural condition for the consistency of the spherical coordinate system at the origin (5.6). The second and third can then be satisfied easily as long as A′ and φ′ are finite at r = 0. Note also p that the areal radius, R = eφ C/A (5.9), vanishes at the origin. All the nontrivial (Riemannian) Christoffel symbols of the metric ansatz (5.6) are, exhaustively [24], t = γ t = 1 A′ A−1 + φ′ , γtr rt 2
γttr = 12 A′ A + φ′ A2 ,
r = 1 A′ A−1 C − 1 C ′ − Cφ′ , γϑϑ 2 2
r = − 1 A′ A−1 + φ′ , γrr 2
r = sin2 ϑγ r , γϕϕ ϑϑ
ϑ = γ ϑ = − 1 A′ A−1 + 1 C ′ C −1 + φ′ , γrϑ ϑr 2 2
ϑ = − sin ϑ cos ϑ , γϕϕ
ϕ ϕ γϑϕ = γϕϑ = cot ϑ ,
ϕ ϕ γrϕ = γϕr = − 21 A′ A−1 + 12 C ′ C −1 + φ′ .
(5.25) From the off-shell conservation of the stringy Einstein tensor, the three equations (5.19), (5.20), (5.21) must imply the on-shell conservation of the stringy Energy-Momentum tensor as in (3.92) and (3.93). For the present spherical and static ansatz, the nontrivial components of (3.92) come from ‘ν = t’ and ‘ν = r’ only. They are, respectively, he−2φ AC −1 F = −
d (CD) , dr
(5.26)
and d r dr (Kr
− 12 T(0) ) =
1 ′ −1 t 2 A A (Kt
+ Kr r − 2Kϑ ϑ ) + φ′ (Kt t − Kr r + 2Kϑ ϑ )
(5.27)
−C ′ C −1 (Kr r − Kϑ ϑ ) − e−4φ A2 BC −2 F . On the other hand, for (3.93), there appears only one nontrivial relation from the choice of ‘ν = t’, d dr
e−2φ A−1 CE = 0 . 44
(5.28)
We will confirm that these relations are indeed satisfied automatically by the three equations (5.19), (5.20), (5.21) which reduce, with the Christoffel symbols (5.25), as follows. Firstly, the scalar equation (5.19) becomes
d (5.29) ln(φ′ C) + 12 e−6φ (A3 B 2 C −2 − h2 AC −2 ) . dr The Ricci curvature, Rµν , and the two derivatives of the string dilaton, ▽µ ∂ν φ, are automatically diagonal, 4πG(T(0) − Kµ µ ) = e−2φ φ′ A
such that the tensorial equation (5.20) is almost diagonal, 8πGKtt = Rtt + 2▽t ∂t φ − 41 Htρσ Ht ρσ = 8πGKrr = Rrr + 2▽r ∂r φ − 41 Hrρσ Hr ρσ =
1 ′ d 2 A A dr
d ln(A′ A−1 C) + φ′ A2 dr ln(φ′ C) − 12 h2 A2 C −2 e−4φ ,
1 ′ −1 d 2A A dr
d ln(A′ A−1 C) − 12 A′ 2 A−2 − C −1/2 dr C ′ C −1/2
d ln(φ′ C) − 12 A2 B 2 C −2 e−4φ , −2φ′ 2 − φ′ dr
8πGKϑϑ = Rϑϑ + 2▽ϑ ∂ϑ φ − 41 Hϑρσ Hϑ ρσ = 1 − 21 C ′′ +
d 1 ′ −1 dr 2 A A C
− φ′ C − 21 (A2 B 2 − h2 )C −1 e−4φ ,
8πGKϕϕ = 8πG sin2 ϑKϑϑ = Rϕϕ + 2▽ϕ ∂ϕ φ − 41 Hϕρσ Hϕ ρσ = sin2 ϑ Rϑϑ + 2▽ϑ ∂ϑ φ − 41 Hϑρσ Hϑ ρσ , (5.30)
with one exception, an off-diagonal component, 8πGK(tr) = 8πGD(r) = Rtr + 2▽t ∂r φ − 41 Htρσ Hr ρσ = − 14 Htρσ Hr ρσ = − 21 hBe−4φ A2 C −2 . (5.31) The last equation for the H-flux (5.21) becomes d − ⋆ d ⋆ e−2φ H(3) = − ⋆ d e−4φ A2 BC −1 dt + e−4φ hC −1 dr = A−1 C sin ϑ dr e−4φ A2 BC −1 dϑ ∧ dϕ = 16πGe−2φ K(2) = 16πGe−2φ E(r)dt ∧ dr + F (r) sin ϑdϑ ∧ dϕ ,
(5.32)
which gives K[tr] (r) = E(r) = 0 ,
16πGe−2φ AC −1 F (r) =
d dr
e−4φ A2 BC −1 .
(5.33)
The former result of (5.33) satisfies the conservation relation (5.28) trivially, while the latter combined with (5.31) implies the conservation relation (5.26). Integrating the latter, we get e−4φ A2 BC −1 = q + V(r) , 45
(5.34)
where we set Z V(r) := −16πG
∞
′
−2φ(r ′ )
dr e
r
Z A(r )F (r )/C(r ) = −16πG ′
′
∞
′
dr e−2d K ϑϕ ,
(5.35)
r
and q is a constant of integration. From our assumption (5.22), when r ≥ rc , both F (r) and V(r) vanish, and consequently e−4φ A2 BC −1 assumes the constant value, q. Now, substituting (5.34) into the second formula in (5.30), we get
8πGKrr =
1 ′ −1 d 2A A dr
d d ln(A′ A−1 C) − 12 A′ 2 A−2 − C −1/2 dr C ′ C −1/2 − 2φ′ 2 − φ′ dr ln(φ′ C)
− 12 e4φ A−2 (q + V)2 .
(5.36) The infinite radius limit of this expression implies, with the conditions of (5.22) and (5.23), that actually q must be trivial: q = 0. Therefore, from (5.34), we are able to fix B(r) and hence Hrϑϕ , B = e4φ A−2 CV ,
(5.37)
Hrϑϕ = e4φ A−2 CV sin ϑ ,
which vanish when r ≥ rc , in agreement with the known vacuum solution [24]. The remaining Einstein Double Field Equations (5.29), (5.30), (5.31) reduce, with
Z Z
ϑ ϕ
from (5.8), to
e−2d ≡ 4πe2φ A−1 C
hV = −16πGK(tr) C , C ′′
(5.38)
Z Z −2d r ϑ e + e4φ A−2 CV 2 , = 2 + 4G(Kr + Kϑ − T(0) )
(5.39)
ϑ ϕ
Z Z d −2d ′ −1 µ t e (A A C) = 2G(Kµ − 2Kt − T(0) ) + e4φ A−2 CV 2 , (5.40) dr ϑ ϕ Z Z d ′ −2d µ 1 2 −4φ −1 e C − G(Kµ − T(0) ) (5.41) − 12 e4φ A−2 CV 2 , (φ C) = 2 h e dr ϑ ϕ Z Z −2d ′ 2 ′ −1 2 ′2 2 −4φ 4φ −2 2 2 r e 4(φ C) +(A A C) +4C−C +h e +e A C V +4CG(2Kr −T(0) ) = 0 . (5.42) ϑ ϕ
46
The first equation (5.38) indicates that h is a proportionality constant relating V = e−4φ A2 BC −1 to K(tr) C. On the other hand, the radial derivative of the entire expression in the last formula (5.42), after substitution of (5.37), (5.39), (5.40), and (5.41), implies the remaining conservation relation (5.27): d dr
=
h
h
R R i 4(φ′ C)2 + (A′ A−1 C)2 + 4C − C ′ 2 + h2 e−4φ + e4φ A−2 C 2 V 2 + 4CG(2Kr r − T(0) ) ϑ ϕ e−2d d r dr (Kr
− 21 T(0) ) − 12 A′ A−1 (Kt t + Kr r − 2Kϑ ϑ ) − φ′ (Kt t − Kr r + 2Kϑ ϑ ) + C ′ C −1 (Kr r − Kϑ ϑ )
i R R +e−4φ A2 BC −2 F × 8CG ϑ ϕ e−2d .
(5.43)
This provides a consistency check for the equations (5.39)–(5.42), and completes our concrete verification that all the Energy-Momentum conservation laws indeed follow from the Einstein Double Field Equations. In order to solve or integrate the second and the third equations, (5.39), (5.40), we prepare the following definitions, W(r) :=
Z
r
4φ
−2
dr e A
0
2
CV =
1 4π
Z
r
dr 0
Z Z
−2d
e
Hrϑϕ H
rϑϕ
,
ϑ ϕ
Z r Z Z e−2d (Kr r + Kϑ ϑ − T(0) ) , X (r) := G dr 0
ϑ ϕ
0
ϑ ϕ
Z r Z Z e−2d (Kr r + Kϑ ϑ + Kϕ ϕ − Kt t − T(0) ) , Y(r) := G dr
c := W(r)
Xb(r) :=
Z
r
dr W ,
0
Z
0
r
dr X ,
(5.44)
which all vanish at the origin, r = 0. Further, when r ≥ rc , the un-hatted functions, W(r), X (r), Y(r),
become constant—for example, X (r) = X (rc ) ≡ Xc for r ≥ rc . Consequently, the hatted functions become linear in the outside region,
c =W cc + (r − rc )Wc , W(r)
Xb(r) = Xbc + (r − rc )Xc
for
r ≥ rc .
(5.45)
Integrating (5.39) twice, we can solve for C(r). There are two constants of integration which we fix by imposing the boundary conditions at the origin: firstly we set C(0) = 0 directly from (5.24) and secondly, with φ0 ≡ φ(0), we fix C ′ (0) = ±he−2φ0 from the consideration of the small r limit of (5.42). We get c . C(r) = r 2 ± he−2φ0 r + 4Xb(r) + W(r) 47
(5.46)
Outside the matter this reduces to a quadratic equation, cc − rc Wc C(r) = (r − α)(r + β) = r 2 + (4Xc + Wc ± he−2φ0 )r + 4Xbc − 4rc Xc + W
r ≥ rc ,
for
(5.47)
where we set two constants, α :=
1 2
β :=
1 2
−(4Xc + Wc ±
4Xc + Wc ±
he−2φ0 ) +
he−2φ0
+
q
q
(4Xc + Wc ±
(4Xc + Wc ±
he−2φ0 )2
he−2φ0 )2
Similarly, (5.40) gives
cc + 16rc Xc − 16Xbc + 4rc Wc − 4W
cc + 16rc Xc − 16Xbc + 4rc Wc − 4W
lim A′ A−1 C = 0 ,
A′ A−1 C = 2Y + W ,
,
.
(5.48)
(5.49)
r→0
for which the trivial constant of integration (zero) has been chosen to meet the boundary condition at the origin (5.24). Eq.(5.49) can be further integrated to determine A(r) with the boundary condition, this time at infinity (5.23), Z A(r) = exp −
∞
r
dr (2Y + W)C
−1
,
lim A(r) = 1 .
(5.50)
r→∞
Away from the matter, with (5.47), this reduces to a closed form,
A(r) =
r−α r+β
a α+β
for
r ≥ rc ,
(5.51)
where we have introduced another constant, a := lim A′ A−1 C = 2Y(rc ) + W(rc ) = 2Yc + Wc , r→∞
(5.52)
such that outside the matter, A′ A−1 C = a
for
48
r ≥ rc .
(5.53)
For A(r) to be real and positive, it is necessary to restrict the range of r. Since α+β is positive semi-definite from (5.48), we are lead to require the cutoff radius to be greater than α, such that r−α > 0. r+β
r ≥ rc > α ≥ −β ,
(5.54)
However, note that the signs of α and β are not yet fixed: in the next subsection 5.3 we shall assume some energy conditions which will ensure both α and β are positive. We now turn to the last differential equation (5.42). Upon substitution of (5.47) and (5.49), it takes the form c − e4φ A−2 C 2 V 2 4(φ′ C)2 + h2 e−4φ = h2 e−4φ0 + 8X W − 4(Y + W)Y + 16(X 2 − Xb) − 4W R R +4 r ± 12 he−2φ0 (4X + W) − 4CG(2Kr r − T(0) ) ϑ ϕ e−2d ,
(5.55)
which, from (5.51), (5.52), reduces outside the matter to 4(φ′ C)2 + h2 e−4φ ≡ b2
for
(5.56)
r ≥ rc .
This new constant, b, meets cc + 4Xc + Wc ± he−2φ0 b2 := (α + β)2 − a2 = 16rc Xc − 16Xbc + 4rc Wc − 4W
2
− 2Yc + Wc
2
. (5.57)
Since the left-hand side of (5.56) is positive, b should be real, and from (5.48), α + β is positive since q p 2 2 cc . α + β = a + b = (4Xc + Wc ± he−2φ0 )2 + 16rc Xc − 16Xbc + 4rc Wc − 4W (5.58) Therefore, outside the matter we have
√ 2φ′ C = ± b2 − h2 e−4φ such that ±
Z
√
2dφ b2
−
h2 e−4φ
=
Z
dr (r − α)(r + β) 49
for
r ≥ rc ,
(5.59)
for
r ≥ rc ,
(5.60)
of which both sides can be integrated to give q 1 r−α 1 2φ 4φ 2 2 ± ln e + e − h /b = ln + constant . |b| α+β r+β
(5.61)
We can determine the constant of integration in (5.61) from the boundary condition at infinity (5.23), lim e2φ = 1 ,
(5.62)
r→∞
to obtain the profile of the string dilaton outside the matter, e2φ = γ+
r−α r+β
√
b a2 +b2
+ γ−
r+β r−α
√
b a2 +b2
for
r ≥ rc ,
(5.63)
where b can be either positive or negative, and γ+ , γ− denote two positive semi-definite constants, γ± := 12 (1 ± For the sake of reality, we require10
p
1 − h2 /b2 ) .
(5.64)
b2 ≥ h2 .
(5.65)
This completes our derivation of the spherically symmetric, static, regular solution to D = 4 Stringy Gravity with a localized stringy matter distribution. We conclude this subsection by summarizing and analyzing our results. • Outside the cutoff radius, r ≥ rc , we recover the spherically symmetric static vacuum solution [24]: e2φ = γ+
r−α r+β
√
b a2 +b2
+ γ−
r+β r−α
√
b a2 +b2
,
B(2) = h cos ϑ dt ∧ dϕ ,
√ a √ a r+β r−α 2 2φ 2 2 2 a2 +b2 a2 +b2 ds = e dr + (r − α)(r + β)dΩ − r+β . dt + r−α
(5.66)
• Moreover, the constants, α, β, a, b, h, are now all determined by the stringy Energy-Momentum tensor 10
In fact, when b = 0, from (5.56), we get h = 0 and φ′ = 0. Although (5.60) and (5.61) would be problematic if b = 0, the
final result (5.63) is still valid, as e2φ = 1.
50
of the matter localized inside the cutoff radius: from (5.48), (5.52), (5.57),
a=
Z
∞
dr
0
Z
π
dϑ 0
q
2
(Zc ± he−2φ0 ) + 4Zec − Zc ± he−2φ0
α=
1 2
β=
1 2
Z
dϕ e−2d
2π
q
(Zc ±
0
b2 = Zc ± he−2φ0
2
2 he−2φ0 )
+ 4Zec + Zc ± he−2φ0
, ,
1 Hrϑϕ H rϑϕ + 2G Kr r + Kϑ ϑ + Kϕ ϕ − Kt t − T(0) , 4π
+ 4Zec − a2 ,
h= Z
∞
Ktr C
,
dr e−2d K ϑϕ
r
(5.67)
where, with (5.44), Z(r) := Zec :=
Z
0
Z
0
rc
r
dr
Z
0
π
dϑ
Z
2π
−2d
dϕ e
0
dr Zc − Z(r) .
1 rϑϕ r ϑ Hrϑϕ H + 4G Kr + Kϑ − T(0) = W(r) + 4X (r) , 4π (5.68)
As before, the subscript index, c, denotes the position at r = rc , such that Zc = Z(rc ) . • Some further comments are in order. – There are two classes of solutions: b =
p p (α + β)2 − a2 ≥ 0 or b = − (α + β)2 − a2 < 0.
– Direct computation from (5.63) shows q √ b √ b r+β ′ 2φ r−α a2 +b2 − γ− r−α a2 +b2 = εbφ b e4φ − h2 /b2 , 2φ Ce = b γ+ r+β
(5.69)
where we define a sign factor,
εbφ :=
+1
−1 +1
if
b > 0 and r ≥ rφ
if
b > 0 and rφ > r ≥ α
if
b < 0 and r ≥ α , 51
(5.70)
with the zero of φ′ given by α+β rφ := 1−
γ− γ+
γ− γ+
√
√
a2 +b2 2b
a2 +b2 2b
φ′ (rφ ) = 0 ,
,
φ(rφ ) =
1 2
ln |h/b| < 0 .
(5.71)
When h is trivial, we have γ− = 0, rφ = α and hence εbφ is fixed to be +1. If h 6= 0 and b > 0, then rφ > α. Otherwise (h 6= 0 and b < 0) we have the opposite, rφ < α. In fact, when b is negative, rφ
also becomes negative and thus unphysical. That is to say, for large enough r, i.e. either r ≥ α with negative b or r ≥ rφ with positive b, the sign of φ′ coincides with that of b, but when b is positive, φ′ becomes negative in the finite interval α ≤ r < rφ .
– Since b is real, the following inequality must be met:
Zc ± he−2φ0
2
+ 4Zec ≥ a2 ,
(5.72)
which imposes a constraint on the stringy Energy-Momentum tensor through (5.67). – From (5.8), outside the matter the integral measure in Stringy Gravity reads e−2d
=
R2 sin ϑ
= γ+
r+β r−α
√ a−b
a2 +b2
+ γ−
r+β r−α
√ a+b a2 +b2
(r − α)(r + β) sin ϑ
for
r ≥ rc . (5.73)
– With the boundary condition at the origin (5.24), the Einstein Double Field Equations (5.40), (5.41) enable us to evaluate, for arbitrary r ≥ 0, 2φ′ C
+ A′ A−1 C
Z
r
d 2φ′ C + A′ A−1 C dr 0 Z r dr h2 e−4φ C −1 − 16πGKt t e2φ A−1 C =
=
dr
0
=
Z
0
r
dr
Z
0
π
dϑ
Z
2π
−2d
dϕ e
0
where Htϑϕ H tϑϕ = −h2 e−6φ AC −2 .
52
1 tϑϕ t Htϑϕ H − 4GKt , 4π
(5.74)
– Combining (5.69) and (5.74) with the boundary condition at infinity (5.23), (5.53), we acquire Z 2π Z ∞ Z π p 1 tϑϕ t −2d 2 2 a + b 1 − h /b = (5.75) dϕ e dϑ dr Htϑϕ H − 4GKt . 4π 0 0 0 We stress that this result is valid irrespective of the sign of b.
– The constant parameter, h, corresponding to the electric H-flux, is given by the formula (5.67) Z ∞ dr e−2d K ϑϕ . h = Ktr (r)C(r) (5.76) r
While this is a nontrivial relation, as the right-hand side of the equality must be constant independent p of r, it is less informative compared to the integral expressions of a in (5.67) or a + b 1 − h2 /b2
in (5.75). We expect a fuller understanding of the h parameter will arise if we solve for the timedependent dynamical Einstein Double Field Equations, allowing h to be time-dependent, h → h(t).
In any case, (5.76) implies that if Ktr is nontrivial somewhere in the interior, there must be electric H-flux everywhere, including outside the matter.
– The small-r radial derivative of the areal radius R (5.9) outside the matter reads, with the sign factor εbφ (5.70), dR dr
i h p √ = e2φ A−1 R−1 r − α + 21 a2 + b2 − 21 a + 12 εbφ b 1 − (h2 /b2 )e−4φ
for
r ≥ rc . (5.77)
– From (5.8), (5.17), (5.33), (5.38), the Noether charge (3.88) for a generic Killing vector reads Z ∞ Z π Z 2π Z 1 −2d t A −2d t t −2 r 1 dϕ e (Kt − 2 T(0) )ξ + dϑ dr e T Aξ = Q[ξ] = hVA ξ sin ϑ . 16πG 0 0 0 Σ (5.78)
5.3 Energy conditions In this subsection we assume that the stringy Energy-Momentum tensor and the stringy graviton fields satisfy the following three conditions: i) the strong energy condition, with magnetic H-flux, Z 2π Z ∞ Z π −2d dϕ e dϑ dr −Kt t + Kr r + Kϑ ϑ + Kϕ ϕ − T(0) + 0
0
0
53
1 rϑϕ Hrϑϕ H ≥ 0 ; 8πG (5.79)
ii) the weak energy condition, with electric H-flux, Z 2π Z ∞ Z π dϕ e−2d −Kt t + dϑ dr 0
0
0
1 tϑϕ H Htϑϕ ≥ 0; 16πG
iii) the pressure condition, with magnetic H-flux and without integration, 1 Kr r + Kϑ ϑ − T(0) + Hrϑϕ H rϑϕ ≥ 0 . 16πG
(5.80)
(5.81)
While the nomenclatures are in analogy with those in General Relativity, the precise expression in each inequality, including the H-flux, is what we shall need in our discussion. Since the magnetic H-flux vanishes outside the matter along with the stringy Energy-Momentum tensor, (5.35), (5.37), the radial integration in Rr the strong energy condition (5.79) is taken effectively from zero to the cutoff radius, i.e. 0 c . In contrast, the electric H-flux has a long tail and the weak energy condition genuinely concerns the infinite volume integral. If the matter comprises point particles only as in (3.118), the above conditions are all clearly met, since Kr
r,
Kϑ ϑ , Kϕ ϕ and −Kt t are individually positive semi-definite, while T(0) is trivial. In such cases, not
only the integrals but also the integrands themselves are positive semi-definite, which would imply −Kt t +
1 Htϑϕ H tϑϕ ≥ 0 16πG
:
weak energy density condition?
(5.82)
and similarly for the strong energy density condition. However, for stringy matter, such as fermions (3.108) or fundamental strings (3.122), the diagonal components, Kµµ , may not be positive definite, and hence the above inequalities appear not to be guaranteed. If the diagonal components are negative, the positively squared H-fluxes need to compete with them. In fact, while we take the energy and the pressure conditions (5.79), (5.80), (5.81) for granted,11 we shall distinguish the energy condition from the energy density condition, and in particular investigate the implications of the relaxation of the latter (5.82). Obviously, from (5.67), the strong energy condition sets the constant, a, to be positive semi-definite. Similarly, the pressure condition ensures Z(r) is a non-decreasing positive function, reaching its maximum value at the cutoff radius, such that 0 ≤ Z(r) ≤ Zc and Z ′ ≥ 0. Consequently, Zec is positive semi-definite and thus α and β are both safely real and positive. In this way, the strong energy and the pressure conditions 11
Strictly speaking, we might relax the pressure condition (5.81) and require Zec ≥ 0 only. However we do not pursue this here.
54
ensure α = |α| ≥ 0 ,
β = |β| ≥ 0 ,
(5.83)
a = |a| ≥ 0 .
Given this positiveness, the string dilaton, φ, outside the matter (5.63), φ=
1 2
ln γ+
r−α r+β
√
b a2 +b2
+ γ−
r+β r−α
√
b a2 +b2
(5.84)
r > rc ,
for
diverges to plus infinity when r → α+ and h 6= 0, irrespective of the sign of b. Specifically, if b > 0 and
h 6= 0, φ decreases from ∞ over the finite interval of α < r < rφ , crossing the horizontal axis of φ = 0 at √ r=
α+β(γ− /γ+ )√ 1−(γ− /γ+ )
1+a2 /b2
1+a2 /b2
, reaches its minimum, φmin =
1 2
ln |h/b| < 0 at r = rφ (5.71), and then increases
to converge to zero over the semi-infinite range, rφ < r ≤ ∞. On the other hand, when b > 0 and h = 0,
the dilaton φ increases monotonically from −∞ to zero over the whole range, α < r ≤ ∞, and if b < 0, for all h, it is the opposite: φ decreases monotonically from ∞ to zero over α < r ≤ ∞.
Now we turn to the weak energy condition. To see its implications, we consider the circular geodesic
motion of a point particle (3.118), which orbits around the central matter with fixed r larger than rc , ν µ d 2 xλ λ dx dx = 0, + γ µν dτ 2 dτ dτ
dr = 0, dτ
ϑ=
π . 2
(5.85)
With the nontrivial Christoffel symbols (5.25), the radial ‘λ = r’ component of the geodesic equation, dϕ 2 d2 r r dt 2 + γ r + γ = 0, determines the angular velocity [24], 2 tt ϕϕ dτ dτ dτ 2 dgtt g′ dgϕϕ −1 2φ′ A2 + A′ A dϕ =− = − tt ′ = ′ . (5.86) dt dr dr 2RR 2φ C − A′ A−1 C + C ′
Associating this with the centripetal acceleration measured by the areal radius through Newtonian gravity, we define and analyze an effective mass, M (r), as a function of the radius, 2 GM (r) dϕ . ≡R 2 R dt From (5.69) and (5.74), we have explicitly, with the sign factor, εbφ (5.70), M (r) = I(r) ×
1 2G
= I(r) ×
1 2G
= I(r) ×
Z
0
(a + b)γ+
r
a + εbφ b
dr
Z
0
π
q
dϑ
r−α r+β
1−
Z
0
2π
√
2b a2 +b2
+ (a − b)γ−
e−4φ h2 /b2 −2d
dϕ e
γ+
(5.87)
r−α r+β
√
2b a2 +b2
1 tϑϕ t Htϑϕ H − 2Kt , 8πG
55
+ γ−
−1 (5.88)
where we have set √ e3φ AC I(r) := ′ = φ C − 12 A′ A−1 C + 12 C ′
p
"
(r − α)(r + β) γ+ r−α+
1 2
√
r−α r+β
√b+a/3
a2 +b2
+ γ−
a2 + b2 − 21 a + 12 εbφ b
which is positive definite for sufficiently large r, having the limit,
p
r+β r−α
√b−a/3
a2 +b2
1 − e−4φ h2 /b2
lim I(r) = 1 .
#3
2
,
(5.89)
(5.90)
r→∞
In fact, for r > α, h 6= 0,12 the numerator in (5.89) is positive-definite, while the denominator, q p 1 1 1 2 2 Ω(r) := r − α + 2 a + b − 2 a + 2 εbφ b 1 − e−4φ h2 /b2 ,
(5.91)
is a monotonically increasing function over α < r ≤ ∞, taking values from a non-positive number to plus
infinity: with (5.69), Ω′ = 1 +
h2 e−4φ > 0, 2(r − α)(r + β)
lim Ω(r) =
r→α
1 2
p
a2 + b2 − |a| − |b| ≤ 0 ,
lim Ω(r) = ∞ .
r→∞
(5.92)
If and only if ab = 0, we have strictly lim Ω(r) = 0. Otherwise, I(r) diverges generically for some finite r→α
r = rΩ > α, as Ω(rΩ ) = 0.
We consider taking the large r limit of (5.88), with the boundary condition (5.23), to obtain p Z ∞ Z π Z 2π a + b 1 − h2 /b2 1 −2d tϑϕ t dϕ e dϑ M∞ ≡ lim M (r) = dr = Htϑϕ H − 2Kt . r→∞ 2G 8πG 0 0 0 (5.93) p While the first equality, or GM∞ = 21 (a + b 1 − h2 /b2 ), is the confirmation of the known result in [24], the second equality reveals the relationship of the mass, M∞ , to the infinite-volume integral of the stringy Energy-Momentum tensor and the electric H-flux. The weak energy condition (5.80) then precisely corresponds to the sufficient and necessary condition for the mass to be positive semi-definite, M∞ ≥ 0.
In fact, the mass, M∞ , appears in the expansion of the metric, specifically the temporal component, gtt ,
12
For the case of h = 0, see (5.97).
56
in the inverse of the areal radius,13 ds2 = gtt dt2 + gRR dR2 + R2 dΩ2 ,
2GM∞ R
gtt (R) = − 1 − gRR (R) = 1 +
√
a−b
+
1−h2 /b2 R
√
b2 +ab
+
1−h2 /b2 − 12 h2 R2
+ ···
√
a2 +b2 − 52 ab
1−h2 /b2 − 14 h2 R2
,
(5.94)
+ ··· ,
and similarly,
p h2 − 2b2 − 2ab 1 − h2 /b2 2GM (R) = 2GM∞ + + ··· . (5.95) R Thus, effectively, the circular geodesic becomes Keplerian for large enough R: that is to say, Stringy Gravity tends to agree with GR at long distances. However, if M (r) is ever negative for some finite r, it means that the gravitational “force” a la (5.87) is repulsive! With (5.92), if M (rΩ ) diverges to plus infinity, the gravitational force is attractive and singular at r = rΩ . On the other hand, if M (rΩ ) = −∞, there appears
an infinite ‘wall’ of repulsive force at the surface of r = rΩ .
We proceed to investigate if M (r) can be negative. For this, we look for the zero of M (r), denoted by rM , which from (5.54) should be greater than α, M (rM ) = 0 ,
rM ≥ rc > α .
(5.96)
Firstly, for the special case of h = 0 (and thus γ+ = 1, γ− = 0), we have outside the matter, p √a+3b (a + b) (r − α)(r + β) r − α 2 a2 +b2 √ . 2GM (r) = r − α + 21 ( a2 + b2 + b − a) r + β
(5.97)
This does not admit any zero which is greater than rc . In particular, if b = 0, we get 2GM = a constant. Henceforth we focus on nontrivial h 6= 0. In this case, if and only if |b| ≥ |a|, M (r) admits a zero, rM , However, the mass, M∞ , does not coincide Noether charge (5.78) for the Killing vector ξ µ ∂µ = ∂t , h with the time-translational √ i a−b 1 2 2 a + b [24] ( c.f. [23, 58]). nor the ADM mass a la Wald, GMADM = 4 a + a+b 13
57
which is uniquely fixed a priori from the first equality of (5.88): with a = |a| (5.83), h h i1√ 2 2 i 1 √1+a2 /b2 2 γ− (|b|−|a|) 2 1+a /b γ− (|b|−|a|) α + β γ+ (|b|+|a|) α + β γ− (|b|−|a|)+2GM∞ i 1 √1+a2 /b2 = i 1 √1+a2 /b2 h h 2 2 γ (|b|−|a|) γ (|b|−|a|) − − 1 − γ− (|b|−|a|)+2GM∞ 1 − γ+ (|b|+|a|) √ h i a2 +b2 2b γ− (b−a) for b > 0 α + β γ+ (a+b) √ √ rM = = h h i 1 1+a2 /b2 i 1 √1+a2 /b2 i a2 +b2 h 2 2 γ (|b|−|a|) γ (|b|−|a|) + + 2b (b−a) α + β γ− (|b|+|a|) α + β γ+ (|b|−|a|)+2GM∞ 1 − γγ− + (a+b) i1√ 2 2 = i 1 √1+a2 /b2 h h 2 γ+ (|b|−|a|) 2 1+a /b γ+ (|b|−|a|) 1 − γ− (|b|+|a|) 1 − γ+ (|b|−|a|)+2GM∞ for b < 0 . (5.98)
Clearly in the case of |b| ≥ |a| (and h 6= 0), rM is real valued and thus can be physical. The second equality of (5.88) implies that the zero of M (r) necessarily meets p |a| − b2 − h2 e−4φ(rM ) = 0 ,
(5.99)
such that, from (5.70), when b > 0, the zero must be between α and rφ , α < rM < rφ
(5.100)
b > 0.
for
Indeed, this can be verified directly as follows. Provided |b| > |a|, the root is greater than α, since √ a2 + b2 for b>0 i 1 √1+a2 /b2 h γ− (|b|−|a|)+2GM∞ 2 −1 γ− (|b|−|a|) rM − α = (5.101) √ 2 2 a +b for b < 0, i 1 √1+a2 /b2 h γ (|b|−|a|)+2GM ∞ 2 + −1 γ+ (|b|−|a|)
and 2GM∞ is positive definite, irrespective of the sign of b, owing to the weak energy condition (5.80).
Further, with b > 0, we have from (5.71), # " 1 √1+a2 /b2 1 √1+a2 /b2 √ 2 2 γ (|b|−|a|) γ − a2 + b2 − γ− γ+ + (|b|+|a|) rφ − rM = "
1−
γ− γ+
1 √1+a2 /b2 2
#"
58
1−
γ− (|b|−|a|) γ+ (|b|+|a|)
1 √1+a2 /b2 2
# ,
(5.102)
which is positive since 0
