An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds

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— journal of September 2011

physics pp. 533–543

An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds

JOTHI MOODLEY^∗and GARETH AMERY

Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

∗Corresponding author. E-mail: sanjana.jm@gmail.com

Abstract. Embeddings into higher dimensions are very important in the study of higher- dimensional theories of our Universe and in high-energy physics. Theorems which have been developed recently guarantee the existence of embeddings of pseudo-Riemannian manifolds into Einstein spaces and more general pseudo-Riemannian spaces. These results provide a technique that can be used to determine solutions for such embeddings. Here we consider local isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate embedded spaces that admit bulks of a specific type. We show that the general Schwarzschild–de Sitter spacetime and Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space of a particular form, and we discuss their five-dimensional solutions.

Keywords. Embedding theory; five-dimensional; spherically symmetric; Einstein bulk.

PACS Nos 04.20.Jb; 04.20.Ex; 04.50.−h

1. Introduction

Since the nineteenth century, isometric embeddings into higher dimensions have been explored extensively in geometry [1–7], with more focus on non-Euclidean spaces in recent times. Theorems given by Dahia and Romero [5,6] prove that there exists a local isometric embedding of any analytic pseudo-Riemannian manifold into an Einstein space and also into a more general pseudo-Riemannian space. Furthermore, it has been shown [8,9] that global embeddings can be constructed from the local ones. These embeddings require one extra dimension, unlike the Euclidean cases where the number of extra dimensions is quite large. Such embedding results are applicable and significant to higher-dimensional theories – string theory [10], Horava–Witten theory [11], D-brane models [12], braneworld models [13,14] and induced matter theory [15] – which rely on the view that our Universe is embedded in a higher dimension. Thus, it has become important to obtain descriptions of objects that are of astrophysical and theoretical interest in the context of higher dimensions – astrophysically-derived constraints on Z₂-symmetric higher-dimensional models [16,17].

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Although solutions to the embedding equations are guaranteed, the task of solving for the metric that embeds a particular space is actually very difficult. Explicit solutions have been found for embeddings of vacuum spacetimes (for example, the exterior Schwarzschild black hole) into Ricci flat [3] and Einstein [4,5] spaces, for embedding Friedmann–Lemaître–Robertson–Walker models into flat space [18], for some embeddings with bulks sourced by massless and self-interacting scalar fields [4,7], and for embedding of a four-dimensional global monopole metric into a five-dimensional Ricci flat bulk [19]. Other work includes ‘stacking’ type results, such as the ‘black string’ [20] and model-dependent numerical analyses (see [21]). Typically, spaces with vanishing energy–

momentum are easier to deal with than spaces which have nonzero energy–momentum, such as the monopole.

The aim here is to investigate embeddings of four-dimensional (4D) spherically symmetric (SS) spacetimes into five-dimensional (5D) Einstein spaces. We consider spherically symmetric spacetimes as they are relevant in astrophysics and cosmology, and we concen- trate on Einstein spaces because of their role in higher-dimensional particle physics, as well as their geometric simplicity. Because the usual method of solving the five-dimensional metric for a chosen four-dimensional space leads to difficulties, we take a slightly different approach to gain insight into this problem. We restrict the metric of the Einstein bulk to a particular form, and we investigate the kind of spherically symmetric spacetimes that may embed into it. Here we begin with a 5D metric whose components are separable with respect to the extra dimension. This form guarantees that the embedding is ‘ener- getically rigid’ and that the Killing geometry of the embedded space is inherited by the higher-dimensional space [17].

This paper is organized as follows. In §2 we review the technique for local isometric embeddings into Einstein spaces and in §3 we apply the formalism to embed spherically symmetric spacetimes that admit Einstein bulks of a specific type. We consistently adopt the following notational conventions: Roman lower case indices label the coordinates of the embedded space, Roman upper case indices label its spatial coordinates and Greek indices label the coordinates of the embedding space. A tilde denotes quantities pertaining to the embedding space and an overbar denotes quantities obtained from the n-dimensional component of the higher-dimensional metric. We use a prime and an overdot to denote partial differentiation with respect to the coordinates r and y, respectively. Note that the lower-dimensional space is referred to as the embedded space and the higher-dimensional space as the embedding space or the bulk.

2. The Dahia–Romero embedding theorem

Consider an n-dimensional analytic pseudo-Riemannian manifold M with metric ds² =gi k(x^j)dxⁱdx^k.

A theorem given by Dahia and Romero [5] proves that M has a local analytic isometric embedding into a(n+1)-dimensional Einstein manifold N with metric in Gaussian normal coordinates (without loss of generality):

ds˜² = ˜g_αβ(x^j,y)dx^αdx^β = ¯gi k(x^j,y)dxⁱdx^k+dy², ²=1, (1)

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along the hypersurface0, defined by y=0. Here y denotes the(n+1)th coordinate and R˜_αβ=(2/(1−n))g˜_αβwhereis a constant. This result means that [22] there exists an analytic map f : U ⊂ M −→ N , with U an open coordinated neighbourhood in M, such that f is a homeomorphism onto its image, the map f_∗: TpM −→Tf(p)N is injective for all points p in U , and gp(V,W)= ˜gf(p)(f_∗(V),f_∗(W)) ∀V,W ∈TpM, ∀p∈U . Here, as usual, TpM denotes the tangent space of M at p∈M.

The metric componentg¯i kis a solution to the field equations for (1) given by R˜i k= ¯Ri k+g¯^{j m}(¯i k¯j m−2¯j k¯i m)−∂¯i k

∂y = 2

1−ng¯i k, (2) R˜i n = ¯g^{j k}(¯∇^j¯i k− ¯∇i¯j k)=0, (3) G˜ⁿ_n = −1

2g¯^{i k}g¯^{j m}[ ¯Ri j km+(¯i k¯j m− ¯j k¯i m)] =, (4)

subject to the conditiong¯i k(x^j,0)=gi k(x^j). By appealing to the Cauchy–Kowalewskaja theorem, it can be shown that a unique analytic solutiong¯i kdoes exist and that eqs (3) and (4), which are known as the Codazzi and Gauss equations respectively, need only be solved on the hypersurface0. Equation (2) is referred to as the propagation equation because it is used to propagate off the hypersurface0so as to specify the whole bulk. ¯i kis the extrinsic curvature of0and is defined by

¯i k= −1 2

∂g¯i k

∂y , ¯i k(x^j,0)=i k.

R¯i kis the Ricci tensor derived fromg¯i kand we assume the sign convention:

R¯i k= ¯^mi m,k− ¯^mi k,m+ ¯^ji m¯^m_{j k}− ¯^mj m¯^j_{i k}.

It can be shown that both the embedded and embedding spaces are well-behaved with regards to stability and causality [23]. Furthermore, the local embeddings can be used to construct an Einstein space that embeds M globally (see [9] for a detailed discussion). This motivates the need for explicit solutions for local embeddings of interest.

3. Embedding spherically symmetric spacetimes

We apply the technique reviewed above to embed a four-dimensional spherically symmetric spacetime gi kwhich has the line element [24]

ds² = −e²^ν(^t^,^r⁾dt²+e²^λ(^t^,^r⁾dr²+r²(dθ²+sin²θdφ²).

The nonzero components of the Ricci tensor calculated from this metric are R00 =e^2(ν−λ)

−ν−ν²+νλ −2 rν

+λtt+λ²t −νtλt, R01 = −2

rλt,

R11 =ν+ν²−νλ −2

rλ −e^2(λ−ν)(λtt+λ²t −νtλt), R22 =r e^−2λν −r e^−2λλ +e^−2λ−1,

R₃₃ =R₂₂sin²θ.

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Equations (2)–(4) for embedding a 4D spherically symmetric spacetime gi k into a 5D Einstein space(R˜_αβ=(−2/3)g˜_αβ)with metric (1) along the y=0 hypersurface are

∂²g¯i k

∂y² = −4g¯i k

3 −g¯^{j m} 2

∂g¯i k

∂y

∂g¯j m

∂y −2∂g¯i m

∂y

∂g¯j k

∂y

−2R¯i k, (5)

0=g^{j k}(∇ji k− ∇ij k), (6)

−2=R+g^{i k}g^{j m}(i kj m−j ki m), (7) with initial conditions

¯

gi k(t,r, θ, φ,0)=gi k(t,r, θ, φ), (8)

∂g¯i k(t,r, θ, φ,0)

∂y = −2i k(t,r, θ, φ), (9)

wherei k = ¯i k(x^j,0)and¯i k = −¹₂(∂g¯i k/∂y). Although the Dahia–Romero theorem [5] guarantees that a solution to the above system exists, the general solution to these equations is not yet known. We have recently [19] considered the embedding of a 4D global monopole metric, which is static and spherically symmetric, and were able to obtain a 5D solution that is Ricci flat. A solution for a non-Ricci flat Einstein space that embeds the monopole is still to be determined. Thus, even for specific SS spacetimes, it is highly difficult to solve the embedding equations. So, we consider another approach: we assume that the 5D Einstein space has a particular form, and we investigate what SS spacetimes may embed into it.

We proceed by making a fairly simple assumption

¯

gi k=diag[A(y)g00,B(y)g11,C(y)g22,D(y)g33], (10) for the 5D metric, where the unknown functions A,B,C and D depend on y only and each metric component is separable in y. This type of metric for the embedding space occurs in the Randall–Sundrum braneworld scenario [14], and allows for a rigid embedding [17].

The initial conditions (8) and (9) become

A(0)=B(0)=C(0)=D(0)=1, (11) A(0)˙ = −200g⁰⁰, B(0)˙ = −211g¹¹,

C(0)˙ = −222g²², D(0)˙ = −233g³³. (12) Condition (12) implies that the extrinsic curvature must have the form

i k=diag[ag00,bg11,cg22,dg33],

where a,b,c and d are constants. We substitute the above expression into the Codazzi (6) and Gauss (7) equations to obtain c=d and

0=(a−b)λt, (13)

0=(b−a)ν +(b−c)2

r , (14)

−2=R+2

ab+2ac+2bc+c²

, (15)

where eq. (15) indicates that the Ricci scalar R of the embedded space must be constant.

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The components of the Ricci tensorR¯i kcalculated from (10) are R¯00 = A

BR00+

1− A

B λtt+λ²t −νtλt

,

R¯01 =R01, R¯₁₁ =R₁₁+

1− B

A λtt+λ²t −νtλt

e²^(λ−ν),

R¯22 =C

BR22+C B −1, R¯33 = D

BR33+ D

B − D C

sin²θ= D

CR¯22sin²θ.

The propagation equation (5) for i = 0 and k = 1 gives 0 = ¯R01, which implies that λt = 0. Thus, eq. (13) is satisfied. Withλt = 0 and the components of R¯i k, the other components of the propagation equation are

A¨+ A˙ 2

−A˙ A + B˙

B +C˙ C + D˙

D

+4

3 A= −2A

BR₀₀g⁰⁰, (16) B¨ + B˙

2 A˙

A − B˙ B +C˙

C + D˙ D

+4

3 B= −2R₁₁g¹¹, (17) C¨ +C˙

2 A˙

A + B˙ B −C˙

C + D˙ D

+4

3 C= −2C BR₂₂g²²

−2 C

B −1 1

r², (18)

D¨ + D˙ 2

A˙ A + B˙

B +C˙ C − D˙

D

+4

3 D= −2D BR₃₃g³³

−2 D

B − D C

1

r². (19) Since the left-hand sides of eqs (16)–(19) depend on y only, we should have

R₀₀g⁰⁰ =e^−2λ

ν+ν²−νλ +2 rν

=α1, (20)

R₁₁g¹¹ =e⁻²^λ

ν+ν²−νλ −2 rλ

=α2, (21)

C

BR22g²²+ C

B −1 1

r² = C Be^−2λ

ν r −λ

r + 1 r²

− 1

r² =α3(y), (22) D

BR33g³³+ D

B −D C

1 r² = D

Be^−2λ ν

r −λ r + 1

r²

− D C

1

r² =α4(y), (23)

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whereα1andα2are constants, andα3andα4are functions of y. Comparing eqs (22) and (23), we deduce thatα4=(D/C)α3, and so these equations are equivalent.

Now subtracting eq. (20) from (21), we obtain ν = −λ −(α2−α1)

2 r e^2λ, (24)

and inserting this into (22) leads to a first-order linear differential equation for e^−2λthat admits the solution

λ(r)= −1 2ln

B

C +α5(y)

r +

B C

α3

3 +α2−α1

6

r²

.

Using the fact thatλ(r)has no y dependence, and applying initial condition (11), we deter- mine that B =C andα˙3 =0 = ˙α5. With B =C, the condition (12) implies that b =c, and eqs (17) and (18) imply that R11g¹¹=R22g²². So, by (21) and (22), we haveα3=α2. Furthermore, eqs (20)–(23) show that the Ricci scalar R=α1+3α2.

Now substituting (24) andλ(r)= −¹₂ln

1+^α_r⁵ +³^α²₆^−α¹r²

into (20) and simplifying the result yields

(α2−α1)(−2α1r³+3α5)

3α2−α1

2 r³+3r+3α5

=0.

For the above equation to hold, we require eitherα1 = α2 or α1 = 0 = α5. In each case, the solution forλ(r)can be substituted into (24), which can then be integrated to provide a solution forν(t,r). The resulting spacetimes satisfy (20)–(23), and are the only 4D spherically symmetric spacetimes that may be embedded into a 5D Einstein space with

¯

gi k given by (10). Note that we still need to solve the Gauss–Codazzi equations (14) and (15), and the propagation equations (16)–(19) subject to the initial conditions (11) and (12), in order to determine the bulk metric explicitly. We consider each case below.

Case I: α1=α2

Whenα1=α2, we have λ(r)= −1

2ln

1+α5

r +α1

3 r² and

ν(t,r)= −λ(r)+g(t),

whereα1, α5∈Rand g(t)is an arbitrary function. So, ds² = −e^2g⁽^t⁾

1+α5

r +α1

3 r² dt²+ dr²

1+^α_r⁵ +^α₃¹r² +r²(dθ²+sin²θdφ²).

This solution is known as the general Schwarzschild–de Sitter spacetime, which has a four- dimensional cosmological constant given by−α1and Ricci scalar R=4α1.

Now we solve the Gauss–Codazzi equations (14) and (15) to obtain a =b=c= ±

−−2α1

6 ,

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and so the initial condition (12) becomes A(0)˙ = ˙B(0)= ˙D(0)= ±

−2−4α1

3 . (25)

The propagation equations (16)–(19) for this case are A¨+ A˙

2

−A˙ A +2B˙

B + D˙ D

+4

3 A= −2A

Bα1, (26)

B¨ + B˙ 2

A˙ A+ D˙

D

+4

3 B = −2α1, (27)

D¨ + D˙ 2

A˙ A+2B˙

B − D˙ D

+4

3 D= −2D

Bα1. (28)

As the Schwarzschild–de Sitter spacetime is an Einstein space, a solution for the Einstein embedding is already known [3,4], and it can be obtained as follows. We set A=B =D so that eqs (26)–(28) reduce to a single equation

A¨+ A˙² A +4

3 A= −2α1, which has the solution

A(y)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

cosh

−

6 y

+

1+2α1

sinh

−

6 y

₂

, =0,

1+ −α1

3 y

2

, =0,

(29) that satisfies conditions (11) and (25). Hence, the 5D Einstein embedding space for the Schwarzschild–de Sitter spacetime is d˜s²= A(y)ds²+dy²with A(y)given by (29).

Case II: α1=0=α5

Whenα1=0=α5, we have λ(r)= −1

2ln

1+α2

2 r² and

ν(t)=g(t),

whereα2∈R, α2=0 and g(t)is an arbitrary function. So, ds² = −e^2g⁽^t⁾dt²+

1+α2

2r² ⁻¹dr²+r²(dθ²+sin²θdφ²),

which has Ricci scalar R=3α2. Here the metric describes the Einstein Universe in which the scale factor is a constant, so the Universe does not expand or contract.

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Sinceν =0, the general solution to the Gauss–Codazzi equations (14) and (15) is given by

c=b= −a 2 ±1

2

a²−4

3 −2α2, where a∈R. The initial condition (12) becomes

A(0)˙ = −2a, B(0)˙ = ˙D(0)=a±

a²−4

3 −2α2. (30)

The functions A, B and D must satisfy the propagation equations A¨+ A˙

2

−A˙ A +2B˙

B + D˙ D

+4

3 A=0, (31)

B¨ + B˙ 2

A˙ A + D˙

D

+4

3 B= −2α2, (32)

D¨ + D˙ 2

A˙ A +2B˙

B − D˙ D

+4

3 D= −2D

Bα2, (33)

with initial conditions (11) and (30). Here we can set B = D so that (33) is equivalent to (32).

Consider=0. By setting A=1, eq. (31) holds trivially and condition (30) implies that a=0. Equation (32) becomes

B¨ + B˙²

2B = −2α2, which admits the solution

B(y)=

1+ −α2

2 y

2

,

that satisfies the initial conditions. Hence, the Einstein Universe can be embedded into a 5D Ricci flat space with metric

ds˜² = −e^2g⁽^t⁾dt²+B(y) dr²

1+^α₂²r² +r²(dθ²+sin²θdφ²)

+dy². By applying the coordinate transformation

e^g⁽^t⁾dt=dT,

−α2

2 r= R Y

1+R² Y²

−1/2

, (34)

1+ −α2

2 y=

−α2

2 Y

1+R² Y²

1/2

, (35)

the above metric can be written in the Minkowski form

ds˜² = −dT²+dR²+R²(dθ²+sin²θdφ²)+dY²,

and so the embedding space is flat. Thus, we regain the result obtained by Wesson [18]

for embedding Einstein Universe into a flat space. However, there the method was to

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start with the 5D Minkowski metric and transform it to a space that embeds the Einstein Universe along a y=constant hypersurface. Here we consider a more general 5D metric and directly solve the field equations to find the embedding.

When=(−3/2)α2, Lie symmetry analysis of eqs (31)–(33) yields a hitherto unknown solution:

A(y)=

cosh

−2

3 y

−a −3

2 sinh

−2

3 y

2

, a∈R, B(y)=C(y)=D(y)=1.

The details of this analysis will appear elsewhere [25]. The solution to eqs (31)–(33) for other cases ofis yet to be determined.

4. Conclusions

In this paper we considered local isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein spaces. We reviewed the technique provided by Dahia and Romero [5] for Einstein embeddings, and then applied it to embed spherically symmetric spacetimes. General solutions to the embedding equations are very difficult to find, even for specific four-dimensional spaces. This motivates us to restrict the bulk metric to a particular form and to investigate what spherically symmetric spacetimes gi kmay embed into it. To begin with, we chose a 5D metric of the form

ds˜² =A(y)g00dt²+B(y)g11dr²+C(y)g22dθ²+D(y)g33dφ²+dy², whose components are separable with respect to the extra dimension y. A metric of this type ensures energetic rigidity and is used in the Randall–Sundrum braneworld scenario [14]. We determined that the general Schwarzschild–de Sitter space and the Einstein Universe are the only 4D spherically symmetric spacetimes that may embed into a 5D Einstein space with this particular form. As the Schwarzschild–de Sitter spacetime is an Einstein space, its Einstein embedding is already known [3,4] and we discussed how it is obtained. In the case of the Einstein Universe, we obtained an embedding of the spacetime that is Ricci flat that can be transformed to Minkowski space via a change of coordinates.

This coincides with a result obtained by Wesson [18] for the embedding of the Einstein Universe into a flat space, although a different method was used there. We also obtained a new solution for the particular case=(−3/2)α2. However, the embedding metric for other non-Ricci flat cases remains to be solved. Additional solutions may be obtained by assuming different functional relationships between A, B, C and D. This is the subject of ongoing work. Other future work would be to investigate what spacetimes can embed into five-dimensional Einstein metrics of other forms.

Acknowledgements

The authors wish to thank Prof. S D Maharaj, Prof. P G L Leach and Prof. N K Dadhich for useful discussions and insightful comments. GA would like to thank the Inter-University Centre for Astronomy and Astrophysics (Pune, India) for their hospitality during a research

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visit which was the genesis of this project. JM was supported by a graduate bursary from the National Institute for Theoretical Physics in South Africa. JM was also supported by conference grants from the University of KwaZulu-Natal and the National Institute for Theoretical Physics.

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