Integrability conditions

In this section we give the accurate version of the results discussed in the previous section. If we apply the general Lie derivative formula (2.16) to the Weyl tensor

components given in (4.3) we obtain

LXC⁰₁₀₁ = −1

3X¹Λ_r− 2

3X¹_rΛ, (4.8a)

LXC⁰₂₀₂ = 1

6(r²e^−2λΛ)_rX¹+2

6r²e^−2λΛX²_θ, (4.8b) LXC⁰₃₀₃ = 1

6X¹(r²sin²θe^−2λΛ)_r+1

6X²(r²sin²θe^−2λΛ)_θ

+2

6r²sin²θe^−2λΛX³_ϕ, (4.8c)

LXC¹₂₁₂ = LXC⁰₂₀₂, (4.8d)

LXC¹₃₁₃ = LXC⁰₃₀₃, (4.8e)

LXC²₃₂₃ = −2LXC⁰₃₀₃. (4.8f)

For the existence of the conformal symmetry (2.18) holds. Hence (4.8d), (4.8e) and (4.8f) are identically satisfied for vanishing tidal forces. Thus we do not utilise them in further calculations.

Now we consider(4.8c). Simplifying the right hand side we get

LXC⁰₃₀₃ = 1

6(r²e^−2λAⁱ_rη_i+A⁴)[2r(1−rλ_r)Λ +r²Λ_r] sin²θe^−2λ

+1

6[−Aⁱ(η_i)_θ+a₁sinϕ+a₂cosϕ](2r²sinθcosθe^−2λΛ)

+2

6r²sin²θe^−2λΛ[−csc²θAⁱ(η_i)_ϕϕ−cotθ(a₁sinϕ+a₂cosϕ)]

= 1 6

{

r²e^−2λAⁱ_r[2r(1−rλ_r)Λ +r²Λ_r] + 2ΛAⁱ−2ΛAⁱ }

r²sin²θe^−2λη_i

+A⁴[2r(1−rλ_r)Λ +r²Λ_r]e^−2λsin²θ

−2

6r²ΛAⁱe^−2λ[sinθcosθ(η_i)_θ+ (η_i)_ϕϕ]

+2

6r²e^−2λΛ(a₁sinϕ+a₂cosϕ)(sinθcosθ−sin²θcotθ)

= sin²θLXC⁰₂₀₂−2

6r²ΛAⁱe^−2λ[sin²θη_i+ sinθcosθ(η_i)_θ+ (η_i)_ϕϕ]. (4.9) The last trigonometric term vanishes identically, so (4.9) has the simple form

LXC⁰303 = sin²θLXC⁰202. (4.10) Equation (4.10) is satisfied identically because of the condition (2.18). For this reason (4.8c) is not used in further calculations.

The remaining conditions to be considered are (4.8a) and (4.8b). With the help of (3.6), (4.8a) and (4.8b) yield

LXC⁰₁₀₁ = −1 3

{

r²Aⁱ_rΛ_r+ 2Λ[2r(1−rλ_r)Aⁱ_r+r²Aⁱ_rr] }

e^−2λη_i

−1

3(A⁴Λ_r+ 2A⁴_rΛ), (4.11)

LXC⁰₂₀₂ = 1

6(r²e^−2λAⁱ_rη_i+A⁴)[2r(1−rλ_r)Λ +r²Λ_r]e^−2λ

+2

6r²e^−2λΛAⁱη_i

= 1 6

{

e^−2λAⁱ_r[2r(1−rλ_r)Λ +r²Λ_r] + 2ΛAⁱ }

r²e^−2λη_i

+[2r(1−rλ_r)Λ +r²Λ_r]e^−2λA⁴. (4.12) By invoking the integrability condition (2.18) we have LXC⁰101= 0 and LXC⁰202 = 0.

Then using the linear independence of trigonometric functions we obtain the following conditions

[r²Λ_r+ 4r(1−rλ_r)Λ]Aⁱ_r+ 2r²Aⁱ_rrΛ = 0, (4.13a)

Λ_rA⁴+ 2ΛA⁴_r = 0, (4.13b)

[2r(1−rλ_r)Λ +r²Λ_r]e^−2λAⁱ_r+ 2ΛAⁱ = 0, (4.13c)

[2r(1−rλ_r)Λ +r²Λ_r]A⁴ = 0. (4.13d) The equations in the system (4.13) are the necessary and sufficient conditions for the integrability conditions (2.18) to be satisfied. The structure of the system (4.13) suggests that there is a natural classification system for the conformal symmetries of the static spherically symmetric spacetimes (3.1). We consider the two cases

i) Λ = 0, ii) Λ̸= 0.

This classification naturally separates the spacetimes into conformally flat (Λ = 0) and non-conformally flat (Λ̸= 0) categories.

Case I : Λ = 0

This case has vanishing Weyl tensor. From (4.2) we obtain

ν_rr+ν_r²−λ_rν_r+r⁻¹(λ_r−ν_r) +r⁻²(1−e^2λ) = 0. (4.14) This second order nonlinear partial differential equation is difficult to solve. However, it can be solved using the procedure adopted by Herreraet al (2001) in their work on conformally flat anisotropic spheres. We first write (4.14) in the following form

[e^−2λν_r r

]

r

+ e^−2(ν+λ) [e^2νν_r

r ]

r−[e^−2λ−1 r²

]

r

= 0. (4.15)

Next we introduce new variables

y= e^−2λ , ν_r = u_r

u. (4.16)

This transforms (4.15) into

yr+ 2[u_rr− ^u_r^r + _r^u2]

[u_r−^u_r] y− 2u

r²[u_r−^u_r] = 0. (4.17) Equation (4.17) is a first order differential equation in y and its integrating factor is given by

exp {

2

∫ [u_rr− ^u_r^r +_r^u2] [u_r− ^u_r] dr

}

= [

u_r− u r

]2

. (4.18)

Multiplying both sides of (4.17) by the integrating factor and simplifying we get

d dr

[(

u_r− u r

)2

y ]

= 2u r²

[

u_r−u r ]

= d dr

[u² r² ]

. (4.19)

Integrating (4.19) we obtain

[

u_r− u r

]2

y = u²

r² +c₁, (4.20)

where c₁ is a constant of integration, which is the same as

[ur

u − 1 r

]2

y = 1 r²

[

1 +c₁r²u⁻² ]

. (4.21)

We can change back to the original variables using u= e^ν to get

[ ν_r− 1

r ]2

e^−2λ = 1 r²

[

1−c²r²e^−2ν ]

(4.22) where we have setc₁ =−c². Equation (4.22) can be written as

νr− 1 r = e^λ

r

√1−c²r²e^−2ν. (4.23)

Rearranging (4.23) and integrating with respect to r we get

∫ ν_r−¹_r

√1−c²r²e^−2νdr =

∫ e^λ

r dr. (4.24)

Now let

α=cre^−ν. (4.25)

Then

dα=−cre^−ν [

ν_r− 1 r ]

dr

=−α [

ν_r− 1 r ]

dr. (4.26)

Thus (4.24) becomes

−

∫ dα

α√

1−α² =

∫ e^λ

r dr. (4.27)

Upon integration this gives

sech⁻¹(α) =

∫ e^λ

r dr+ ˜C, (4.28)

where ˜C is a constant. Simplifying (4.28) we obtain

e^ν =crcosh

[ ∫ e^λ

r dr+ ˜C ]

. (4.29)

The result (4.29) is the general solution of the nonlinear equation (4.14).

Gathering all this information leads to the theorem:

Theorem 4.4.1. In a static spherically symmetric spacetime which is conformally flat, i.e. Λ = 0, the gravitational potentials ν and λ are related by the equation

e^ν =crcosh

[ ∫ e^λ

r dr+ ˜C ]

,

where C˜ is a constant. The conformal Killing vector X is given by (3.6) and the functions A⁰(t, r), A⁴(t, r) and Aⁱ(t, r) are arbitrary.

Case II : Λ ̸ = 0

This case is not conformally flat. We can write (4.13a) in the form

(Aⁱ_r)_r+ [Λ_r

2Λ+ 2 r −2λ_r

]

Aⁱ_r = 0, (4.30)

If we consider this as a first order equation inAⁱ_r then we can integrate to get

Aⁱ_r = e^2λ

r²Λ^1/2h₁(t). (4.31)

whereh₁(t) is a function that arises in the integration process. Substituting (4.31) into (4.13c) we obtain

Aⁱ =− 1 2Λ^1/2

[ 2(1

r −λ_r) + Λ_r Λ

]

h₁(t), (4.32)

and we have an explicit form for Aⁱ. From (4.13b), we have

A⁴ = g₁(t)

Λ^1/2, (4.33)

where g₁(t) is a function of integration. It remains to solve (4.13d). Substitute (4.33) in (4.13d) to get

[2r(1−rλ_r)Λ +r²Λ_r]g₁(t) = 0.

This leads to two subcases:

Case II(a) :g₁(t) = 0 For this subcase we have

A⁴ = 0 ,

Aⁱ =− 1 2Λ^1/2

[ 2

(1 r −λ_r

) + Λ_r

Λ ]

h₁(t).

The above forms, together with (3.7e) and (3.7f), imply

A⁰_r = 0, A⁰_t= 0.

Hence

A⁰ =C, where C is a constant.

Case II(b) :g₁(t)̸= 0 This case implies that

2r(1−rλr)Λ +r²Λr = 0.

We can solve the above equation to obtain

Λ = e^2λ

r² k. (4.34)

where k is a constant. Then (4.2) and (4.34) give

ν_rr+ν_r²−λ_rν_r+r⁻¹(λ_r−ν_r) +r⁻²[1−(1 +k)e^2λ] = 0. (4.35) Equation (4.35) is a highly nonlinear equation. Fortunately the method utilised in Case I may also be applied to solve (4.35). We believe that (4.35) is a new equa- tion that arises in conformal symmetries of static spacetimes and has not been solved previously. We first write (4.35) in the following form

[e^−2λνr

r ]

r+ e^−2(ν+λ) [e^2ννr

r ]

r−[e^−2λ−(1 +k) r²

]

r = 0. (4.36)

We introduce the variables

y= e^−2λ , ν_r = u_r u.

Then (4.36) becomes

y_r+ 2[u_rr− ^u_r^r + _r^u2]

[u_r−^u_r] y− 2(1 +k)u

r²[u_r−^u_r] = 0. (4.37) We multiply (4.37) by the integrating factor.

exp {

2

∫ [u_rr− ^u_r^r +_r^u2] [u_r− ^u_r] dr

}

= [

u_r− u r

]2

.

Then (4.37) can be expressed as

d dr

[(

ur− u r

)2

y ]

= (1 +k) d dr

[u² r² ]

, (4.38)

which can be integrated to give

[

ur− u r

]2

y= (1 +k)u²

r² +c2, (4.39)

where c₂ is a constant of integration. If we transform back to the original variables using u= e^ν then (4.39) becomes

[ ν_r− 1

r ]2

e^−2λ = (1 +k)1 r²

[

1−˜c²r²e^−2ν ]

, (4.40)

where we have set _1+k^c2 =−˜c². This equation gives

ν_r− 1 r =√

1 +ke^λ r

√1−˜c²r²e^−2ν. (4.41)

Rearranging (4.41) and integrating with respect to r we get

∫ √ ν_r−¹_r

1−˜c²r²e^−2νdr =√ 1 +k

∫ e^λ

r dr. (4.42)

Now let

β = ˜cre^−ν. (4.43)

Then

dβ =−˜cre^−ν [

ν_r− 1 r ]

dr

=−β [

νr− 1 r ]

dr. (4.44)

Thus (4.42) becomes

−

∫ dβ

β√

1−β² =√ 1 +k

∫ e^λ

r dr. (4.45)

Upon integrating this gives

sech⁻¹(β) = √ 1 +k

∫ e^λ

r dr+ ˜C₁, (4.46)

where ˜C₁ is a constant. Simplifying (4.46) we obtain

e^ν = ˜crcosh [√

1 +k

∫ e^λ

r dr+ ˜C₁ ]

, (4.47)

where ˜C₁ is a constant of integration.

We conclude from (4.13c), (4.33) and (4.34) that

Aⁱ = 0 , A⁴ =re^−λg₁(t).

Using (3.7f) we see that

A⁰_t= (1−rν_r)e^−λg₁(t).

Integrating this we obtain

A⁰ = (1−rν_r)e^−λg₂(t) +g₃(r), where g₂(t) = ∫

g₁(t)dt, and g₃(r) is a function of integration.

We gather all these results in the following theorem:

Theorem 4.4.2. For static spherically symmetric spacetimes which are non-conformally flat, i.e. Λ̸= 0, we have either A⁴ or Aⁱ must vanish. The two cases are given by:

a)

A⁴ = 0,

A⁰ =C, a constant

Aⁱ(t, r) =− 1 2Λ^1/2

[ 2

(1 r −λr

) + Λ_r

Λ ]

h1(t), where h₁(t) is arbitrary.

b)

A⁴(t, r) = re^−λg₁(t),

Aⁱ(t, r) = 0,

A⁰(t, r) = (1−rν_r)e^−λg₂(t) +g₃(r), where g₁(t) and g₃(r) are arbitrary functions and g₂(t) = ∫

g₁(t)dt.

Furthermore, the gravitational potentials are related by the equation e^ν = ˜crcosh

[√ 1 +k

∫ e^λ

r dr+ ˜C₁ ]

,

where k, ˜c and C˜1 are constants.

Theorem 4.4.1 and Theorem 4.4.2 represent the classification of the conformal geometry in static spherically symmetric spacetimes in terms of the Weyl tensor. We emphasize that this classification is the most general in terms of the Weyl tensor.

It is remarkable that the nonlinear differential equations relating the gravitational potentials can be solved in general. The potentialsνandλare related by the hyperbolic function cosh. Also note that the theorems correct the Maartens et al (1995, 1996) result. Their analysis is simpler because the nonlinear differential equations (4.14) and (4.35) do not arise due to the mistake in their integrability conditions. As we have demonstrated these nonlinear equations lead to a more rich structure in the conformal geometry directly involving the gravitational potentials. Our analysis completes the general solution and classification of conformal motions in static spherically symmetric spacetimes.

Chapter 5

Classification of Conformal motions in non-static spacetimes

5.1 Introduction

In this chapter we consider the classification of symmetries in terms of conformal motions for non-static spherical spacetimes. We classify the conformal symmetries in terms of the Weyl tensor which represents tidal forces. We first calculate the Weyl tensor components for the non-static line element (3.8) in §5.2. We have checked the correctness of these expressions with Mathematica (Wolfram 2010) and Maple (2009).

We then use these Weyl tensor components to calculate the Lie derivatives in §5.3.

We apply the integrability condition (2.18) to classify the conformal motions for these spacetimes. Two cases arise which are distinguished by the vanishing or non-vanishing of the Weyl tensor. In the first case we explicitly obtain a relationship relating the metric functions by solving a nonlinear second order partial differential. In the second case we find that the metric functions are related via a nonlinear third order partial differential equation for which particular solutions exist.