A Scalar Polynomial Singularity Without an Event Horizon. - CiteSeerX

A Scalar Polynomial Singularity Without an Event Horizon. Mark D.Roberts April 10, 2004 Published: Gen.Rel.Grav. 17(1985)913-926. Eprint: None Comments: 13 pages 30172 bytes, five diagrams, LaTex. 2 KEYWORDS: Cosmic Censorship: Singularities. 1999 PACS Classification Scheme: http://publish.aps.org/eprint/gateway/pacslist 98.50.-v,90.80.-k,04.90.+e 1991 Mathematics Subject Classification: http://www.ams.org/msc 85A40,83F05,85A05,70F99. Abstract It is shown that the solution of the field equations for a static spherically symmetric scalar field has a scalar polynomial singularity and no event horizon. The solution does not develop from nonsingular data on any Cauchy surface. The possible existence of a universal scalar field, the conformal diagram and the geodesics of the soltuion, and the energy and momentum of the field present are discussed.

1

1

Introduction.

The singularity theorems (see for example Hawking and Ellis [1]) suggest that in a large number of space-times singularities occur; because these might have an unpredictable influence on space-time Penrose [2] put forward the cosmic censorship hypothesis that singularities are hidden by an event horizon. Here it is shown that the general solution of the field equations for a static spherically symmetric scalar field found by Wyman [3] is a scalar polynomial singularity with no horizon; however, as the singularity does not arise from a development of nonsingular data on any partial Cauchy surface it does not provide a counterexample to usual statements of he hypothesis. The solution is of interest because it has two parameters - the mass m and the scalar charge σ, provided σ 6= 0 there is no horizon, if σ = 0 the Schwarzschild solution is recovered. In the 1950s there were many gravitational theories such as those of Yilmaz [4] and Szekeres [5] which involved scalar fields. This led to work on spherically symmetric scalar fields by Buchdahl [6] and Bergman and Leipnik [7]. It is interesting to note that Bergman and Leipnik remarked that these solutions were similar to Schwarzschild’s “except for the disappearance of the famous singularity”, singularity here meaning the coordinate singularity at r = 2m, to which Buchdahl [replied] “this remark is difficult to understand as one can always make the finite singularity of the Schwarzschild solution disappear by simply choosing the wrong (i.e. unphysical) sign for the constant of integration m”. It is indeed unphysical as it does not agree with Newtonian theory in the weak field limit. The Reissner-Nordstr¨om solution also has no horizon for q 2 > m2 ; however, charges of this magnitude are unlikely to exist in macroscopic objects, whereas the solution discussed here has no horizon provided [the] scalar charge [has] σ 6= 0. Furthermore, as shown by Agnese and LaCamera [8], this solution has the same Eddington-Robertson parameters as the Schwarzschild solution. Gravit[at]ional theories involving scalar fields are supposed to be more in keeping with Mach’s principle and are still viable, see for example Will [9]. The question naturally arises as to whether black holes exist in these theories; recently Duriusseau [10] has shown that they do not in the theories of Just, Brans-Dicke, and Tonnelat. In Wyman’s solution the presence of any scalar “charge” no matter how small prevents the existence of an event horizon and so it is apposite to speculate on the existence of universal scalar fields. Zel’dovich [11] suggests that the vacuum polarization of quantum field theory endows the vacuum with a nonzero energymomentum tensor so that the field equations become (vac)

Gij = 8π(Tij + Tij

),

(1)

(vac)

and he suggests that by putting Tij = (−Λ/8π)gij we have justified a nonzero cosmological constant. however, we can interpret such fluctuations as giving rise to a universal scalar field by setting (vac)

Tij

1 = k(ϕ,i ϕ,j − gij ϕ,k ϕ,k ). 2 2

(2)

An argument for universal scalar fields working from the uncertainty principle has been put forward by Roberts [12]. Black holes are supposed to radiate particles (Hawking [13]), among which are scalar particles. If we choose to incorporate the scalar particles into the solution of the field equations we would use Wyman’s solution, then the horizon would disappear. Yilmaz’s solution is a particular case of Wyman’s solution and arises in the gauge theories of Chang and Johnson [14], where usually we would expect the Schwarzschild solution.

2

The Line Element and Conformal Diagram.

To solve the field equations for a scalar field we consider th energy density of a scalar field in flat space and then set this equal to Ttt . In this case there is still a horizon and the horizon has been used to explain the size of elementary particles by Ross [15] and quark confinement by Nagy [16]. Alternatively we can consider the energy momentum tensor for a scalar field and equate this to the field equations. The conformal case has been derived by Froyland [17] and is qualitively similar to the nonconformal case for which the energy momentum tensor is 1 1 Tij = (ϕ,i ϕ,j − ϕ,k ϕ,k ), (3) 4π 2 in the static spherically symmetric case this give the line element [3]     η 2m η4 2m 2 2 ds = exp − dt − 4 exp cosech4 dr2 r r r r   η 2m cosech2 d2 Σ, (4) −η 2 exp r r where η 2 = m2 +σ 2 and m is interpreted as the mass and σ the scalar charge. In this coordinate system volume does not increase as a two-sphere as measured by r, this make comparison with Schwarzschild spacetime difficult. A coordinate system in which volume increases as a two-sphere is found by defining a new radial coordinate [8] R(r) = exp(m/r)cosech(η/r),

(5)

which does not possess an explicit algebraic inverse r(R), and is well behaved for real σ; for complex σ there is a point of inflection at r = η[arccoth(m/η]−1 . Using R the line element becomes    −2 2m m η η 2 2 dR2 − R2 d2 Σ, (6) ds = exp − dt − − sinh + cosh r η r r which possess no event horizon unless σ 2 = 0, in which case the Schwarzschild solution is recovered. Defining a new radial coordinate, r=

2η , 1 − exp(−2η/r) 3

(7)

we obtain a form of the line element recently found by Agnese and LaCamera [18] 

2

ds =

2η 1− r

m/η



2η dt − 1 − r 2

−m/η



2η d r− 1− r 2

1−m/η

r2 dΣ2 . (8)

Although this form of the line element looks similar to the standard line element for the Schwarzschild space-time it is very different because r = 2η is apoint as can be seen from the angular term and also at r = 2η (where r = 0) there is a curvature singularity as we shall see. This form permits construction of the conformal diagram by a method analogous to that found in [1]. Defining r∗ =

Z  1−

2η r

−m/η

v = t + r∗ ,

dr,

w = t − r8

we arrive at the double null form:  m/η  1−m/η 2η 2η ds2 = 1 − dv dw − 1 − r2 dΣ2 . r r

(9)

(10)

Defining v 0 = α exp βv,

w0 = −α exp(−βw),

x0 =

1 0 (v − w0 ), 2

t0 =

1 0 (v + w0 ), (11) 2

we have a form corresponding to Minkowski space-time:   1−m/η  2η 2η exp(−2βr∗ )(dt02 − dx02 ) − 1 − r2 dΣ2 , ds2 = (αβ)−2 1 − r r (12) where r is determined implicitly by x02 − t02 = α2 exp 2βr∗ ,

(13)

so that x02 > t02 as exponeation is always positive, x0 = (α/2)[exp βv+exp(−βw)] is always positive so that x0 > |t0 | showing that in this case the space-time has not been extended. Defining v 00 = arctan v 0 ,

w00 = arctan w0 ,

(14)

we can construct the conformal diagram.

3

The Singularity

There are spherically symmetric space-times, such as the interior Schwarzschild solution, which do no have curvature singularities. We now explicitly demonstrate that Wyman’s solution has a curvature singularity and compare its rate

4

x

o

2.0

x gRR

o o 1.0

o

1

-1.0

2

3

4

5

x

Fig.1

5

g RR: o=Wyman; x=Schwarzschild.

1.0 gtt

0.5

0 1

o

o x

x 2

3

o x

4

R

5

-0.5

Fig.2. gtt ;0=Wyman; x=Schwarzschild.

50 o I

o x x -2

x o

-1

log R 1 o x

o x

2 o x

-30

3

o x

Fig.3. The Curvature invariant I. o=Wyman; x=Schwarzschild.

6

20

o

-1 log -g sin 2

o

10

x x x

o log R

-3

-2

x

-1

1

2

x o

x o

-10 Fig.4. log -g

-1

2 sin . o=Wyman; x=Schwarzschild.

of divergence to the Schwarzschild curvature singularity. We work in the r coordinate system as it suffices to show that there is a curvature singularity in this system. Using Takeno’s formula [19] we obtain   η  nm η ηo Rrϕrϕ η3 2m Rrθrθ = = exp cosech2 coth − , 2 3 r r r r r r sin θ   η m 2mη 2m Rrtrt = exp − coth − , (15) r4 r r η     η  m2 2m 2m Rθϕθϕ 2 = η exp − coth + 2 +1 , sin2 θ r η r η     η m Rϕtϕt m 2m 2 η Rθtθt = = − coth exp − sinh − . η r r r η sin2 θ These give the curvature invariant: IR

= Rijkl R

×

( h

ijkl

8m2 = 6 exp η



−4m r



sinh8

η r

 2 η η m η i2 η coth − + 2 coth − − r m r 2η m

)   η  m 2 +3 coth − . r η

7

(16)

i+

Scri+

r=infinity or v"=pi/2 r=0 02 R=2eta io

r=infinity or w"=-pi/2 t=constant

Scri-

r=constant>0 or R=constant>2eta

iFig.5. The conformal diagram.

8

Also, g

η8 = − 4 exp r = −



4m r



cosech8

η r

sin2 θ

(17)

  η4 4 4 η R cosech sin2 θ r4 r

In the Schwarzschild case IR = 48mr−6 and g = −r4 sin2 θ and IR diverges as r−6 , g −1 as r−4 , and gIR as r−2 . In the Wyman case IR diverges as exp[4(2η − m)/r] because coth becomes of the order unity for large arguments; g −1 diverges as r4 exp[4(2η − m)/r] and gIR ar r−2 . A numerical comparison of the rate of divergence for η = 1.5 and m = 1 in the R coordinate system and η = m = 1 showed faster divergence for η = 1.5.

4

The Geodesics

The geodesics of Wyman’s space-time can be constructed in the usual manner; see for example Chandrasekhar [20]. For all geodesics there is the equation   dt 2m = E exp , (18) dτ r where ta is the affine parameter along the geodesic and E is a constant. From this we see that ta does not change sign with respect to the global coordinate t, for a space-time with horizon it changes sign upon going through the horizon. There is the equation of motion:        η  L2 η η4 2m 2m 2m E 2 − 4 exp cosech4 r˙ 2 − 2 exp − sinh2 exp r r r r η r r = 2L = 0 or ∞,

(19)

for null and timelike geodesics, respectively. For radial (L = 0) null geodesics η η τ = ± coth (20) E r as τ does not take on values between +η/E and −η/E the space-time is geodesically incomplete. Giving τ the value η/E and then increasing it the geodesic goes from r − 0 to r = ∞ thus demonstrating that the space-time is asymptotically unpredictable. For radial (L = 0) timelike geodesics 19 reduces to   2     η dr η 2m 2m · 4 exp cosech4 = exp · E 2 − 1, (21) dτ r r r r this is zero at rc = −2m/ ln(E 2 ), 9

(22)

for E 2 < 1 there is a turning point in rc and the timelike geodesics do not reach the singularity, for E 2 > 1 the geodesics do. Note that E 2 < 1 is the condition for a nonradial geodesic to have a bound orbit. The corresponding equation for the Reissner-Nordstr¨ om solution is (Reference [20], equation 88, p.219)  2 2m q 2 dr = (E 2 − 1) + − 2, (23) dτ r r This equals zero at rc = m ± [m2 − q 2 (1 − E 2 )]1/2 ,

(24)

and provided rc is real there is a turning point at rc and the geodesics do not reach the singularity. Here in contrast to Wyman’s case the radial timelike geodesics with E 2 > 1 never reach the singularity.

5

The Energy and Momentum of the Field Present

The line element was derived using 3 for the energy-momentum tensor. from the line element and using Takeno’s formula [19] we can write down the Ricci and Einstein tensors in a form without any derivatives: Rrr Rθθ = Rtt R

=

2σ 2 , r4

= 0,   η 2σ 2 2m = − 4 exp − sinh4 , η r r σ2 , r4

Grr

=

Gθθ

= −

Gtt

=

(25)

  σ2 2 η sinh , η2 r     σ2 4m 4 η exp − sinh , η4 r r

when σ = 0 the vacuum is recovered. From 26 and 3 it follows that σ ϕ=− . r

(26)

Using 3 and 26 and letting V i be a future-directed timelike vector, for the weak condition we have 1 4πTij V i V j = ϕ,i V i ϕ,j V j − gij V i V j ϕ,k ϕ,k , 2 =

1 (ϕ,r V r )2 − gij V i V j · g rr (ϕ,r )2 , 2

≥ 0

for real σ. 10

(27)

Now 2

i

16π Tij V T

kj

Vk

=

    σ4 4m 8 η exp − sinh (V r Vr + V t Vt + V θ Vθ + V ϕ Vϕ ) η8 r r

> 0,

(28)

for all σ thus showing that Tij V i is a timelike vector and hence the dominant condition is satisfied. Equation 3 gives Rij = 2ϕ,i ϕ,j ,

(29)

therefore Rij V i V j

= 2(ϕ,i V i )2 ≥ 0

for real σ.

(30)

Thus the nonspacelike convergence condition is also satisfied.

6

Conclusion

The Schwarzschild solution is a vacuum solution. Vacuum space-times are valid at large distances from sources, but the event horizon occurs at a short distance. Choosing small enough σ the space-time discussed here [can] be made experimentally indistinguishable from the Schwarzschild space-time; however it has different global properties. Like the Schwarzschild solution it is geodesically incomplete and has a scalar polynomial singularity. Like the naked RiessnerNoerdstr¨ om solution it is inextensible and has no event horizon, but it differs from the Reissner-Nordstr¨ om in that geodesics with different energies reach the singularity.

7

Acknowledgements

I would like to thank the referee for his detailed criticisms and C.S.Sharma for encouragement.

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